העתק (פיזיקה) – ויקיפדיה

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href="/w/index.php?title=%D7%9E%D7%99%D7%95%D7%97%D7%93:%D7%9B%D7%A0%D7%99%D7%A1%D7%94_%D7%9C%D7%97%D7%A9%D7%91%D7%95%D7%9F&returnto=%D7%94%D7%A2%D7%AA%D7%A7+%28%D7%A4%D7%99%D7%96%D7%99%D7%A7%D7%94%29" title="מומלץ להיכנס לחשבון, אך אין חובה לעשות זאת [o]" accesskey="o" class=""><span>כניסה לחשבון</span></a> </li> </ul> </div> </div> </div> <div id="vector-user-links-dropdown" class="vector-dropdown vector-user-menu vector-button-flush-right vector-user-menu-logged-out" title="אפשרויות נוספות" > <input type="checkbox" id="vector-user-links-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-user-links-dropdown" class="vector-dropdown-checkbox " aria-label="כלים אישיים" > <label id="vector-user-links-dropdown-label" for="vector-user-links-dropdown-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-ellipsis mw-ui-icon-wikimedia-ellipsis"></span> <span class="vector-dropdown-label-text">כלים אישיים</span> </label> <div class="vector-dropdown-content"> <div id="p-personal" class="vector-menu mw-portlet mw-portlet-personal user-links-collapsible-item" title="תפריט משתמש" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-sitesupport" class="user-links-collapsible-item mw-list-item"><a href="//donate.wikimedia.org/wiki/Special:FundraiserRedirector?utm_source=donate&utm_medium=sidebar&utm_campaign=C13_he.wikipedia.org&uselang=he"><span>תרומה לוויקיפדיה</span></a></li><li id="pt-createaccount" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=%D7%9E%D7%99%D7%95%D7%97%D7%93:%D7%94%D7%A8%D7%A9%D7%9E%D7%94_%D7%9C%D7%97%D7%A9%D7%91%D7%95%D7%9F&returnto=%D7%94%D7%A2%D7%AA%D7%A7+%28%D7%A4%D7%99%D7%96%D7%99%D7%A7%D7%94%29" title="מומלץ ליצור חשבון ולהיכנס אליו, אך אין חובה לעשות זאת"><span class="vector-icon mw-ui-icon-userAdd mw-ui-icon-wikimedia-userAdd"></span> <span>יצירת חשבון</span></a></li><li id="pt-login" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=%D7%9E%D7%99%D7%95%D7%97%D7%93:%D7%9B%D7%A0%D7%99%D7%A1%D7%94_%D7%9C%D7%97%D7%A9%D7%91%D7%95%D7%9F&returnto=%D7%94%D7%A2%D7%AA%D7%A7+%28%D7%A4%D7%99%D7%96%D7%99%D7%A7%D7%94%29" title="מומלץ להיכנס לחשבון, אך אין חובה לעשות זאת [o]" accesskey="o"><span class="vector-icon mw-ui-icon-logIn mw-ui-icon-wikimedia-logIn"></span> <span>כניסה לחשבון</span></a></li> </ul> </div> </div> <div id="p-user-menu-anon-editor" class="vector-menu mw-portlet mw-portlet-user-menu-anon-editor" > <div class="vector-menu-heading"> דפים לעורכים שלא נכנסו לחשבון <a href="/wiki/%D7%A2%D7%96%D7%A8%D7%94:%D7%91%D7%A8%D7%95%D7%9B%D7%99%D7%9D_%D7%94%D7%91%D7%90%D7%99%D7%9D" aria-label="מידע נוסף על עריכה"><span>מידע נוסף</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/%D7%9E%D7%99%D7%95%D7%97%D7%93:%D7%94%D7%AA%D7%A8%D7%95%D7%9E%D7%95%D7%AA_%D7%A9%D7%9C%D7%99" title="רשימת העריכות שנעשו מכתובת IP זו [y]" accesskey="y"><span>תרומות</span></a></li><li id="pt-anontalk" class="mw-list-item"><a href="/wiki/%D7%9E%D7%99%D7%95%D7%97%D7%93:%D7%94%D7%A9%D7%99%D7%97%D7%94_%D7%A9%D7%9C%D7%99" title="דיון על העריכות שנעשו מכתובת IP זו [n]" accesskey="n"><span>שיחה</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 id="mw-dismissablenotice-anonplace"></div><script>(function(){var node=document.getElementById("mw-dismissablenotice-anonplace");if(node){node.outerHTML="\u003Cdiv class=\"mw-dismissable-notice\"\u003E\u003Cdiv class=\"mw-dismissable-notice-close\"\u003E[\u003Ca tabindex=\"0\" role=\"button\"\u003Eהסתרה\u003C/a\u003E]\u003C/div\u003E\u003Cdiv class=\"mw-dismissable-notice-body\"\u003E\u003C!