加速度 - 维基百科，自由的百科全书

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未登录编辑者的页面 <a href="/wiki/Help:%E6%96%B0%E6%89%8B%E5%85%A5%E9%97%A8" aria-label="了解有关编辑的更多信息"><span>了解详情</span></a> </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-anoncontribs" class="mw-list-item"><a href="/wiki/Special:%E6%88%91%E7%9A%84%E8%B4%A1%E7%8C%AE" title="来自此IP地址的编辑列表[y]" accesskey="y"><span>贡献</span></a></li><li id="pt-anontalk" class="mw-list-item"><a href="/wiki/Special:%E6%88%91%E7%9A%84%E8%AE%A8%E8%AE%BA%E9%A1%B5" title="对于来自此IP地址编辑的讨论[n]" accesskey="n"><span>讨论</span></a></li> </ul> </div> </div> </div> </div> </nav> </div> </header> </div> <div class="mw-page-container"> <div class="mw-page-container-inner"> <div class="vector-sitenotice-container"> <div id="siteNotice"></div> </div> <div class="vector-column-start"> <div class="vector-main-menu-container"> <div id="mw-navigation"> <nav id="mw-panel" class="vector-main-menu-landmark" aria-label="站点"> <div id="vector-main-menu-pinned-container" class="vector-pinned-container"> </div> </nav> </div> </div> <div class="vector-sticky-pinned-container"> <nav id="mw-panel-toc" aria-label="目录" data-event-name="ui.sidebar-toc" class="mw-table-of-contents-container vector-toc-landmark"> <div id="vector-toc-pinned-container" class="vector-pinned-container"> <div id="vector-toc" class="vector-toc vector-pinnable-element"> <div class="vector-pinnable-header vector-toc-pinnable-header vector-pinnable-header-pinned" data-feature-name="toc-pinned" data-pinnable-element-id="vector-toc" > <h2 class="vector-pinnable-header-label">目录</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">移至侧栏</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">隐藏</button> </div> <ul class="vector-toc-contents" id="mw-panel-toc-list"> <li id="toc-mw-content-text" class="vector-toc-list-item vector-toc-level-1"> <a href="#" class="vector-toc-link"> <div class="vector-toc-text">序言</div> </a> </li> <li id="toc-简述" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#简述"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>简述</span> </div> </a> <button aria-controls="toc-简述-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>开关简述子章节</span> </button> <ul id="toc-简述-sublist" class="vector-toc-list"> <li id="toc-直线运动中的平均加速度和瞬时加速度" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#直线运动中的平均加速度和瞬时加速度"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1</span> <span>直线运动中的平均加速度和瞬时加速度</span> </div> </a> <ul id="toc-直线运动中的平均加速度和瞬时加速度-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-曲线运动中的加速度" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#曲线运动中的加速度"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.2</span> <span>曲线运动中的加速度</span> </div> </a> <ul id="toc-曲线运动中的加速度-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-伽利略变换" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#伽利略变换"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.3</span> <span>伽利略变换</span> </div> </a> <ul id="toc-伽利略变换-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-牛顿第二定律" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#牛顿第二定律"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.4</span> <span>牛顿第二定律</span> </div> </a> <ul id="toc-牛顿第二定律-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-惯性力" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#惯性力"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.5</span> <span>惯性力</span> </div> </a> <ul id="toc-惯性力-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-加加速度" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#加加速度"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.6</span> <span>加加速度</span> </div> </a> <ul id="toc-加加速度-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-角加速度" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#角加速度"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.7</span> <span>角加速度</span> </div> </a> <ul id="toc-角加速度-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-加速度的分解" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#加速度的分解"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>加速度的分解</span> </div> </a> <button aria-controls="toc-加速度的分解-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>开关加速度的分解子章节</span> </button> <ul id="toc-加速度的分解-sublist" class="vector-toc-list"> <li id="toc-按坐标系分解" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#按坐标系分解"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>按坐标系分解</span> </div> </a> <ul id="toc-按坐标系分解-sublist" class="vector-toc-list"> <li id="toc-平面直角坐标系" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#平面直角坐标系"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1.1</span> <span>平面直角坐标系</span> </div> </a> <ul id="toc-平面直角坐标系-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-极坐标系" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#极坐标系"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1.2</span> <span>极坐标系</span> </div> </a> <ul id="toc-极坐标系-sublist" class="vector-toc-list"> <li id="toc-极坐标系分解的推导" class="vector-toc-list-item vector-toc-level-4"> <a class="vector-toc-link" href="#极坐标系分解的推导"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1.2.1</span> <span>极坐标系分解的推导</span> </div> </a> <ul id="toc-极坐标系分解的推导-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-按自然坐标系分解" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#按自然坐标系分解"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1.3</span> <span>按自然坐标系分解</span> </div> </a> <ul id="toc-按自然坐标系分解-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-按功能分解" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#按功能分解"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>按功能分解</span> </div> </a> <ul id="toc-按功能分解-sublist" class="vector-toc-list"> <li id="toc-匀速圆周运动_向心加速度" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#匀速圆周运动_向心加速度"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2.1</span> <span>匀速圆周运动向心加速度</span> </div> </a> <ul id="toc-匀速圆周运动_向心加速度-sublist" class="vector-toc-list"> <li id="toc-向心加速度的推导" class="vector-toc-list-item vector-toc-level-4"> <a class="vector-toc-link" href="#向心加速度的推导"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2.1.1</span> <span>向心加速度的推导</span> </div> </a> <ul id="toc-向心加速度的推导-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-科里奥利效应" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#科里奥利效应"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2.2</span> <span>科里奥利效应</span> </div> </a> <ul id="toc-科里奥利效应-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-欧拉力" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#欧拉力"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2.3</span> <span>欧拉力</span> </div> </a> <ul id="toc-欧拉力-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </li> <li id="toc-几种特殊的运动" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#几种特殊的运动"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>几种特殊的运动</span> </div> </a> <button aria-controls="toc-几种特殊的运动-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>开关几种特殊的运动子章节</span> </button> <ul id="toc-几种特殊的运动-sublist" class="vector-toc-list"> <li id="toc-匀速直线运动" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#匀速直线运动"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>匀速直线运动</span> </div> </a> <ul id="toc-匀速直线运动-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-匀变速直线运动" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#匀变速直线运动"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>匀变速直线运动</span> </div> </a> <ul id="toc-匀变速直线运动-sublist" class="vector-toc-list"> <li id="toc-自由落体运动_重力加速度" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#自由落体运动_重力加速度"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2.1</span> <span>自由落体运动重力加速度</span> </div> </a> <ul id="toc-自由落体运动_重力加速度-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-加速度恒定的运动" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#加速度恒定的运动"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.3</span> <span>加速度恒定的运动</span> </div> </a> <ul id="toc-加速度恒定的运动-sublist" class="vector-toc-list"> <li id="toc-抛体运动" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#抛体运动"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.3.1</span> <span>抛体运动</span> </div> </a> <ul id="toc-抛体运动-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-简谐运动" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#简谐运动"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.4</span> <span>简谐运动</span> </div> </a> <ul id="toc-简谐运动-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-加速度的应用" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#加速度的应用"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>加速度的应用</span> </div> </a> <button aria-controls="toc-加速度的应用-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>开关加速度的应用子章节</span> </button> <ul id="toc-加速度的应用-sublist" class="vector-toc-list"> <li id="toc-电磁辐射" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#电磁辐射"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>电磁辐射</span> </div> </a> <ul id="toc-电磁辐射-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-狭义相对论" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#狭义相对论"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.2</span> <span>狭义相对论</span> </div> </a> <ul id="toc-狭义相对论-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-广义相对论" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#广义相对论"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.3</span> <span>广义相对论</span> </div> </a> <ul id="toc-广义相对论-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-参见" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#参见"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>参见</span> </div> </a> <ul id="toc-参见-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-參考文獻" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#參考文獻"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>參考文獻</span> </div> </a> <ul id="toc-參考文獻-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="目录" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="开关目录" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">开关目录</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">加速度</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="前往另一种语言写成的文章。138种语言可用" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-138" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">138种语言</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-af mw-list-item"><a href="https://af.wikipedia.org/wiki/Versnelling" title="Versnelling – 南非荷兰语" lang="af" hreflang="af" data-title="Versnelling" data-language-autonym="Afrikaans" data-language-local-name="南非荷兰语" class="interlanguage-link-target"><span>Afrikaans</span></a></li><li class="interlanguage-link interwiki-als mw-list-item"><a href="https://als.wikipedia.org/wiki/Beschleunigung" title="Beschleunigung – 瑞士德语" lang="gsw" hreflang="gsw" data-title="Beschleunigung" data-language-autonym="Alemannisch" data-language-local-name="瑞士德语" class="interlanguage-link-target"><span>Alemannisch</span></a></li><li class="interlanguage-link interwiki-am mw-list-item"><a href="https://am.wikipedia.org/wiki/%E1%8D%8D%E1%8C%A5%E1%8A%95%E1%8C%A5%E1%8A%90%E1%89%B5" title="ፍጥንጥነት – 阿姆哈拉语" lang="am" hreflang="am" data-title="ፍጥንጥነት" data-language-autonym="አማርኛ" data-language-local-name="阿姆哈拉语" class="interlanguage-link-target"><span>አማርኛ</span></a></li><li class="interlanguage-link interwiki-an mw-list-item"><a href="https://an.wikipedia.org/wiki/Aceleraci%C3%B3n" title="Aceleración – 阿拉贡语" lang="an" hreflang="an" data-title="Aceleración" data-language-autonym="Aragonés" data-language-local-name="阿拉贡语" class="interlanguage-link-target"><span>Aragonés</span></a></li><li class="interlanguage-link interwiki-anp mw-list-item"><a href="https://anp.wikipedia.org/wiki/%E0%A4%A4%E0%A5%8D%E0%A4%B5%E0%A4%B0%E0%A4%A3" title="त्वरण – 昂加语" lang="anp" hreflang="anp" data-title="त्वरण" data-language-autonym="अंगिका" data-language-local-name="昂加语" class="interlanguage-link-target"><span>अंगिका</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%AA%D8%B3%D8%A7%D8%B1%D8%B9" title="تسارع – 阿拉伯语" lang="ar" hreflang="ar" data-title="تسارع" data-language-autonym="العربية" data-language-local-name="阿拉伯语" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-arz mw-list-item"><a href="https://arz.wikipedia.org/wiki/%D8%AA%D8%B3%D8%B1%D9%8A%D8%B9" title="تسريع – 埃及阿拉伯文" lang="arz" hreflang="arz" data-title="تسريع" data-language-autonym="مصرى" data-language-local-name="埃及阿拉伯文" class="interlanguage-link-target"><span>مصرى</span></a></li><li class="interlanguage-link interwiki-as mw-list-item"><a href="https://as.wikipedia.org/wiki/%E0%A6%A4%E0%A7%8D%E0%A6%AC%E0%A7%B0%E0%A6%A3" title="ত্বৰণ – 阿萨姆语" lang="as" hreflang="as" data-title="ত্বৰণ" data-language-autonym="অসমীয়া" data-language-local-name="阿萨姆语" class="interlanguage-link-target"><span>অসমীয়া</span></a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/Aceleraci%C3%B3n" title="Aceleración – 阿斯图里亚斯语" lang="ast" hreflang="ast" data-title="Aceleración" data-language-autonym="Asturianu" data-language-local-name="阿斯图里亚斯语" class="interlanguage-link-target"><span>Asturianu</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/T%C9%99cil" title="Təcil – 阿塞拜疆语" lang="az" hreflang="az" data-title="Təcil" data-language-autonym="Azərbaycanca" data-language-local-name="阿塞拜疆语" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-azb mw-list-item"><a href="https://azb.wikipedia.org/wiki/%D8%A7%DB%8C%D9%88%D9%85%D9%87" title="ایومه – South Azerbaijani" lang="azb" hreflang="azb" data-title="ایومه" data-language-autonym="تۆرکجه" data-language-local-name="South Azerbaijani" class="interlanguage-link-target"><span>تۆرکجه</span></a></li><li class="interlanguage-link interwiki-ba mw-list-item"><a href="https://ba.wikipedia.org/wiki/%D0%A2%D0%B8%D2%99%D0%BB%D3%99%D0%BD%D0%B5%D1%88" title="Тиҙләнеш – 巴什基尔语" lang="ba" hreflang="ba" data-title="Тиҙләнеш" data-language-autonym="Башҡортса" data-language-local-name="巴什基尔语" class="interlanguage-link-target"><span>Башҡортса</span></a></li><li class="interlanguage-link interwiki-bcl mw-list-item"><a href="https://bcl.wikipedia.org/wiki/Akselerasyon" title="Akselerasyon – Central Bikol" lang="bcl" hreflang="bcl" data-title="Akselerasyon" data-language-autonym="Bikol Central" data-language-local-name="Central Bikol" class="interlanguage-link-target"><span>Bikol Central</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%9F%D0%B0%D1%81%D0%BA%D0%B0%D1%80%D1%8D%D0%BD%D0%BD%D0%B5" title="Паскарэнне – 白俄罗斯语" lang="be" hreflang="be" data-title="Паскарэнне" data-language-autonym="Беларуская" data-language-local-name="白俄罗斯语" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-be-x-old mw-list-item"><a href="https://be-tarask.wikipedia.org/wiki/%D0%9F%D0%B0%D1%81%D0%BA%D0%B0%D1%80%D1%8D%D0%BD%D1%8C%D0%BD%D0%B5" title="Паскарэньне – Belarusian (Taraškievica orthography)" lang="be-tarask" hreflang="be-tarask" data-title="Паскарэньне" data-language-autonym="Беларуская (тарашкевіца)" data-language-local-name="Belarusian (Taraškievica orthography)" class="interlanguage-link-target"><span>Беларуская (тарашкевіца)</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%A3%D1%81%D0%BA%D0%BE%D1%80%D0%B5%D0%BD%D0%B8%D0%B5" title="Ускорение – 保加利亚语" lang="bg" hreflang="bg" data-title="Ускорение" data-language-autonym="Български" data-language-local-name="保加利亚语" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bh mw-list-item"><a href="https://bh.wikipedia.org/wiki/%E0%A4%AA%E0%A4%B0%E0%A4%B5%E0%A5%87%E0%A4%97" title="परवेग – Bhojpuri" lang="bh" hreflang="bh" data-title="परवेग" data-language-autonym="भोजपुरी" data-language-local-name="Bhojpuri" class="interlanguage-link-target"><span>भोजपुरी</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%A4%E0%A7%8D%E0%A6%AC%E0%A6%B0%E0%A6%A3" title="ত্বরণ – 孟加拉语" lang="bn" hreflang="bn" data-title="ত্বরণ" data-language-autonym="বাংলা" data-language-local-name="孟加拉语" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-bo mw-list-item"><a href="https://bo.wikipedia.org/wiki/%E0%BD%98%E0%BE%B1%E0%BD%B4%E0%BD%A2%E0%BC%8B%E0%BD%A6%E0%BE%A3%E0%BD%BC%E0%BD%93%E0%BC%8D" title="མྱུར་སྣོན། – 藏语" lang="bo" hreflang="bo" data-title="མྱུར་སྣོན།" data-language-autonym="བོད་ཡིག" data-language-local-name="藏语" class="interlanguage-link-target"><span>བོད་ཡིག</span></a></li><li class="interlanguage-link interwiki-bpy mw-list-item"><a href="https://bpy.wikipedia.org/wiki/%E0%A6%A4%E0%A7%8D%E0%A6%AC%E0%A6%B0%E0%A6%A3" title="ত্বরণ – 比什奴普萊利亞文" lang="bpy" hreflang="bpy" data-title="ত্বরণ" data-language-autonym="বিষ্ণুপ্রিয়া মণিপুরী" data-language-local-name="比什奴普萊利亞文" class="interlanguage-link-target"><span>বিষ্ণুপ্রিয়া মণিপুরী</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Ubrzanje" title="Ubrzanje – 波斯尼亚语" lang="bs" hreflang="bs" data-title="Ubrzanje" data-language-autonym="Bosanski" data-language-local-name="波斯尼亚语" class="interlanguage-link-target"><span>Bosanski</span></a></li><li class="interlanguage-link interwiki-bxr mw-list-item"><a href="https://bxr.wikipedia.org/wiki/%D0%A5%D1%83%D1%80%D0%B4%D0%B0%D0%B4%D1%85%D0%B0%D0%BB" title="Хурдадхал – Russia Buriat" lang="bxr" hreflang="bxr" data-title="Хурдадхал" data-language-autonym="Буряад" data-language-local-name="Russia Buriat" class="interlanguage-link-target"><span>Буряад</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Acceleraci%C3%B3" title="Acceleració – 加泰罗尼亚语" lang="ca" hreflang="ca" data-title="Acceleració" data-language-autonym="Català" data-language-local-name="加泰罗尼亚语" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cdo mw-list-item"><a href="https://cdo.wikipedia.org/wiki/G%C4%83-s%C3%B3k-d%C3%B4" title="Gă-sók-dô – Mindong" lang="cdo" hreflang="cdo" data-title="Gă-sók-dô" data-language-autonym="閩東語 / Mìng-dĕ̤ng-ngṳ̄" data-language-local-name="Mindong" class="interlanguage-link-target"><span>閩東語 / Mìng-dĕ̤ng-ngṳ̄</span></a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%D9%84%DB%95%D8%B2" title="لەز – 中库尔德语" lang="ckb" hreflang="ckb" data-title="لەز" data-language-autonym="کوردی" data-language-local-name="中库尔德语" class="interlanguage-link-target"><span>کوردی</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Zrychlen%C3%AD" title="Zrychlení – 捷克语" lang="cs" hreflang="cs" data-title="Zrychlení" data-language-autonym="Čeština" data-language-local-name="捷克语" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%A5%C4%83%D0%B2%C4%83%D1%80%D1%82%D0%BB%D0%B0%D0%BD%D1%83" title="Хăвăртлану – 楚瓦什语" lang="cv" hreflang="cv" data-title="Хăвăртлану" data-language-autonym="Чӑвашла" data-language-local-name="楚瓦什语" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-cy mw-list-item"><a href="https://cy.wikipedia.org/wiki/Cyflymiad" title="Cyflymiad – 威尔士语" lang="cy" hreflang="cy" data-title="Cyflymiad" data-language-autonym="Cymraeg" data-language-local-name="威尔士语" class="interlanguage-link-target"><span>Cymraeg</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Acceleration" title="Acceleration – 丹麦语" lang="da" hreflang="da" data-title="Acceleration" data-language-autonym="Dansk" data-language-local-name="丹麦语" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Beschleunigung" title="Beschleunigung – 德语" lang="de" hreflang="de" data-title="Beschleunigung" data-language-autonym="Deutsch" data-language-local-name="德语" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%95%CF%80%CE%B9%CF%84%CE%AC%CF%87%CF%85%CE%BD%CF%83%CE%B7" title="Επιτάχυνση – 希腊语" lang="el" hreflang="el" data-title="Επιτάχυνση" data-language-autonym="Ελληνικά" data-language-local-name="希腊语" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-eml mw-list-item"><a href="https://eml.wikipedia.org/wiki/Azelerazi%C3%A5n" title="Azeleraziån – Emiliano-Romagnolo" lang="egl" hreflang="egl" data-title="Azeleraziån" data-language-autonym="Emiliàn e rumagnòl" data-language-local-name="Emiliano-Romagnolo" class="interlanguage-link-target"><span>Emiliàn e rumagnòl</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Acceleration" title="Acceleration – 英语" lang="en" hreflang="en" data-title="Acceleration" data-language-autonym="English" data-language-local-name="英语" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Akcelo" title="Akcelo – 世界语" lang="eo" hreflang="eo" data-title="Akcelo" data-language-autonym="Esperanto" data-language-local-name="世界语" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Aceleraci%C3%B3n" title="Aceleración – 西班牙语" lang="es" hreflang="es" data-title="Aceleración" data-language-autonym="Español" data-language-local-name="西班牙语" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Kiirendus" title="Kiirendus – 爱沙尼亚语" lang="et" hreflang="et" data-title="Kiirendus" data-language-autonym="Eesti" data-language-local-name="爱沙尼亚语" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Azelerazio" title="Azelerazio – 巴斯克语" lang="eu" hreflang="eu" data-title="Azelerazio" data-language-autonym="Euskara" data-language-local-name="巴斯克语" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%B4%D8%AA%D8%A7%D8%A8" title="شتاب – 波斯语" lang="fa" hreflang="fa" data-title="شتاب" data-language-autonym="فارسی" data-language-local-name="波斯语" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Kiihtyvyys" title="Kiihtyvyys – 芬兰语" lang="fi" hreflang="fi" data-title="Kiihtyvyys" data-language-autonym="Suomi" data-language-local-name="芬兰语" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-fiu-vro mw-list-item"><a href="https://fiu-vro.wikipedia.org/wiki/Kip%C3%B5ndus" title="Kipõndus – 佛羅文" lang="vro" hreflang="vro" data-title="Kipõndus" data-language-autonym="Võro" data-language-local-name="佛羅文" class="interlanguage-link-target"><span>Võro</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Acc%C3%A9l%C3%A9ration" title="Accélération – 法语" lang="fr" hreflang="fr" data-title="Accélération" data-language-autonym="Français" data-language-local-name="法语" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-frr mw-list-item"><a href="https://frr.wikipedia.org/wiki/Faardferanrang" title="Faardferanrang – 北弗里西亚语" lang="frr" hreflang="frr" data-title="Faardferanrang" data-language-autonym="Nordfriisk" data-language-local-name="北弗里西亚语" class="interlanguage-link-target"><span>Nordfriisk</span></a></li><li class="interlanguage-link interwiki-fy mw-list-item"><a href="https://fy.wikipedia.org/wiki/Fersnelling_(natuerkunde)" title="Fersnelling (natuerkunde) – 西弗里西亚语" lang="fy" hreflang="fy" data-title="Fersnelling (natuerkunde)" data-language-autonym="Frysk" data-language-local-name="西弗里西亚语" class="interlanguage-link-target"><span>Frysk</span></a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/Luasgh%C3%A9ar%C3%BA" title="Luasghéarú – 爱尔兰语" lang="ga" hreflang="ga" data-title="Luasghéarú" data-language-autonym="Gaeilge" data-language-local-name="爱尔兰语" class="interlanguage-link-target"><span>Gaeilge</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Aceleraci%C3%B3n" title="Aceleración – 加利西亚语" lang="gl" hreflang="gl" data-title="Aceleración" data-language-autonym="Galego" data-language-local-name="加利西亚语" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-gn mw-list-item"><a href="https://gn.wikipedia.org/wiki/%C3%91emoag%CC%83e" title="Ñemoag̃e – 瓜拉尼语" lang="gn" hreflang="gn" data-title="Ñemoag̃e" data-language-autonym="Avañe'ẽ" data-language-local-name="瓜拉尼语" class="interlanguage-link-target"><span>Avañe'ẽ</span></a></li><li class="interlanguage-link interwiki-gv mw-list-item"><a href="https://gv.wikipedia.org/wiki/Siyraghey" title="Siyraghey – 马恩语" lang="gv" hreflang="gv" data-title="Siyraghey" data-language-autonym="Gaelg" data-language-local-name="马恩语" class="interlanguage-link-target"><span>Gaelg</span></a></li><li class="interlanguage-link interwiki-hak mw-list-item"><a href="https://hak.wikipedia.org/wiki/K%C3%A2-suk-thu" title="Kâ-suk-thu – 客家语" lang="hak" hreflang="hak" data-title="Kâ-suk-thu" data-language-autonym="客家語 / Hak-kâ-ngî" data-language-local-name="客家语" class="interlanguage-link-target"><span>客家語 / Hak-kâ-ngî</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%AA%D7%90%D7%95%D7%A6%D7%94" title="תאוצה – 希伯来语" lang="he" hreflang="he" data-title="תאוצה" data-language-autonym="עברית" data-language-local-name="希伯来语" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%A4%E0%A5%8D%E0%A4%B5%E0%A4%B0%E0%A4%A3" title="त्वरण – 印地语" lang="hi" hreflang="hi" data-title="त्वरण" data-language-autonym="हिन्दी" data-language-local-name="印地语" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hif mw-list-item"><a href="https://hif.wikipedia.org/wiki/Acceleration" title="Acceleration – 斐濟印地文" lang="hif" hreflang="hif" data-title="Acceleration" data-language-autonym="Fiji Hindi" data-language-local-name="斐濟印地文" class="interlanguage-link-target"><span>Fiji Hindi</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Ubrzanje" title="Ubrzanje – 克罗地亚语" lang="hr" hreflang="hr" data-title="Ubrzanje" data-language-autonym="Hrvatski" data-language-local-name="克罗地亚语" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-ht mw-list-item"><a href="https://ht.wikipedia.org/wiki/Akselerasyon_santrip%C3%A8d" title="Akselerasyon santripèd – 海地克里奥尔语" lang="ht" hreflang="ht" data-title="Akselerasyon santripèd" data-language-autonym="Kreyòl ayisyen" data-language-local-name="海地克里奥尔语" class="interlanguage-link-target"><span>Kreyòl ayisyen</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Gyorsul%C3%A1s" title="Gyorsulás – 匈牙利语" lang="hu" hreflang="hu" data-title="Gyorsulás" data-language-autonym="Magyar" data-language-local-name="匈牙利语" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D4%B1%D6%80%D5%A1%D5%A3%D5%A1%D6%81%D5%B8%D6%82%D5%B4" title="Արագացում – 亚美尼亚语" lang="hy" hreflang="hy" data-title="Արագացում" data-language-autonym="Հայերեն" data-language-local-name="亚美尼亚语" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-ia mw-list-item"><a href="https://ia.wikipedia.org/wiki/Acceleration" title="Acceleration – 国际语" lang="ia" hreflang="ia" data-title="Acceleration" data-language-autonym="Interlingua" data-language-local-name="国际语" class="interlanguage-link-target"><span>Interlingua</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Percepatan" title="Percepatan – 印度尼西亚语" lang="id" hreflang="id" data-title="Percepatan" data-language-autonym="Bahasa Indonesia" data-language-local-name="印度尼西亚语" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-io mw-list-item"><a href="https://io.wikipedia.org/wiki/Acelero" title="Acelero – 伊多语" lang="io" hreflang="io" data-title="Acelero" data-language-autonym="Ido" data-language-local-name="伊多语" class="interlanguage-link-target"><span>Ido</span></a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/Hr%C3%B6%C3%B0un" title="Hröðun – 冰岛语" lang="is" hreflang="is" data-title="Hröðun" data-language-autonym="Íslenska" data-language-local-name="冰岛语" class="interlanguage-link-target"><span>Íslenska</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Accelerazione" title="Accelerazione – 意大利语" lang="it" hreflang="it" data-title="Accelerazione" data-language-autonym="Italiano" data-language-local-name="意大利语" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E5%8A%A0%E9%80%9F%E5%BA%A6" title="加速度 – 日语" lang="ja" hreflang="ja" data-title="加速度" data-language-autonym="日本語" data-language-local-name="日语" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/%E1%83%90%E1%83%A9%E1%83%A5%E1%83%90%E1%83%A0%E1%83%94%E1%83%91%E1%83%90" title="აჩქარება – 格鲁吉亚语" lang="ka" hreflang="ka" data-title="აჩქარება" data-language-autonym="ქართული" data-language-local-name="格鲁吉亚语" class="interlanguage-link-target"><span>ქართული</span></a></li><li class="interlanguage-link interwiki-kbp mw-list-item"><a href="https://kbp.wikipedia.org/wiki/Ki%C9%96i_ki%C9%96i_wonuu_(acc%C3%A9l%C3%A9ration)" title="Kiɖi kiɖi wonuu (accélération) – Kabiye" lang="kbp" hreflang="kbp" data-title="Kiɖi kiɖi wonuu (accélération)" data-language-autonym="Kabɩyɛ" data-language-local-name="Kabiye" class="interlanguage-link-target"><span>Kabɩyɛ</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D2%AE%D0%B4%D0%B5%D1%83" title="Үдеу – 哈萨克语" lang="kk" hreflang="kk" data-title="Үдеу" data-language-autonym="Қазақша" data-language-local-name="哈萨克语" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-km mw-list-item"><a href="https://km.wikipedia.org/wiki/%E1%9E%9F%E1%9F%86%E1%9E%91%E1%9E%BB%E1%9F%87" title="សំទុះ – 高棉语" lang="km" hreflang="km" data-title="សំទុះ" data-language-autonym="ភាសាខ្មែរ" data-language-local-name="高棉语" class="interlanguage-link-target"><span>ភាសាខ្មែរ</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EA%B0%80%EC%86%8D%EB%8F%84" title="가속도 – 韩语" lang="ko" hreflang="ko" data-title="가속도" data-language-autonym="한국어" data-language-local-name="韩语" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-krc badge-Q17437798 badge-goodarticle mw-list-item" title="优良条目"><a href="https://krc.wikipedia.org/wiki/%D0%A2%D0%B5%D1%80%D0%BA%D0%BB%D0%B5%D0%BD%D0%B8%D1%83" title="Терклениу – 卡拉恰伊巴尔卡尔语" lang="krc" hreflang="krc" data-title="Терклениу" data-language-autonym="Къарачай-малкъар" data-language-local-name="卡拉恰伊巴尔卡尔语" class="interlanguage-link-target"><span>Къарачай-малкъар</span></a></li><li class="interlanguage-link interwiki-ku mw-list-item"><a href="https://ku.wikipedia.org/wiki/Lezdan_(fiz%C3%AEk)" title="Lezdan (fizîk) – 库尔德语" lang="ku" hreflang="ku" data-title="Lezdan (fizîk)" data-language-autonym="Kurdî" data-language-local-name="库尔德语" class="interlanguage-link-target"><span>Kurdî</span></a></li><li class="interlanguage-link interwiki-kw mw-list-item"><a href="https://kw.wikipedia.org/wiki/Uskisheans" title="Uskisheans – 康沃尔语" lang="kw" hreflang="kw" data-title="Uskisheans" data-language-autonym="Kernowek" data-language-local-name="康沃尔语" class="interlanguage-link-target"><span>Kernowek</span></a></li><li class="interlanguage-link interwiki-ky mw-list-item"><a href="https://ky.wikipedia.org/wiki/%D0%AB%D0%BB%D0%B4%D0%B0%D0%BC%D0%B4%D0%B0%D0%BD%D1%83%D1%83" title="Ылдамдануу – 柯尔克孜语" lang="ky" hreflang="ky" data-title="Ылдамдануу" data-language-autonym="Кыргызча" data-language-local-name="柯尔克孜语" class="interlanguage-link-target"><span>Кыргызча</span></a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Acceleratio" title="Acceleratio – 拉丁语" lang="la" hreflang="la" data-title="Acceleratio" data-language-autonym="Latina" data-language-local-name="拉丁语" class="interlanguage-link-target"><span>Latina</span></a></li><li class="interlanguage-link interwiki-lfn mw-list-item"><a href="https://lfn.wikipedia.org/wiki/Aselera" title="Aselera – 新共同語言" lang="lfn" hreflang="lfn" data-title="Aselera" data-language-autonym="Lingua Franca Nova" data-language-local-name="新共同語言" class="interlanguage-link-target"><span>Lingua Franca Nova</span></a></li><li class="interlanguage-link interwiki-lmo mw-list-item"><a href="https://lmo.wikipedia.org/wiki/Acelerazion" title="Acelerazion – 倫巴底文" lang="lmo" hreflang="lmo" data-title="Acelerazion" data-language-autonym="Lombard" data-language-local-name="倫巴底文" class="interlanguage-link-target"><span>Lombard</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Pagreitis" title="Pagreitis – 立陶宛语" lang="lt" hreflang="lt" data-title="Pagreitis" data-language-autonym="Lietuvių" data-language-local-name="立陶宛语" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Pa%C4%81trin%C4%81jums" title="Paātrinājums – 拉脱维亚语" lang="lv" hreflang="lv" data-title="Paātrinājums" data-language-autonym="Latviešu" data-language-local-name="拉脱维亚语" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%97%D0%B0%D0%B1%D1%80%D0%B7%D1%83%D0%B2%D0%B0%D1%9A%D0%B5" title="Забрзување – 马其顿语" lang="mk" hreflang="mk" data-title="Забрзување" data-language-autonym="Македонски" data-language-local-name="马其顿语" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%A4%E0%B5%8D%E0%B4%B5%E0%B4%B0%E0%B4%A3%E0%B4%82" title="ത്വരണം – 马拉雅拉姆语" lang="ml" hreflang="ml" data-title="ത്വരണം" data-language-autonym="മലയാളം" data-language-local-name="马拉雅拉姆语" class="interlanguage-link-target"><span>മലയാളം</span></a></li><li class="interlanguage-link interwiki-mn mw-list-item"><a href="https://mn.wikipedia.org/wiki/%D0%A5%D1%83%D1%80%D0%B4%D0%B0%D1%82%D0%B3%D0%B0%D0%BB" title="Хурдатгал – 蒙古语" lang="mn" hreflang="mn" data-title="Хурдатгал" data-language-autonym="Монгол" data-language-local-name="蒙古语" class="interlanguage-link-target"><span>Монгол</span></a></li><li class="interlanguage-link interwiki-mr mw-list-item"><a href="https://mr.wikipedia.org/wiki/%E0%A4%AA%E0%A5%8D%E0%A4%B0%E0%A4%B5%E0%A5%87%E0%A4%97" title="प्रवेग – 马拉地语" lang="mr" hreflang="mr" data-title="प्रवेग" data-language-autonym="मराठी" data-language-local-name="马拉地语" class="interlanguage-link-target"><span>मराठी</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Pecutan" title="Pecutan – 马来语" lang="ms" hreflang="ms" data-title="Pecutan" data-language-autonym="Bahasa Melayu" data-language-local-name="马来语" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-my mw-list-item"><a href="https://my.wikipedia.org/wiki/%E1%80%A1%E1%80%9C%E1%80%BB%E1%80%84%E1%80%BA%E1%80%95%E1%80%BC%E1%80%B1%E1%80%AC%E1%80%84%E1%80%BA%E1%80%B8%E1%80%94%E1%80%BE%E1%80%AF%E1%80%94%E1%80%BA%E1%80%B8" title="အလျင်ပြောင်းနှုန်း – 缅甸语" lang="my" hreflang="my" data-title="အလျင်ပြောင်းနှုန်း" data-language-autonym="မြန်မာဘာသာ" data-language-local-name="缅甸语" class="interlanguage-link-target"><span>မြန်မာဘာသာ</span></a></li><li class="interlanguage-link interwiki-myv mw-list-item"><a href="https://myv.wikipedia.org/wiki/%D0%9A%D0%B0%D0%BF%D1%88%D0%B0%D0%B2%D1%82%D0%BE%D0%BC%D0%B0" title="Капшавтома – 厄尔兹亚语" lang="myv" hreflang="myv" data-title="Капшавтома" data-language-autonym="Эрзянь" data-language-local-name="厄尔兹亚语" class="interlanguage-link-target"><span>Эрзянь</span></a></li><li class="interlanguage-link interwiki-nap mw-list-item"><a href="https://nap.wikipedia.org/wiki/Accellerazione" title="Accellerazione – 那不勒斯语" lang="nap" hreflang="nap" data-title="Accellerazione" data-language-autonym="Napulitano" data-language-local-name="那不勒斯语" class="interlanguage-link-target"><span>Napulitano</span></a></li><li class="interlanguage-link interwiki-nds mw-list-item"><a href="https://nds.wikipedia.org/wiki/Versnellen" title="Versnellen – 低地德语" lang="nds" hreflang="nds" data-title="Versnellen" data-language-autonym="Plattdüütsch" data-language-local-name="低地德语" class="interlanguage-link-target"><span>Plattdüütsch</span></a></li><li class="interlanguage-link interwiki-ne mw-list-item"><a href="https://ne.wikipedia.org/wiki/%E0%A4%AA%E0%A5%8D%E0%A4%B0%E0%A4%B5%E0%A5%87%E0%A4%97" title="प्रवेग – 尼泊尔语" lang="ne" hreflang="ne" data-title="प्रवेग" data-language-autonym="नेपाली" data-language-local-name="尼泊尔语" class="interlanguage-link-target"><span>नेपाली</span></a></li><li class="interlanguage-link interwiki-new mw-list-item"><a href="https://new.wikipedia.org/wiki/%E0%A4%AA%E0%A5%8D%E0%A4%B0%E0%A4%B5%E0%A5%87%E0%A4%97" title="प्रवेग – 尼瓦尔语" lang="new" hreflang="new" data-title="प्रवेग" data-language-autonym="नेपाल भाषा" data-language-local-name="尼瓦尔语" class="interlanguage-link-target"><span>नेपाल भाषा</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Versnelling_(natuurkunde)" title="Versnelling (natuurkunde) – 荷兰语" lang="nl" hreflang="nl" data-title="Versnelling (natuurkunde)" data-language-autonym="Nederlands" data-language-local-name="荷兰语" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Akselerasjon" title="Akselerasjon – 挪威尼诺斯克语" lang="nn" hreflang="nn" data-title="Akselerasjon" data-language-autonym="Norsk nynorsk" data-language-local-name="挪威尼诺斯克语" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Akselerasjon" title="Akselerasjon – 书面挪威语" lang="nb" hreflang="nb" data-title="Akselerasjon" data-language-autonym="Norsk bokmål" data-language-local-name="书面挪威语" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-nov mw-list-item"><a href="https://nov.wikipedia.org/wiki/Akseleratione" title="Akseleratione – 諾維亞文" lang="nov" hreflang="nov" data-title="Akseleratione" data-language-autonym="Novial" data-language-local-name="諾維亞文" class="interlanguage-link-target"><span>Novial</span></a></li><li class="interlanguage-link interwiki-oc mw-list-item"><a href="https://oc.wikipedia.org/wiki/Acceleracion" title="Acceleracion – 奥克语" lang="oc" hreflang="oc" data-title="Acceleracion" data-language-autonym="Occitan" data-language-local-name="奥克语" class="interlanguage-link-target"><span>Occitan</span></a></li><li class="interlanguage-link interwiki-pa mw-list-item"><a href="https://pa.wikipedia.org/wiki/%E0%A8%AA%E0%A9%8D%E0%A8%B0%E0%A8%B5%E0%A9%87%E0%A8%97" title="ਪ੍ਰਵੇਗ – 旁遮普语" lang="pa" hreflang="pa" data-title="ਪ੍ਰਵੇਗ" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="旁遮普语" class="interlanguage-link-target"><span>ਪੰਜਾਬੀ</span></a></li><li class="interlanguage-link interwiki-pag mw-list-item"><a href="https://pag.wikipedia.org/wiki/Akpelusyon" title="Akpelusyon – 邦阿西南语" lang="pag" hreflang="pag" data-title="Akpelusyon" data-language-autonym="Pangasinan" data-language-local-name="邦阿西南语" class="interlanguage-link-target"><span>Pangasinan</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Przyspieszenie" title="Przyspieszenie – 波兰语" lang="pl" hreflang="pl" data-title="Przyspieszenie" data-language-autonym="Polski" data-language-local-name="波兰语" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pms mw-list-item"><a href="https://pms.wikipedia.org/wiki/Acelerassion" title="Acelerassion – 皮埃蒙特文" lang="pms" hreflang="pms" data-title="Acelerassion" data-language-autonym="Piemontèis" data-language-local-name="皮埃蒙特文" class="interlanguage-link-target"><span>Piemontèis</span></a></li><li class="interlanguage-link interwiki-pnb mw-list-item"><a href="https://pnb.wikipedia.org/wiki/%D8%A7%D8%B3%D8%B1%D8%A7%D8%B9" title="اسراع – Western Punjabi" lang="pnb" hreflang="pnb" data-title="اسراع" data-language-autonym="پنجابی" data-language-local-name="Western Punjabi" class="interlanguage-link-target"><span>پنجابی</span></a></li><li class="interlanguage-link interwiki-ps mw-list-item"><a href="https://ps.wikipedia.org/wiki/%D8%B3%D8%B1%D8%B9%D8%AA" title="سرعت – 普什图语" lang="ps" hreflang="ps" data-title="سرعت" data-language-autonym="پښتو" data-language-local-name="普什图语" class="interlanguage-link-target"><span>پښتو</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Acelera%C3%A7%C3%A3o" title="Aceleração – 葡萄牙语" lang="pt" hreflang="pt" data-title="Aceleração" data-language-autonym="Português" data-language-local-name="葡萄牙语" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-qu mw-list-item"><a href="https://qu.wikipedia.org/wiki/P%27ikwachiy" title="P'ikwachiy – 克丘亚语" lang="qu" hreflang="qu" data-title="P'ikwachiy" data-language-autonym="Runa Simi" data-language-local-name="克丘亚语" class="interlanguage-link-target"><span>Runa Simi</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Accelera%C8%9Bie" title="Accelerație – 罗马尼亚语" lang="ro" hreflang="ro" data-title="Accelerație" data-language-autonym="Română" data-language-local-name="罗马尼亚语" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%A3%D1%81%D0%BA%D0%BE%D1%80%D0%B5%D0%BD%D0%B8%D0%B5" title="Ускорение – 俄语" lang="ru" hreflang="ru" data-title="Ускорение" data-language-autonym="Русский" data-language-local-name="俄语" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-rue mw-list-item"><a href="https://rue.wikipedia.org/wiki/%D0%90%D0%BA%D1%86%D0%B5%D0%BB%D0%B5%D1%80%D0%B0%D1%86%D1%96%D1%8F" title="Акцелерація – 盧森尼亞文" lang="rue" hreflang="rue" data-title="Акцелерація" data-language-autonym="Русиньскый" data-language-local-name="盧森尼亞文" class="interlanguage-link-target"><span>Русиньскый</span></a></li><li class="interlanguage-link interwiki-sc mw-list-item"><a href="https://sc.wikipedia.org/wiki/Atzellerada" title="Atzellerada – 萨丁语" lang="sc" hreflang="sc" data-title="Atzellerada" data-language-autonym="Sardu" data-language-local-name="萨丁语" class="interlanguage-link-target"><span>Sardu</span></a></li><li class="interlanguage-link interwiki-scn mw-list-item"><a href="https://scn.wikipedia.org/wiki/Accilirazzioni" title="Accilirazzioni – 西西里语" lang="scn" hreflang="scn" data-title="Accilirazzioni" data-language-autonym="Sicilianu" data-language-local-name="西西里语" class="interlanguage-link-target"><span>Sicilianu</span></a></li><li class="interlanguage-link interwiki-sd mw-list-item"><a href="https://sd.wikipedia.org/wiki/%D8%A7%D8%B3%D8%B1%D8%A7%D8%B9" title="اسراع – 信德语" lang="sd" hreflang="sd" data-title="اسراع" data-language-autonym="سنڌي" data-language-local-name="信德语" class="interlanguage-link-target"><span>سنڌي</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Ubrzanje" title="Ubrzanje – 塞尔维亚-克罗地亚语" lang="sh" hreflang="sh" data-title="Ubrzanje" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="塞尔维亚-克罗地亚语" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-si mw-list-item"><a href="https://si.wikipedia.org/wiki/%E0%B6%AD%E0%B7%8A%E0%B7%80%E0%B6%BB%E0%B6%AB%E0%B6%BA" title="ත්වරණය – 僧伽罗语" lang="si" hreflang="si" data-title="ත්වරණය" data-language-autonym="සිංහල" data-language-local-name="僧伽罗语" class="interlanguage-link-target"><span>සිංහල</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Acceleration" title="Acceleration – Simple English" lang="en-simple" hreflang="en-simple" data-title="Acceleration" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Zr%C3%BDchlenie" title="Zrýchlenie – 斯洛伐克语" lang="sk" hreflang="sk" data-title="Zrýchlenie" data-language-autonym="Slovenčina" data-language-local-name="斯洛伐克语" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Pospe%C5%A1ek" title="Pospešek – 斯洛文尼亚语" lang="sl" hreflang="sl" data-title="Pospešek" data-language-autonym="Slovenščina" data-language-local-name="斯洛文尼亚语" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sn mw-list-item"><a href="https://sn.wikipedia.org/wiki/Rufangu" title="Rufangu – 绍纳语" lang="sn" hreflang="sn" data-title="Rufangu" data-language-autonym="ChiShona" data-language-local-name="绍纳语" class="interlanguage-link-target"><span>ChiShona</span></a></li><li class="interlanguage-link interwiki-so mw-list-item"><a href="https://so.wikipedia.org/wiki/Dardargelinta" title="Dardargelinta – 索马里语" lang="so" hreflang="so" data-title="Dardargelinta" data-language-autonym="Soomaaliga" data-language-local-name="索马里语" class="interlanguage-link-target"><span>Soomaaliga</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Nxitimi" title="Nxitimi – 阿尔巴尼亚语" lang="sq" hreflang="sq" data-title="Nxitimi" data-language-autonym="Shqip" data-language-local-name="阿尔巴尼亚语" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%A3%D0%B1%D1%80%D0%B7%D0%B0%D1%9A%D0%B5" title="Убрзање – 塞尔维亚语" lang="sr" hreflang="sr" data-title="Убрзање" data-language-autonym="Српски / srpski" data-language-local-name="塞尔维亚语" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-su mw-list-item"><a href="https://su.wikipedia.org/wiki/Aksel%C3%A9rasi" title="Akselérasi – 巽他语" lang="su" hreflang="su" data-title="Akselérasi" data-language-autonym="Sunda" data-language-local-name="巽他语" class="interlanguage-link-target"><span>Sunda</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Acceleration" title="Acceleration – 瑞典语" lang="sv" hreflang="sv" data-title="Acceleration" data-language-autonym="Svenska" data-language-local-name="瑞典语" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-sw mw-list-item"><a href="https://sw.wikipedia.org/wiki/Mchapuko" title="Mchapuko – 斯瓦希里语" lang="sw" hreflang="sw" data-title="Mchapuko" data-language-autonym="Kiswahili" data-language-local-name="斯瓦希里语" class="interlanguage-link-target"><span>Kiswahili</span></a></li><li class="interlanguage-link interwiki-szl mw-list-item"><a href="https://szl.wikipedia.org/wiki/Szw%C5%AFng" title="Szwůng – 西里西亚语" lang="szl" hreflang="szl" data-title="Szwůng" data-language-autonym="Ślůnski" data-language-local-name="西里西亚语" class="interlanguage-link-target"><span>Ślůnski</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%AE%E0%AF%81%E0%AE%9F%E0%AF%81%E0%AE%95%E0%AF%8D%E0%AE%95%E0%AE%AE%E0%AF%8D" title="முடுக்கம் – 泰米尔语" lang="ta" hreflang="ta" data-title="முடுக்கம்" data-language-autonym="தமிழ்" data-language-local-name="泰米尔语" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-te mw-list-item"><a href="https://te.wikipedia.org/wiki/%E0%B0%A4%E0%B1%8D%E0%B0%B5%E0%B0%B0%E0%B0%A3%E0%B0%AE%E0%B1%81" title="త్వరణము – 泰卢固语" lang="te" hreflang="te" data-title="త్వరణము" data-language-autonym="తెలుగు" data-language-local-name="泰卢固语" class="interlanguage-link-target"><span>తెలుగు</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%84%E0%B8%A7%E0%B8%B2%E0%B8%A1%E0%B9%80%E0%B8%A3%E0%B9%88%E0%B8%87" title="ความเร่ง – 泰语" lang="th" hreflang="th" data-title="ความเร่ง" data-language-autonym="ไทย" data-language-local-name="泰语" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tl mw-list-item"><a href="https://tl.wikipedia.org/wiki/Arangkada" title="Arangkada – 他加禄语" lang="tl" hreflang="tl" data-title="Arangkada" data-language-autonym="Tagalog" data-language-local-name="他加禄语" class="interlanguage-link-target"><span>Tagalog</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/%C4%B0vme" title="İvme – 土耳其语" lang="tr" hreflang="tr" data-title="İvme" data-language-autonym="Türkçe" data-language-local-name="土耳其语" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-tt mw-list-item"><a href="https://tt.wikipedia.org/wiki/%D0%A2%D0%B8%D0%B7%D0%BB%D3%99%D0%BD%D0%B5%D1%88" title="Тизләнеш – 鞑靼语" lang="tt" hreflang="tt" data-title="Тизләнеш" data-language-autonym="Татарча / tatarça" data-language-local-name="鞑靼语" class="interlanguage-link-target"><span>Татарча / tatarça</span></a></li><li class="interlanguage-link interwiki-tyv mw-list-item"><a href="https://tyv.wikipedia.org/wiki/%D0%94%D2%AF%D1%80%D0%B3%D0%B5%D0%B4%D1%8D%D1%8D%D1%88%D0%BA%D0%B8%D0%BD" title="Дүргедээшкин – 图瓦语" lang="tyv" hreflang="tyv" data-title="Дүргедээшкин" data-language-autonym="Тыва дыл" data-language-local-name="图瓦语" class="interlanguage-link-target"><span>Тыва дыл</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%9F%D1%80%D0%B8%D1%81%D0%BA%D0%BE%D1%80%D0%B5%D0%BD%D0%BD%D1%8F" title="Прискорення – 乌克兰语" lang="uk" hreflang="uk" data-title="Прискорення" data-language-autonym="Українська" data-language-local-name="乌克兰语" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-ur mw-list-item"><a href="https://ur.wikipedia.org/wiki/%D8%A7%D8%B3%D8%B1%D8%A7%D8%B9" title="اسراع – 乌尔都语" lang="ur" hreflang="ur" data-title="اسراع" data-language-autonym="اردو" data-language-local-name="乌尔都语" class="interlanguage-link-target"><span>اردو</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Tezlanish" title="Tezlanish – 乌兹别克语" lang="uz" hreflang="uz" data-title="Tezlanish" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="乌兹别克语" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></a></li><li class="interlanguage-link interwiki-vec mw-list-item"><a href="https://vec.wikipedia.org/wiki/Acelerasion" title="Acelerasion – 威尼斯语" lang="vec" hreflang="vec" data-title="Acelerasion" data-language-autonym="Vèneto" data-language-local-name="威尼斯语" class="interlanguage-link-target"><span>Vèneto</span></a></li><li class="interlanguage-link interwiki-vep mw-list-item"><a href="https://vep.wikipedia.org/wiki/Pigustuz" title="Pigustuz – 维普森语" lang="vep" hreflang="vep" data-title="Pigustuz" data-language-autonym="Vepsän kel’" data-language-local-name="维普森语" class="interlanguage-link-target"><span>Vepsän kel’</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/Gia_t%E1%BB%91c" title="Gia tốc – 越南语" lang="vi" hreflang="vi" data-title="Gia tốc" data-language-autonym="Tiếng Việt" data-language-local-name="越南语" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-war mw-list-item"><a href="https://war.wikipedia.org/wiki/Akselerasyon" title="Akselerasyon – 瓦瑞语" lang="war" hreflang="war" data-title="Akselerasyon" data-language-autonym="Winaray" data-language-local-name="瓦瑞语" class="interlanguage-link-target"><span>Winaray</span></a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E5%8A%A0%E9%80%9F%E5%BA%A6" title="加速度 – 吴语" lang="wuu" hreflang="wuu" data-title="加速度" data-language-autonym="吴语" data-language-local-name="吴语" class="interlanguage-link-target"><span>吴语</span></a></li><li class="interlanguage-link interwiki-yi mw-list-item"><a href="https://yi.wikipedia.org/wiki/%D7%A4%D7%90%D7%A8%D7%92%D7%99%D7%9B%D7%A2%D7%A8%D7%95%D7%A0%D7%92" title="פארגיכערונג – 意第绪语" lang="yi" hreflang="yi" data-title="פארגיכערונג" data-language-autonym="ייִדיש" data-language-local-name="意第绪语" class="interlanguage-link-target"><span>ייִדיש</span></a></li><li class="interlanguage-link interwiki-yo mw-list-item"><a href="https://yo.wikipedia.org/wiki/%C3%8Cy%C3%A1ra" title="Ìyára – 约鲁巴语" lang="yo" hreflang="yo" data-title="Ìyára" data-language-autonym="Yorùbá" data-language-local-name="约鲁巴语" class="interlanguage-link-target"><span>Yorùbá</span></a></li><li class="interlanguage-link interwiki-zh-classical mw-list-item"><a href="https://zh-classical.wikipedia.org/wiki/%E5%8A%A0%E9%80%9F%E5%BA%A6" title="加速度 – 文言文" lang="lzh" hreflang="lzh" data-title="加速度" data-language-autonym="文言" data-language-local-name="文言文" class="interlanguage-link-target"><span>文言</span></a></li><li class="interlanguage-link interwiki-zh-min-nan mw-list-item"><a href="https://zh-min-nan.wikipedia.org/wiki/Ka-sok-t%C5%8D%CD%98" title="Ka-sok-tō͘ – 闽南语" lang="nan" hreflang="nan" data-title="Ka-sok-tō͘" data-language-autonym="閩南語 / Bân-lâm-gú" data-language-local-name="闽南语" class="interlanguage-link-target"><span>閩南語 / Bân-lâm-gú</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E5%8A%A0%E9%80%9F%E5%BA%A6" title="加速度 – 粤语" lang="yue" hreflang="yue" data-title="加速度" data-language-autonym="粵語" data-language-local-name="粤语" class="interlanguage-link-target"><span>粵語</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q11376#sitelinks-wikipedia" title="编辑跨语言链接" class="wbc-editpage">编辑链接</a></span></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div class="vector-page-toolbar-container"> <div id="left-navigation"> <nav aria-label="命名空间"> <div id="p-associated-pages" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-associated-pages" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-nstab-main" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/%E5%8A%A0%E9%80%9F%E5%BA%A6" title="浏览条目正文[c]" accesskey="c"><span>条目</span></a></li><li id="ca-talk" class="vector-tab-noicon mw-list-item"><a href="/wiki/Talk:%E5%8A%A0%E9%80%9F%E5%BA%A6" rel="discussion" title="关于此页面的讨论[t]" accesskey="t"><span>讨论</span></a></li> </ul> </div> </div> <div id="vector-variants-dropdown" class="vector-dropdown " > <input type="checkbox" id="vector-variants-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-variants-dropdown" class="vector-dropdown-checkbox " aria-label="更改语言变体" > <label id="vector-variants-dropdown-label" for="vector-variants-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" ><span class="vector-dropdown-label-text">不转换</span> </label> <div class="vector-dropdown-content"> <div id="p-variants" class="vector-menu mw-portlet mw-portlet-variants" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-varlang-0" class="selected ca-variants-zh mw-list-item"><a href="/zh/%E5%8A%A0%E9%80%9F%E5%BA%A6" lang="zh" hreflang="zh"><span>不转换</span></a></li><li id="ca-varlang-1" class="ca-variants-zh-Hans mw-list-item"><a href="/zh-hans/%E5%8A%A0%E9%80%9F%E5%BA%A6" lang="zh-Hans" hreflang="zh-Hans"><span>简体</span></a></li><li id="ca-varlang-2" class="ca-variants-zh-Hant mw-list-item"><a href="/zh-hant/%E5%8A%A0%E9%80%9F%E5%BA%A6" lang="zh-Hant" hreflang="zh-Hant"><span>繁體</span></a></li><li id="ca-varlang-3" class="ca-variants-zh-Hans-CN mw-list-item"><a href="/zh-cn/%E5%8A%A0%E9%80%9F%E5%BA%A6" lang="zh-Hans-CN" hreflang="zh-Hans-CN"><span>大陆简体</span></a></li><li id="ca-varlang-4" class="ca-variants-zh-Hant-HK mw-list-item"><a href="/zh-hk/%E5%8A%A0%E9%80%9F%E5%BA%A6" lang="zh-Hant-HK" hreflang="zh-Hant-HK"><span>香港繁體</span></a></li><li id="ca-varlang-5" class="ca-variants-zh-Hant-MO mw-list-item"><a href="/zh-mo/%E5%8A%A0%E9%80%9F%E5%BA%A6" lang="zh-Hant-MO" hreflang="zh-Hant-MO"><span>澳門繁體</span></a></li><li id="ca-varlang-6" class="ca-variants-zh-Hans-MY mw-list-item"><a href="/zh-my/%E5%8A%A0%E9%80%9F%E5%BA%A6" lang="zh-Hans-MY" hreflang="zh-Hans-MY"><span>大马简体</span></a></li><li id="ca-varlang-7" class="ca-variants-zh-Hans-SG mw-list-item"><a href="/zh-sg/%E5%8A%A0%E9%80%9F%E5%BA%A6" lang="zh-Hans-SG" hreflang="zh-Hans-SG"><span>新加坡简体</span></a></li><li id="ca-varlang-8" class="ca-variants-zh-Hant-TW mw-list-item"><a href="/zh-tw/%E5%8A%A0%E9%80%9F%E5%BA%A6" lang="zh-Hant-TW" hreflang="zh-Hant-TW"><span>臺灣正體</span></a></li> </ul> </div> </div> </div> </div> </nav> </div> <div id="right-navigation" class="vector-collapsible"> <nav aria-label="查看"> <div id="p-views" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-views" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-view" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/%E5%8A%A0%E9%80%9F%E5%BA%A6"><span>阅读</span></a></li><li id="ca-edit" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=%E5%8A%A0%E9%80%9F%E5%BA%A6&action=edit" title="编辑该页面[e]" accesskey="e"><span>编辑</span></a></li><li id="ca-history" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=%E5%8A%A0%E9%80%9F%E5%BA%A6&action=history" title="本页面的早前版本。[h]" accesskey="h"><span>查看历史</span></a></li> </ul> </div> </div> </nav> <nav class="vector-page-tools-landmark" aria-label="页面工具"> <div id="vector-page-tools-dropdown" class="vector-dropdown vector-page-tools-dropdown" > <input type="checkbox" id="vector-page-tools-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-tools-dropdown" class="vector-dropdown-checkbox " aria-label="工具" > <label id="vector-page-tools-dropdown-label" for="vector-page-tools-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" ><span class="vector-dropdown-label-text">工具</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-tools-unpinned-container" class="vector-unpinned-container"> <div id="vector-page-tools" class="vector-page-tools vector-pinnable-element"> <div class="vector-pinnable-header vector-page-tools-pinnable-header vector-pinnable-header-unpinned" data-feature-name="page-tools-pinned" data-pinnable-element-id="vector-page-tools" data-pinned-container-id="vector-page-tools-pinned-container" data-unpinned-container-id="vector-page-tools-unpinned-container" > <div class="vector-pinnable-header-label">工具</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-page-tools.pin">移至侧栏</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-page-tools.unpin">隐藏</button> </div> <div id="p-cactions" class="vector-menu mw-portlet mw-portlet-cactions emptyPortlet vector-has-collapsible-items" title="更多选项" > <div class="vector-menu-heading"> 操作 </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-more-view" class="selected vector-more-collapsible-item mw-list-item"><a href="/wiki/%E5%8A%A0%E9%80%9F%E5%BA%A6"><span>阅读</span></a></li><li id="ca-more-edit" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=%E5%8A%A0%E9%80%9F%E5%BA%A6&action=edit" title="编辑该页面[e]" accesskey="e"><span>编辑</span></a></li><li id="ca-more-history" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=%E5%8A%A0%E9%80%9F%E5%BA%A6&action=history"><span>查看历史</span></a></li> </ul> </div> </div> <div id="p-tb" class="vector-menu mw-portlet mw-portlet-tb" > <div class="vector-menu-heading"> 常规 </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="t-whatlinkshere" class="mw-list-item"><a href="/wiki/Special:%E9%93%BE%E5%85%A5%E9%A1%B5%E9%9D%A2/%E5%8A%A0%E9%80%9F%E5%BA%A6" title="列出所有与本页相链的页面[j]" accesskey="j"><span>链入页面</span></a></li><li id="t-recentchangeslinked" class="mw-list-item"><a href="/wiki/Special:%E9%93%BE%E5%87%BA%E6%9B%B4%E6%94%B9/%E5%8A%A0%E9%80%9F%E5%BA%A6" rel="nofollow" title="页面链出所有页面的更改[k]" accesskey="k"><span>相关更改</span></a></li><li id="t-upload" class="mw-list-item"><a href="/wiki/Project:%E4%B8%8A%E4%BC%A0" title="上传图像或多媒体文件[u]" accesskey="u"><span>上传文件</span></a></li><li id="t-specialpages" class="mw-list-item"><a href="/wiki/Special:%E7%89%B9%E6%AE%8A%E9%A1%B5%E9%9D%A2" title="全部特殊页面的列表[q]" accesskey="q"><span>特殊页面</span></a></li><li id="t-permalink" class="mw-list-item"><a href="/w/index.php?title=%E5%8A%A0%E9%80%9F%E5%BA%A6&oldid=83276899" title="此页面该修订版本的固定链接"><span>固定链接</span></a></li><li id="t-info" class="mw-list-item"><a href="/w/index.php?title=%E5%8A%A0%E9%80%9F%E5%BA%A6&action=info" title="关于此页面的更多信息"><span>页面信息</span></a></li><li id="t-cite" class="mw-list-item"><a href="/w/index.php?title=Special:%E5%BC%95%E7%94%A8%E6%AD%A4%E9%A1%B5%E9%9D%A2&page=%E5%8A%A0%E9%80%9F%E5%BA%A6&id=83276899&wpFormIdentifier=titleform" title="有关如何引用此页面的信息"><span>引用此页</span></a></li><li id="t-urlshortener" class="mw-list-item"><a href="/w/index.php?title=Special:URL%E7%BC%A9%E7%9F%AD%E7%A8%8B%E5%BA%8F&url=https%3A%2F%2Fzh.wikipedia.org%2Fwiki%2F%25E5%258A%25A0%25E9%2580%259F%25E5%25BA%25A6"><span>获取短链接</span></a></li><li id="t-urlshortener-qrcode" class="mw-list-item"><a href="/w/index.php?title=Special:QrCode&url=https%3A%2F%2Fzh.wikipedia.org%2Fwiki%2F%25E5%258A%25A0%25E9%2580%259F%25E5%25BA%25A6"><span>下载二维码</span></a></li> </ul> </div> </div> <div id="p-electronpdfservice-sidebar-portlet-heading" class="vector-menu mw-portlet mw-portlet-electronpdfservice-sidebar-portlet-heading" > <div class="vector-menu-heading"> 打印/导出 </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="electron-print_pdf" class="mw-list-item"><a href="/w/index.php?title=Special:DownloadAsPdf&page=%E5%8A%A0%E9%80%9F%E5%BA%A6&action=show-download-screen"><span>下载为PDF</span></a></li><li id="t-print" class="mw-list-item"><a href="javascript:print();" rel="alternate" title="本页面的可打印版本[p]" accesskey="p"><span>打印页面</span></a></li> </ul> </div> </div> <div id="p-wikibase-otherprojects" class="vector-menu mw-portlet mw-portlet-wikibase-otherprojects" > <div class="vector-menu-heading"> 在其他项目中 </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="wb-otherproject-link wb-otherproject-commons mw-list-item"><a href="https://commons.wikimedia.org/wiki/Category:Acceleration" hreflang="en"><span>维基共享资源</span></a></li><li id="t-wikibase" class="wb-otherproject-link wb-otherproject-wikibase-dataitem mw-list-item"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q11376" title="链接到连接的数据仓库项目[g]" accesskey="g"><span>维基数据项目</span></a></li> </ul> </div> </div> </div> </div> </div> </div> </nav> </div> </div> </div> <div class="vector-column-end"> <div class="vector-sticky-pinned-container"> <nav class="vector-page-tools-landmark" aria-label="页面工具"> <div id="vector-page-tools-pinned-container" class="vector-pinned-container"> </div> </nav> <nav class="vector-appearance-landmark" aria-label="外观"> <div id="vector-appearance-pinned-container" class="vector-pinned-container"> <div id="vector-appearance" class="vector-appearance vector-pinnable-element"> <div class="vector-pinnable-header vector-appearance-pinnable-header vector-pinnable-header-pinned" data-feature-name="appearance-pinned" data-pinnable-element-id="vector-appearance" data-pinned-container-id="vector-appearance-pinned-container" data-unpinned-container-id="vector-appearance-unpinned-container" > <div class="vector-pinnable-header-label">外观</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">移至侧栏</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">隐藏</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> <div id="mw-indicator-noteTA-5ff6fd68" class="mw-indicator"><div class="mw-parser-output"><span class="skin-invert" typeof="mw:File"><span title="本页使用了标题或全文手工转换"><img alt="本页使用了标题或全文手工转换" src="//upload.wikimedia.org/wikipedia/commons/thumb/c/cd/Zh_conversion_icon_m.svg/35px-Zh_conversion_icon_m.svg.png" decoding="async" width="35" height="22" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/cd/Zh_conversion_icon_m.svg/53px-Zh_conversion_icon_m.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/cd/Zh_conversion_icon_m.svg/70px-Zh_conversion_icon_m.svg.png 2x" data-file-width="32" data-file-height="20" /></span></span></div></div> </div> <div id="siteSub" class="noprint">维基百科，自由的百科全书</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="zh" dir="ltr"><style data-mw-deduplicate="TemplateStyles:r84833064">.mw-parser-output .hatnote{font-size:small}.mw-parser-output div.hatnote{padding-left:2em;margin-bottom:0.8em;margin-top:0.8em}.mw-parser-output .hatnote-notice-img::after{content:"\202f \202f \202f \202f "}.mw-parser-output .hatnote-notice-img-small::after{content:"\202f \202f "}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}body.skin-minerva .mw-parser-output .hatnote-notice-img,body.skin-minerva .mw-parser-output .hatnote-notice-img-small{display:none}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable"><span typeof="mw:File"><a href="/wiki/Wikipedia:%E6%B6%88%E6%AD%A7%E4%B9%89" title="Wikipedia:消歧义"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/5/5f/Disambig_gray.svg/25px-Disambig_gray.svg.png" decoding="async" width="25" height="19" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/5f/Disambig_gray.svg/38px-Disambig_gray.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/5f/Disambig_gray.svg/50px-Disambig_gray.svg.png 2x" data-file-width="220" data-file-height="168" /></a></span><style data-mw-deduplicate="TemplateStyles:r74069148">body:not(.skin-minerva) .mw-parser-output .ifmobile>.mobile{display:none}body.skin-minerva .mw-parser-output .ifmobile>.nomobile{display:inherit;display:initial}</style><span class="ifmobile"><span class="nomobile">  </span><span class="mobile"></span></span>「<b>加速</b>」重定向至此。关于社会理论，请见「<b><a href="/wiki/%E5%8A%A0%E9%80%9F%E4%B8%BB%E4%B9%89" title="加速主义">加速主义</a></b>」。</div> <style data-mw-deduplicate="TemplateStyles:r83732972">.mw-parser-output .ambox{border:1px solid #a2a9b1;border-left:10px solid #36c;background-color:#fbfbfb;box-sizing:border-box}.mw-parser-output .ambox+link+.ambox,.mw-parser-output .ambox+link+style+.ambox,.mw-parser-output .ambox+link+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+style+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+link+.ambox{margin-top:-1px}html body.mediawiki .mw-parser-output .ambox.mbox-small-left{margin:4px 1em 4px 0;overflow:hidden;width:238px;border-collapse:collapse;font-size:88%;line-height:1.25em}.mw-parser-output .ambox-speedy{border-left:10px solid #b32424;background-color:#fee7e6}.mw-parser-output .ambox-delete{border-left:10px solid #b32424}.mw-parser-output .ambox-content{border-left:10px solid #f28500}.mw-parser-output .ambox-style{border-left:10px solid #fc3}.mw-parser-output .ambox-move{border-left:10px solid #9932cc}.mw-parser-output .ambox-protection{border-left:10px solid #a2a9b1}.mw-parser-output .ambox .mbox-text{border:none;padding:0.25em 0.5em;width:100%}.mw-parser-output .ambox .mbox-image{border:none;padding:2px 0 2px 0.5em;text-align:center}.mw-parser-output .ambox .mbox-imageright{border:none;padding:2px 0.5em 2px 0;text-align:center}.mw-parser-output .ambox .mbox-empty-cell{border:none;padding:0;width:1px}.mw-parser-output .ambox .mbox-image-div{width:52px}html.client-js body.skin-minerva .mw-parser-output .mbox-text-span{margin-left:23px!important}@media(min-width:720px){.mw-parser-output .ambox{margin:0 10%}}@media screen{html.skin-theme-clientpref-night .mw-parser-output .ambox{border-left-color:#36c!important}html.skin-theme-clientpref-night .mw-parser-output .ambox-speedy,html.skin-theme-clientpref-night .mw-parser-output .ambox-delete{border-left-color:#b32424!important}html.skin-theme-clientpref-night .mw-parser-output .ambox-speedy{background-color:#300!important}html.skin-theme-clientpref-night .mw-parser-output .ambox-content{border-left-color:#f28500!important}html.skin-theme-clientpref-night .mw-parser-output .ambox-style{border-left-color:#fc3!important}html.skin-theme-clientpref-night .mw-parser-output .ambox-move{border-left-color:#9932cc!important}html.skin-theme-clientpref-night .mw-parser-output .ambox-protection{border-left-color:#a2a9b1!important}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .ambox{border-left-color:#36c!important}html.skin-theme-clientpref-os .mw-parser-output .ambox-speedy,html.skin-theme-clientpref-os .mw-parser-output .ambox-delete{border-left-color:#b32424!important}html.skin-theme-clientpref-os .mw-parser-output .ambox-speedy{background-color:#300!important}html.skin-theme-clientpref-os .mw-parser-output .ambox-content{border-left-color:#f28500!important}html.skin-theme-clientpref-os .mw-parser-output .ambox-style{border-left-color:#fc3!important}html.skin-theme-clientpref-os .mw-parser-output .ambox-move{border-left-color:#9932cc!important}html.skin-theme-clientpref-os .mw-parser-output .ambox-protection{border-left-color:#a2a9b1!important}}</style><table class="box-Citation_style plainlinks metadata ambox ambox-style ambox-citation_style" role="presentation"><tbody><tr><td class="mbox-image"><div style="width:52px"><span typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f2/Edit-clear.svg/40px-Edit-clear.svg.png" decoding="async" width="40" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f2/Edit-clear.svg/60px-Edit-clear.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f2/Edit-clear.svg/80px-Edit-clear.svg.png 2x" data-file-width="48" data-file-height="48" /></span></span></div></td><td class="mbox-text"><div class="mbox-text-span">此條目的<b><a href="/wiki/Wikipedia:%E5%88%97%E6%98%8E%E6%9D%A5%E6%BA%90#文獻參考的格式" title="Wikipedia:列明来源">引用需要清理，使其符合格式</a></b>。<span class="hide-when-compact"></span> <small class="date-container"><i>(<span class="date">2017年7月4日</span>)</i></small><span class="hide-when-compact"><br /><small>参考文献应符合正确的<a href="/wiki/Wikipedia:%E5%88%97%E6%98%8E%E6%9D%A5%E6%BA%90" title="Wikipedia:列明来源">引用</a>、<a href="/wiki/Help:%E8%84%9A%E6%B3%A8" title="Help:脚注">脚注</a>及<a href="/wiki/Wikipedia:%E5%A4%96%E9%83%A8%E9%93%BE%E6%8E%A5" title="Wikipedia:外部链接">外部链接</a>格式。</small></span><span class="hide-when-compact"></span></div></td></tr></tbody></table> <dl><dd><small>本条目中，<a href="/wiki/%E5%90%91%E9%87%8F" title="向量">向量</a>與<a href="/wiki/%E6%A0%87%E9%87%8F_(%E6%95%B0%E5%AD%A6)" title="标量 (数学)">标量</a>分別用<a href="/wiki/Help:%E6%95%B8%E5%AD%B8%E5%85%AC%E5%BC%8F#字體#正粗體" class="mw-redirect" title="Help:數學公式">粗體</a>與<a href="/wiki/Help:%E6%95%B8%E5%AD%B8%E5%85%AC%E5%BC%8F#斜體" class="mw-redirect" title="Help:數學公式">斜體</a>顯示。例如，位置向量通常用 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {r} \,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {r} \,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3a1ffb0f7dee0c7f11111906fe55becd542d4d82" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-right: -0.387ex; width:1.489ex; height:1.676ex;" alt="{\displaystyle \mathbf {r} \,\!}"></span> 表示；而其大小則用 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86937851b7cb70e8802197a7b5ddcf39c0b02445" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-right: -0.387ex; width:1.436ex; height:1.676ex;" alt="{\displaystyle r\,\!}"></span> 來表示。</small></dd></dl> <div id="noteTA-5ff6fd68" class="noteTA"><div class="noteTA-group"><div data-noteta-group-source="module" data-noteta-group="Physics"></div></div></div> <style data-mw-deduplicate="TemplateStyles:r83732082">.mw-parser-output .infobox-subbox{padding:0;border:none;margin:-3px;width:auto;min-width:100%;font-size:100%;clear:none;float:none;background-color:transparent}.mw-parser-output .infobox-3cols-child{margin:auto}.mw-parser-output .infobox .navbar{font-size:100%}body.skin-minerva .mw-parser-output .infobox-header,body.skin-minerva .mw-parser-output .infobox-subheader,body.skin-minerva .mw-parser-output .infobox-above,body.skin-minerva .mw-parser-output .infobox-title,body.skin-minerva .mw-parser-output .infobox-image,body.skin-minerva .mw-parser-output .infobox-full-data,body.skin-minerva .mw-parser-output .infobox-below{text-align:center}@media screen{html.skin-theme-clientpref-night .mw-parser-output .infobox-full-data:not(.notheme)>div:not(.notheme)[style]{background:#1f1f23!important;color:#f8f9fa}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .infobox-full-data:not(.notheme) div:not(.notheme){background:#1f1f23!important;color:#f8f9fa}}html.skin-theme-clientpref-night .mw-parser-output .infobox td div:not(.notheme)[style]{background:transparent!important;color:var(--color-base,#202122)}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .infobox td div:not(.notheme)[style]{background:transparent!important;color:var(--color-base,#202122)}}html.skin-theme-clientpref-night .mw-parser-output .infobox td div.NavHead:not(.notheme)[style]{background:transparent!important}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .infobox td div.NavHead:not(.notheme)[style]{background:transparent!important}}@media(min-width:640px){body.skin--responsive .mw-parser-output .infobox-table{display:table!important}body.skin--responsive .mw-parser-output .infobox-table>caption{display:table-caption!important}body.skin--responsive .mw-parser-output .infobox-table>tbody{display:table-row-group}body.skin--responsive .mw-parser-output .infobox-table tr{display:table-row!important}body.skin--responsive .mw-parser-output .infobox-table th,body.skin--responsive .mw-parser-output .infobox-table td{padding-left:inherit;padding-right:inherit}}</style><table class="infobox"><tbody><tr><th colspan="2" class="infobox-above">加速度</th></tr><tr><td colspan="2" class="infobox-image"><span typeof="mw:File"><a href="/wiki/File:Gravity_gravita_grave.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/7d/Gravity_gravita_grave.gif/100px-Gravity_gravita_grave.gif" decoding="async" width="100" height="138" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/7d/Gravity_gravita_grave.gif/150px-Gravity_gravita_grave.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/7d/Gravity_gravita_grave.gif/200px-Gravity_gravita_grave.gif 2x" data-file-width="289" data-file-height="400" /></a></span></td></tr><tr><th scope="row" class="infobox-label" style=""><div style="padding:0.1em 0;line-height:1.2em;">常見符號</div></th><td class="infobox-data" style=""><b>a</b></td></tr><tr><th scope="row" class="infobox-label" style=""><a href="/wiki/%E5%9B%BD%E9%99%85%E5%8D%95%E4%BD%8D" title="国际单位">国际单位</a></th><td class="infobox-data" style=""><a href="/wiki/%E7%B1%B3%E6%AF%8F%E4%BA%8C%E6%AC%A1%E6%96%B9%E7%A7%92" title="米每二次方秒">米每二次方秒</a></td></tr><tr><th scope="row" class="infobox-label" style=""><a href="/wiki/%E5%9B%A0%E6%AC%A1%E5%88%86%E6%9E%90#定義" title="因次分析">因次</a></th><td class="infobox-data" style=""><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {L}}{\mathsf {T}}^{-2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">L</mi> </mrow> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">T</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {L}}{\mathsf {T}}^{-2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e2639355e0e107e11e7d06c0f772f155c24a0ed1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.176ex; height:2.676ex;" alt="{\displaystyle {\mathsf {L}}{\mathsf {T}}^{-2}}"></span></td></tr></tbody></table> <p><b>加速度（英語：Acceleration）</b>是<a href="/wiki/%E7%89%A9%E7%90%86%E5%AD%A6" title="物理学">物理学</a>中的一个物理量，是一个<a href="/wiki/%E7%9F%A2%E9%87%8F" class="mw-redirect" title="矢量">矢量</a>，主要应用于<a href="/wiki/%E7%B6%93%E5%85%B8%E7%89%A9%E7%90%86" class="mw-redirect" title="經典物理">经典物理</a>当中，一般用字母<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {a} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {a} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1a957216653a9ee0d0133dcefd13fb75e36b8b9d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.299ex; height:1.676ex;" alt="{\displaystyle \mathbf {a} }"></span>表示，在<a href="/wiki/%E5%9B%BD%E9%99%85%E5%8D%95%E4%BD%8D%E5%88%B6" title="国际单位制">国际单位制</a>中的单位为米每二次方秒（<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {m/s^{2}} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">m</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mi mathvariant="normal">s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {m/s^{2}} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/44dce6eb81495ac1cae4dc87a0c694c5b36fb5ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.069ex; height:3.176ex;" alt="{\displaystyle \mathrm {m/s^{2}} }"></span>）。加速度是<a href="/wiki/%E9%80%9F%E5%BA%A6" title="速度">速度</a>矢量對于<a href="/wiki/%E6%97%B6%E9%97%B4" title="时间">时间</a>的变化率，描述速度的方向和大小变化的快慢。 </p><p>在<a href="/wiki/%E7%BB%8F%E5%85%B8%E5%8A%9B%E5%AD%A6" title="经典力学">经典力学</a>中，<a href="/wiki/%E7%89%9B%E9%A1%BF%E7%AC%AC%E4%BA%8C%E5%AE%9A%E5%BE%8B" class="mw-redirect" title="牛顿第二定律">牛顿第二定律</a>说明了力和加速度成正比，這定律又稱為「加速度定律」。假設施加於物體的淨外力為零，則加速度為零，速度為常數，由於動量是質量與速度的乘積，所以<a href="/wiki/%E5%8B%95%E9%87%8F%E5%AE%88%E6%81%86" class="mw-redirect" title="動量守恆">動量守恆</a>。在<a href="/wiki/%E9%9B%BB%E5%8B%95%E5%8A%9B%E5%AD%B8" class="mw-redirect" title="電動力學">電動力學</a>裏，呈加速度運動的<a href="/wiki/%E5%B8%B6%E9%9B%BB%E7%B2%92%E5%AD%90" title="帶電粒子">帶電粒子</a>會發射<a href="/wiki/%E7%94%B5%E7%A3%81%E8%BE%90%E5%B0%84" class="mw-redirect" title="电磁辐射">电磁辐射</a>。 </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="简述"><span id=".E7.AE.80.E8.BF.B0"></span>简述</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%8A%A0%E9%80%9F%E5%BA%A6&action=edit&section=1" title="编辑章节：简述"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>简单地说，<a href="/wiki/%E9%80%9F%E5%BA%A6" title="速度">速度</a>描述了位置是如何变化的，而加速度描述了速度是如何变化的。比如，水平地向前扔出一个物体，起初它的速度朝向正前，然而由於<a href="/wiki/%E9%87%8D%E5%8A%9B" class="mw-redirect" title="重力">重力</a>它开始在向前的同时向下坠落，即其速度改变了。这里改变物体速度的主要是地球的<a href="/wiki/%E9%87%8D%E5%8A%9B" class="mw-redirect" title="重力">重力</a>引起的<a href="/wiki/%E9%87%8D%E5%8A%9B%E5%8A%A0%E9%80%9F%E5%BA%A6" title="重力加速度">重力加速度</a>。 </p><p>加速度具有<a href="/wiki/%E5%90%91%E9%87%8F" title="向量">向量</a>性质，即需要用大小和方向同时描述一个加速度。在光滑水平面上向前运动的物体，如果向左或向右施以力，即给予了不同的加速度，则其速度会发生变化（包含了速率及方向），然而向左的加速度和向右的加速度显然引起了不同的效果。同样，施力的大小不同，引起的加速度不同，最终的结果也不一样，亦可以從向量的加成性來看。作为一个矢量，加速度的叠加和分解分别遵循<a href="/wiki/%E5%B9%B3%E8%A1%8C%E5%9B%9B%E9%82%8A%E5%BD%A2%E6%81%86%E7%AD%89%E5%BC%8F" title="平行四邊形恆等式">平行四边形法则</a>和<a href="/wiki/%E7%9F%A2%E9%87%8F#加法与减法" class="mw-redirect" title="矢量">三角形法则</a>。 </p><p>具體而言，加速度描述的是速度随时间的变化率。需要注意的是，由于速度也是矢量，因此加速度不为零的物体速度的大小（称之为速率）也不一定会发生变化，实际上，如果加速度保持与速度垂直，速度大小就一直不会改变，同时方向一直改变。这种情况在生活中最常见的是圆周运动，比如在被拴在一端固定的线的另一端的一个小物体在线保持绷直时做的运动，又比如带电粒子在仅受静磁场的<a href="/wiki/%E6%B4%9B%E4%BC%A6%E5%85%B9%E5%8A%9B" title="洛伦兹力">洛伦兹力</a><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {F} =q\mathbf {v} \times \mathbf {B} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mo>=</mo> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {F} =q\mathbf {v} \times \mathbf {B} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8208e984160e97a12a3837a497561fd379bb048e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.003ex; height:2.509ex;" alt="{\displaystyle \mathbf {F} =q\mathbf {v} \times \mathbf {B} }"></span>时做的运动。 </p> <div class="mw-heading mw-heading3"><h3 id="直线运动中的平均加速度和瞬时加速度"><span id=".E7.9B.B4.E7.BA.BF.E8.BF.90.E5.8A.A8.E4.B8.AD.E7.9A.84.E5.B9.B3.E5.9D.87.E5.8A.A0.E9.80.9F.E5.BA.A6.E5.92.8C.E7.9E.AC.E6.97.B6.E5.8A.A0.E9.80.9F.E5.BA.A6"></span>直线运动中的平均加速度和瞬时加速度</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%8A%A0%E9%80%9F%E5%BA%A6&action=edit&section=2" title="编辑章节：直线运动中的平均加速度和瞬时加速度"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>设<a href="/wiki/%E8%B4%A8%E7%82%B9" class="mw-redirect" title="质点">质点</a>A呈一維运动，<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65658b7b223af9e1acc877d848888ecdb4466560" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}"></span>时刻位于<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x(t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d54c275db3a1e620737b58e143b0818107fa5f5c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.979ex; height:2.843ex;" alt="{\displaystyle x(t)}"></span>处，经过<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8c28867ecd34e2caed12cf38feadf6a81a7ee542" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.775ex; height:2.176ex;" alt="{\displaystyle \Delta t}"></span>时间后位于<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x(t+\Delta t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo>+</mo> <mi mathvariant="normal">Δ</mi> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x(t+\Delta t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8c0af01ba8ca4c071e3c1aa2a59b2c2a92b8ba51" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.594ex; height:2.843ex;" alt="{\displaystyle x(t+\Delta t)}"></span>处，则定义质点A在<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65658b7b223af9e1acc877d848888ecdb4466560" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}"></span>时刻的瞬时速度（简称速度）为 </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v(t)=\lim _{\Delta t\to 0}{\frac {x(t+\Delta t)-x(t)}{\Delta t}}={\frac {\mathrm {d} x}{\mathrm {d} t}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Δ</mi> <mi>t</mi> <mo stretchy="false">→</mo> <mn>0</mn> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>x</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo>+</mo> <mi mathvariant="normal">Δ</mi> <mi>t</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>x</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="normal">Δ</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v(t)=\lim _{\Delta t\to 0}{\frac {x(t+\Delta t)-x(t)}{\Delta t}}={\frac {\mathrm {d} x}{\mathrm {d} t}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/180b9a6b472d4ed816a02c104acee618c7e707f9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:35.496ex; height:6.009ex;" alt="{\displaystyle v(t)=\lim _{\Delta t\to 0}{\frac {x(t+\Delta t)-x(t)}{\Delta t}}={\frac {\mathrm {d} x}{\mathrm {d} t}}}"></span></dd></dl> <p>其中，<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\mathrm {d} x}{\mathrm {d} t}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\mathrm {d} x}{\mathrm {d} t}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/05be89f48a6b1496a7477ec53eb53b27c44bcd9e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:3.458ex; height:5.509ex;" alt="{\displaystyle {\frac {\mathrm {d} x}{\mathrm {d} t}}}"></span>表示位移對时间的一阶<a href="/wiki/%E5%AF%BC%E6%95%B0" title="导数">导数</a>，在时间-位移图上表现为求斜率。 </p><p>首先，定义<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65658b7b223af9e1acc877d848888ecdb4466560" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}"></span>时刻到<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t+\Delta t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> <mo>+</mo> <mi mathvariant="normal">Δ</mi> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t+\Delta t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e896baa706e07d8100439dc56c5b27771e354e4a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.456ex; height:2.343ex;" alt="{\displaystyle t+\Delta t}"></span>时刻之间的<b>平均加速度</b>为 </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bar {a}}={\frac {v(t+\Delta t)-v(t)}{\Delta t}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>v</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo>+</mo> <mi mathvariant="normal">Δ</mi> <mi>t</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>v</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="normal">Δ</mi> <mi>t</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bar {a}}={\frac {v(t+\Delta t)-v(t)}{\Delta t}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5749c553aef0c913532c2a56dbf8f49c2d9e7325" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:21.174ex; height:5.843ex;" alt="{\displaystyle {\bar {a}}={\frac {v(t+\Delta t)-v(t)}{\Delta t}}}"></span></dd></dl> <p>平均加速度粗略地表示了在该段时间内物体速度的变化情况。如果<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8c28867ecd34e2caed12cf38feadf6a81a7ee542" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.775ex; height:2.176ex;" alt="{\displaystyle \Delta t}"></span>越小，该段时间内速度的波动就越小，描述的速度变化情况也就越精细，从而定义质点A在<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65658b7b223af9e1acc877d848888ecdb4466560" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}"></span>时刻的<b>瞬时加速度</b>为 </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a(t)=\lim _{\Delta t\to 0}{\frac {v(t+\Delta t)-v(t)}{\Delta t}}={\frac {\mathrm {d} v}{\mathrm {d} t}}={\frac {\mathrm {d^{2}} x}{\mathrm {d} t^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Δ</mi> <mi>t</mi> <mo stretchy="false">→</mo> <mn>0</mn> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>v</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo>+</mo> <mi mathvariant="normal">Δ</mi> <mi>t</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>v</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="normal">Δ</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>v</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi mathvariant="normal">d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mi>x</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a(t)=\lim _{\Delta t\to 0}{\frac {v(t+\Delta t)-v(t)}{\Delta t}}={\frac {\mathrm {d} v}{\mathrm {d} t}}={\frac {\mathrm {d^{2}} x}{\mathrm {d} t^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/55e45e2436fea25a9989f8c7d8eb48e4b69dfa6d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:42.603ex; height:6.009ex;" alt="{\displaystyle a(t)=\lim _{\Delta t\to 0}{\frac {v(t+\Delta t)-v(t)}{\Delta t}}={\frac {\mathrm {d} v}{\mathrm {d} t}}={\frac {\mathrm {d^{2}} x}{\mathrm {d} t^{2}}}}"></span></dd></dl> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Graph_of_acceleration_of_positive_zero_negative.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/24/Graph_of_acceleration_of_positive_zero_negative.jpg/200px-Graph_of_acceleration_of_positive_zero_negative.jpg" decoding="async" width="200" height="141" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/24/Graph_of_acceleration_of_positive_zero_negative.jpg/300px-Graph_of_acceleration_of_positive_zero_negative.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/24/Graph_of_acceleration_of_positive_zero_negative.jpg/400px-Graph_of_acceleration_of_positive_zero_negative.jpg 2x" data-file-width="717" data-file-height="504" /></a><figcaption>三个质点从坐标原点以相同的速度出发，由于分别拥有正、零、负的加速度而导致其位置和关于时间的曲线。</figcaption></figure> <p>瞬时加速度，简称加速度<sup id="cite_ref-赵凯华力学_1-0" class="reference"><a href="#cite_note-赵凯华力学-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup class="reference" style="white-space:nowrap;">:21</sup>。进而有 </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v(t_{1})=\int _{t_{0}}^{t_{1}}a(t)\mathrm {d} t+v(t_{0})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> <mo stretchy="false">(</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </msubsup> <mi>a</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> <mo>+</mo> <mi>v</mi> <mo stretchy="false">(</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v(t_{1})=\int _{t_{0}}^{t_{1}}a(t)\mathrm {d} t+v(t_{0})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9d7a4e121aaee16f364b48c8afacc0b73954d65" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:26.12ex; height:6.509ex;" alt="{\displaystyle v(t_{1})=\int _{t_{0}}^{t_{1}}a(t)\mathrm {d} t+v(t_{0})}"></span></dd></dl> <p>在直线运动时，矢量約化为带符号的标量，其绝对值表示该物理量的大小。速度为正表示向右，速度为负表示向左（二維空間座標中）。加速度与速度方向相同（即符号相同）时表示物体不断加速，不同则表示物体不断减速。 </p><p>右图画出了三个质点在<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/43469ec032d858feae5aa87029e22eaaf0109e9c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.101ex; height:2.176ex;" alt="{\displaystyle t=0}"></span>从坐标原点以相同的速度<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/60faad24775635f4722ccc438093dbbfe05f34ae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.182ex; height:2.009ex;" alt="{\displaystyle v_{0}}"></span>出发，由于分别拥有正、零、负的加速度而导致其位置<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span>关于时间的曲线。可以将其想象为在光滑桌面上，三个木块以相同初速度，沿斜面向下、沿水平面、沿斜面向上滑行。 </p><p>在位移-时间图上，加速度由曲线的<a href="/wiki/%E5%87%B9%E5%87%B8%E6%80%A7" class="mw-disambig" title="凹凸性">凹凸性</a>表示，加速度为正的部分表现为<a href="/wiki/%E5%87%B8%E5%87%BD%E6%95%B0" title="凸函数">凸函数</a>，反之为<a href="/wiki/%E5%87%B9%E5%87%BD%E6%95%B0" title="凹函数">凹函数</a>，亦可以從微分的角度來分析。 </p> <div class="mw-heading mw-heading3"><h3 id="曲线运动中的加速度"><span id=".E6.9B.B2.E7.BA.BF.E8.BF.90.E5.8A.A8.E4.B8.AD.E7.9A.84.E5.8A.A0.E9.80.9F.E5.BA.A6"></span>曲线运动中的加速度</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%8A%A0%E9%80%9F%E5%BA%A6&action=edit&section=3" title="编辑章节：曲线运动中的加速度"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Acceleration_difference.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/Acceleration_difference.svg/200px-Acceleration_difference.svg.png" decoding="async" width="200" height="223" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/Acceleration_difference.svg/300px-Acceleration_difference.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/Acceleration_difference.svg/400px-Acceleration_difference.svg.png 2x" data-file-width="700" data-file-height="780" /></a><figcaption>用两次差分表示如何从位移矢量近似地得到加速度矢量，在數學表示中以粗體或是上方標註箭號為向量。