-- CentralNotice --\u003E\u003Cdiv id=\"localNotice\" data-nosnippet=\"\"\u003E\u003Cdiv class=\"anonnotice\" lang=\"he\" dir=\"rtl\"\u003E\u003Cp\u003E\u003Cb\u003Eתמיד רציתם לכתוב בוויקיפדיה אבל לא ידעתם איך? אתם מוזמנים לסדנת עריכה בוויקיפדיה. הסדנה תתקיים בספרייה הלאומית (בבניינה החדש) בירושלים ביום שישי, 06.12.24, בשעה 09:00. להרשמה לחצו \u003Ca href=\"/wiki/%D7%95%D7%99%D7%A7%D7%99%D7%A4%D7%93%D7%99%D7%94:%D7%9E%D7%99%D7%96%D7%9E%D7%99_%D7%95%D7%99%D7%A7%D7%99%D7%A4%D7%93%D7%99%D7%94/%D7%92%D7%9C%D7%90%D7%9D/%D7%94%D7%A1%D7%A4%D7%A8%D7%99%D7%99%D7%94_%D7%94%D7%9C%D7%90%D7%95%D7%9E%D7%99%D7%AA/%D7%90%D7%99%D7%A8%D7%95%D7%A2%D7%99%D7%9D/%D7%A1%D7%93%D7%A0%D7%AA_%D7%A2%D7%A8%D7%99%D7%9B%D7%94_%D7%93%D7%A6%D7%9E%D7%91%D7%A8_2024\" title=\"ויקיפדיה:מיזמי ויקיפדיה/גלאם/הספרייה הלאומית/אירועים/סדנת עריכה דצמבר 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class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">העברה לסרגל הצד</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">הסתרה</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">התחלה</div> </a> </li> <li id="toc-מהירות_ותאוצה" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#מהירות_ותאוצה"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>מהירות ותאוצה</span> </div> </a> <ul id="toc-מהירות_ותאוצה-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-הגדרת_אורך_עקום_התנועה" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#הגדרת_אורך_עקום_התנועה"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>הגדרת אורך עקום התנועה</span> </div> </a> <ul id="toc-הגדרת_אורך_עקום_התנועה-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-גוף_קשיח" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#גוף_קשיח"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>גוף קשיח</span> </div> </a> <ul id="toc-גוף_קשיח-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-מערכות_ייחוס" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#מערכות_ייחוס"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>מערכות ייחוס</span> </div> </a> <ul id="toc-מערכות_ייחוס-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-קישורים_חיצוניים" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#קישורים_חיצוניים"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>קישורים חיצוניים</span> </div> </a> <ul id="toc-קישורים_חיצוניים-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="תוכן עניינים" 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="מצב תוכן העניינים" > <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">מצב תוכן העניינים</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">העתק (פיזיקה)</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="מעבר לערך בשפה אחרת. זמין ב־57 שפות" > <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-57" 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">57 שפות</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Displacement_(geometry)" title="Displacement (geometry) – אנגלית" lang="en" hreflang="en" data-title="Displacement (geometry)" data-language-autonym="English" data-language-local-name="אנגלית" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-am mw-list-item"><a href="https://am.wikipedia.org/wiki/%E1%88%9D%E1%8B%95%E1%88%AB%E1%8D%8D" title="ምዕራፍ – אמהרית" lang="am" hreflang="am" data-title="ምዕራፍ" data-language-autonym="አማርኛ" data-language-local-name="אמהרית" class="interlanguage-link-target"><span>አማርኛ</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%A5%D8%B2%D8%A7%D8%AD%D8%A9_(%D9%81%D9%8A%D8%B2%D9%8A%D8%A7%D8%A1)" title="إزاحة (فيزياء) – ערבית" lang="ar" hreflang="ar" data-title="إزاحة (فيزياء)" data-language-autonym="العربية" data-language-local-name="ערבית" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-as mw-list-item"><a