</figcaption></figure> <p>设质点A在空间中运动，原点O指向A的矢量<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {r} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {r} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eca0f46511c4c986c48b254073732c0bd98ae0c1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.102ex; height:1.676ex;" alt="{\displaystyle \mathbf {r} }"></span>为其矢径，则可类似定义其速度矢量和加速度矢量为<sup id="cite_ref-赵凯华力学_1-1" class="reference"><a href="#cite_note-赵凯华力学-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup class="reference" style="white-space:nowrap;">:24</sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {v} (t)={\frac {\mathrm {d} \mathbf {r} (t)}{\mathrm {d} t}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {v} (t)={\frac {\mathrm {d} \mathbf {r} (t)}{\mathrm {d} t}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87b7172bafb98cd6ba02c2d947cc98d527d5ce3f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:13.038ex; height:5.843ex;" alt="{\displaystyle \mathbf {v} (t)={\frac {\mathrm {d} \mathbf {r} (t)}{\mathrm {d} t}}}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {a} (t)={\frac {\mathrm {d} \mathbf {v} (t)}{\mathrm {d} t}}={\frac {\mathrm {d^{2}} \mathbf {r} (t)}{\mathrm {d} t^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi mathvariant="normal">d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {a} (t)={\frac {\mathrm {d} \mathbf {v} (t)}{\mathrm {d} t}}={\frac {\mathrm {d^{2}} \mathbf {r} (t)}{\mathrm {d} t^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01d4e10fa9ff5eec2b2a72be88f82055af1e854f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:23.268ex; height:6.176ex;" alt="{\displaystyle \mathbf {a} (t)={\frac {\mathrm {d} \mathbf {v} (t)}{\mathrm {d} t}}={\frac {\mathrm {d^{2}} \mathbf {r} (t)}{\mathrm {d} t^{2}}}}"></span></dd></dl> <p>右图表现的是一个质点沿一曲线运动的轨迹，表示出了两次微分的过程，为了清晰表示，这里使用差分（<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8c28867ecd34e2caed12cf38feadf6a81a7ee542" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.775ex; height:2.176ex;" alt="{\displaystyle \Delta t}"></span>并不趋于0）近似代替了微分，因此表现的是平均速度和平均加速度。可以看出，加速度与速度都具有方向和大小，并且即使在同一时刻两者方向也不一定相同。加速度与速度方向平行的分量表示速度大小的变化率（相同则加速，相反则减速），而与速度垂直的分量表示速度方向的变化率（速度矢量转动的<a href="/wiki/%E8%A7%92%E9%80%9F%E5%BA%A6" title="角速度">角速度</a>）。 </p><p>在<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8c28867ecd34e2caed12cf38feadf6a81a7ee542" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.775ex; height:2.176ex;" alt="{\displaystyle \Delta t}"></span>足够小时，可以将那一小段曲线运动（称作元弧）近似看作直线运动或圆周运动<sup id="cite_ref-赵凯华力学_1-2" class="reference"><a href="#cite_note-赵凯华力学-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup class="reference" style="white-space:nowrap;">:30</sup>。 </p> <div class="mw-heading mw-heading3"><h3 id="伽利略变换"><span id=".E4.BC.BD.E5.88.A9.E7.95.A5.E5.8F.98.E6.8D.A2"></span>伽利略变换</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%8A%A0%E9%80%9F%E5%BA%A6&action=edit&section=4" title="编辑章节：伽利略变换"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>在经典物理下，即速度远小于光速、研究宏观物体时，可以使用<a href="/wiki/%E4%BC%BD%E5%88%A9%E7%95%A5%E5%8F%98%E6%8D%A2" title="伽利略变换">伽利略变换</a>来研究不同参考系间的加速度的联系，簡單來說就是座標間的轉換，但仍有保持一定的不變量<sup id="cite_ref-郑永令力学_2-0" class="reference"><a href="#cite_note-郑永令力学-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup class="reference" style="white-space:nowrap;">:32</sup>： </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {a} =\mathbf {a} '+\mathbf {a} _{\text{rel}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>=</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>′</mo> </msup> <mo>+</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>rel</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {a} =\mathbf {a} '+\mathbf {a} _{\text{rel}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d8a03bc797928b4863f53c3100d114d6001d1c28" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.586ex; height:2.843ex;" alt="{\displaystyle \mathbf {a} =\mathbf {a} '+\mathbf {a} _{\text{rel}}}"></span></dd></dl> <p>其中，<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {a} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {a} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1a957216653a9ee0d0133dcefd13fb75e36b8b9d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.299ex; height:1.676ex;" alt="{\displaystyle \mathbf {a} }"></span>为物体在参考系<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}"></span>下的加速度，<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {a} '}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {a} '}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f64f8d8a25dea36b30a58a3695448d7af314757a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.984ex; height:2.509ex;" alt="{\displaystyle \mathbf {a} '}"></span>为物体在参考系<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>S</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S'}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf9961844d1f539adee019e432dc18aa2a7ede59" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.206ex; height:2.509ex;" alt="{\displaystyle S'}"></span>下的加速度，<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {a} _{\text{rel}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>rel</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {a} _{\text{rel}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3f53eb9c32019a58e0b6b7638b76489d3091d980" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.364ex; height:2.009ex;" alt="{\displaystyle \mathbf {a} _{\text{rel}}}"></span>为参考系<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>S</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S'}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf9961844d1f539adee019e432dc18aa2a7ede59" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.206ex; height:2.509ex;" alt="{\displaystyle S'}"></span>在参考系<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}"></span>下的加速度。 </p><p>考虑站在地面看火车上的人抛出一个小球，这个公式表达：小球相对于地面的加速度<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {a} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {a} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1a957216653a9ee0d0133dcefd13fb75e36b8b9d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.299ex; height:1.676ex;" alt="{\displaystyle \mathbf {a} }"></span>，等于小球相对于火车的加速度<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {a} '}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {a} '}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f64f8d8a25dea36b30a58a3695448d7af314757a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.984ex; height:2.509ex;" alt="{\displaystyle \mathbf {a} '}"></span>加上火车相对于地面的加速度<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {a} _{\text{rel}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>rel</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {a} _{\text{rel}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3f53eb9c32019a58e0b6b7638b76489d3091d980" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.364ex; height:2.009ex;" alt="{\displaystyle \mathbf {a} _{\text{rel}}}"></span>。这个式子是矢量表达式，即三个加速度矢量的方向不在同一条直线上时，要使用<a href="/wiki/%E7%9F%A2%E9%87%8F#加法与减法" class="mw-redirect" title="矢量">矢量加法</a>计算。 </p> <div class="mw-heading mw-heading3"><h3 id="牛顿第二定律"><span id=".E7.89.9B.E9.A1.BF.E7.AC.AC.E4.BA.8C.E5.AE.9A.E5.BE.8B"></span>牛顿第二定律</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%8A%A0%E9%80%9F%E5%BA%A6&action=edit&section=5" title="编辑章节：牛顿第二定律"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r84833064"><div role="note" class="hatnote navigation-not-searchable">主条目：<a href="/wiki/%E7%89%9B%E9%A1%BF%E7%AC%AC%E4%BA%8C%E5%AE%9A%E5%BE%8B" class="mw-redirect" title="牛顿第二定律">牛顿第二定律</a></div> <p>加速度最主要的应用之一是牛顿第二定律。简单地说，牛顿第二定律表明<sup id="cite_ref-郑永令力学_2-1" class="reference"><a href="#cite_note-郑永令力学-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup class="reference" style="white-space:nowrap;">:57</sup>，感受到合外力的作用，物体的加速度与合外力成正比，与质量成反比，加速度方向沿合力方向，在国际单位制中表示为 </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {F} =m\mathbf {a} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mo>=</mo> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {F} =m\mathbf {a} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d0c60fab89e8c3193952047dc565bcf8d233d115" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.121ex; height:2.176ex;" alt="{\displaystyle \mathbf {F} =m\mathbf {a} }"></span></dd></dl> <p>其中<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {F} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {F} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/da18bef8c979f3548bb0d8976f5844012d7b8256" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.683ex; height:2.176ex;" alt="{\displaystyle \mathbf {F} }"></span>表示物体所受合外力，<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}"></span>为物体质量，<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {a} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {a} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1a957216653a9ee0d0133dcefd13fb75e36b8b9d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.299ex; height:1.676ex;" alt="{\displaystyle \mathbf {a} }"></span>为物体的加速度。 </p><p>在<a href="/wiki/%E5%8F%A4%E5%85%B8%E7%89%A9%E7%90%86" class="mw-redirect" title="古典物理">古典物理</a>下，牛顿第二定律廣泛适用。此外，牛顿第二定律要求所处参考系为<a href="/wiki/%E6%83%AF%E6%80%A7%E5%8F%82%E8%80%83%E7%B3%BB" title="惯性参考系">惯性参考系</a>。由于经典物理的研究几乎都会或多或少地涉及到物体在力的作用下的运动，又由于牛顿第二定律和伽利略变换极具简洁性，所以，牛顿第二定律是經典動力學裏的重要基礎定律，質點運動亦然。 </p> <div class="mw-heading mw-heading3"><h3 id="惯性力"><span id=".E6.83.AF.E6.80.A7.E5.8A.9B"></span>惯性力</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%8A%A0%E9%80%9F%E5%BA%A6&action=edit&section=6" title="编辑章节：惯性力"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r84833064"><div role="note" class="hatnote navigation-not-searchable">主条目：<a href="/wiki/%E6%85%A3%E6%80%A7%E5%8A%9B" title="慣性力">惯性力</a></div> <p>當帶質量物體加速時，<a href="/wiki/%E6%85%A3%E6%80%A7" title="慣性">慣性</a>是物體維持原有<a href="/wiki/%E7%8B%80%E6%85%8B" class="mw-redirect mw-disambig" title="狀態">運動狀態</a>的傾向，慣性力是對於這傾向的衡量<sup id="cite_ref-Beer_3-0" class="reference"><a href="#cite_note-Beer-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup>。因為慣性力實際上並不存在，只有原本將該物體加速的作用力實際存在，因此慣性力又稱為<a href="/wiki/%E5%81%87%E6%83%B3%E5%8A%9B" class="mw-redirect" title="假想力">假想力</a>。更具體而言，根據牛顿第二定律， </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{i}\mathbf {F} _{i}=m\mathbf {a} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{i}\mathbf {F} _{i}=m\mathbf {a} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/31a3002406cb3ace75aec1e725ad08f1e495adf9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:12.663ex; height:5.509ex;" alt="{\displaystyle \sum _{i}\mathbf {F} _{i}=m\mathbf {a} }"></span></dd></dl> <p>其中，<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {F} _{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {F} _{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5d49039f831a156e0c33270962d8d159d553a8bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.482ex; height:2.509ex;" alt="{\displaystyle \mathbf {F} _{i}}"></span>是第<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}"></span>個作用力，<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {a} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {a} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1a957216653a9ee0d0133dcefd13fb75e36b8b9d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.299ex; height:1.676ex;" alt="{\displaystyle \mathbf {a} }"></span>是物體的加速度。 </p><p>重新編排，可以得到 </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{i}\mathbf {F} _{i}-m\mathbf {a} =0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{i}\mathbf {F} _{i}-m\mathbf {a} =0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a4f8d068638205dd55e3856215b78ae0b1f149cf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:16.666ex; height:5.509ex;" alt="{\displaystyle \sum _{i}\mathbf {F} _{i}-m\mathbf {a} =0}"></span></dd></dl> <p>設定慣性向量<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {I} =-m\mathbf {a} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">I</mi> </mrow> <mo>=</mo> <mo>−</mo> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {I} =-m\mathbf {a} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8957fc5b42798008d1aca06439dacc29251769a2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.26ex; height:2.343ex;" alt="{\displaystyle \mathbf {I} =-m\mathbf {a} }"></span>，這物體的動力系統滿足方程式 </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{i}\mathbf {F} +\mathbf {I} =0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">I</mi> </mrow> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{i}\mathbf {F} +\mathbf {I} =0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c147ee0c34875749e6379b9e01704cbcf03c70ee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:13.54ex; height:5.509ex;" alt="{\displaystyle \sum _{i}\mathbf {F} +\mathbf {I} =0}"></span></dd></dl> <p>想像這慣性向量為由於加速度運動而產生的一種力，稱為慣性力。因為慣性力與所有作用於這物體的外力的向量總和為零，這動力系統可以視為處於動力平衡狀態。藉著這機制，可以將動力系統約化為靜力系統，用靜力學發展出的方法來解析動力問題<sup id="cite_ref-Beer_3-1" class="reference"><a href="#cite_note-Beer-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Lanczos_4-0" class="reference"><a href="#cite_note-Lanczos-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup><sup class="reference" style="white-space:nowrap;">:88ff</sup>。 </p><p>惯性力在平日生活中其实很常见，例如，停止不動的火车突然向前方加速，则所有站立乘客都会向後方倾移，這便是惯性力效應，從另外一個角度而言也是為了提供乘客們有足夠的摩擦力來進行移動。 </p> <div class="mw-heading mw-heading3"><h3 id="加加速度"><span id=".E5.8A.A0.E5.8A.A0.E9.80.9F.E5.BA.A6"></span>加加速度</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%8A%A0%E9%80%9F%E5%BA%A6&action=edit&section=7" title="编辑章节：加加速度"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r84833064"><div role="note" class="hatnote navigation-not-searchable">主条目：<a href="/wiki/%E5%8A%A0%E5%8A%A0%E9%80%9F%E5%BA%A6" title="加加速度">加加速度</a></div> <p>将位移對于时间进行一阶求导得到了速度，二阶求导得到了加速度。可能会想到，可以通过进行三阶求导来得到一个诸如加加速度的物理量。 </p><p>在<a href="/wiki/%E5%B7%A5%E7%A8%8B%E5%AD%A6" title="工程学">工程学</a>中经常需要用到加加速度，特别是在<a href="/wiki/%E4%BA%A4%E9%80%9A%E5%B7%A5%E5%85%B7" class="mw-redirect" title="交通工具">交通工具</a>设计以及材料等问题<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup>。交通工具在加速时将使乘客产生不适感，这种不适感不仅来自于<b>加速度</b>，也与加加速度有关。在这种情况中，加速度反应人体器官在加速度運動時所感受到的力（见<a href="/wiki/%E7%89%9B%E9%A0%93%E7%AC%AC%E4%BA%8C%E5%AE%9A%E5%BE%8B" class="mw-redirect" title="牛頓第二定律">牛頓第二定律</a>），加加速度则反应这作用力的变化快慢。较大的加加速度将会使人体产生相当的不适感，例如在<a href="/wiki/%E7%94%B5%E6%A2%AF" class="mw-redirect" title="电梯">电梯</a>升降，<a href="/wiki/%E6%B1%BD%E8%BD%A6" title="汽车">汽车</a>、<a href="/wiki/%E7%81%AB%E8%BD%A6" class="mw-redirect" title="火车">火车</a>等加速和转弯的过程中（在这些情况中加速度和加加速度的效应一般会同时存在）。人體需要時間適應加速度的變化，假若加加速度超過安全標準，則可能會發生像車禍造成的<a href="/w/index.php?title=%E9%A0%B8%E9%83%A8%E6%89%AD%E5%82%B7&action=edit&redlink=1" class="new" title="頸部扭傷（页面不存在）">頸部扭傷</a>（whiplash）一類的人體傷害。因而在设计交通工具时加加速度是必须考虑的因素。 </p><p>在物理學裏，加加速度现在主要应用在<a href="/wiki/%E6%B7%B7%E6%B2%8C%E7%90%86%E8%AE%BA" title="混沌理论">混沌理论</a>領域<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup>。 </p> <div class="mw-heading mw-heading3"><h3 id="角加速度"><span id=".E8.A7.92.E5.8A.A0.E9.80.9F.E5.BA.A6"></span>角加速度</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%8A%A0%E9%80%9F%E5%BA%A6&action=edit&section=8" title="编辑章节：角加速度"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>角加速度涉及繞著固定轴轉動的物体，例如，想象一个圆盘和一个垂直固定於其中心的木棍，两隻手合拍住木棍并前後磨擦，造成木棍與圓盤共同轉動（例如，在地上高速旋转的<a href="/wiki/%E9%99%80%E8%9E%BA" title="陀螺">陀螺</a>，绕著固定点轉動）。在圆盘上做一个标记（如一条半径），则繞著固定轴轉動的物体可以简单地用<a href="/wiki/%E6%A0%87%E9%87%8F_(%E7%89%A9%E7%90%86%E5%AD%A6)" title="标量 (物理学)">标量</a><a href="/wiki/%E8%A7%92%E5%BA%A6" class="mw-redirect" title="角度">角弧</a>（即该标记转动的角弧）來做定量描述。 </p><p>旋轉運動可以與直线运动相类比：位移、速度、加速度，分别对应於角弧、角速度、角加速度。直线运动中已有的定律和方法可以直接带入，用於旋轉運動，例如，使用已有的<a href="#匀加速直线运动">匀加速直线运动</a>理论来研究匀角加速度固定轴转动<sup id="cite_ref-郑永令力学_2-2" class="reference"><a href="#cite_note-郑永令力学-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup class="reference" style="white-space:nowrap;">:249</sup>。 </p><p>在国际单位制中，角加速度的单位为弧度每二次方秒（<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {rad/s^{2}} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">d</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mi mathvariant="normal">s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {rad/s^{2}} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e877c3a100345c585c9106a68a79f3c303445af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.5ex; height:3.176ex;" alt="{\displaystyle \mathrm {rad/s^{2}} }"></span>）。其定义式为 </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha (t)={\frac {\mathrm {d} \omega (t)}{\mathrm {d} t}}={\frac {\mathrm {d^{2}} \theta (t)}{\mathrm {d} t^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>ω</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi mathvariant="normal">d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mi>θ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha (t)={\frac {\mathrm {d} \omega (t)}{\mathrm {d} t}}={\frac {\mathrm {d^{2}} \theta (t)}{\mathrm {d} t^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4c1ce5e25b617d35b81ad87f47d3e12d6dbbc384" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:23.479ex; height:6.176ex;" alt="{\displaystyle \alpha (t)={\frac {\mathrm {d} \omega (t)}{\mathrm {d} t}}={\frac {\mathrm {d^{2}} \theta (t)}{\mathrm {d} t^{2}}}}"></span></dd></dl> <p>其中，<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }"></span>為物體角加速度，<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ω</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48eff443f9de7a985bb94ca3bde20813ea737be8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.446ex; height:1.676ex;" alt="{\displaystyle \omega }"></span>为物体<a href="/wiki/%E8%A7%92%E9%80%9F%E5%BA%A6" title="角速度">角速度</a>，<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e5ab2664b422d53eb0c7df3b87e1360d75ad9af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:2.176ex;" alt="{\displaystyle \theta }"></span>为物体转过的角弧。 </p> <div class="mw-heading mw-heading2"><h2 id="加速度的分解"><span id=".E5.8A.A0.E9.80.9F.E5.BA.A6.E7.9A.84.E5.88.86.E8.A7.A3"></span>加速度的分解</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%8A%A0%E9%80%9F%E5%BA%A6&action=edit&section=9" title="编辑章节：加速度的分解"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>处理关于空间加速度矢量的问题，除了直接计算矢量之外，更多的时候可以将加速度按照适当坐标轴分解，即将矢量形式的加速度表示为相互独立的不同方向上的标量的形式。因为标量的计算要容易很多，因此这是解决问题常用的方法。 </p> <div class="mw-heading mw-heading3"><h3 id="按坐标系分解"><span id=".E6.8C.89.E5.9D.90.E6.A0.87.E7.B3.BB.E5.88.86.E8.A7.A3"></span>按坐标系分解</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%8A%A0%E9%80%9F%E5%BA%A6&action=edit&section=10" title="编辑章节：按坐标系分解"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading4"><h4 id="平面直角坐标系"><span id=".E5.B9.B3.E9.9D.A2.E7.9B.B4.E8.A7.92.E5.9D.90.E6.A0.87.E7.B3.BB"></span>平面直角坐标系</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%8A%A0%E9%80%9F%E5%BA%A6&action=edit&section=11" title="编辑章节：平面直角坐标系"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>在<a href="/wiki/%E7%AC%9B%E5%8D%A1%E5%B0%94%E5%9D%90%E6%A0%87%E7%B3%BB" title="笛卡尔坐标系">平面直角坐标系</a>中， </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {a} (t)=a_{x}(t)\mathbf {i} +a_{y}(t)\mathbf {j} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">i</mi> </mrow> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">j</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {a} (t)=a_{x}(t)\mathbf {i} +a_{y}(t)\mathbf {j} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fafec5549a63f170b29373950ddddf45d5bb62bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:21.425ex; height:3.009ex;" alt="{\displaystyle \mathbf {a} (t)=a_{x}(t)\mathbf {i} +a_{y}(t)\mathbf {j} }"></span></dd></dl> <p>其中<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {i} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">i</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {i} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dc909dd0c91a68cdb9cda4ae133fa1f82c987c01" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.742ex; height:2.176ex;" alt="{\displaystyle \mathbf {i} }"></span>和<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {j} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">j</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {j} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca5d874dbb32b8b33e83ca521f592808387a486e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.164ex; width:0.98ex; height:2.509ex;" alt="{\displaystyle \mathbf {j} }"></span>分别为<i>x</i>、<i>y</i><a href="/wiki/%E5%9D%90%E6%A0%87%E8%BD%B4" class="mw-redirect" title="坐标轴">坐标轴</a>上的<a href="/wiki/%E5%8D%95%E4%BD%8D%E7%9F%A2%E9%87%8F" class="mw-redirect" title="单位矢量">单位矢量</a>，皆为常矢量。 </p><p>这种分解方式的优点在于，形式简便，思维简单；因为单位矢量不会变化，故质点在二个方向上的投影等价于直线运动，并将其叠加，使得问题完全化为代数问题，并且可以直接使用直线运动的已有结论<sup id="cite_ref-郑永令力学_2-3" class="reference"><a href="#cite_note-郑永令力学-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup class="reference" style="white-space:nowrap;">:18</sup>。 </p> <div class="mw-heading mw-heading4"><h4 id="极坐标系"><span id=".E6.9E.81.E5.9D.90.E6.A0.87.E7.B3.BB"></span>极坐标系</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%8A%A0%E9%80%9F%E5%BA%A6&action=edit&section=12" title="编辑章节：极坐标系"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:CircularCoordinates.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/11/CircularCoordinates.svg/200px-CircularCoordinates.svg.png" decoding="async" width="200" height="165" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/11/CircularCoordinates.svg/300px-CircularCoordinates.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/11/CircularCoordinates.svg/400px-CircularCoordinates.svg.png 2x" data-file-width="320" data-file-height="264" /></a><figcaption>在極點為O、極軸為L的极坐标系裏，點 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (3,60^{\circ })}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>3</mn> <mo>,</mo> <msup> <mn>60</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (3,60^{\circ })}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1646be315adca8e6e9efd32eb029c4910ba6c95f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.385ex; height:2.843ex;" alt="{\displaystyle (3,60^{\circ })}"></span> 、點 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (4,210^{\circ })}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>4</mn> <mo>,</mo> <msup> <mn>210</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (4,210^{\circ })}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/14e62cac932df41ccf9d2ff8e5186dc39a34abcf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.547ex; height:2.843ex;" alt="{\displaystyle (4,210^{\circ })}"></span> 的坐標分別以綠色、藍色展示。</figcaption></figure> <p>在二维空间裡，极坐标系用半径坐标 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"></span> 、角坐标 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e5ab2664b422d53eb0c7df3b87e1360d75ad9af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:2.176ex;" alt="{\displaystyle \theta }"></span> 来表示質點的位置。半径坐标是極點与質點的直线距离；角坐标是極點与質點的连线对於极轴的角弧。在任意点的两个<a href="/wiki/%E5%8D%95%E4%BD%8D%E7%9F%A2%E9%87%8F" class="mw-redirect" title="单位矢量">单位矢量</a>分别为沿半径向外的<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {e} _{r}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {e} _{r}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/887d07a86b50894ee5e5d16b76e8fbaf5f5eaa11" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.199ex; height:2.009ex;" alt="{\displaystyle \mathbf {e} _{r}}"></span>和垂直于半径指向角坐标正方向的<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {e} _{\theta }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>θ</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {e} _{\theta }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c8df567db25da5831ccd6615329591bf02901893" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.228ex; height:2.009ex;" alt="{\displaystyle \mathbf {e} _{\theta }}"></span>。不論是直角坐標或是極坐標都可以互相來駔變換，坐標和坐標之間有一定的轉換量，使用以方便所在的坐標系為主。 </p><p>從极坐标 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"></span> 和 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e5ab2664b422d53eb0c7df3b87e1360d75ad9af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:2.176ex;" alt="{\displaystyle \theta }"></span> 可以计算出<a href="/wiki/%E7%9B%B4%E8%A7%92%E5%9D%90%E6%A0%87" class="mw-redirect" title="直角坐标">直角坐标</a>： </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=r\cos \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mi>r</mi> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=r\cos \theta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f27adec1ac1792288cba78a125b22893a59507a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.453ex; height:2.176ex;" alt="{\displaystyle x=r\cos \theta }"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y=r\sin \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>=</mo> <mi>r</mi> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y=r\sin \theta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e29d62630263d5354736fb536f72ca956b65382" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.023ex; height:2.509ex;" alt="{\displaystyle y=r\sin \theta }"></span></dd></dl> <p>在极坐标系中，位置<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {r} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {r} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eca0f46511c4c986c48b254073732c0bd98ae0c1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.102ex; height:1.676ex;" alt="{\displaystyle \mathbf {r} }"></span>、速度<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {v} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {v} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35c1866e359fbfd2e0f606c725ba5cc37a5195d6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.411ex; height:1.676ex;" alt="{\displaystyle \mathbf {v} }"></span>、加速度<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {a} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {a} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1a957216653a9ee0d0133dcefd13fb75e36b8b9d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.299ex; height:1.676ex;" alt="{\displaystyle \mathbf {a} }"></span>分别为 </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {r} =r\mathbf {e} _{r}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>=</mo> <mi>r</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {r} =r\mathbf {e} _{r}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6fbb321cd619160efac791a982382a2b0cc05f57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.448ex; height:2.009ex;" alt="{\displaystyle \mathbf {r} =r\mathbf {e} _{r}}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {v} =r{\frac {\mathrm {d} \theta }{\mathrm {d} t}}\mathbf {e} _{\theta }+{\frac {dr}{\mathrm {d} t}}\mathbf {e} _{r}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>=</mo> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>θ</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>θ</mi> </mrow> </msub> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>r</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {v} =r{\frac {\mathrm {d} \theta }{\mathrm {d} t}}\mathbf {e} _{\theta }+{\frac {dr}{\mathrm {d} t}}\mathbf {e} _{r}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bb13d3ef41ee859e7f477f3d60f8b04fbaab36ca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:19.146ex; height:5.509ex;" alt="{\displaystyle \mathbf {v} =r{\frac {\mathrm {d} \theta }{\mathrm {d} t}}\mathbf {e} _{\theta }+{\frac {dr}{\mathrm {d} t}}\mathbf {e} _{r}}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {a} =\left[{\frac {\mathrm {d} ^{2}r}{\mathrm {d} t^{2}}}-r\left({\frac {\mathrm {d} \theta }{\mathrm {d} t}}\right)^{2}\right]\mathbf {e} _{r}+\left(2{\frac {\mathrm {d} r}{\mathrm {d} t}}{\frac {\mathrm {d} \theta }{\mathrm {d} t}}+r{\frac {\mathrm {d} ^{2}\theta }{\mathrm {d} t^{2}}}\right)\mathbf {e} _{\theta }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>=</mo> <mrow> <mo>[</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>r</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>−</mo> <mi>r</mi> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>θ</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>]</mo> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> <mo>+</mo> <mrow> <mo>(</mo> <mrow> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>r</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>θ</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>θ</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>θ</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {a} =\left[{\frac {\mathrm {d} ^{2}r}{\mathrm {d} t^{2}}}-r\left({\frac {\mathrm {d} \theta }{\mathrm {d} t}}\right)^{2}\right]\mathbf {e} _{r}+\left(2{\frac {\mathrm {d} r}{\mathrm {d} t}}{\frac {\mathrm {d} \theta }{\mathrm {d} t}}+r{\frac {\mathrm {d} ^{2}\theta }{\mathrm {d} t^{2}}}\right)\mathbf {e} _{\theta }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87eee0bf811b2a68b0ee5d7a131c966427daf7d8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:50.108ex; height:7.509ex;" alt="{\displaystyle \mathbf {a} =\left[{\frac {\mathrm {d} ^{2}r}{\mathrm {d} t^{2}}}-r\left({\frac {\mathrm {d} \theta }{\mathrm {d} t}}\right)^{2}\right]\mathbf {e} _{r}+\left(2{\frac {\mathrm {d} r}{\mathrm {d} t}}{\frac {\mathrm {d} \theta }{\mathrm {d} t}}+r{\frac {\mathrm {d} ^{2}\theta }{\mathrm {d} t^{2}}}\right)\mathbf {e} _{\theta }}"></span></dd></dl> <div class="mw-heading mw-heading5"><h5 id="极坐标系分解的推导"><span id=".E6.9E.81.E5.9D.90.E6.A0.87.E7.B3.BB.E5.88.86.E8.A7.A3.E7.9A.84.E6.8E.A8.E5.AF.BC"></span>极坐标系分解的推导</h5><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%8A%A0%E9%80%9F%E5%BA%A6&action=edit&section=13" title="编辑章节：极坐标系分解的推导"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>两个单位矢量<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {e} _{r}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {e} _{r}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/887d07a86b50894ee5e5d16b76e8fbaf5f5eaa11" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.199ex; height:2.009ex;" alt="{\displaystyle \mathbf {e} _{r}}"></span>、<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {e} _{\theta }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>θ</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {e} _{\theta }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c8df567db25da5831ccd6615329591bf02901893" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.228ex; height:2.009ex;" alt="{\displaystyle \mathbf {e} _{\theta }}"></span>会随质点所处位置不同而变化，并可通过分析得出结论，在质点运动的时候<sup id="cite_ref-郑永令力学_2-4" class="reference"><a href="#cite_note-郑永令力学-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup class="reference" style="white-space:nowrap;">:27</sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {d} \mathbf {e} _{r}=\mathrm {d} \theta \mathbf {e} _{\theta }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>θ</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>θ</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {d} \mathbf {e} _{r}=\mathrm {d} \theta \mathbf {e} _{\theta }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b20056e435995b02e5eb1c0add887719b72ba7e9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.201ex; height:2.509ex;" alt="{\displaystyle \mathrm {d} \mathbf {e} _{r}=\mathrm {d} \theta \mathbf {e} _{\theta }}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {d} \mathbf {e} _{\theta }=-\mathrm {d} \theta \mathbf {e} _{r}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>θ</mi> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>θ</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {d} \mathbf {e} _{\theta }=-\mathrm {d} \theta \mathbf {e} _{r}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4e4bacefd9d4086bbf53c0eeeed384d7d082fe46" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.01ex; height:2.509ex;" alt="{\displaystyle \mathrm {d} \mathbf {e} _{\theta }=-\mathrm {d} \theta \mathbf {e} _{r}}"></span></dd></dl> <p>其中，经过微小时间<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {d} t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {d} t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/588a981eb3c6f32c01153f8710a7f70029b5e553" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.132ex; height:2.176ex;" alt="{\displaystyle \mathrm {d} t}"></span>後，单位矢量<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {e} _{r}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {e} _{r}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/887d07a86b50894ee5e5d16b76e8fbaf5f5eaa11" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.199ex; height:2.009ex;" alt="{\displaystyle \mathbf {e} _{r}}"></span>、<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {e} _{\theta }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>θ</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {e} _{\theta }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c8df567db25da5831ccd6615329591bf02901893" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.228ex; height:2.009ex;" alt="{\displaystyle \mathbf {e} _{\theta }}"></span>的微小变化分別為<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {d} \mathbf {e} _{r}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {d} \mathbf {e} _{r}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/26af818e870d9b1375e1998fd1750277500c18c2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.491ex; height:2.509ex;" alt="{\displaystyle \mathrm {d} \mathbf {e} _{r}}"></span>、<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {d} \mathbf {e} _{\theta }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>θ</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {d} \mathbf {e} _{\theta }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9c2744f1c448d9d578d415536b4a96b2bfe6e11c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.521ex; height:2.509ex;" alt="{\displaystyle \mathrm {d} \mathbf {e} _{\theta }}"></span>。 </p><p>應用微分的<a href="/wiki/%E8%8E%B1%E5%B8%83%E5%B0%BC%E5%85%B9%E6%B3%95%E5%88%99" class="mw-redirect" title="莱布尼兹法则">莱布尼兹法则</a> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}(a\ b)=a\ {\frac {\mathrm {d} b}{\mathrm {d} t}}+b\ {\frac {\mathrm {d} a}{\mathrm {d} t}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <mo stretchy="false">(</mo> <mi>a</mi> <mtext> </mtext> <mi>b</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>a</mi> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>b</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mi>b</mi> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>a</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}(a\ b)=a\ {\frac {\mathrm {d} b}{\mathrm {d} t}}+b\ {\frac {\mathrm {d} a}{\mathrm {d} t}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/89ae29ea2b2276c42e86ca686017ebdf4c99f782" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:23.398ex; height:5.509ex;" alt="{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}(a\ b)=a\ {\frac {\mathrm {d} b}{\mathrm {d} t}}+b\ {\frac {\mathrm {d} a}{\mathrm {d} t}}}"></span></dd></dl> <p>在极坐标系下，速度<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {v} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {v} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35c1866e359fbfd2e0f606c725ba5cc37a5195d6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.411ex; height:1.676ex;" alt="{\displaystyle \mathbf {v} }"></span>、加速度<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {a} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {a} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1a957216653a9ee0d0133dcefd13fb75e36b8b9d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.299ex; height:1.676ex;" alt="{\displaystyle \mathbf {a} }"></span>分別為 </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\mathbf {v} &={\frac {\mathrm {d} }{\mathrm {d} t}}(r\mathbf {e} _{r})\\&=r{\frac {\mathrm {d} \theta }{\mathrm {d} t}}\mathbf {e} _{\theta }+{\frac {\mathrm {d} r}{\mathrm {d} t}}\mathbf {e} _{r}\\\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <mo stretchy="false">(</mo> <mi>r</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>θ</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>θ</mi> </mrow> </msub> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>r</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\mathbf {v} &={\frac {\mathrm {d} }{\mathrm {d} t}}(r\mathbf {e} _{r})\\&=r{\frac {\mathrm {d} \theta }{\mathrm {d} t}}\mathbf {e} _{\theta }+{\frac {\mathrm {d} r}{\mathrm {d} t}}\mathbf {e} _{r}\\\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/71a8844d4dd0358d5e2b9b2f10df048ae91f098d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.657ex; margin-bottom: -0.181ex; width:19.974ex; height:10.843ex;" alt="{\displaystyle {\begin{aligned}\mathbf {v} &={\frac {\mathrm {d} }{\mathrm {d} t}}(r\mathbf {e} _{r})\\&=r{\frac {\mathrm {d} \theta }{\mathrm {d} t}}\mathbf {e} _{\theta }+{\frac {\mathrm {d} r}{\mathrm {d} t}}\mathbf {e} _{r}\\\end{aligned}}}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\mathbf {a} &={\frac {\mathrm {d} \mathbf {v} }{\mathrm {d} t}}\\&={\frac {\mathrm {d} }{\mathrm {d} t}}\left(r{\frac {\mathrm {d} \theta }{\mathrm {d} t}}\mathbf {e} _{\theta }\right)+{\frac {\mathrm {d} }{\mathrm {d} t}}\left({\frac {\mathrm {d} r}{\mathrm {d} t}}\mathbf {e} _{r}\right)\\&=r{\frac {\mathrm {d} \theta }{\mathrm {d} t}}{\frac {\mathrm {d} \mathbf {e} _{\theta }}{\mathrm {d} t}}+r{\frac {\mathrm {d} ^{2}\theta }{\mathrm {d} t^{2}}}\mathbf {e} _{\theta }+{\frac {\mathrm {d} r}{\mathrm {d} t}}{\frac {\mathrm {d} \theta }{\mathrm {d} t}}\mathbf {e} _{\theta }+{\frac {\mathrm {d} ^{2}r}{\mathrm {d} t^{2}}}\mathbf {e} _{r}+{\frac {\mathrm {d} r}{\mathrm {d} t}}{\frac {\mathrm {d} \mathbf {e} _{r}}{\mathrm {d} t}}\\&=-r\left({\frac {\mathrm {d} \theta }{\mathrm {d} t}}\right)^{2}\mathbf {e} _{r}+r{\frac {\mathrm {d} ^{2}\theta }{\mathrm {d} t^{2}}}\mathbf {e} _{\theta }+{\frac {\mathrm {d} r}{\mathrm {d} t}}{\frac {\mathrm {d} \theta }{\mathrm {d} t}}\mathbf {e} _{\theta }+{\frac {\mathrm {d} ^{2}r}{\mathrm {d} t^{2}}}\mathbf {e} _{r}+{\frac {\mathrm {d} r}{\mathrm {d} t}}{\frac {\mathrm {d} \theta }{\mathrm {d} t}}\mathbf {e} _{\theta }\\&=\left[{\frac {\mathrm {d} ^{2}r}{\mathrm {d} t^{2}}}-r\left({\frac {\mathrm {d} \theta }{\mathrm {d} t}}\right)^{2}\right]\mathbf {e} _{r}+\left(2{\frac {\mathrm {d} r}{\mathrm {d} t}}{\frac {\mathrm {d} \theta }{\mathrm {d} t}}+r{\frac {\mathrm {d} ^{2}\theta }{\mathrm {d} t^{2}}}\right)\mathbf {e} _{\theta }\\\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>θ</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>θ</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>r</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>θ</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>θ</mi> </mrow> </msub> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>θ</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>θ</mi> </mrow> </msub> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>r</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>θ</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>θ</mi> </mrow> </msub> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>r</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>r</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mo>−</mo> <mi>r</mi> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>θ</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> <mo>+</mo> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>θ</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>θ</mi> </mrow> </msub> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>r</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>θ</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>θ</mi> </mrow> </msub> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>r</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>r</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>θ</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>θ</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow> <mo>[</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>r</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>−</mo> <mi>r</mi> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>θ</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>]</mo> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> <mo>+</mo> <mrow> <mo>(</mo> <mrow> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>r</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>θ</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>θ</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>θ</mi> </mrow> </msub> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\mathbf {a} &={\frac {\mathrm {d} \mathbf {v} }{\mathrm {d} t}}\\&={\frac {\mathrm {d} }{\mathrm {d} t}}\left(r{\frac {\mathrm {d} \theta }{\mathrm {d} t}}\mathbf {e} _{\theta }\right)+{\frac {\mathrm {d} }{\mathrm {d} t}}\left({\frac {\mathrm {d} r}{\mathrm {d} t}}\mathbf {e} _{r}\right)\\&=r{\frac {\mathrm {d} \theta }{\mathrm {d} t}}{\frac {\mathrm {d} \mathbf {e} _{\theta }}{\mathrm {d} t}}+r{\frac {\mathrm {d} ^{2}\theta }{\mathrm {d} t^{2}}}\mathbf {e} _{\theta }+{\frac {\mathrm {d} r}{\mathrm {d} t}}{\frac {\mathrm {d} \theta }{\mathrm {d} t}}\mathbf {e} _{\theta }+{\frac {\mathrm {d} ^{2}r}{\mathrm {d} t^{2}}}\mathbf {e} _{r}+{\frac {\mathrm {d} r}{\mathrm {d} t}}{\frac {\mathrm {d} \mathbf {e} _{r}}{\mathrm {d} t}}\\&=-r\left({\frac {\mathrm {d} \theta }{\mathrm {d} t}}\right)^{2}\mathbf {e} _{r}+r{\frac {\mathrm {d} ^{2}\theta }{\mathrm {d} t^{2}}}\mathbf {e} _{\theta }+{\frac {\mathrm {d} r}{\mathrm {d} t}}{\frac {\mathrm {d} \theta }{\mathrm {d} t}}\mathbf {e} _{\theta }+{\frac {\mathrm {d} ^{2}r}{\mathrm {d} t^{2}}}\mathbf {e} _{r}+{\frac {\mathrm {d} r}{\mathrm {d} t}}{\frac {\mathrm {d} \theta }{\mathrm {d} t}}\mathbf {e} _{\theta }\\&=\left[{\frac {\mathrm {d} ^{2}r}{\mathrm {d} t^{2}}}-r\left({\frac {\mathrm {d} \theta }{\mathrm {d} t}}\right)^{2}\right]\mathbf {e} _{r}+\left(2{\frac {\mathrm {d} r}{\mathrm {d} t}}{\frac {\mathrm {d} \theta }{\mathrm {d} t}}+r{\frac {\mathrm {d} ^{2}\theta }{\mathrm {d} t^{2}}}\right)\mathbf {e} _{\theta }\\\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/27510f25b4ac9bdfb43ff0a70fb7f164c36b99fe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -15.288ex; margin-bottom: -0.217ex; width:60.492ex; height:32.176ex;" alt="{\displaystyle {\begin{aligned}\mathbf {a} &={\frac {\mathrm {d} \mathbf {v} }{\mathrm {d} t}}\\&={\frac {\mathrm {d} }{\mathrm {d} t}}\left(r{\frac {\mathrm {d} \theta }{\mathrm {d} t}}\mathbf {e} _{\theta }\right)+{\frac {\mathrm {d} }{\mathrm {d} t}}\left({\frac {\mathrm {d} r}{\mathrm {d} t}}\mathbf {e} _{r}\right)\\&=r{\frac {\mathrm {d} \theta }{\mathrm {d} t}}{\frac {\mathrm {d} \mathbf {e} _{\theta }}{\mathrm {d} t}}+r{\frac {\mathrm {d} ^{2}\theta }{\mathrm {d} t^{2}}}\mathbf {e} _{\theta }+{\frac {\mathrm {d} r}{\mathrm {d} t}}{\frac {\mathrm {d} \theta }{\mathrm {d} t}}\mathbf {e} _{\theta }+{\frac {\mathrm {d} ^{2}r}{\mathrm {d} t^{2}}}\mathbf {e} _{r}+{\frac {\mathrm {d} r}{\mathrm {d} t}}{\frac {\mathrm {d} \mathbf {e} _{r}}{\mathrm {d} t}}\\&=-r\left({\frac {\mathrm {d} \theta }{\mathrm {d} t}}\right)^{2}\mathbf {e} _{r}+r{\frac {\mathrm {d} ^{2}\theta }{\mathrm {d} t^{2}}}\mathbf {e} _{\theta }+{\frac {\mathrm {d} r}{\mathrm {d} t}}{\frac {\mathrm {d} \theta }{\mathrm {d} t}}\mathbf {e} _{\theta }+{\frac {\mathrm {d} ^{2}r}{\mathrm {d} t^{2}}}\mathbf {e} _{r}+{\frac {\mathrm {d} r}{\mathrm {d} t}}{\frac {\mathrm {d} \theta }{\mathrm {d} t}}\mathbf {e} _{\theta }\\&=\left[{\frac {\mathrm {d} ^{2}r}{\mathrm {d} t^{2}}}-r\left({\frac {\mathrm {d} \theta }{\mathrm {d} t}}\right)^{2}\right]\mathbf {e} _{r}+\left(2{\frac {\mathrm {d} r}{\mathrm {d} t}}{\frac {\mathrm {d} \theta }{\mathrm {d} t}}+r{\frac {\mathrm {d} ^{2}\theta }{\mathrm {d} t^{2}}}\right)\mathbf {e} _{\theta }\\\end{aligned}}}"></span></dd></dl> <div class="mw-heading mw-heading4"><h4 id="按自然坐标系分解"><span id=".E6.8C.89.E8.87.AA.E7.84.B6.E5.9D.90.E6.A0.87.E7.B3.BB.E5.88.86.E8.A7.A3"></span>按自然坐标系分解</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%8A%A0%E9%80%9F%E5%BA%A6&action=edit&section=14" title="编辑章节：按自然坐标系分解"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Acceleration_components_modified.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/82/Acceleration_components_modified.svg/200px-Acceleration_components_modified.svg.png" decoding="async" width="200" height="105" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/82/Acceleration_components_modified.svg/300px-Acceleration_components_modified.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/82/Acceleration_components_modified.svg/400px-Acceleration_components_modified.svg.png 2x" data-file-width="512" data-file-height="269" /></a><figcaption>加速度按自然坐标系分解</figcaption></figure> <p>假設一個質點移動於二維平面。在質點軌道的任意位置，二維自然坐标系的一个坐标轴方向（<a href="/wiki/%E6%9B%B2%E7%BA%BF%E7%9A%84%E5%BE%AE%E5%88%86%E5%87%A0%E4%BD%95" title="曲线的微分几何">切向</a>）保持与軌道<a href="/wiki/%E5%88%87%E7%B7%9A" class="mw-redirect" title="切線">切線</a>方向平行，另一个坐标轴方向（<a href="/wiki/%E6%B3%95%E5%90%91" class="mw-redirect" title="法向">法向</a>）則與軌道<a href="/wiki/%E6%B3%95%E7%BA%BF" title="法线">法線</a>平行。分解按右图。向量可以無限地做拆解，所以只需要選擇對於分析最有利的為主！通常以切線方向和法線方向來分解。 </p><p>简单地说，加速度的切向分量<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{\text{t}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>t</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{\text{t}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6a062b1c1f2742dad1eff3002c4b9443a1ce3b72" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.102ex; height:2.009ex;" alt="{\displaystyle a_{\text{t}}}"></span>表示速度大小的变化，加速度的法向分量<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{\text{n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>n</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{\text{n}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2479202f13604573da76057b4b9657a2e71e2401" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.376ex; height:2.009ex;" alt="{\displaystyle a_{\text{n}}}"></span>表示速度方向的变化，即 </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{\text{t}}={\frac {\mathrm {d} v}{\mathrm {d} t}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>t</mtext> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>v</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{\text{t}}={\frac {\mathrm {d} v}{\mathrm {d} t}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2a1eb36834f9a49636002b985d3259737329156d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:8.456ex; height:5.509ex;" alt="{\displaystyle a_{\text{t}}={\frac {\mathrm {d} v}{\mathrm {d} t}}}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{\text{n}}={\frac {v^{2}}{\rho }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>n</mtext> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>ρ</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{\text{n}}={\frac {v^{2}}{\rho }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6208aa94b61ed62d49f6735eb6470143c5d161ea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:8.492ex; height:6.176ex;" alt="{\displaystyle a_{\text{n}}={\frac {v^{2}}{\rho }}}"></span></dd></dl> <p>其中，<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e07b00e7fc0847fbd16391c778d65bc25c452597" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle v}"></span>为此时刻的速度大小，<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f7d439671d1289b6a816e6af7a304be40608d64" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.202ex; height:2.176ex;" alt="{\displaystyle \rho }"></span>为此时刻的曲率半径<sup id="cite_ref-郑永令力学_2-5" class="reference"><a href="#cite_note-郑永令力学-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup class="reference" style="white-space:nowrap;">:24</sup>。 </p> <div class="mw-heading mw-heading3"><h3 id="按功能分解"><span id=".E6.8C.89.E5.8A.9F.E8.83.BD.E5.88.86.E8.A7.A3"></span>按功能分解</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%8A%A0%E9%80%9F%E5%BA%A6&action=edit&section=15" title="编辑章节：按功能分解"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading4"><h4 id="匀速圆周运动_向心加速度"><span id=".E5.8C.80.E9.80.9F.E5.9C.86.E5.91.A8.E8.BF.90.E5.8A.A8_.E5.90.91.E5.BF.83.E5.8A.A0.E9.80.9F.E5.BA.A6"></span>匀速圆周运动向心加速度</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%8A%A0%E9%80%9F%E5%BA%A6&action=edit&section=16" title="编辑章节：匀速圆周运动向心加速度"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r84833064"><div role="note" class="hatnote navigation-not-searchable">主条目：<a href="/wiki/%E5%8C%80%E9%80%9F%E5%9C%86%E5%91%A8%E8%BF%90%E5%8A%A8" class="mw-redirect" title="匀速圆周运动">匀速圆周运动</a></div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Aequalsvsquareoverr.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b6/Aequalsvsquareoverr.png/200px-Aequalsvsquareoverr.png" decoding="async" width="200" height="140" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b6/Aequalsvsquareoverr.png/300px-Aequalsvsquareoverr.png 1.5x, //upload.wikimedia.org/wikipedia/commons/b/b6/Aequalsvsquareoverr.png 2x" data-file-width="324" data-file-height="227" /></a><figcaption>在匀速圆周运动中，速度大小不改变，方向不停改变，需要保持垂直于其切向的加速度来改变方向。</figcaption></figure> <p>若质点以不变的速率（速度大小）沿著圆周繞著圆心运动，则质点呈匀速圆周运动，质点具有向心加速度<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {a} _{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {a} _{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/907e8b9a45f7f587f88b8b2655aab13fabd7a0a5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.518ex; height:2.009ex;" alt="{\displaystyle \mathbf {a} _{n}}"></span>，其方向保持沿半径方向向裏（因此不断变化），大小为 </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{\text{n}}=\omega ^{2}r={\frac {v^{2}}{r}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>n</mtext> </mrow> </msub> <mo>=</mo> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>r</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>r</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{\text{n}}=\omega ^{2}r={\frac {v^{2}}{r}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5b6f84f7cfdc6f34f41df3c65b9128c3a4c3a5f1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:15.14ex; height:5.676ex;" alt="{\displaystyle a_{\text{n}}=\omega ^{2}r={\frac {v^{2}}{r}}}"></span></dd></dl> <p>其中，<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ω</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48eff443f9de7a985bb94ca3bde20813ea737be8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.446ex; height:1.676ex;" alt="{\displaystyle \omega }"></span>是角速度。 </p><p>這公式也可以从极坐标系分解中，代入与匀速圆周运动相关的特殊值得到。更一般的情况下（非匀速圆周运动），以矢量来表示， </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {a} _{\text{n}}=\mathbf {\omega } \times (\mathbf {\omega } \times \mathbf {r} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>n</mtext> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>ω</mi> </mrow> <mo>×</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>ω</mi> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {a} _{\text{n}}=\mathbf {\omega } \times (\mathbf {\omega } \times \mathbf {r} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c1ba2452df6adab66209795052c1c82fc64af4b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.028ex; height:2.843ex;" alt="{\displaystyle \mathbf {a} _{\text{n}}=\mathbf {\omega } \times (\mathbf {\omega } \times \mathbf {r} )}"></span></dd></dl> <p>在矢量式中，令沿半径向外为正。在平面的情况下，该矢量式約化为上述标量式，这时会得到一个负号，通常以圓座標來表示最為合適。 </p><p>假设，在一根绳子的一端系上一个小物体（比如石頭），另一端握在手中，大致保持手不动而水平旋转，则手会明確地感受到绳子的拉力，该拉力的反作用力在绳子的另一端表現為向心力，提供小物体的向心加速度。当转得越快，向心力会越大，可以定性地验证上述向量式。从这个实验，可以看出，向心加速度总是使物体趋向于朝著圆心做运动；如果没有绳子施予向心力，物体一定会飞奔出去。 </p><p>再舉一个例子，在游乐场的巨大旋转圆盘上，大部分遊客都會站立不稳，总是会向外摔倒，這是因為缺乏向心力施予於遊客。在旋转圆盘的非惯性系中，遊客會感受到<a href="/wiki/%E6%85%A3%E6%80%A7%E5%8A%9B" title="慣性力">慣性力</a>，但由於缺乏向心力，無法達成平衡狀態，因此被向外“甩”出去<sup id="cite_ref-Lanczos_4-1" class="reference"><a href="#cite_note-Lanczos-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup><sup class="reference" style="white-space:nowrap;">:96-100</sup>，這慣性力又稱為<a href="/wiki/%E7%A6%BB%E5%BF%83%E5%8A%9B" class="mw-redirect" title="离心力">离心力</a>，人们以这个原理制成了<a href="/wiki/%E9%9B%A2%E5%BF%83%E6%A9%9F" title="離心機">离心机</a>。 </p><p>上述公式不但對於从匀速圆周运动成立，也可以應用於各种圆周运动、甚至任意曲线运动，只是上述的<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e07b00e7fc0847fbd16391c778d65bc25c452597" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle v}"></span>和<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ω</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48eff443f9de7a985bb94ca3bde20813ea737be8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.446ex; height:1.676ex;" alt="{\displaystyle \omega }"></span>应理解为该时刻的瞬时物理量，<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"></span>应以<a href="/wiki/%E6%9B%B2%E7%8E%87%E5%8D%8A%E5%BE%84" title="曲率半径">曲率半径</a><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f7d439671d1289b6a816e6af7a304be40608d64" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.202ex; height:2.176ex;" alt="{\displaystyle \rho }"></span>替代，表示的是物体的加速度在垂直于路径方向的分量。 </p> <div class="mw-heading mw-heading5"><h5 id="向心加速度的推导"><span id=".E5.90.91.E5.BF.83.E5.8A.A0.E9.80.9F.E5.BA.A6.E7.9A.84.E6.8E.A8.E5.AF.BC"></span>向心加速度的推导</h5><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%8A%A0%E9%80%9F%E5%BA%A6&action=edit&section=17" title="编辑章节：向心加速度的推导"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>如勻速圓周運動圖所示，某一时刻质点速度为<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {v} _{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {v} _{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/282458bb19c231f94697405bddd93af04a34cabe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.465ex; height:2.009ex;" alt="{\displaystyle \mathbf {v} _{1}}"></span>，极短时间<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle dt}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle dt}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ebee76a835701fd1f26047a09855f2ea36bb08fc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.055ex; height:2.176ex;" alt="{\displaystyle dt}"></span>过后，质点沿圆周前行，速度变成<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {v} _{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {v} _{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/498720fbe6f897f2b86d2cf0f37498d682932aa0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.465ex; height:2.009ex;" alt="{\displaystyle \mathbf {v} _{2}}"></span>。因为是匀速圆周运动，速度大小保持为<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e07b00e7fc0847fbd16391c778d65bc25c452597" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle v}"></span>，但方向会保持与圆相切（垂直于半径<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"></span>），不断改变。关注图的左上部分，当<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ed48a5e36207156fb792fa79d29925d2f7901e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.332ex; height:2.509ex;" alt="{\displaystyle \beta }"></span>使用<a href="/wiki/%E5%BC%A7%E5%BA%A6" title="弧度">弧度值</a>时，速度大小的改变<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle dv}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mi>v</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle dv}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9e5f695cb4931398dd078f0335e6de6905abf748" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.343ex; height:2.176ex;" alt="{\displaystyle dv}"></span>在<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d\beta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mi>β</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d\beta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2d013a6b8153d430d38493445c3e8f03c9332d12" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.548ex; height:2.509ex;" alt="{\displaystyle d\beta }"></span>极小时近似于<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle vd\beta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> <mi>d</mi> <mi>β</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle vd\beta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9abe3c42e6edc885cb82d196d28887187ef35160" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.675ex; height:2.509ex;" alt="{\displaystyle vd\beta }"></span>，方向大约沿半径向裏，则该段时间内的平均加速度大小为<sup id="cite_ref-郑永令力学_2-6" class="reference"><a href="#cite_note-郑永令力学-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup class="reference" style="white-space:nowrap;">:23</sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{\text{n}}=v{\frac {\mathrm {d} \beta }{\mathrm {d} t}}=v\omega }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>n</mtext> </mrow> </msub> <mo>=</mo> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>β</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mi>v</mi> <mi>ω</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{\text{n}}=v{\frac {\mathrm {d} \beta }{\mathrm {d} t}}=v\omega }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/108ff31dd7475b9ff0e7a12ea15c571f8b368efc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:15.735ex; height:5.509ex;" alt="{\displaystyle a_{\text{n}}=v{\frac {\mathrm {d} \beta }{\mathrm {d} t}}=v\omega }"></span></dd></dl> <p>其中，角速度<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega ={\frac {\mathrm {d} \beta }{\mathrm {d} t}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ω</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>β</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega ={\frac {\mathrm {d} \beta }{\mathrm {d} t}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/84d926439af38a220c26e2ac5acc68f09cc585ac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:8.005ex; height:5.509ex;" alt="{\displaystyle \omega ={\frac {\mathrm {d} \beta }{\mathrm {d} t}}}"></span>。 </p><p>以上“近似”在<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle dt\to 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mi>t</mi> <mo stretchy="false">→</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle dt\to 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/02f957c6106281a61de6adf0ec77aaf6ea42aba9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.832ex; height:2.176ex;" alt="{\displaystyle dt\to 0}"></span>时精确成立。又因为 </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v=\omega r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> <mo>=</mo> <mi>ω</mi> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v=\omega r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b5edc54db08559dc3e33cb3cc568126e12b46521" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.721ex; height:1.676ex;" alt="{\displaystyle v=\omega r}"></span></dd></dl> <p>所以， </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{\text{n}}=\omega ^{2}r={\frac {v^{2}}{r}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>n</mtext> </mrow> </msub> <mo>=</mo> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>r</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>r</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{\text{n}}=\omega ^{2}r={\frac {v^{2}}{r}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5b6f84f7cfdc6f34f41df3c65b9128c3a4c3a5f1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:15.14ex; height:5.676ex;" alt="{\displaystyle a_{\text{n}}=\omega ^{2}r={\frac {v^{2}}{r}}}"></span></dd></dl> <div class="mw-heading mw-heading4"><h4 id="科里奥利效应"><span id=".E7.A7.91.E9.87.8C.E5.A5.A5.E5.88.A9.E6.95.88.E5.BA.94"></span>科里奥利效应</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%8A%A0%E9%80%9F%E5%BA%A6&action=edit&section=18" title="编辑章节：科里奥利效应"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Winston_2016-02-12_1200Z.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e5/Winston_2016-02-12_1200Z.png/200px-Winston_2016-02-12_1200Z.png" decoding="async" width="200" height="200" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e5/Winston_2016-02-12_1200Z.png/300px-Winston_2016-02-12_1200Z.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e5/Winston_2016-02-12_1200Z.png/400px-Winston_2016-02-12_1200Z.png 2x" data-file-width="600" data-file-height="600" /></a><figcaption><a href="/wiki/%E6%B0%A3%E6%97%8B%E6%BA%AB%E6%96%AF%E9%A0%93" title="氣旋溫斯頓">氣旋溫斯頓</a>接近初始峰值强度時的紅外線衛星雲圖。由於科里奧利力影響，風暴以順時針方向旋轉</figcaption></figure> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r84833064"><div role="note" class="hatnote navigation-not-searchable">主条目：<a href="/wiki/%E7%A7%91%E9%87%8C%E5%A5%A5%E5%88%A9%E5%8A%9B" title="科里奥利力">科里奥利力</a></div> <p>給定固定参考系<i>S</i>與旋轉参考系<i>S'</i>，從固定参考系<i>S</i>觀察，旋轉参考系<i>S'</i>以勻角速度轉動。移動於旋轉参考系<i>S'</i>的質點因為運動速度而產生的偏轉效應，稱為「科里奥利效应」，這是為紀念法国科学家<a href="/wiki/%E8%B4%BE%E6%96%AF%E5%B8%95-%E5%8F%A4%E6%96%AF%E5%A1%94%E5%A4%AB%C2%B7%E7%A7%91%E9%87%8C%E5%A5%A5%E5%88%A9" class="mw-redirect" title="贾斯帕-古斯塔夫·科里奥利">贾斯帕-古斯塔夫·科里奥利</a>而命名。 </p><p>舉例而言，設想一个巨大的圆盘在地上绕著定點匀角速度轉動（定點運動），而這定點為圆盤的圆心，在圆盘上沿半径方向有一个直导轨，一个物体被限制在导轨上运动，从圆心匀速向外移动。從地上（固定參考系）觀察，物體的軌跡不是一条直线，而是一条弧形或者螺旋形路线，物体也會感受到導軌的<a href="/wiki/%E7%B4%84%E6%9D%9F%E5%8A%9B" class="mw-redirect" title="約束力">約束力</a>，其方向垂直於導軌，並且指向圆盘旋轉方向（不是角速度向量的方向），這約束力促使物體朝著圆盘旋轉方向加速，使物體的軌跡呈弧形或螺旋形。從圓盤（旋轉坐標系）觀察，物體所感受到的科里奥利力會與導軌施予的約束力相抵消，因此，物體只會呈直線運動。假若，導軌不存在，則物體會逆著圆盘旋轉方向以科里奥利加速度運動。 </p><p>在科里奥利效应裏，参考系S'的物體的柯里奧利加速度與感受到的科里奥利力分別為<sup id="cite_ref-Lanczos_4-2" class="reference"><a href="#cite_note-Lanczos-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup><sup class="reference" style="white-space:nowrap;">:100-103</sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {a} _{\text{cor}}=-2{\boldsymbol {\Omega }}\times \mathbf {v} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>cor</mtext> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Ω</mi> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {a} _{\text{cor}}=-2{\boldsymbol {\Omega }}\times \mathbf {v} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ba55598dbd7e118ae7530393fef36dd01b996057" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.98ex; height:2.509ex;" alt="{\displaystyle \mathbf {a} _{\text{cor}}=-2{\boldsymbol {\Omega }}\times \mathbf {v} }"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {F} _{\text{cor}}=-2m{\boldsymbol {\Omega }}\times \mathbf {v} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>cor</mtext> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mn>2</mn> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Ω</mi> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {F} _{\text{cor}}=-2m{\boldsymbol {\Omega }}\times \mathbf {v} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ea3a34397d81fc3e0204a8f0187b1431d6477ac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:18.404ex; height:2.509ex;" alt="{\displaystyle \mathbf {F} _{\text{cor}}=-2m{\boldsymbol {\Omega }}\times \mathbf {v} }"></span></dd></dl> <p>其中<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {a} _{\text{cor}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>cor</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {a} _{\text{cor}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4d88792266f3e68b9c0cd89be3ac0c00a24642ef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.728ex; height:2.009ex;" alt="{\displaystyle \mathbf {a} _{\text{cor}}}"></span>为科里奥利加速度，<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {\Omega }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Ω</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {\Omega }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9c95a28793dac7413de04992d822822f7e2dba16" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.931ex; height:2.176ex;" alt="{\displaystyle {\boldsymbol {\Omega }}}"></span>为参考系S'在参考系S中的<a href="/wiki/%E8%A7%92%E9%80%9F%E5%BA%A6" title="角速度">角速度矢量</a>，<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {v} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {v} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35c1866e359fbfd2e0f606c725ba5cc37a5195d6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.411ex; height:1.676ex;" alt="{\displaystyle \mathbf {v} }"></span>为物体在参考系<i>S'</i>中的速度矢量。 </p><p>在<a href="/wiki/%E6%B0%94%E8%B1%A1%E5%AD%A6" title="气象学">气象学</a>裏，科里奥利力使得<a href="/wiki/%E7%86%B1%E5%B8%B6%E6%B0%A3%E6%97%8B" title="熱帶氣旋">熱帶氣旋</a>在沒有強<a href="/wiki/%E7%86%B1%E5%B8%B6%E6%B0%A3%E6%97%8B#引導氣流（駛流）" title="熱帶氣旋">引導氣流</a>影響下移向兩極<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup>。熱帶氣旋靠近<a href="/wiki/%E5%85%A9%E6%A5%B5" class="mw-redirect" title="兩極">兩極</a>部分含有東風，科里奥利力會將東風拉向兩極；靠近赤道部分含有西風，科里奥利力會將西風拉向赤道。在地球上越接近赤道科里奥利力會越弱，所以科里奥利力影響熱帶氣旋靠近兩極部分會較靠近赤道部分為多。因此，在沒有其他引導氣流抵消科里奥利力的情況下，北半球的熱帶氣旋一般會向北移動，而南半球的熱帶氣旋則會向南移動<sup id="cite_ref-BritTCtrackcoriolis_9-0" class="reference"><a href="#cite_note-BritTCtrackcoriolis-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup>。 </p><p>科里奥利力也會開啟氣旋系統的旋轉，但驅動高速度旋轉的主要因素，不是科里奥利力，而是<a href="/wiki/%E6%B1%BD%E5%8C%96%E7%86%B1" class="mw-redirect" title="汽化熱">凝結熱</a><sup id="cite_ref-NOAA_Question_of_the_Month_10-0" class="reference"><a href="#cite_note-NOAA_Question_of_the_Month-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup>。 </p> <div class="mw-heading mw-heading4"><h4 id="欧拉力"><span id=".E6.AC.A7.E6.8B.89.E5.8A.9B"></span>欧拉力</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%8A%A0%E9%80%9F%E5%BA%A6&action=edit&section=19" title="编辑章节：欧拉力"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r84833064"><div role="note" class="hatnote navigation-not-searchable">主条目：<a href="/wiki/%E6%AC%A7%E6%8B%89%E5%8A%9B" title="欧拉力">欧拉力</a></div> <p>給定固定参考系<i>S</i>與旋轉参考系<i>S'</i>，從固定参考系<i>S</i>觀察，旋轉参考系<i>S'</i>以非勻角速度<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {\Omega }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Ω</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {\Omega }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9c95a28793dac7413de04992d822822f7e2dba16" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.931ex; height:2.176ex;" alt="{\displaystyle {\boldsymbol {\Omega }}}"></span>轉動。從旋轉参考系<i>S'</i>觀察，物體因這非勻角速度而感受到的<a href="/wiki/%E8%99%9B%E8%A8%AD%E5%8A%9B" class="mw-redirect" title="虛設力">虛設力</a>（fictitious force）稱為「歐拉力」，產生的加速度稱為「歐拉加速度」。歐拉力<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {F} _{\text{Euler}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>Euler</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {F} _{\text{Euler}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/024719abbcc5fb16901f7b7acfebe2f5c735a0a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.78ex; height:2.509ex;" alt="{\displaystyle \mathbf {F} _{\text{Euler}}}"></span>與歐拉加速度<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {a} _{\text{Euler}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>Euler</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {a} _{\text{Euler}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fab638c6c6f61285d85b96c84100f69c261265cd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.397ex; height:2.009ex;" alt="{\displaystyle \mathbf {a} _{\text{Euler}}}"></span>之間的關係式為 </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {F} _{\text{Euler}}=m\mathbf {a} _{\text{Euler}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>Euler</mtext> </mrow> </msub> <mo>=</mo> <mi>m</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>Euler</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {F} _{\text{Euler}}=m\mathbf {a} _{\text{Euler}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1912fbeadd70f675a19de2ee0da7def8a81b1422" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.316ex; height:2.509ex;" alt="{\displaystyle \mathbf {F} _{\text{Euler}}=m\mathbf {a} _{\text{Euler}}}"></span></dd></dl> <p>設想一个巨大的圆盘在地上绕著定點轉動，而這定點為圆盤的圆心。在圓盤的非圆心位置固定一个物体。圆盘呈非勻角速度運動，則從旋轉参考系S'觀察，延著物体的圓形軌跡切向（不是角速度向量的方向），此物体的受力是歐拉力。 </p><p>欧拉加速度的一般公式为<sup id="cite_ref-Lanczos_4-3" class="reference"><a href="#cite_note-Lanczos-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup><sup class="reference" style="white-space:nowrap;">:100-103</sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {a} _{\text{Euler}}=-{\frac {\mathrm {d} {\boldsymbol {\Omega }}}{\mathrm {d} t}}\times \mathbf {r} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>Euler</mtext> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Ω</mi> </mrow> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {a} _{\text{Euler}}=-{\frac {\mathrm {d} {\boldsymbol {\Omega }}}{\mathrm {d} t}}\times \mathbf {r} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72241055148b4fba203a517f2a324e3bf44fb3f7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:18.306ex; height:5.509ex;" alt="{\displaystyle \mathbf {a} _{\text{Euler}}=-{\frac {\mathrm {d} {\boldsymbol {\Omega }}}{\mathrm {d} t}}\times \mathbf {r} }"></span></dd></dl> <p>其中，<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {r} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {r} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eca0f46511c4c986c48b254073732c0bd98ae0c1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.102ex; height:1.676ex;" alt="{\displaystyle \mathbf {r} }"></span>是物體在旋轉参考系S'的位置。 </p><p>欧拉力为<sup id="cite_ref-Lanczos_4-4" class="reference"><a href="#cite_note-Lanczos-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup><sup class="reference" style="white-space:nowrap;">:100-103</sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {F} _{\text{Euler}}=-m{\frac {\mathrm {d} \mathbf {\Omega } }{\mathrm {d} t}}\times \mathbf {r} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>Euler</mtext> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Ω</mi> </mrow> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {F} _{\text{Euler}}=-m{\frac {\mathrm {d} \mathbf {\Omega } }{\mathrm {d} t}}\times \mathbf {r} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0245577fbf8dd2d40c6859a8025f72f5c703fc1e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:20.73ex; height:5.509ex;" alt="{\displaystyle \mathbf {F} _{\text{Euler}}=-m{\frac {\mathrm {d} \mathbf {\Omega } }{\mathrm {d} t}}\times \mathbf {r} }"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="几种特殊的运动"><span id=".E5.87.A0.E7.A7.8D.E7.89.B9.E6.AE.8A.E7.9A.84.E8.BF.90.E5.8A.A8"></span>几种特殊的运动</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%8A%A0%E9%80%9F%E5%BA%A6&action=edit&section=20" title="编辑章节：几种特殊的运动"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>以下为几种特殊的运动，因为在不同的模型下质点常被不同地近似处理，并且可以得出的结论较之上面的积分式常能极大地简省计算量，故有研究的价值。最常運用的就是拋體運動，以及自由落體。 </p> <div class="mw-heading mw-heading3"><h3 id="匀速直线运动"><span id=".E5.8C.80.E9.80.9F.E7.9B.B4.E7.BA.BF.E8.BF.90.E5.8A.A8"></span>匀速直线运动</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%8A%A0%E9%80%9F%E5%BA%A6&action=edit&section=21" title="编辑章节：匀速直线运动"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r84833064"><div role="note" class="hatnote navigation-not-searchable">主条目：<a href="/wiki/%E5%8C%80%E9%80%9F%E7%9B%B4%E7%BA%BF%E8%BF%90%E5%8A%A8" class="mw-redirect" title="匀速直线运动">匀速直线运动</a></div> <p>若某质点保持加速度<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a=0\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>=</mo> <mn>0</mn> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a=0\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/916725854445eb0914114934985f95d94b9cc384" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-right: -0.387ex; width:5.878ex; height:2.176ex;" alt="{\displaystyle a=0\,\!}"></span>，则其速度<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {v} \,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {v} \,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9df6e8492015c331122565c580264dcc31144461" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-right: -0.387ex; width:1.798ex; height:1.676ex;" alt="{\displaystyle \mathbf {v} \,\!}"></span>的大小和方向不会变化，质点将保持在同一直线上以同一速率（速度大小）运动，这种运动被称作匀速直线运动。特殊地，若速度<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v=0\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> <mo>=</mo> <mn>0</mn> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v=0\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6214b315e4f6628288933c84ceee2535a33147a0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-right: -0.387ex; width:5.776ex; height:2.176ex;" alt="{\displaystyle v=0\,\!}"></span>，则质点静止。 </p><p>匀速直线运动主要出现在<a href="/wiki/%E7%89%9B%E9%A1%BF%E7%AC%AC%E4%B8%80%E5%AE%9A%E5%BE%8B" class="mw-redirect" title="牛顿第一定律">牛顿第一定律</a>中，该定律表示：“不受任何力或受合力为零的物体作匀速直线运动。”由于<a href="/wiki/%E5%9F%BA%E6%9C%AC%E7%9B%B8%E4%BA%92%E4%BD%9C%E7%94%A8" title="基本相互作用">自然界中大部分力</a>的随距离增大而减小，故离所有其它物体足够远的某一物体的运动能够在足够的<a href="/wiki/%E7%B2%BE%E5%BA%A6" class="mw-redirect" title="精度">精度</a>下被近似为匀速直线运动。这种近似常被用于寻找<a href="/wiki/%E6%83%AF%E6%80%A7%E7%B3%BB" class="mw-redirect" title="惯性系">惯性参考系</a>和<a href="/wiki/%E7%B2%92%E5%AD%90%E7%89%A9%E7%90%86%E5%AD%A6" class="mw-redirect" title="粒子物理学">粒子物理学</a>的运算当中。 </p> <div class="mw-heading mw-heading3"><h3 id="匀变速直线运动"><span id=".E5.8C.80.E5.8F.98.E9.80.9F.E7.9B.B4.E7.BA.BF.E8.BF.90.E5.8A.A8"></span>匀变速直线运动</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%8A%A0%E9%80%9F%E5%BA%A6&action=edit&section=22" title="编辑章节：匀变速直线运动"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Free_Fall_(Six_Flags_Over_Georgia).jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/58/Free_Fall_%28Six_Flags_Over_Georgia%29.jpg/170px-Free_Fall_%28Six_Flags_Over_Georgia%29.jpg" decoding="async" width="170" height="227" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/58/Free_Fall_%28Six_Flags_Over_Georgia%29.jpg/255px-Free_Fall_%28Six_Flags_Over_Georgia%29.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/58/Free_Fall_%28Six_Flags_Over_Georgia%29.jpg/340px-Free_Fall_%28Six_Flags_Over_Georgia%29.jpg 2x" data-file-width="768" data-file-height="1024" /></a><figcaption>位于乔治亚州的六旗主题公园的自由落体机，从高达数十米的地方由静止释放，长长的途中几乎只受到重力，近似为自由落体运动，使得乘客落到地面附近时拥有极高的速度。</figcaption></figure> <p>若某作质点作直线运动并保持加速度<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/99a94f96d2455b9d7faf3cec3eb02ab3c455aec1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-right: -0.387ex; width:1.617ex; height:1.676ex;" alt="{\displaystyle a\,\!}"></span>恒定，则质点作匀变速直线运动。在这种情况下，若<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t=0\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> <mo>=</mo> <mn>0</mn> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t=0\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e59305766bd22dc9622f040d92d974849e0bc6e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-right: -0.387ex; width:5.488ex; height:2.176ex;" alt="{\displaystyle t=0\,\!}"></span>时刻速度为<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v_{0}\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v_{0}\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/79348a856802c3a5e81ba53d5472425c0b0deedc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-right: -0.387ex; width:2.569ex; height:2.009ex;" alt="{\displaystyle v_{0}\,\!}"></span>，<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/60ca0e6d5640c7a6280feeaa20db96e2c00adf94" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-right: -0.387ex; width:1.227ex; height:2.009ex;" alt="{\displaystyle t\,\!}"></span>时刻速度为<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v(t)\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v(t)\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/15a7e1d2c8a3fcc645706d522e25884e0d8b3284" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-right: -0.387ex; width:4.164ex; height:2.843ex;" alt="{\displaystyle v(t)\,\!}"></span>，<a href="/wiki/%E4%BD%8D%E7%A7%BB" title="位移">位移</a>为<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s(x)\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s(x)\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6d87da32d21c6cb4cbf7575886a3267634f3addf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-right: -0.387ex; width:4.617ex; height:2.843ex;" alt="{\displaystyle s(x)\,\!}"></span>，则可由上面积分式得出 </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v(t)=v_{0}+at\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <mi>a</mi> <mi>t</mi> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v(t)=v_{0}+at\,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cafd20540cf12589d9362bc9c821fb7308bd252f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.001ex; height:2.843ex;" alt="{\displaystyle v(t)=v_{0}+at\,.}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}s(t)&=v_{0}t+{\frac {1}{2}}at^{2}\\&={\frac {v(t)+v_{0}}{2}}t\\\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>s</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mi>t</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>a</mi> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>v</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>+</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> <mn>2</mn> </mfrac> </mrow> <mi>t</mi> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}s(t)&=v_{0}t+{\frac {1}{2}}at^{2}\\&={\frac {v(t)+v_{0}}{2}}t\\\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32ea6a29bf8044bddb8775888cb693dbb8726965" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.691ex; margin-bottom: -0.313ex; width:18.573ex; height:11.176ex;" alt="{\displaystyle {\begin{aligned}s(t)&=v_{0}t+{\frac {1}{2}}at^{2}\\&={\frac {v(t)+v_{0}}{2}}t\\\end{aligned}}}"></span></dd></dl> <p>以及得出 </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a={\frac {v(t)^{2}-v_{0}^{2}}{2s(t)}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>v</mi> <mo stretchy="false">(</mo> <mi>t</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msubsup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mrow> <mrow> <mn>2</mn> <mi>s</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a={\frac {v(t)^{2}-v_{0}^{2}}{2s(t)}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d11e110543d38e5137778972645b6becc8bc689f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:15.017ex; height:6.843ex;" alt="{\displaystyle a={\frac {v(t)^{2}-v_{0}^{2}}{2s(t)}}}"></span></dd></dl> <div class="mw-heading mw-heading4"><h4 id="自由落体运动_重力加速度"><span id=".E8.87.AA.E7.94.B1.E8.90.BD.E4.BD.93.E8.BF.90.E5.8A.A8_.E9.87.8D.E5.8A.9B.E5.8A.A0.E9.80.9F.E5.BA.A6"></span>自由落体运动重力加速度</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%8A%A0%E9%80%9F%E5%BA%A6&action=edit&section=23" title="编辑章节：自由落体运动重力加速度"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r84833064"><div role="note" class="hatnote navigation-not-searchable">主条目：<a href="/wiki/%E8%87%AA%E7%94%B1%E8%90%BD%E4%BD%93" class="mw-redirect" title="自由落体">自由落体</a></div> <p>自由落体运动是指初速度为0，加速度恒为竖直向下 <sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> 的<a href="/wiki/%E9%87%8D%E5%8A%9B%E5%8A%A0%E9%80%9F%E5%BA%A6" title="重力加速度">重力加速度</a>g的运动，在地球上大约有<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g=9.8\operatorname {m\cdot s^{-2}} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo>=</mo> <mn>9.8</mn> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mi mathvariant="normal">m</mi> <mo>⋅</mo> <msup> <mi mathvariant="normal">s</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>-</mtext> <mn>2</mn> </mrow> </msup> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g=9.8\operatorname {m\cdot s^{-2}} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/421b7f97904fad3b72c81639abd9d16ad24a8cb1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.707ex; height:3.009ex;" alt="{\displaystyle g=9.8\operatorname {m\cdot s^{-2}} }"></span><sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup>。自由落体运动是匀变速直线运动的一种特殊情况。自由落体运动是将地球上的物体下落的状况进行理想化的抽象模型，当物体在地面附近，且所受<a href="/wiki/%E7%A9%BA%E6%B0%94%E9%98%BB%E5%8A%9B" class="mw-redirect" title="空气阻力">空气阻力</a>远小于其<a href="/wiki/%E9%87%8D%E5%8A%9B" class="mw-redirect" title="重力">重力</a>时，在一定精度内可被视作自由落体运动。 </p> <div class="mw-heading mw-heading3"><h3 id="加速度恒定的运动"><span id=".E5.8A.A0.E9.80.9F.E5.BA.A6.E6.81.92.E5.AE.9A.E7.9A.84.E8.BF.90.E5.8A.A8"></span>加速度恒定的运动</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%8A%A0%E9%80%9F%E5%BA%A6&action=edit&section=24" title="编辑章节：加速度恒定的运动"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>加速度是一个矢量，因此“加速度恒定”暗示加速度的大小和方向都不随时间变化。 </p> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Bouncing_ball_strobe_edit.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3c/Bouncing_ball_strobe_edit.jpg/200px-Bouncing_ball_strobe_edit.jpg" decoding="async" width="200" height="129" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3c/Bouncing_ball_strobe_edit.jpg/300px-Bouncing_ball_strobe_edit.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3c/Bouncing_ball_strobe_edit.jpg/400px-Bouncing_ball_strobe_edit.jpg 2x" data-file-width="1800" data-file-height="1159" /></a><figcaption>一个从左向右被抛出的篮球是如何在重力下运动的（抛体运动）。相邻两个球影之间有相同的时间间隔。</figcaption></figure> <p>当加速度<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {a} \,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {a} \,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fd4df2e5485c9ffc33b197bdc2e8d93bbe559206" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.687ex; height:1.676ex;" alt="{\displaystyle \mathbf {a} \,}"></span>与速度<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {v} \,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {v} \,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4f9a0b99f33337761cdf1577409d136ec3719a50" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.798ex; height:1.676ex;" alt="{\displaystyle \mathbf {v} \,}"></span>不在同一条直线上时，选取适当的坐标系，可以将其按照<a href="/wiki/%E5%B9%B3%E9%9D%A2%E7%9B%B4%E8%A7%92%E5%9D%90%E6%A0%87%E7%B3%BB" class="mw-redirect" title="平面直角坐标系">平面直角坐标系</a>分解，使质点的运动在其中一个坐标轴上的投影为匀速直线运动，另一个方向上为匀变速直线运动。根据<a href="/w/index.php?title=%E7%8B%AC%E7%AB%8B%E4%BD%9C%E7%94%A8%E5%8E%9F%E7%90%86&action=edit&redlink=1" class="new" title="独立作用原理（页面不存在）">独立作用原理</a>，两者的合运动（即质点的轨迹）为一条抛物线的一部分。 </p> <div class="mw-heading mw-heading4"><h4 id="抛体运动"><span id=".E6.8A.9B.E4.BD.93.E8.BF.90.E5.8A.A8"></span>抛体运动</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%8A%A0%E9%80%9F%E5%BA%A6&action=edit&section=25" title="编辑章节：抛体运动"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r84833064"><div role="note" class="hatnote navigation-not-searchable">主条目：<a href="/wiki/%E5%B9%B3%E6%8A%9B%E8%BF%90%E5%8A%A8" title="平抛运动">平抛运动</a></div> <p>抛体运动具体包括平抛运动和斜（上、下）抛运动，和自由落体运动类似，它是在地球上向不同方向抛出的物体在忽略空气阻力的情况下的运动状况进行理想化的抽象模型。物体拥有一个非竖直方向的不为零初速度<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {v_{0}} \,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="bold">v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">0</mn> </mrow> </msub> </mrow> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {v_{0}} \,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b61b87fd240e4072ff6f7e4d62952dc9204bbeeb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-right: -0.387ex; width:2.975ex; height:2.009ex;" alt="{\displaystyle \mathbf {v_{0}} \,\!}"></span>，和竖直向下、大小恒定为重力加速度g的加速度，落地前的轨迹为一条抛物线的一部分。这也正是抛物线名字的由来。 </p> <div class="mw-heading mw-heading3"><h3 id="简谐运动"><span id=".E7.AE.80.E8.B0.90.E8.BF.90.E5.8A.A8"></span>简谐运动</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%8A%A0%E9%80%9F%E5%BA%A6&action=edit&section=26" title="编辑章节：简谐运动"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r84833064"><div role="note" class="hatnote navigation-not-searchable">主条目：<a href="/wiki/%E7%AE%80%E8%B0%90%E8%BF%90%E5%8A%A8" class="mw-redirect" title="简谐运动">简谐运动</a></div> <p>再一个例子是简谐运动，即质点在正弦或余弦函数形式下的一维运动，一般形式为 </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=A\cos(\omega t+\phi _{0})\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mi>A</mi> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>ω</mi> <mi>t</mi> <mo>+</mo> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=A\cos(\omega t+\phi _{0})\,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0714c2e08be84dc29f1a45a30d4362caaafc7a3d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.078ex; height:2.843ex;" alt="{\displaystyle x=A\cos(\omega t+\phi _{0})\,.}"></span></dd></dl> <p>其中，<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c6aaf5ce10d6add44b973e28fb3d95f37abf3721" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.13ex; height:2.176ex;" alt="{\displaystyle A\,}"></span>为振幅，<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega \,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ω</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega \,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1caedcdf3f920c9e1271a9500bf2621df8a46c0c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.833ex; height:1.676ex;" alt="{\displaystyle \omega \,}"></span>为角频率，<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi _{0}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi _{0}\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b9c17d2d8422c93f363bda7c493a284bbc57e9c2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.827ex; height:2.509ex;" alt="{\displaystyle \phi _{0}\,}"></span>为初相位。将其对时间求导后可得出 </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v=-A\omega \sin(\omega t+\phi _{0})\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> <mo>=</mo> <mo>−</mo> <mi>A</mi> <mi>ω</mi> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>ω</mi> <mi>t</mi> <mo>+</mo> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v=-A\omega \sin(\omega t+\phi _{0})\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/db5bf5f1ebddde7b4ee9a74ee9df767a401af264" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.228ex; height:2.843ex;" alt="{\displaystyle v=-A\omega \sin(\omega t+\phi _{0})\,}"></span></dd></dl> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a=-A\omega ^{2}\cos(\omega t+\phi _{0})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>=</mo> <mo>−</mo> <mi>A</mi> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>ω</mi> <mi>t</mi> <mo>+</mo> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a=-A\omega ^{2}\cos(\omega t+\phi _{0})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4088e2a812245b6e20ea609a92c971e6000d671c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.253ex; height:3.176ex;" alt="{\displaystyle a=-A\omega ^{2}\cos(\omega t+\phi _{0})}"></span></dd></dl> <p>由此也可以得出一些有趣的结论，如在任一时刻， </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a=-x\omega ^{2}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>=</mo> <mo>−</mo> <mi>x</mi> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a=-x\omega ^{2}\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/beeb4b8c6c3498ba48709ac3e7b97257ab7e1181" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.353ex; height:2.843ex;" alt="{\displaystyle a=-x\omega ^{2}\,}"></span></dd></dl> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A^{2}=\left({\frac {a}{\omega ^{2}}}\right)^{2}+\left({\frac {v}{\omega }}\right)^{2}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>a</mi> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>v</mi> <mi>ω</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A^{2}=\left({\frac {a}{\omega ^{2}}}\right)^{2}+\left({\frac {v}{\omega }}\right)^{2}\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2eead108cc0946ea00cbcdcbff732ceaf09aa576" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:23.047ex; height:6.509ex;" alt="{\displaystyle A^{2}=\left({\frac {a}{\omega ^{2}}}\right)^{2}+\left({\frac {v}{\omega }}\right)^{2}\,}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="加速度的应用"><span id=".E5.8A.A0.E9.80.9F.E5.BA.A6.E7.9A.84.E5.BA.94.E7.94.A8"></span>加速度的应用</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%8A%A0%E9%80%9F%E5%BA%A6&action=edit&section=27" title="编辑章节：加速度的应用"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="电磁辐射"><span id=".E7.94.B5.E7.A3.81.E8.BE.90.E5.B0.84"></span>电磁辐射</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%8A%A0%E9%80%9F%E5%BA%A6&action=edit&section=28" title="编辑章节：电磁辐射"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r84833064"><div role="note" class="hatnote navigation-not-searchable">主条目：<a href="/wiki/%E7%94%B5%E7%A3%81%E8%BE%90%E5%B0%84" class="mw-redirect" title="电磁辐射">电磁辐射</a></div> <p>加速度的另一个重要应用之处是带电粒子的电磁辐射（即手机和收音机使用所需要的信号来源）。通过对<a href="/wiki/%E9%BA%A6%E5%85%8B%E6%96%AF%E9%9F%A6%E6%96%B9%E7%A8%8B%E7%BB%84" class="mw-redirect" title="麦克斯韦方程组">麦克斯韦方程组</a>的研究，可以将带电粒子产生电磁辐射的规律概括性地定性总结为：带电粒子的加速度产生电磁辐射，并且电磁辐射的强度和加速度大小正相关<sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup>。电磁辐射常见于用带电粒子的碰撞实验中。这类实验的一个早期著名例子是卢瑟福用电子碰撞金箔的实验，这个实验导致了对原子结构的深入探索。而这类实验至今广泛见于在各种大型<a href="/w/index.php?title=%E7%B2%92%E5%AD%90%E5%B0%8D%E6%92%9E%E6%A9%9F&action=edit&redlink=1" class="new" title="粒子對撞機（页面不存在）">粒子對撞機</a>中，带电粒子以很高的速度运动，经受撞击后变慢、静止甚至反弹回来，这个过程中显然速度发生剧烈改变，一定经受了加速度不为零的过程，也一定会放出辐射。这样产生的辐射被称为<a href="/wiki/%E8%BD%AB%E8%87%B4%E8%BE%90%E5%B0%84" title="轫致辐射">轫致辐射</a>。 </p><p>加速度产生电磁辐射的另一个很典型的例子是回旋加速器（<a href="/wiki/%E5%9B%9E%E6%97%8B%E8%BE%90%E5%B0%84" title="回旋辐射">回旋辐射</a>）。带电粒子在回旋加速器中作圆周运动，每半圈加速一次，同时运动半径增大从而形成螺旋轨道，最后以很高的速度射出。圆周运动需要向心加速度来维持，当速度相当高时，加速度太大以至于因为电磁辐射损失的能量过多，导致回旋加速器实际对粒子的加速作用有上限。光速可比拟时，这时因为相对论效应而需使用<a href="/wiki/%E5%90%8C%E6%AD%A5%E5%8A%A0%E9%80%9F%E5%99%A8" title="同步加速器">同步加速器</a>。这样产生的光能量高、偏振高，并且集中在一个很小的锥角里（相对论效应导致的<a href="/w/index.php?title=%E5%89%8D%E7%81%AF%E6%95%88%E5%BA%94&action=edit&redlink=1" class="new" title="前灯效应（页面不存在）">前灯效应</a>），因此是很好的大型物理用<a href="/wiki/%E5%90%8C%E6%AD%A5%E8%BE%90%E5%B0%84%E5%85%89%E6%BA%90" title="同步辐射光源">同步輻射光源</a>。 </p> <div class="mw-heading mw-heading3"><h3 id="狭义相对论"><span id=".E7.8B.AD.E4.B9.89.E7.9B.B8.E5.AF.B9.E8.AE.BA"></span>狭义相对论</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%8A%A0%E9%80%9F%E5%BA%A6&action=edit&section=29" title="编辑章节：狭义相对论"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r84833064"><div role="note" class="hatnote navigation-not-searchable">主条目：<a href="/wiki/%E6%B4%9B%E4%BC%A6%E5%85%B9%E5%8F%98%E6%8D%A2" title="洛伦兹变换">洛伦兹变换</a></div> <p><a href="/wiki/%E7%8B%AD%E4%B9%89%E7%9B%B8%E5%AF%B9%E8%AE%BA" title="狭义相对论">狭义相对论</a>用于速度可以和<a href="/wiki/%E5%85%89%E9%80%9F" title="光速">光速</a>相比拟时的运动，并且要求参考系是<a href="/wiki/%E6%83%AF%E6%80%A7%E7%B3%BB" class="mw-redirect" title="惯性系">惯性系</a>。在狭义相对论中，加速度的定义没有改变。然而，由于在狭义相对论中，不同的参考系有不同的时间和空间度量标准，即当前参考系中的加速度为<b>当前</b>参考系中的位移对<b>当前</b>参考系中的时间的二阶导数，因此在参考系变换（<a href="/wiki/%E6%B4%9B%E4%BC%A6%E5%85%B9%E5%8F%98%E6%8D%A2" title="洛伦兹变换">洛伦兹变换</a>）时变得复杂很多。 </p><p>设有两个参考系<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}"></span>、<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>S</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S'}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf9961844d1f539adee019e432dc18aa2a7ede59" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.206ex; height:2.509ex;" alt="{\displaystyle S'}"></span>，在空间直角坐标系中，三个坐标轴相对应平行，在<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t=t'=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> <mo>=</mo> <msup> <mi>t</mi> <mo>′</mo> </msup> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t=t'=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d80fd67edc034373b5b453322ca454bc9e20e651" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.723ex; height:2.509ex;" alt="{\displaystyle t=t'=0}"></span>时刻两坐标系原点对齐，在<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}"></span>中<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>S</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S'}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf9961844d1f539adee019e432dc18aa2a7ede59" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.206ex; height:2.509ex;" alt="{\displaystyle S'}"></span>以速率<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e07b00e7fc0847fbd16391c778d65bc25c452597" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle v}"></span>沿<i>x</i>正方向运动。 </p><p>同一事件在两个参考系中的時空坐標<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x,y,z,t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x,y,z,t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6167b5ade60932159390fbe64f2690daea4f7697" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.324ex; height:2.843ex;" alt="{\displaystyle (x,y,z,t)}"></span>、<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x',y',z',t')}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mo>′</mo> </msup> <mo>,</mo> <msup> <mi>y</mi> <mo>′</mo> </msup> <mo>,</mo> <msup> <mi>z</mi> <mo>′</mo> </msup> <mo>,</mo> <msup> <mi>t</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x',y',z',t')}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c758b416d05206f505fb7aa11b838e75cdb05c94" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.07ex; height:3.009ex;" alt="{\displaystyle (x',y',z',t')}"></span>變换如下： </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{cases}x=\gamma (x'+vt)\\y=y'\\z=z'\\t=\gamma \left(t'+{\cfrac {v}{c^{2}}}x\right)\\\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mi>x</mi> <mo>=</mo> <mi>γ</mi> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mo>′</mo> </msup> <mo>+</mo> <mi>v</mi> <mi>t</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mi>y</mi> <mo>=</mo> <msup> <mi>y</mi> <mo>′</mo> </msup> </mtd> </mtr> <mtr> <mtd> <mi>z</mi> <mo>=</mo> <msup> <mi>z</mi> <mo>′</mo> </msup> </mtd> </mtr> <mtr> <mtd> <mi>t</mi> <mo>=</mo> <mi>γ</mi> <mrow> <mo>(</mo> <mrow> <msup> <mi>t</mi> <mo>′</mo> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi>v</mi> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mstyle> </mrow> </mfrac> </mrow> <mi>x</mi> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{cases}x=\gamma (x'+vt)\\y=y'\\z=z'\\t=\gamma \left(t'+{\cfrac {v}{c^{2}}}x\right)\\\end{cases}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3e184f7a8898032655521db00960c981516c766" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -7.338ex; width:20.678ex; height:15.843ex;" alt="{\displaystyle {\begin{cases}x=\gamma (x'+vt)\\y=y'\\z=z'\\t=\gamma \left(t'+{\cfrac {v}{c^{2}}}x\right)\\\end{cases}}}"></span></dd></dl> <p>其中，單撇符號 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle '}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi></mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle '}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/761bf83727fb142497cfbee036fd61b368d4fbf0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:0.685ex; height:2.343ex;" alt="{\displaystyle '}"></span> 标示该物理量是在 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>S</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S'}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf9961844d1f539adee019e432dc18aa2a7ede59" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.206ex; height:2.509ex;" alt="{\displaystyle S'}"></span>下的测量，<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma ={\tfrac {1}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>γ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <msqrt> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> </msqrt> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma ={\tfrac {1}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f98c24d1cd20d3d0286cc943b4a0dae8567df612" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.171ex; width:11.225ex; height:6.509ex;" alt="{\displaystyle \gamma ={\tfrac {1}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}}"></span> 是<a href="/wiki/%E5%8B%9E%E4%BE%96%E8%8C%B2%E5%9B%A0%E5%AD%90" title="勞侖茲因子">勞侖茲因子</a>，用<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(u_{x},u_{y},u_{z}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mrow> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(u_{x},u_{y},u_{z}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c8cfc7a17349933ae2c5c04385a403bfefe06287" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.09ex; height:3.009ex;" alt="{\displaystyle \left(u_{x},u_{y},u_{z}\right)}"></span>表示一质点的速度，<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(a_{x},a_{y},a_{z}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mrow> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(a_{x},a_{y},a_{z}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/161bc44870aa4eb09a0c2ff5634edea17acfbaa3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.79ex; height:3.009ex;" alt="{\displaystyle \left(a_{x},a_{y},a_{z}\right)}"></span>表示其加速度。表示定义式如下 </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{cases}{\cfrac {\mathrm {d} x}{\mathrm {d} t}}=u_{x}\\{\cfrac {\mathrm {d} u_{x}}{\mathrm {d} t}}=a_{x}\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mstyle> </mrow> </mfrac> </mrow> <mo>=</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mstyle> </mrow> </mfrac> </mrow> <mo>=</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{cases}{\cfrac {\mathrm {d} x}{\mathrm {d} t}}=u_{x}\\{\cfrac {\mathrm {d} u_{x}}{\mathrm {d} t}}=a_{x}\end{cases}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01a1ab0ece765e8fc64ee7ff323436f3c49ed6b3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.171ex; width:12.949ex; height:13.509ex;" alt="{\displaystyle {\begin{cases}{\cfrac {\mathrm {d} x}{\mathrm {d} t}}=u_{x}\\{\cfrac {\mathrm {d} u_{x}}{\mathrm {d} t}}=a_{x}\end{cases}}}"></span></dd></dl> <p>y、z方向的定义式与之类似。综合该定义式，利用坐标转换的t部分，将坐标转换的x、y、z连续两次进行一阶求导<sup id="cite_ref-郑永令力学_2-7" class="reference"><a href="#cite_note-郑永令力学-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup class="reference" style="white-space:nowrap;">:501</sup>。 </p><p>通过展开，可以得到<sup id="cite_ref-French1968_14-0" class="reference"><a href="#cite_note-French1968-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{cases}a_{x}={\cfrac {a'_{x}}{\gamma ^{3}\left(1+{\frac {vu'_{x}}{c^{2}}}\right)^{3}}}\\a_{y}={\cfrac {1}{\gamma ^{2}}}\left[{\cfrac {a'_{y}}{\left(1+{\frac {vu'_{x}}{c^{2}}}\right)^{2}}}-{\cfrac {{\frac {vu'_{y}}{c^{2}}}a'_{x}}{\left(1+{\frac {vu'_{x}}{c^{2}}}\right)^{3}}}\right]\\a_{z}={\cfrac {1}{\gamma ^{2}}}\left[{\cfrac {a'_{z}}{\left(1+{\frac {vu'_{x}}{c^{2}}}\right)^{2}}}-{\cfrac {{\frac {vu'_{z}}{c^{2}}}a'_{x}}{\left(1+{\frac {vu'_{x}}{c^{2}}}\right)^{3}}}\right]\\\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msubsup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> <mo>′</mo> </msubsup> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <msup> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>v</mi> <msubsup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> <mo>′</mo> </msubsup> </mrow> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mstyle> </mrow> </mfrac> </mrow> <mrow> <mo>[</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msubsup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> <mo>′</mo> </msubsup> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msup> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>v</mi> <msubsup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> <mo>′</mo> </msubsup> </mrow> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mstyle> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>v</mi> <msubsup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> <mo>′</mo> </msubsup> </mrow> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <msubsup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> <mo>′</mo> </msubsup> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msup> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>v</mi> <msubsup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> <mo>′</mo> </msubsup> </mrow> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mrow> <mo>]</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mstyle> </mrow> </mfrac> </mrow> <mrow> <mo>[</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msubsup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> <mo>′</mo> </msubsup> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msup> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>v</mi> <msubsup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> <mo>′</mo> </msubsup> </mrow> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mstyle> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>v</mi> <msubsup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> <mo>′</mo> </msubsup> </mrow> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <msubsup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> <mo>′</mo> </msubsup> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msup> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>v</mi> <msubsup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> <mo>′</mo> </msubsup> </mrow> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mrow> <mo>]</mo> </mrow> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{cases}a_{x}={\cfrac {a'_{x}}{\gamma ^{3}\left(1+{\frac {vu'_{x}}{c^{2}}}\right)^{3}}}\\a_{y}={\cfrac {1}{\gamma ^{2}}}\left[{\cfrac {a'_{y}}{\left(1+{\frac {vu'_{x}}{c^{2}}}\right)^{2}}}-{\cfrac {{\frac {vu'_{y}}{c^{2}}}a'_{x}}{\left(1+{\frac {vu'_{x}}{c^{2}}}\right)^{3}}}\right]\\a_{z}={\cfrac {1}{\gamma ^{2}}}\left[{\cfrac {a'_{z}}{\left(1+{\frac {vu'_{x}}{c^{2}}}\right)^{2}}}-{\cfrac {{\frac {vu'_{z}}{c^{2}}}a'_{x}}{\left(1+{\frac {vu'_{x}}{c^{2}}}\right)^{3}}}\right]\\\end{cases}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8198c2635d3862c19606b42d1bb6e445756a49e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -14.505ex; width:42.033ex; height:30.176ex;" alt="{\displaystyle {\begin{cases}a_{x}={\cfrac {a'_{x}}{\gamma ^{3}\left(1+{\frac {vu'_{x}}{c^{2}}}\right)^{3}}}\\a_{y}={\cfrac {1}{\gamma ^{2}}}\left[{\cfrac {a'_{y}}{\left(1+{\frac {vu'_{x}}{c^{2}}}\right)^{2}}}-{\cfrac {{\frac {vu'_{y}}{c^{2}}}a'_{x}}{\left(1+{\frac {vu'_{x}}{c^{2}}}\right)^{3}}}\right]\\a_{z}={\cfrac {1}{\gamma ^{2}}}\left[{\cfrac {a'_{z}}{\left(1+{\frac {vu'_{x}}{c^{2}}}\right)^{2}}}-{\cfrac {{\frac {vu'_{z}}{c^{2}}}a'_{x}}{\left(1+{\frac {vu'_{x}}{c^{2}}}\right)^{3}}}\right]\\\end{cases}}}"></span></dd></dl> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Elevator_gravity1.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/0d/Elevator_gravity1.svg/150px-Elevator_gravity1.svg.png" decoding="async" width="150" height="323" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/0d/Elevator_gravity1.svg/225px-Elevator_gravity1.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/0d/Elevator_gravity1.svg/300px-Elevator_gravity1.svg.png 2x" data-file-width="230" data-file-height="495" /></a><figcaption>假想实验：站在两种封闭电梯厢中两个人，无法分辨球的加速度是由惯性力还是真正的引力施加的。</figcaption></figure><p>其中，<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma =1/{\sqrt {1-v^{2}/c^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>γ</mi> <mo>=</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma =1/{\sqrt {1-v^{2}/c^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a0efffd10b8047241e9d9f4498b77760af9457bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:18.418ex; height:4.843ex;" alt="{\displaystyle \gamma =1/{\sqrt {1-v^{2}/c^{2}}}}"></span> 是<a href="/wiki/%E5%8B%9E%E4%BE%96%E8%8C%B2%E5%9B%A0%E5%AD%90" title="勞侖茲因子">勞侖茲因子</a>。 </p><p>可以看出，在狭义相对论中，加速度的变换公式冗长而复杂，各分量的公式也极不相似。再加上如果要考虑到力，虽然<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {F} =m\mathbf {a} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mo>=</mo> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {F} =m\mathbf {a} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d0c60fab89e8c3193952047dc565bcf8d233d115" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.121ex; height:2.176ex;" alt="{\displaystyle \mathbf {F} =m\mathbf {a} }"></span>仍然成立，但质量也会变得随参考系的不同而不同。以上原因导致加速度在牛顿力学中那种因为简洁而具有的优越性，在狭义相对论中不复存在，必須使用更有功能的數學工具，即<a href="/wiki/%E5%BC%A0%E9%87%8F%E5%88%86%E6%9E%90" class="mw-redirect" title="张量分析">張量分析</a><sup id="cite_ref-郑永令力学_2-8" class="reference"><a href="#cite_note-郑永令力学-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup class="reference" style="white-space:nowrap;">:501</sup>。 </p> <div class="mw-heading mw-heading3"><h3 id="广义相对论"><span id=".E5.B9.BF.E4.B9.89.E7.9B.B8.E5.AF.B9.E8.AE.BA"></span>广义相对论</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%8A%A0%E9%80%9F%E5%BA%A6&action=edit&section=30" title="编辑章节：广义相对论"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r84833064"><div role="note" class="hatnote navigation-not-searchable">主条目：<a href="/wiki/%E5%BB%A3%E7%BE%A9%E7%9B%B8%E5%B0%8D%E8%AB%96%E5%85%A5%E9%96%80" title="廣義相對論入門">广义相对论入门</a>和<a href="/wiki/%E7%AD%89%E6%95%88%E5%8E%9F%E7%90%86" title="等效原理">等效原理</a></div> <p>在广义相对论中和在量子力学中，大都是从能量、动量等的角度出发（类似于<a href="/wiki/%E5%88%86%E6%9E%90%E5%8A%9B%E5%AD%A6" title="分析力学">分析力学</a>），而很少会像牛顿第二定律一样涉及到作用力；实际上，即使在需要表示出“位移的二阶导数”这一个量的时候，会趋向于直接使用<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\ddot {x}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo>¨</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\ddot {x}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06e0e705ddda28c6cd06cdc6e18be9abf88bb395" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:2.176ex;" alt="{\displaystyle {\ddot {x}}}"></span>，等价于<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\mathrm {d} ^{2}x}{\mathrm {d} t^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>x</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\mathrm {d} ^{2}x}{\mathrm {d} t^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6be64f83b6dea2a9d380194847e3974da6b362bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:4.513ex; height:6.009ex;" alt="{\displaystyle {\frac {\mathrm {d} ^{2}x}{\mathrm {d} t^{2}}}}"></span>，来求解<a href="/wiki/%E5%BE%AE%E5%88%86%E6%96%B9%E7%A8%8B" title="微分方程">微分方程</a>。因此，加速度在进階理论中較少被用到。 </p><p>运用到加速度的其中一个例子是<a href="/wiki/%E7%AD%89%E6%95%88%E5%8E%9F%E7%90%86" title="等效原理">等效原理</a>，简单地说<sup id="cite_ref-郑永令力学_2-9" class="reference"><a href="#cite_note-郑永令力学-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup class="reference" style="white-space:nowrap;">:523</sup>，它叙述了观测者不能在局部的区域内分辨出由加速度所产生的惯性力或由物体所产生的引力。比如，观测者站在地球上静止的电梯厢中向前方抛球，球会向下坠落，是因为<a href="#抛体运动">地球的引力</a>；而在远离任何星体的宇宙中的一个电梯厢，在以重力加速度<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3556280e66fe2c0d0140df20935a6f057381d77" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}"></span>向上（定义观测者踩到的地面为下）加速运动时，观测者抛出一个球，仍然会向“下”坠落，是因为<a href="#惯性力">惯性力</a>。作为在封闭电梯厢中的观测者无法分辨这两种情况，爱因斯坦据此提出，引力与惯性力等价。等效原理是广义相对论中的支柱性原理之一。 </p> <div class="mw-heading mw-heading2"><h2 id="参见"><span id=".E5.8F.82.E8.A7.81"></span>参见</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%8A%A0%E9%80%9F%E5%BA%A6&action=edit&section=31" title="编辑章节：参见"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/%E6%85%A3%E6%80%A7%E5%8A%9B" title="慣性力">惯性力</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="參考文獻"><span id=".E5.8F.83.E8.80.83.E6.96.87.E7.8D.BB"></span>參考文獻</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%8A%A0%E9%80%9F%E5%BA%A6&action=edit&section=32" title="编辑章节：參考文獻"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="reflist columns references-column-count references-column-count-2" style="-moz-column-count: 2; -webkit-column-count: 2; column-count: 2; list-style-type: decimal;"> <ol class="references"> <li id="cite_note-赵凯华力学-1"><span class="mw-cite-backlink">^ <a href="#cite_ref-赵凯华力学_1-0"><sup><b>1.0</b></sup></a> <a href="#cite_ref-赵凯华力学_1-1"><sup><b>1.1</b></sup></a> <a href="#cite_ref-赵凯华力学_1-2"><sup><b>1.2</b></sup></a></span> <span class="reference-text"><cite class="citation book">赵凯华，罗蔚茵. 《新概念物理教程·力学（第二版）》. 北京: 高等教育出版社. 2004. <a href="/wiki/Special:%E7%BD%91%E7%BB%9C%E4%B9%A6%E6%BA%90/7-04-015201-0" title="Special:网络书源/7-04-015201-0"><span title="国际标准书号">ISBN</span> 7-04-015201-0</a>.</cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fzh.wikipedia.org%3A%E5%8A%A0%E9%80%9F%E5%BA%A6&rft.au=%E8%B5%B5%E5%87%AF%E5%8D%8E%EF%BC%8C%E7%BD%97%E8%94%9A%E8%8C%B5&rft.btitle=%E3%80%8A%E6%96%B0%E6%A6%82%E5%BF%B5%E7%89%A9%E7%90%86%E6%95%99%E7%A8%8B%C2%B7%E5%8A%9B%E5%AD%A6%EF%BC%88%E7%AC%AC%E4%BA%8C%E7%89%88%EF%BC%89%E3%80%8B&rft.date=2004&rft.genre=book&rft.isbn=7-04-015201-0&rft.place=%E5%8C%97%E4%BA%AC&rft.pub=%E9%AB%98%E7%AD%89%E6%95%99%E8%82%B2%E5%87%BA%E7%89%88%E7%A4%BE&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-郑永令力学-2"><span class="mw-cite-backlink">^ <a href="#cite_ref-郑永令力学_2-0"><sup><b>2.00</b></sup></a> <a href="#cite_ref-郑永令力学_2-1"><sup><b>2.01</b></sup></a> <a href="#cite_ref-郑永令力学_2-2"><sup><b>2.02</b></sup></a> <a href="#cite_ref-郑永令力学_2-3"><sup><b>2.03</b></sup></a> <a href="#cite_ref-郑永令力学_2-4"><sup><b>2.04</b></sup></a> <a href="#cite_ref-郑永令力学_2-5"><sup><b>2.05</b></sup></a> <a href="#cite_ref-郑永令力学_2-6"><sup><b>2.06</b></sup></a> <a href="#cite_ref-郑永令力学_2-7"><sup><b>2.07</b></sup></a> <a href="#cite_ref-郑永令力学_2-8"><sup><b>2.08</b></sup></a> <a href="#cite_ref-郑永令力学_2-9"><sup><b>2.09</b></sup></a></span> <span class="reference-text"><cite class="citation book">郑永令，贾起民，方小敏. 《力学（第二版）》. 北京: 高等教育出版社. 2002. <a href="/wiki/Special:%E7%BD%91%E7%BB%9C%E4%B9%A6%E6%BA%90/978-7-04-011084-5" title="Special:网络书源/978-7-04-011084-5"><span title="国际标准书号">ISBN</span> 978-7-04-011084-5</a>.</cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fzh.wikipedia.org%3A%E5%8A%A0%E9%80%9F%E5%BA%A6&rft.au=%E9%83%91%E6%B0%B8%E4%BB%A4%EF%BC%8C%E8%B4%BE%E8%B5%B7%E6%B0%91%EF%BC%8C%E6%96%B9%E5%B0%8F%E6%95%8F&rft.btitle=%E3%80%8A%E5%8A%9B%E5%AD%A6%EF%BC%88%E7%AC%AC%E4%BA%8C%E7%89%88%EF%BC%89%E3%80%8B&rft.date=2002&rft.genre=book&rft.isbn=978-7-04-011084-5&rft.place=%E5%8C%97%E4%BA%AC&rft.pub=%E9%AB%98%E7%AD%89%E6%95%99%E8%82%B2%E5%87%BA%E7%89%88%E7%A4%BE&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-Beer-3"><span class="mw-cite-backlink">^ <a href="#cite_ref-Beer_3-0"><sup><b>3.0</b></sup></a> <a href="#cite_ref-Beer_3-1"><sup><b>3.1</b></sup></a></span> <span class="reference-text"><cite id="CITEREFBeerJohnston,_Jr.2004" class="citation">Beer, Ferdinand; Johnston, Jr., E. Russ, Vector Mechanics for Engineers:dynamics 7th: pp. 699, 2004, <a href="/wiki/Special:%E7%BD%91%E7%BB%9C%E4%B9%A6%E6%BA%90/978-0-072-93079-5" title="Special:网络书源/978-0-072-93079-5"><span title="国际标准书号">ISBN</span> 978-0-072-93079-5</a></cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fzh.wikipedia.org%3A%E5%8A%A0%E9%80%9F%E5%BA%A6&rft.au=Johnston%2C+Jr.%2C+E.+Russ&rft.aufirst=Ferdinand&rft.aulast=Beer&rft.btitle=Vector+Mechanics+for+Engineers%3Adynamics&rft.date=2004&rft.edition=7th&rft.genre=book&rft.isbn=978-0-072-93079-5&rft.pages=pp.+699&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span><span class="citation-comment" style="display:none; color:#33aa33"> 引文格式1维护：冗余文本 (<a href="/wiki/Category:%E5%BC%95%E6%96%87%E6%A0%BC%E5%BC%8F1%E7%BB%B4%E6%8A%A4%EF%BC%9A%E5%86%97%E4%BD%99%E6%96%87%E6%9C%AC" title="Category:引文格式1维护：冗余文本">link</a>)</span></span> </li> <li id="cite_note-Lanczos-4"><span class="mw-cite-backlink">^ <a href="#cite_ref-Lanczos_4-0"><sup><b>4.0</b></sup></a> <a href="#cite_ref-Lanczos_4-1"><sup><b>4.1</b></sup></a> <a href="#cite_ref-Lanczos_4-2"><sup><b>4.2</b></sup></a> <a href="#cite_ref-Lanczos_4-3"><sup><b>4.3</b></sup></a> <a href="#cite_ref-Lanczos_4-4"><sup><b>4.4</b></sup></a></span> <span class="reference-text"><cite id="CITEREFLanczos1970" class="citation">Lanczos, Cornelius, The Variational Principles of Mechanics, Dovers Publications, Inc, 1970, <a href="/wiki/Special:%E7%BD%91%E7%BB%9C%E4%B9%A6%E6%BA%90/978-0-486-65067-8" title="Special:网络书源/978-0-486-65067-8"><span title="国际标准书号">ISBN</span> 978-0-486-65067-8</a></cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fzh.wikipedia.org%3A%E5%8A%A0%E9%80%9F%E5%BA%A6&rft.aufirst=Cornelius&rft.aulast=Lanczos&rft.btitle=The+Variational+Principles+of+Mechanics&rft.date=1970&rft.genre=book&rft.isbn=978-0-486-65067-8&rft.pub=Dovers+Publications%2C+Inc&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><cite class="citation web"><a rel="nofollow" class="external text" href="http://thetartan.org/2007/4/16/scitech/work">存档副本</a>. <span class="reference-accessdate"> [<span class="nowrap">2013-02-19</span>]</span>. （原始内容<a rel="nofollow" class="external text" href="https://web.archive.org/web/20130518051700/http://thetartan.org/2007/4/16/scitech/work">存档</a>于2013-05-18）.</cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fzh.wikipedia.org%3A%E5%8A%A0%E9%80%9F%E5%BA%A6&rft.btitle=%E5%AD%98%E6%A1%A3%E5%89%AF%E6%9C%AC&rft.genre=unknown&rft_id=http%3A%2F%2Fthetartan.org%2F2007%2F4%2F16%2Fscitech%2Fwork&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text">黄沛天，马善钧，徐学翔，胡利云．变加速动力学纵横，2010年7月5日查询</span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"><cite class="citation journal">Sprott JC. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20100613054556/http://sprott.physics.wisc.edu/pubs/paper229.pdf">Some simple chaotic jerk functions</a> <span style="font-size:85%;">(PDF)</span>. Am J Phys. 1997, <b>65</b> (6): 537–43 <span class="reference-accessdate"> [<span class="nowrap">2009-09-28</span>]</span>. <a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1997AmJPh..65..537S"><span title="Bibcode">Bibcode:1997AmJPh..65..537S</span></a>. <a rel="nofollow" class="external text" href="https://doi.org/10.1119%2F1.18585"><span title="數位物件識別號">doi:10.1119/1.18585</span></a>. （<a rel="nofollow" class="external text" href="http://sprott.physics.wisc.edu/pubs/paper229.pdf">原始内容</a> <span style="font-size:85%;">(PDF)</span>存档于2010-06-13）.</cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fzh.wikipedia.org%3A%E5%8A%A0%E9%80%9F%E5%BA%A6&rft.atitle=Some+simple+chaotic+jerk+functions&rft.au=Sprott+JC&rft.date=1997&rft.genre=article&rft.issue=6&rft.jtitle=Am+J+Phys&rft.pages=537-43&rft.volume=65&rft_id=http%3A%2F%2Fsprott.physics.wisc.edu%2Fpubs%2Fpaper229.pdf&rft_id=info%3Abibcode%2F1997AmJPh..65..537S&rft_id=info%3Adoi%2F10.1119%2F1.18585&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text"><span lang="en">Baum, Steven K</span>，1997年1月20日，<span lang="en"><a rel="nofollow" class="external text" href="http://stommel.tamu.edu/~baum/paleo/paleogloss/node10.html">The Glossary: Cn-Cz..</a> Glossary of Oceanography and the Related Geosciences with References. Texas A&M University</span>，於2006年11月29日。<span style="font-family: sans-serif; cursor: default; color:var(--color-subtle, #54595d); font-size: 0.8em; bottom: 0.1em; font-weight: bold;" title="英語">（英文）</span></span> </li> <li id="cite_note-BritTCtrackcoriolis-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-BritTCtrackcoriolis_9-0">^</a></b></span> <span class="reference-text"><cite class="citation web"><a rel="nofollow" class="external text" href="http://www.britannica.com/EBchecked/topic/606551/tropical-cyclone#849004.hook">Tropical cyclone: Tropical cyclone tracks.</a>. <a href="/wiki/Encyclop%C3%A6dia_Britannica" class="mw-redirect" title="Encyclopædia Britannica">Encyclopædia Britannica</a>. 