href="https://as.wikipedia.org/wiki/%E0%A6%B8%E0%A7%B0%E0%A6%A3" title="সৰণ – אסאמית" lang="as" hreflang="as" data-title="সৰণ" data-language-autonym="অসমীয়া" data-language-local-name="אסאמית" class="interlanguage-link-target"><span>অসমীয়া</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%9F%D0%B5%D1%80%D0%B0%D0%BC%D1%8F%D1%88%D1%87%D1%8D%D0%BD%D0%BD%D0%B5" title="Перамяшчэнне – בלארוסית" lang="be" hreflang="be" data-title="Перамяшчэнне" data-language-autonym="Беларуская" data-language-local-name="בלארוסית" 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%B5%D1%80%D0%B0%D0%BC%D1%8F%D1%88%D1%87%D1%8D%D0%BD%D1%8C%D0%BD%D0%B5" title="Перамяшчэньне – בלארוסית טרשקביץ׳" lang="be-tarask" hreflang="be-tarask" data-title="Перамяшчэньне" data-language-autonym="Беларуская (тарашкевіца)" data-language-local-name="בלארוסית טרשקביץ׳" 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%9F%D1%80%D0%B5%D0%BC%D0%B5%D1%81%D1%82%D0%B2%D0%B0%D0%BD%D0%B5_(%D0%B2%D0%B5%D0%BA%D1%82%D0%BE%D1%80)" title="Преместване (вектор) – בולגרית" lang="bg" hreflang="bg" data-title="Преместване (вектор)" data-language-autonym="Български" data-language-local-name="בולגרית" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%B8%E0%A6%B0%E0%A6%A3" title="সরণ – בנגלית" lang="bn" hreflang="bn" data-title="সরণ" data-language-autonym="বাংলা" data-language-local-name="בנגלית" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Despla%C3%A7ament_(cinem%C3%A0tica)" title="Desplaçament (cinemàtica) – קטלאנית" lang="ca" hreflang="ca" data-title="Desplaçament (cinemàtica)" data-language-autonym="Català" data-language-local-name="קטלאנית" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%9A%D1%83%D2%AB%D0%BD%C4%83%D0%BB%C4%83%D1%85" title="Куҫнăлăх – צ׳ובשית" lang="cv" hreflang="cv" data-title="Куҫнăлăх" data-language-autonym="Чӑвашла" data-language-local-name="צ׳ובשית" 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/Dadleoliad" title="Dadleoliad – ולשית" lang="cy" hreflang="cy" data-title="Dadleoliad" data-language-autonym="Cymraeg" data-language-local-name="ולשית" class="interlanguage-link-target"><span>Cymraeg</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%9C%CE%B5%CF%84%CE%B1%CF%84%CF%8C%CF%80%CE%B9%CF%83%CE%B7" title="Μετατόπιση – יוונית" lang="el" hreflang="el" data-title="Μετατόπιση" data-language-autonym="Ελληνικά" data-language-local-name="יוונית" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Desplazamiento_(vector)" title="Desplazamiento (vector) – ספרדית" lang="es" hreflang="es" data-title="Desplazamiento (vector)" data-language-autonym="Español" data-language-local-name="ספרדית" 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/Siire_(mehaanika)" title="Siire (mehaanika) – אסטונית" lang="et" hreflang="et" data-title="Siire (mehaanika)" data-language-autonym="Eesti" data-language-local-name="אסטונית" 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/Desplazamendu_(fisika)" title="Desplazamendu (fisika) – בסקית" lang="eu" hreflang="eu" data-title="Desplazamendu (fisika)" data-language-autonym="Euskara" data-language-local-name="בסקית" 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%A8%D8%B1%D8%AF%D8%A7%D8%B1_%D8%AC%D8%A7%D8%A8%D9%87%E2%80%8C%D8%AC%D8%A7%DB%8C%DB%8C" title="بردار جابه‌جایی – פרסית" lang="fa" hreflang="fa" data-title="بردار جابه‌جایی" data-language-autonym="فارسی" data-language-local-name="פרסית" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/D%C3%A9placement_(g%C3%A9om%C3%A9trie)" title="Déplacement (géométrie) – צרפתית" lang="fr" hreflang="fr" data-title="Déplacement (géométrie)" data-language-autonym="Français" data-language-local-name="צרפתית" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%B5%E0%A4%BF%E0%A4%B8%E0%A5%8D%E0%A4%A5%E0%A4%BE%E0%A4%AA%E0%A4%A8_(%E0%A4%B8%E0%A4%A6%E0%A4%BF%E0%A4%B6)" title="विस्थापन (सदिश) – הינדי" lang="hi" hreflang="hi" data-title="विस्थापन (सदिश)" data-language-autonym="हिन्दी" data-language-local-name="הינדי" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-ht mw-list-item"><a href="https://ht.wikipedia.org/wiki/Deplasman" title="Deplasman – קראולית האיטית" lang="ht" hreflang="ht" data-title="Deplasman" data-language-autonym="Kreyòl ayisyen" data-language-local-name="קראולית האיטית" class="interlanguage-link-target"><span>Kreyòl ayisyen</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D5%8F%D5%A5%D5%B2%D5%A1%D6%83%D5%B8%D5%AD%D5%B8%D6%82%D5%A9%D5%B5%D5%B8%D6%82%D5%B6_(%D6%86%D5%AB%D5%A6%D5%AB%D5%AF%D5%A1)" title="Տեղափոխություն (ֆիզիկա) – ארמנית" lang="hy" hreflang="hy" data-title="Տեղափոխություն (ֆիզիկա)" data-language-autonym="Հայերեն" data-language-local-name="ארמנית" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Perpindahan" title="Perpindahan – אינדונזית" lang="id" hreflang="id" data-title="Perpindahan" data-language-autonym="Bahasa Indonesia" data-language-local-name="אינדונזית" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Spostamento_(cinematica)" title="Spostamento (cinematica) – איטלקית" lang="it" hreflang="it" data-title="Spostamento (cinematica)" data-language-autonym="Italiano" data-language-local-name="איטלקית" 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%A4%89%E4%BD%8D" title="変位 – יפנית" lang="ja" hreflang="ja" data-title="変位" data-language-autonym="日本語" data-language-local-name="יפנית" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%90%D1%83%D1%8B%D1%81%D1%82%D1%8B%D1%80%D1%83" title="Ауыстыру – קזחית" lang="kk" hreflang="kk" data-title="Ауыстыру" data-language-autonym="Қазақша" data-language-local-name="קזחית" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EB%B3%80%EC%9C%84" title="변위 – קוריאנית" lang="ko" hreflang="ko" data-title="변위" data-language-autonym="한국어" data-language-local-name="קוריאנית" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Poslinkis" title="Poslinkis – ליטאית" lang="lt" hreflang="lt" data-title="Poslinkis" data-language-autonym="Lietuvių" data-language-local-name="ליטאית" 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/P%C4%81rvietojums" title="Pārvietojums – לטבית" lang="lv" hreflang="lv" data-title="Pārvietojums" data-language-autonym="Latviešu" data-language-local-name="לטבית" 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%9F%D0%BE%D0%BC%D0%B5%D1%81%D1%82%D1%83%D0%B2%D0%B0%D1%9A%D0%B5_(%D0%B2%D0%B5%D0%BA%D1%82%D0%BE%D1%80)" title="Поместување (вектор) – מקדונית" lang="mk" hreflang="mk" data-title="Поместување (вектор)" data-language-autonym="Македонски" data-language-local-name="מקדונית" 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%B8%E0%B5%8D%E0%B4%A5%E0%B4%BE%E0%B4%A8%E0%B4%BE%E0%B4%A8%E0%B5%8D%E0%B4%A4%E0%B4%B0%E0%B4%82" title="സ്ഥാനാന്തരം – מליאלאם" lang="ml" hreflang="ml" data-title="സ്ഥാനാന്തരം" data-language-autonym="മലയാളം" data-language-local-name="מליאלאם" class="interlanguage-link-target"><span>മലയാളം</span></a></li><li class="interlanguage-link interwiki-mr mw-list-item"><a href="https://mr.wikipedia.org/wiki/%E0%A4%B5%E0%A4%BF%E0%A4%B8%E0%A5%8D%E0%A4%A5%E0%A4%BE%E0%A4%AA%E0%A4%A8_(%E0%A4%B8%E0%A4%A6%E0%A4%BF%E0%A4%B6)" title="विस्थापन (सदिश) – מראטהית" lang="mr" hreflang="mr" data-title="विस्थापन (सदिश)" data-language-autonym="मराठी" data-language-local-name="מראטהית" 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/Sesaran" title="Sesaran – מלאית" lang="ms" hreflang="ms" data-title="Sesaran" data-language-autonym="Bahasa Melayu" data-language-local-name="מלאית" 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%9B%E1%80%BD%E1%80%B1%E1%80%B7" title="အရွေ့ – בורמזית" lang="my" hreflang="my" data-title="အရွေ့" data-language-autonym="မြန်မာဘာသာ" data-language-local-name="בורמזית" class="interlanguage-link-target"><span>မြန်မာဘာသာ</span></a></li><li class="interlanguage-link interwiki-om mw-list-item"><a href="https://om.wikipedia.org/wiki/Qaxxaamura" title="Qaxxaamura – אורומו" lang="om" hreflang="om" data-title="Qaxxaamura" data-language-autonym="Oromoo" data-language-local-name="אורומו" class="interlanguage-link-target"><span>Oromoo</span></a></li><li class="interlanguage-link interwiki-pa mw-list-item"><a href="https://pa.wikipedia.org/wiki/%E0%A8%B5%E0%A8%BF%E0%A9%B1%E0%A8%A5" title="ਵਿੱਥ – פנג׳אבי" lang="pa" hreflang="pa" data-title="ਵਿੱਥ" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="פנג׳אבי" class="interlanguage-link-target"><span>ਪੰਜਾਬੀ</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Przemieszczenie_(fizyka)" title="Przemieszczenie (fizyka) – פולנית" lang="pl" hreflang="pl" data-title="Przemieszczenie (fizyka)" data-language-autonym="Polski" data-language-local-name="פולנית" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pnb mw-list-item"><a href="https://pnb.wikipedia.org/wiki/%D9%88%D8%AA%DA%BE%D9%91" title="وتھّ – פנג׳בית מערבית" lang="pnb" hreflang="pnb" data-title="وتھّ" data-language-autonym="پنجابی" data-language-local-name="פנג׳בית מערבית" class="interlanguage-link-target"><span>پنجابی</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Deslocamento" title="Deslocamento – פורטוגזית" lang="pt" hreflang="pt" data-title="Deslocamento" data-language-autonym="Português" data-language-local-name="פורטוגזית" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9F%D0%B5%D1%80%D0%B5%D0%BC%D0%B5%D1%89%D0%B5%D0%BD%D0%B8%D0%B5" title="Перемещение – רוסית" lang="ru" hreflang="ru" data-title="Перемещение" data-language-autonym="Русский" data-language-local-name="רוסית" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Pomeraj" title="Pomeraj – סרבו-קרואטית" lang="sh" hreflang="sh" data-title="Pomeraj" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="סרבו-קרואטית" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Displacement_vector" title="Displacement vector – אנגלית פשוטה" lang="en-simple" hreflang="en-simple" data-title="Displacement vector" data-language-autonym="Simple English" data-language-local-name="אנגלית פשוטה" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sn mw-list-item"><a href="https://sn.wikipedia.org/wiki/Nhambwe_yasudurukwa" title="Nhambwe yasudurukwa – שונה" lang="sn" hreflang="sn" data-title="Nhambwe yasudurukwa" data-language-autonym="ChiShona" data-language-local-name="שונה" class="interlanguage-link-target"><span>ChiShona</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Zhvendosja_(fizik%C3%AB)" title="Zhvendosja (fizikë) – אלבנית" lang="sq" hreflang="sq" data-title="Zhvendosja (fizikë)" data-language-autonym="Shqip" data-language-local-name="אלבנית" 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%92%D0%B5%D0%BA%D1%82%D0%BE%D1%80_%D0%BF%D0%BE%D0%BC%D0%B5%D1%80%D0%B0%D1%98%D0%B0" title="Вектор помераја – סרבית" lang="sr" hreflang="sr" data-title="Вектор помераја" data-language-autonym="Српски / srpski" data-language-local-name="סרבית" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%87%E0%AE%9F%E0%AE%AA%E0%AF%8D%E0%AE%AA%E0%AF%86%E0%AE%AF%E0%AE%B0%E0%AF%8D%E0%AE%9A%E0%AF%8D%E0%AE%9A%E0%AE%BF" title="இடப்பெயர்ச்சி – טמילית" lang="ta" hreflang="ta" data-title="இடப்பெயர்ச்சி" data-language-autonym="தமிழ்" data-language-local-name="טמילית" 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%B8%E0%B1%8D%E0%B0%A5%E0%B0%BE%E0%B0%A8%E0%B0%AD%E0%B1%8D%E0%B0%B0%E0%B0%82%E0%B0%B6%E0%B0%AE%E0%B1%81" title="స్థానభ్రంశము – טלוגו" lang="te" hreflang="te" data-title="స్థానభ్రంశము" data-language-autonym="తెలుగు" data-language-local-name="טלוגו" 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%81%E0%B8%B2%E0%B8%A3%E0%B8%81%E0%B8%A3%E0%B8%B0%E0%B8%88%E0%B8%B1%E0%B8%94_(%E0%B9%80%E0%B8%A7%E0%B8%81%E0%B9%80%E0%B8%95%E0%B8%AD%E0%B8%A3%E0%B9%8C)" title="การกระจัด (เวกเตอร์) – תאית" lang="th" hreflang="th" data-title="การกระจัด (เวกเตอร์)" data-language-autonym="ไทย" data-language-local-name="תאית" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tk mw-list-item"><a href="https://tk.wikipedia.org/wiki/%C3%9Dol_we_orun_%C3%BC%C3%BDtgetme" title="Ýol we orun üýtgetme – טורקמנית" lang="tk" hreflang="tk" data-title="Ýol we orun üýtgetme" data-language-autonym="Türkmençe" data-language-local-name="טורקמנית" class="interlanguage-link-target"><span>Türkmençe</span></a></li><li class="interlanguage-link interwiki-tl mw-list-item"><a href="https://tl.wikipedia.org/wiki/Displasmento_(Heometriya)" title="Displasmento (Heometriya) – טאגאלוג" lang="tl" hreflang="tl" data-title="Displasmento (Heometriya)" data-language-autonym="Tagalog" data-language-local-name="טאגאלוג" 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/Yer_de%C4%9Fi%C5%9Ftirme_(fizik)" title="Yer değiştirme (fizik) – טורקית" lang="tr" hreflang="tr" data-title="Yer değiştirme (fizik)" data-language-autonym="Türkçe" data-language-local-name="טורקית" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%9F%D0%B5%D1%80%D0%B5%D0%BC%D1%96%D1%89%D0%B5%D0%BD%D0%BD%D1%8F" title="Переміщення – אוקראינית" lang="uk" hreflang="uk" data-title="Переміщення" data-language-autonym="Українська" data-language-local-name="אוקראינית" 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/%DB%81%D9%B9%D8%A7%D8%A4" title="ہٹاؤ – אורדו" lang="ur" hreflang="ur" data-title="ہٹاؤ" data-language-autonym="اردو" data-language-local-name="אורדו" 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/Ko%27chish_(Fizika)" title="Ko'chish (Fizika) – אוזבקית" lang="uz" hreflang="uz" data-title="Ko'chish (Fizika)" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="אוזבקית" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/Li_%C4%91%E1%BB%99" title="Li độ – וייטנאמית" lang="vi" hreflang="vi" data-title="Li độ" data-language-autonym="Tiếng Việt" data-language-local-name="וייטנאמית" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-xh mw-list-item"><a href="https://xh.wikipedia.org/wiki/I-displacement_vector" title="I-displacement vector – קוסה" lang="xh" hreflang="xh" data-title="I-displacement vector" data-language-autonym="IsiXhosa" data-language-local-name="קוסה" class="interlanguage-link-target"><span>IsiXhosa</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E4%BD%8D%E7%A7%BB" title="位移 – סינית" lang="zh" hreflang="zh" data-title="位移" data-language-autonym="中文" data-language-local-name="סינית" class="interlanguage-link-target"><span>中文</span></a></li><li class="interlanguage-link interwiki-zh-classical mw-list-item"><a href="https://zh-classical.wikipedia.org/wiki/%E4%BD%8D%E7%A7%BB" title="位移 – סינית קלאסית" lang="lzh" hreflang="lzh" data-title="位移" data-language-autonym="文言" data-language-local-name="סינית קלאסית" class="interlanguage-link-target"><span>文言</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E4%BD%8D%E7%A7%BB" title="位移 – קנטונזית" lang="yue" hreflang="yue" data-title="位移" data-language-autonym="粵語" data-language-local-name="קנטונזית" 