2008-02-25 <span class="reference-accessdate"> [<span class="nowrap">2009-05-07</span>]</span>. （原始内容<a rel="nofollow" class="external text" href="https://www.webcitation.org/68cEI2DRL?url=http://www.britannica.com/EBchecked/topic/606551/tropical-cyclone#849004.hook">存档</a>于2012-06-22）.</cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fzh.wikipedia.org%3A%E5%8A%A0%E9%80%9F%E5%BA%A6&rft.btitle=Tropical+cyclone%3A+Tropical+cyclone+tracks.&rft.date=2008-02-25&rft.genre=unknown&rft.pub=Encyclop%C3%A6dia+Britannica&rft_id=http%3A%2F%2Fwww.britannica.com%2FEBchecked%2Ftopic%2F606551%2Ftropical-cyclone%23849004.hook&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-NOAA_Question_of_the_Month-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-NOAA_Question_of_the_Month_10-0">^</a></b></span> <span class="reference-text"><cite class="citation web"><a rel="nofollow" class="external text" href="http://www.aoml.noaa.gov/hrd/tcfaq/D7.html">NOAA FAQ: How much energy does a hurricane release?</a>. National Oceanic & Atmospheric Administration. August 2001 <span class="reference-accessdate"> [<span class="nowrap">2009-06-30</span>]</span>. （原始内容<a rel="nofollow" class="external text" href="https://web.archive.org/web/20171102212903/http://www.aoml.noaa.gov/hrd/tcfaq/D7.html">存档</a>于2017-11-02）.</cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fzh.wikipedia.org%3A%E5%8A%A0%E9%80%9F%E5%BA%A6&rft.btitle=NOAA+FAQ%3A+How+much+energy+does+a+hurricane+release%3F&rft.date=2001-08&rft.genre=unknown&rft.pub=National+Oceanic+%26+Atmospheric+Administration&rft_id=http%3A%2F%2Fwww.aoml.noaa.gov%2Fhrd%2Ftcfaq%2FD7.html&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text"><b>竖直向下</b>，又称<b>铅直向下</b>，被定义为重力加速度的方向。但其具体方向因重力加速度的两种定义不同而异，分别为<i>指向地心</i>和<i>与纬度有关</i>，参见<a href="/wiki/%E4%B8%87%E6%9C%89%E5%BC%95%E5%8A%9B#两者的微妙差别" class="mw-redirect" title="万有引力">万有引力#两者的微妙差别</a>。</span> </li> <li id="cite_note-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-12">^</a></b></span> <span class="reference-text">g在不同地区稍有不同，并且g有两种不同的定义（见上一条注释）。一般需要更精确的计算中g可近似的取作标准重力加速度，即g=g<sub>n</sub>=9.80665 ms<sup>-2</sup>，这个值是已经包括了和地球自转的向心力的。该数值来自气象港，<a rel="nofollow" class="external free" href="http://qxg.com.cn/n/?cid=44&nid=764&fc=nd，2010年5月18日查阅。">http://qxg.com.cn/n/?cid=44&nid=764&fc=nd，2010年5月18日查阅。</a> （<a rel="nofollow" class="external text" href="//web.archive.org/web/20090224000809/http://qxg.com.cn/n/?cid=44&nid=764&fc=nd%EF%BC%8C2010%E5%B9%B45%E6%9C%8818%E6%97%A5%E6%9F%A5%E9%98%85">页面存档备份</a>，存于<a href="/wiki/%E4%BA%92%E8%81%94%E7%BD%91%E6%A1%A3%E6%A1%88%E9%A6%86" title="互联网档案馆">互联网档案馆</a>）</span> </li> <li id="cite_note-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-13">^</a></b></span> <span class="reference-text">这里并没有用到准确的物理术语。准确地说，是辐射的<a href="/w/index.php?title=%E8%83%BD%E6%B5%81%E5%AF%86%E5%BA%A6&action=edit&redlink=1" class="new" title="能流密度（页面不存在）">能流密度</a>与粒子加速度的平方成正比。<cite class="citation book">赵凯华，陈熙谋. 《新概念物理教程·电磁学》. 北京: 高等教育出版社. 2003: pp.417–419. <a href="/wiki/Special:%E7%BD%91%E7%BB%9C%E4%B9%A6%E6%BA%90/7-04-011693-6" title="Special:网络书源/7-04-011693-6"><span title="国际标准书号">ISBN</span> 7-04-011693-6</a>.</cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fzh.wikipedia.org%3A%E5%8A%A0%E9%80%9F%E5%BA%A6&rft.au=%E8%B5%B5%E5%87%AF%E5%8D%8E%EF%BC%8C%E9%99%88%E7%86%99%E8%B0%8B&rft.btitle=%E3%80%8A%E6%96%B0%E6%A6%82%E5%BF%B5%E7%89%A9%E7%90%86%E6%95%99%E7%A8%8B%C2%B7%E7%94%B5%E7%A3%81%E5%AD%A6%E3%80%8B&rft.date=2003&rft.genre=book&rft.isbn=7-04-011693-6&rft.pages=pp.417-419&rft.place=%E5%8C%97%E4%BA%AC&rft.pub=%E9%AB%98%E7%AD%89%E6%95%99%E8%82%B2%E5%87%BA%E7%89%88%E7%A4%BE&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span><span class="citation-comment" style="display:none; color:#33aa33"> 引文格式1维护：冗余文本 (<a href="/wiki/Category:%E5%BC%95%E6%96%87%E6%A0%BC%E5%BC%8F1%E7%BB%B4%E6%8A%A4%EF%BC%9A%E5%86%97%E4%BD%99%E6%96%87%E6%9C%AC" title="Category:引文格式1维护：冗余文本">link</a>)</span></span> </li> <li id="cite_note-French1968-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-French1968_14-0">^</a></b></span> <span class="reference-text"><cite id="CITEREFFrench1968" class="citation">French, Anthony, Special Relativity (Mit Introductory Physics Series), United States of America: W. W. Norton: pp. 153–154, 1968, <a href="/wiki/Special:%E7%BD%91%E7%BB%9C%E4%B9%A6%E6%BA%90/978-0748764228" title="Special:网络书源/978-0748764228"><span title="国际标准书号">ISBN</span> 978-0748764228</a> <span style="font-family: sans-serif; cursor: default; color:var(--color-subtle, #54595d); font-size: 0.8em; bottom: 0.1em; font-weight: bold;" title="连接到英语网页">（英语）</span></cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fzh.wikipedia.org%3A%E5%8A%A0%E9%80%9F%E5%BA%A6&rft.aufirst=Anthony&rft.aulast=French&rft.btitle=Special+Relativity+%28Mit+Introductory+Physics+Series%29&rft.date=1968&rft.genre=book&rft.isbn=978-0748764228&rft.pages=pp.+153-154&rft.place=United+States+of+America&rft.pub=W.+W.+Norton&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span><span class="citation-comment" style="display:none; color:#33aa33"> 引文格式1维护：冗余文本 (<a href="/wiki/Category:%E5%BC%95%E6%96%87%E6%A0%BC%E5%BC%8F1%E7%BB%B4%E6%8A%A4%EF%BC%9A%E5%86%97%E4%BD%99%E6%96%87%E6%9C%AC" title="Category:引文格式1维护：冗余文本">link</a>)</span></span> </li> </ol></div> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r84265675">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output .hlist ol ul,.mw-parser-output .hlist ul dl,.mw-parser-output .hlist ul ol,.mw-parser-output .hlist ul ul{display:inline}.mw-parser-output .hlist .mw-empty-li{display:none}.mw-parser-output .hlist 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.navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:%E7%BB%8F%E5%85%B8%E5%8A%9B%E5%AD%A6%E5%9B%BD%E9%99%85%E5%8D%95%E4%BD%8D" title="Template:经典力学国际单位"><abbr title="查看该模板">查</abbr></a></li><li class="nv-talk"><a href="/w/index.php?title=Template_talk:%E7%BB%8F%E5%85%B8%E5%8A%9B%E5%AD%A6%E5%9B%BD%E9%99%85%E5%8D%95%E4%BD%8D&action=edit&redlink=1" class="new" title="Template talk:经典力学国际单位（页面不存在）"><abbr title="讨论该模板">论</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:%E7%BC%96%E8%BE%91%E9%A1%B5%E9%9D%A2/Template:%E7%BB%8F%E5%85%B8%E5%8A%9B%E5%AD%A6%E5%9B%BD%E9%99%85%E5%8D%95%E4%BD%8D" title="Special:编辑页面/Template:经典力学国际单位"><abbr title="编辑该模板">编</abbr></a></li></ul></div><div id="经典力学国际单位" style="font-size:110%;margin:0 5em"><a href="/wiki/%E7%BB%8F%E5%85%B8%E5%8A%9B%E5%AD%A6" title="经典力学">经典力学</a><a href="/wiki/%E5%9B%BD%E9%99%85%E5%8D%95%E4%BD%8D%E5%88%B6" title="国际单位制">国际单位</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0px"><div style="padding:0;"><table class="wikitable" style="text-align:center;line-height:0.9;border-collapse:collapse;margin:auto;border:none;background:none;"> <tbody><tr> <th colspan="4">线性（平动）的量</th> <td rowspan="12" style="border:none;backgound:none;"></td> <th colspan="4">角度（转动）的量</th> </tr> <tr> <th style="font-weight:normal;font-size:80%;">量纲</th> <th style="font-weight:normal;">—</th> <th style="font-weight:normal;">L</th> <th style="font-weight:normal;">L<sup>2</sup></th> <th style="font-weight:normal;font-size:80%;">量纲</th> <th style="font-weight:normal;">—</th> <th style="font-weight:normal;">—</th> <th style="font-weight:normal;">—</th> </tr> <tr> <th style="font-weight:normal;">T</th> <td><a href="/wiki/%E6%97%B6%E9%97%B4" title="时间">时间</a>: <span class="serif"><span class="texhtml mvar" style="font-style:italic;margin-left:2px;margin-right:2px;">t</span></span><br /><a href="/wiki/%E7%A7%92" title="秒">s</a></td> <td><a href="/wiki/%E4%BD%8D%E7%A7%BB%E7%A7%AF%E5%88%86" title="位移积分">位移积分</a>: <span class="serif"><span class="texhtml"><b>A</b></span></span><br /><span class="ilh-all" data-orig-title="m s" data-lang-code="en" data-lang-name="英语" data-foreign-title="meter second"><span class="ilh-page"><a href="/w/index.php?title=M_s&action=edit&redlink=1" class="new" title="M s（页面不存在）">m s</a></span><span class="noprint ilh-comment">（<span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/meter_second" class="extiw" title="en:meter second"><span lang="en" dir="auto">meter second</span></a></span>）</span></span></td> <td></td> <th style="font-weight:normal;">T</th> <td><a href="/wiki/%E6%97%B6%E9%97%B4" title="时间">时间</a>: <span class="serif"><span class="texhtml mvar" style="font-style:italic;margin-left:2px;margin-right:2px;">t</span></span><br /><a href="/wiki/%E7%A7%92" title="秒">s</a></td> <td></td> <td></td> </tr> <tr> <th style="font-weight:normal;">—</th> <td></td> <td><a href="/wiki/%E8%B7%9D%E7%A6%BB" title="距离">距离</a>: <span class="serif"><span class="texhtml mvar" style="font-style:italic;margin-left:2px;margin-right:2px;">d</span></span>, <span class="nowrap"><a href="/wiki/%E4%BD%8D%E7%BD%AE%E5%90%91%E9%87%8F" title="位置向量">位矢</a>: <span class="serif"><span class="texhtml"><b>r</b></span></span>, <span class="serif"><span class="texhtml"><b>s</b></span></span>, <span class="serif"><span class="texhtml"><b>x</b></span></span></span>, <a href="/wiki/%E4%BD%8D%E7%A7%BB" title="位移">位移</a><br /><a href="/wiki/%E7%B1%B3_(%E5%8D%95%E4%BD%8D)" title="米 (单位)">m</a></td> <td><a href="/wiki/%E9%9D%A2%E7%A7%AF" title="面积">面积</a>: <span class="serif"><span class="texhtml"><i>A</i></span></span><br /><a href="/wiki/%E5%B9%B3%E6%96%B9%E7%B1%B3" title="平方米">m<sup>2</sup></a></td> <th style="font-weight:normal;">—</th> <td></td> <td><a href="/wiki/%E8%A7%92" title="角">角度</a>: <span class="serif"><span class="texhtml mvar" style="font-style:italic;margin-left:2px;margin-right:2px;">θ</span></span>, <a href="/wiki/%E8%A7%92%E7%A7%BB" title="角移">角移</a>: <span class="serif"><span class="texhtml"><b>θ</b></span></span><br /><a href="/wiki/%E5%BC%A7%E5%BA%A6" title="弧度">rad</a></td> <td><span class="nowrap"><a href="/wiki/%E7%AB%8B%E9%AB%94%E8%A7%92" title="立體角">立體角</a>: <span class="serif"><span class="texhtml">Ω</span></span><br /><a href="/wiki/%E7%90%83%E9%9D%A2%E5%BA%A6" title="球面度">rad<sup>2</sup>, sr</a></span></td> </tr> <tr> <th style="font-weight:normal;">T<sup>−1</sup></th> <td><span class="nowrap"><a href="/wiki/%E9%A0%BB%E7%8E%87_(%E7%89%A9%E7%90%86%E5%AD%B8)" title="頻率 (物理學)">頻率</a>: <span class="serif"><span class="texhtml mvar" style="font-style:italic;margin-left:2px;margin-right:2px;">f</span></span></span><br /><span class="ilh-all" data-orig-title="每秒" data-lang-code="en" data-lang-name="英语" data-foreign-title="inverse second"><span class="ilh-page"><a href="/w/index.php?title=%E6%AF%8F%E7%A7%92&action=edit&redlink=1" class="new" title="每秒（页面不存在）">s<sup>−1</sup></a></span><span class="noprint ilh-comment">（<span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/inverse_second" class="extiw" title="en:inverse second"><span lang="en" dir="auto">inverse second</span></a></span>）</span></span>, <a href="/wiki/%E8%B5%AB%E5%85%B9" title="赫兹">Hz</a></td> <td><a href="/wiki/%E9%80%9F%E7%8E%87" title="速率">速率</a>: <span class="serif"><span class="texhtml mvar" style="font-style:italic;margin-left:2px;margin-right:2px;">v</span></span>, <a href="/wiki/%E9%80%9F%E5%BA%A6" title="速度">速度</a>: <span class="serif"><span class="texhtml"><b>v</b></span></span><br /><a href="/wiki/%E7%B1%B3%E6%AF%8F%E7%A7%92" title="米每秒">m s<sup>−1</sup></a></td> <td><a href="/wiki/%E9%BB%8F%E5%BA%A6" title="黏度">面積速率</a>: <span class="serif"><span class="texhtml mvar" style="font-style:italic;margin-left:2px;margin-right:2px;">ν</span></span><br /><a href="https://en.wikipedia.org/wiki/metre_squared_per_second" class="extiw" title="en:metre squared per second">m<sup>2</sup> s<sup>−1</sup></a></td> <th style="font-weight:normal;">T<sup>−1</sup></th> <td><span class="nowrap"><a href="/wiki/%E9%A0%BB%E7%8E%87_(%E7%89%A9%E7%90%86%E5%AD%B8)" title="頻率 (物理學)">頻率</a>: <span class="serif"><span class="texhtml mvar" style="font-style:italic;margin-left:2px;margin-right:2px;">f</span></span></span><br /><span class="ilh-all" data-orig-title="每秒" data-lang-code="en" data-lang-name="英语" data-foreign-title="inverse second"><span class="ilh-page"><a href="/w/index.php?title=%E6%AF%8F%E7%A7%92&action=edit&redlink=1" class="new" title="每秒（页面不存在）">s<sup>−1</sup></a></span><span class="noprint ilh-comment">（<span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/inverse_second" class="extiw" title="en:inverse second"><span lang="en" dir="auto">inverse second</span></a></span>）</span></span>, <a href="/wiki/%E8%B5%AB%E5%85%B9" title="赫兹">Hz</a></td> <td><a href="/wiki/%E8%A7%92%E9%80%9F%E5%BA%A6" title="角速度">角速率</a>: <span class="serif"><span class="texhtml mvar" style="font-style:italic;margin-left:2px;margin-right:2px;">ω</span></span>, <a href="/wiki/%E8%A7%92%E9%80%9F%E5%BA%A6" title="角速度">角速度</a>: <span class="serif"><span class="texhtml"><b>ω</b></span></span><br /><a href="/wiki/%E5%BC%A7%E5%BA%A6%E6%AF%8F%E7%A7%92" title="弧度每秒">rad<span style="letter-spacing:0.1em"> </span>s<sup>−1</sup></a></td> <td></td> </tr> <tr> <th style="font-weight:normal;">T<sup>−2</sup></th> <td></td> <td><a class="mw-selflink selflink">加速度</a>: <span class="serif"><span class="texhtml"><b>a</b></span></span><br /><a href="/wiki/%E7%B1%B3%E6%AF%8F%E4%BA%8C%E6%AC%A1%E6%96%B9%E7%A7%92" title="米每二次方秒">m s<sup>−2</sup></a></td> <td></td> <th style="font-weight:normal;">T<sup>−2</sup></th> <td></td> <td><a href="/wiki/%E8%A7%92%E5%8A%A0%E9%80%9F%E5%BA%A6" title="角加速度">角加速度</a>: <span class="serif"><span class="texhtml"><b>α</b></span></span><br />rad<span style="letter-spacing:0.1em"> </span>s<sup>−2</sup></td> <td></td> </tr> <tr> <th style="font-weight:normal;">T<sup>−3</sup></th> <td></td> <td><a href="/wiki/%E5%8A%A0%E5%8A%A0%E9%80%9F%E5%BA%A6" title="加加速度">加加速度</a>: <span class="serif"><span class="texhtml"><b>j</b></span></span><br />m s<sup>−3</sup></td> <td></td> <th style="font-weight:normal;">T<sup>−3</sup></th> <td></td> <td><a href="/wiki/%E5%8A%A0%E5%8A%A0%E9%80%9F%E5%BA%A6" title="加加速度">角加加速度</a>: <span class="serif"><span class="texhtml"><b>ζ</b></span></span><br />rad<span style="letter-spacing:0.1em"> </span>s<sup>−3</sup></td> <td></td> </tr> <tr> <td colspan="4" style="padding-top:0;"></td> <td colspan="4" style="padding-top:0;"></td> </tr> <tr><th style="font-weight:normal;">M</th> <td><a href="/wiki/%E8%B4%A8%E9%87%8F" title="质量">质量</a>: <span class="serif"><span class="texhtml mvar" style="font-style:italic;margin-left:2px;margin-right:2px;">m</span></span><br /><a href="/wiki/%E5%8D%83%E5%85%8B" title="千克">kg</a></td> <td></td> <td></td> <th style="font-weight:normal;">ML<sup>2</sup></th> <td><a href="/wiki/%E8%BD%89%E5%8B%95%E6%85%A3%E9%87%8F" title="轉動慣量">轉動慣量</a>: <span class="serif"><span class="texhtml mvar" style="font-style:italic;margin-left:2px;margin-right:2px;">I</span></span><br />kg<span style="letter-spacing:0.1em"> </span>m<sup>2</sup></td> <td></td> <td></td> </tr> <tr> <th style="font-weight:normal;">MT<sup>−1</sup></th> <td></td> <td><a href="/wiki/%E5%8A%A8%E9%87%8F" title="动量">动量</a>: <span class="serif"><span class="texhtml"><b>p</b></span></span>, <a href="/wiki/%E5%86%B2%E9%87%8F" title="冲量">冲量</a>: <span class="serif"><span class="texhtml"><b>J</b></span></span><br />kg<span style="letter-spacing:0.1em"> </span>m s<sup>−1</sup>, <span class="ilh-all" data-orig-title="N s" data-lang-code="en" data-lang-name="英语" data-foreign-title="newton second"><span class="ilh-page"><a href="/w/index.php?title=N_s&action=edit&redlink=1" class="new" title="N s（页面不存在）">N s</a></span><span class="noprint ilh-comment">（<span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/newton_second" class="extiw" title="en:newton second"><span lang="en" dir="auto">newton second</span></a></span>）</span></span></td> <td><a href="/wiki/%E4%BD%9C%E7%94%A8%E9%87%8F" title="作用量">作用量</a>: <span class="serif"><span class="texhtml">𝒮</span></span>, <a href="/wiki/Actergy" class="mw-redirect" title="Actergy">actergy</a>: <span class="serif"><span class="texhtml">ℵ</span></span><br />kg<span style="letter-spacing:0.1em"> </span>m<sup>2</sup> s<sup>−1</sup>, <span class="ilh-all" data-orig-title="J s" data-lang-code="en" data-lang-name="英语" data-foreign-title="joule-second"><span class="ilh-page"><a href="/w/index.php?title=J_s&action=edit&redlink=1" class="new" title="J s（页面不存在）">J s</a></span><span class="noprint ilh-comment">（<span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/joule-second" class="extiw" title="en:joule-second"><span lang="en" dir="auto">joule-second</span></a></span>）</span></span></td> <th style="font-weight:normal;">ML<sup>2</sup>T<sup>−1</sup></th> <td></td> <td><a href="/wiki/%E8%A7%92%E5%8A%A8%E9%87%8F" title="角动量">角动量</a>: <span class="serif"><span class="texhtml"><b>L</b></span></span>, <a href="/wiki/%E7%B6%93%E5%85%B8%E5%8A%9B%E5%AD%B8%E6%96%B9%E7%A8%8B%E5%88%97%E8%A1%A8#導出的動力學物理量" title="經典力學方程列表">角衝量</a>: <span class="serif"><span class="texhtml"><b>ι</b></span></span><br />kg<span style="letter-spacing:0.1em"> </span>m<sup>2</sup> s<sup>−1</sup></td> <td><a href="/wiki/%E4%BD%9C%E7%94%A8%E9%87%8F" title="作用量">作用量</a>: <span class="serif"><span class="texhtml">𝒮</span></span>, <a href="/wiki/Actergy" class="mw-redirect" title="Actergy">actergy</a>: <span class="serif"><span class="texhtml">ℵ</span></span><br />kg<span style="letter-spacing:0.1em"> </span>m<sup>2</sup> s<sup>−1</sup>, <span class="ilh-all" data-orig-title="J s" data-lang-code="en" data-lang-name="英语" data-foreign-title="joule-second"><span class="ilh-page"><a href="/w/index.php?title=J_s&action=edit&redlink=1" class="new" title="J s（页面不存在）">J s</a></span><span class="noprint ilh-comment">（<span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/joule-second" class="extiw" title="en:joule-second"><span lang="en" dir="auto">joule-second</span></a></span>）</span></span></td> </tr> <tr> <th style="font-weight:normal;">MT<sup>−2</sup></th> <td></td> <td><a href="/wiki/%E5%8A%9B" title="力">力</a>: <span class="serif"><span class="texhtml"><b>F</b></span></span>, <a href="/wiki/%E9%87%8D%E9%87%8F" title="重量">重量</a>: <span class="serif"><span class="texhtml"><b>F</b><sub>g</sub></span></span><br /><span style="margin-right:0.1em;">kg </span> m s<sup>−2</sup>, <a href="/wiki/%E7%89%9B%E9%A0%93_(%E5%96%AE%E4%BD%8D)" title="牛頓 (單位)">N</a></td> <td><a href="/wiki/%E8%83%BD%E9%87%8F" title="能量">能量</a>: <span class="serif"><span class="texhtml mvar" style="font-style:italic;margin-left:2px;margin-right:2px;">E</span></span>, <a href="/wiki/%E5%8A%9F" title="功">功</a>: <span class="serif"><span class="texhtml mvar" style="font-style:italic;margin-left:2px;margin-right:2px;">W</span></span><br /><span style="margin-right:0.1em;">kg</span> m<sup>2</sup> s<sup>−2</sup>, <a href="/wiki/%E7%84%A6%E8%80%B3" title="焦耳">J</a></td> <th style="font-weight:normal;">ML<sup>2</sup>T<sup>−2</sup></th> <td></td> <td><a href="/wiki/%E5%8A%9B%E7%9F%A9" title="力矩">力矩</a>: <span class="serif"><span class="texhtml"><b>τ</b></span></span>, <span class="ilh-all" data-orig-title="Moment (physics)" data-lang-code="en" data-lang-name="英语" data-foreign-title="Moment (physics)"><span class="ilh-page"><a href="/w/index.php?title=Moment_(physics)&action=edit&redlink=1" class="new" title="Moment (physics)（页面不存在）">moment</a></span><span class="noprint ilh-comment">（<span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/Moment_(physics)" class="extiw" title="en:Moment (physics)"><span lang="en" dir="auto">Moment (physics)</span></a></span>）</span></span>: <span class="serif"><span class="texhtml"><b>M</b></span></span><br /><span style="margin-right:0.1em;">kg</span> m<sup>2</sup> s<sup>−2</sup>, <a href="/wiki/%E7%89%9B%E9%A0%93%E7%B1%B3" title="牛頓米">N m</a></td> <td><a href="/wiki/%E8%83%BD%E9%87%8F" title="能量">能量</a>: <span class="serif"><span class="texhtml mvar" style="font-style:italic;margin-left:2px;margin-right:2px;">E</span></span>, <a href="/wiki/%E5%8A%9F" title="功">功</a>: <span class="serif"><span class="texhtml mvar" style="font-style:italic;margin-left:2px;margin-right:2px;">W</span></span><br /><span style="margin-right:0.1em;">kg</span> m<sup>2</sup> s<sup>−2</sup>, <a href="/wiki/%E7%84%A6%E8%80%B3" title="焦耳">J</a></td> </tr> <tr> <th style="font-weight:normal;">MT<sup>−3</sup></th> <td></td> <td><a href="/wiki/%E5%8A%A0%E5%8A%9B" title="加力">加力</a>: <span class="serif"><span class="texhtml"><b>Y</b></span></span><br /><span style="margin-right:0.1em;">kg</span> m s<sup>−3</sup>, N s<sup>−1</sup></td> <td><a href="/wiki/%E5%8A%9F%E7%8E%87" title="功率">功率</a>: <span class="serif"><span class="texhtml mvar" style="font-style:italic;margin-left:2px;margin-right:2px;">P</span></span><br /><span style="margin-right:0.1em;">kg</span> m<sup>2</sup> s<sup>−3</sup>, <a href="/wiki/%E7%93%A6%E7%89%B9" title="瓦特">W</a></td> <th style="font-weight:normal;">ML<sup>2</sup>T<sup>−3</sup></th> <td></td> <td><span class="ilh-all" data-orig-title="rotatum" data-lang-code="en" data-lang-name="英语" data-foreign-title="rotatum"><span class="ilh-page"><a href="/w/index.php?title=Rotatum&action=edit&redlink=1" class="new" title="Rotatum（页面不存在）">rotatum</a></span><span class="noprint ilh-comment">（<span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/rotatum" class="extiw" title="en:rotatum"><span lang="en" dir="auto">rotatum</span></a></span>）</span></span>: <span class="serif"><span class="texhtml"><b>P</b></span></span><br /><span style="margin-right:0.1em;">kg</span> m<sup>2</sup> s<sup>−3</sup>, N m s<sup>−1</sup></td> <td><a href="/wiki/%E5%8A%9F%E7%8E%87" title="功率">功率</a>: <span class="serif"><span class="texhtml mvar" style="font-style:italic;margin-left:2px;margin-right:2px;">P</span></span><br /><span style="margin-right:0.1em;">kg</span> m<sup>2</sup> s<sup>−3</sup>, <a href="/wiki/%E7%93%A6%E7%89%B9" title="瓦特">W</a></td> </tr> </tbody></table></div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r84265675"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r84261037"></div><div role="navigation" class="navbox" aria-labelledby="经典力学" style="padding:3px"><table class="nowraplinks mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="collapsible-title navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r84265675"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r84244141"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:%E7%B6%93%E5%85%B8%E5%8A%9B%E5%AD%B8" title="Template:經典力學"><abbr title="查看该模板">查</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:%E7%B6%93%E5%85%B8%E5%8A%9B%E5%AD%B8" title="Template talk:經典力學"><abbr title="讨论该模板">论</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:%E7%BC%96%E8%BE%91%E9%A1%B5%E9%9D%A2/Template:%E7%B6%93%E5%85%B8%E5%8A%9B%E5%AD%B8" title="Special:编辑页面/Template:經典力學"><abbr title="编辑该模板">编</abbr></a></li></ul></div><div id="经典力学" style="font-size:110%;margin:0 5em"><a href="/wiki/%E7%BB%8F%E5%85%B8%E5%8A%9B%E5%AD%A6" title="经典力学">经典力学</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">表述形式</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><a href="/wiki/%E7%BB%8F%E5%85%B8%E5%8A%9B%E5%AD%A6" title="经典力学">矢量力学</a></li> <li><a href="/wiki/%E5%88%86%E6%9E%90%E5%8A%9B%E5%AD%A6" title="分析力学">分析力学</a>（<a href="/wiki/%E6%8B%89%E6%A0%BC%E6%9C%97%E6%97%A5%E5%8A%9B%E5%AD%A6" title="拉格朗日力学">拉格朗日力学</a></li> <li><a href="/wiki/%E5%93%88%E5%AF%86%E9%A1%BF%E5%8A%9B%E5%AD%A6" title="哈密顿力学">哈密頓力學</a>）</li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">基础概念</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><a href="/wiki/%E7%A9%BA%E9%96%93" title="空間">空间</a></li> <li><a href="/wiki/%E6%97%B6%E9%97%B4" title="时间">时间</a></li> <li><a href="/wiki/%E9%80%9F%E5%BA%A6" title="速度">速度</a></li> <li><a class="mw-selflink selflink">加速度</a></li> <li><a href="/wiki/%E8%B4%A8%E9%87%8F" title="质量">质量</a></li> <li><a href="/wiki/%E5%BC%95%E5%8A%9B" title="引力">引力</a></li> <li><a href="/wiki/%E5%8A%9B%E7%9F%A9" title="力矩">力矩</a></li> <li><a href="/wiki/%E5%8F%82%E8%80%83%E7%B3%BB" title="参考系">參考系</a></li> <li><a href="/wiki/%E5%8A%9B" title="力">力</a></li> <li><a href="/wiki/%E5%8A%9B%E5%81%B6" title="力偶">力偶</a></li> <li><a href="/wiki/%E5%86%B2%E9%87%8F" title="冲量">冲量</a></li> <li><a href="/wiki/%E5%8A%A8%E9%87%8F" title="动量">动量</a></li> <li><a href="/wiki/%E5%88%9A%E4%BD%93" title="刚体">刚体</a></li> <li><a href="/wiki/%E8%A7%92%E5%8A%A8%E9%87%8F" title="角动量">角动量</a></li> <li><a href="/wiki/%E6%85%A3%E6%80%A7" title="慣性">慣性</a></li> <li><a href="/wiki/%E8%BD%89%E5%8B%95%E6%85%A3%E9%87%8F" title="轉動慣量">轉動慣量</a></li> <li><a href="/wiki/%E8%83%BD%E9%87%8F" title="能量">能量</a></li> <li><a href="/wiki/%E5%8A%A8%E8%83%BD" title="动能">动能</a></li> <li><a href="/wiki/%E5%8A%BF%E8%83%BD" title="势能">位能</a></li> <li><a href="/wiki/%E8%99%9B%E5%8A%9F" title="虛功">虛功</a></li> <li><a href="/wiki/%E4%BD%9C%E7%94%A8%E9%87%8F" title="作用量">作用量</a></li> <li><a href="/wiki/%E6%8B%89%E6%A0%BC%E6%9C%97%E6%97%A5%E9%87%8F" title="拉格朗日量">拉格朗日量</a></li> <li><a href="/wiki/%E5%93%88%E5%AF%86%E9%A1%BF%E5%8A%9B%E5%AD%A6" title="哈密顿力学">哈密頓量</a></li> <li><a href="/wiki/%E5%8A%9F" title="功">功</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">重要理论</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><a href="/wiki/%E7%89%9B%E9%A1%BF%E8%BF%90%E5%8A%A8%E5%AE%9A%E5%BE%8B" title="牛顿运动定律">牛顿运动定律</a></li> <li><a href="/wiki/%E8%83%A1%E5%85%8B%E5%AE%9A%E5%BE%8B" title="胡克定律">胡克定律</a></li> <li><a href="/wiki/%E4%B8%87%E6%9C%89%E5%BC%95%E5%8A%9B%E5%AE%9A%E5%BE%8B" title="万有引力定律">牛顿万有引力定律</a></li> <li><a href="/wiki/%E7%B0%A1%E8%AB%A7%E9%81%8B%E5%8B%95" title="簡諧運動">簡諧運動</a></li> <li><a href="/wiki/%E9%81%94%E6%9C%97%E8%B2%9D%E7%88%BE%E5%8E%9F%E7%90%86" title="達朗貝爾原理">達朗貝爾原理</a></li> <li><a href="/wiki/%E6%AC%A7%E6%8B%89%E6%96%B9%E7%A8%8B_(%E5%88%9A%E4%BD%93%E8%BF%90%E5%8A%A8)" title="欧拉方程 (刚体运动)">歐拉方程式</a></li> <li><a href="/wiki/%E5%93%88%E5%AF%86%E9%A0%93%E5%8E%9F%E7%90%86" title="哈密頓原理">哈密頓原理</a></li> <li><a href="/wiki/%E6%8B%89%E6%A0%BC%E6%9C%97%E6%97%A5%E6%96%B9%E7%A8%8B%E5%BC%8F" title="拉格朗日方程式">拉格朗日方程式</a></li> <li><a href="/wiki/%E6%9C%80%E5%B0%8F%E4%BD%9C%E7%94%A8%E9%87%8F%E5%8E%9F%E7%90%86" title="最小作用量原理">最小作用量原理</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">应用</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><a href="/wiki/%E7%AE%80%E5%8D%95%E6%9C%BA%E6%A2%B0" title="简单机械">简单机械</a></li> <li><a href="/wiki/%E6%96%9C%E9%9D%A2" title="斜面">斜面</a></li> <li><a href="/wiki/%E6%9D%A0%E6%9D%86" title="杠杆">杠杆</a></li> <li><a href="/wiki/%E6%BB%91%E8%BD%AE" title="滑轮">滑轮</a></li> <li><a href="/wiki/%E8%9E%BA%E6%97%8B_(%E7%B0%A1%E5%96%AE%E6%A9%9F%E6%A2%B0)" title="螺旋 (簡單機械)">螺旋</a></li> <li><a href="/wiki/%E6%A5%94%E5%AD%90" title="楔子">楔子</a></li> <li><a href="/wiki/%E8%BC%AA%E8%BB%B8" title="輪軸">輪軸</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">科学史</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><a href="/wiki/%E7%89%A9%E7%90%86%E5%AD%A6%E5%8F%B2#力学的发展史" title="物理学史">发展史</a></li> <li><a href="/wiki/%E7%BA%A6%E7%BF%B0%E5%86%85%E6%96%AF%C2%B7%E5%BC%80%E6%99%AE%E5%8B%92" title="约翰内斯·开普勒">开普勒</a></li> <li><a href="/wiki/%E8%89%BE%E8%96%A9%E5%85%8B%C2%B7%E7%89%9B%E9%A0%93" class="mw-redirect" title="艾薩克·牛頓">牛頓</a></li> <li><a href="/wiki/%E8%90%8A%E6%98%82%E5%93%88%E5%BE%B7%C2%B7%E6%AD%90%E6%8B%89" title="萊昂哈德·歐拉">歐拉</a></li> <li><a href="/wiki/%E8%AE%A9%C2%B7%E5%8B%92%E6%9C%97%C2%B7%E8%BE%BE%E6%9C%97%E8%B4%9D%E5%B0%94" title="让·勒朗·达朗贝尔">達朗貝爾</a></li> <li><a href="/wiki/%E5%A8%81%E5%BB%89%C2%B7%E5%93%88%E5%AF%86%E9%A0%93" title="威廉·哈密頓">哈密頓</a></li> <li><a href="/wiki/%E6%B5%B7%E5%9B%A0%E9%87%8C%E5%B8%8C%C2%B7%E8%B5%AB%E5%85%B9" title="海因里希·赫兹">赫茲</a></li> <li><a href="/wiki/%E7%BA%A6%E7%91%9F%E5%A4%AB%C2%B7%E6%8B%89%E6%A0%BC%E6%9C%97%E6%97%A5" title="约瑟夫·拉格朗日">拉格朗日</a></li> <li><a href="/wiki/%E7%9A%AE%E5%9F%83%E5%B0%94-%E8%A5%BF%E8%92%99%C2%B7%E6%8B%89%E6%99%AE%E6%8B%89%E6%96%AF" title="皮埃尔-西蒙·拉普拉斯">拉普拉斯</a></li> <li><a href="/wiki/%E4%BC%BD%E5%88%A9%E7%95%A5%C2%B7%E4%BC%BD%E5%88%A9%E8%8E%B1" title="伽利略·伽利莱">伽利略</a></li> <li><a href="/wiki/%E5%8D%A1%E7%88%BE%C2%B7%E9%9B%85%E5%8F%AF%E6%AF%94" title="卡爾·雅可比">雅可比</a></li> <li><a href="/wiki/%E5%9F%83%E7%B1%B3%C2%B7%E8%AF%BA%E7%89%B9" title="埃米·诺特">諾特</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">分支</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><a href="/wiki/%E9%9D%99%E5%8A%9B%E5%AD%A6" title="静力学">静力学</a></li> <li><a href="/wiki/%E5%8B%95%E5%8A%9B%E5%AD%B8" title="動力學">動力學</a></li> <li><a href="/wiki/%E8%BF%90%E5%8A%A8%E5%AD%A6" title="运动学">运动学</a></li> <li><a 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