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/Q190291#sitelinks-wikipedia" title="עריכת קישורים בין־לשוניים" class="wbc-editpage">עריכת הקישורים</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="מרחבי שם"> <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 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כ־PDF</span></a></li><li id="t-print" class="mw-list-item"><a href="/w/index.php?title=%D7%94%D7%A2%D7%AA%D7%A7_(%D7%A4%D7%99%D7%96%D7%99%D7%A7%D7%94)&printable=yes" title="גרסה להדפסה של הדף הזה [p]" accesskey="p"><span>גרסה להדפסה</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"> במיזמים אחרים </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:Displacement_vector" hreflang="en"><span>ויקישיתוף</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/Q190291" title="קישור לפריט המשויך במאגר הנתונים [g]" accesskey="g"><span>פריט ויקינתונים</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="כלי דף"> <div id="vector-page-tools-pinned-container" class="vector-pinned-container"> </div> </nav> <nav class="vector-appearance-landmark" aria-label="מראה"> <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">מראה</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">העברה לסרגל הצד</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">הסתרה</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> <div id="siteSub" class="noprint">מתוך ויקיפדיה, האנציקלופדיה החופשית</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-rtl mw-parser-output" lang="he" dir="rtl"><table class="toccolours portalSumm" style="clear:left;float:left; width:250px; margin:0.5em 0.5em 0.5em 0em; border:1px solid #999; padding:5px;"> <tbody><tr> <td> <div style="text-align: center;"> <p><big><b>עיינו גם בפורטל</b></big> </p> <div align="center"> <p><span typeof="mw:File"><a href="/wiki/%D7%A4%D7%95%D7%A8%D7%98%D7%9C:%D7%A4%D7%99%D7%96%D7%99%D7%A7%D7%94" title="פורטל:פיזיקה"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4b/P_physics_Bohr_model.svg/90px-P_physics_Bohr_model.svg.png" decoding="async" width="90" height="81" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4b/P_physics_Bohr_model.svg/135px-P_physics_Bohr_model.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4b/P_physics_Bohr_model.svg/180px-P_physics_Bohr_model.svg.png 2x" data-file-width="400" data-file-height="360" /></a></span> </p> </div> <p><b><a href="/wiki/%D7%A4%D7%95%D7%A8%D7%98%D7%9C:%D7%A4%D7%99%D7%96%D7%99%D7%A7%D7%94" title="פורטל:פיזיקה">פורטל הפיזיקה</a></b> מהווה שער לחובבי הפיזיקה ולמתעניינים בתחום. בין היתר, בפורטל תוכלו למצוא מידע על פיזיקאים חשובים, על ענפי הפיזיקה ועל תאוריות פיזיקליות. </p> </div> </td></tr></tbody></table><figure class="mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/%D7%A7%D7%95%D7%91%D7%A5:Distancedisplacement-he.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/83/Distancedisplacement-he.svg/270px-Distancedisplacement-he.svg.png" decoding="async" width="270" height="167" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/83/Distancedisplacement-he.svg/405px-Distancedisplacement-he.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/83/Distancedisplacement-he.svg/540px-Distancedisplacement-he.svg.png 2x" data-file-width="323" data-file-height="200" /></a><figcaption>וקטור ההעתק לעומת הדרך שעשה גוף לאורך מסלול</figcaption></figure> <p>ב<a href="/wiki/%D7%9E%D7%9B%D7%A0%D7%99%D7%A7%D7%94_%D7%A7%D7%9C%D7%90%D7%A1%D7%99%D7%AA" title="מכניקה קלאסית">מכניקה קלאסית</a>, וקטור ה<b>העתק</b> הוא ה<a href="/wiki/%D7%95%D7%A7%D7%98%D7%95%D7%A8_(%D7%A4%D7%99%D7%96%D7%99%D7%A7%D7%94)" title="וקטור (פיזיקה)">ווקטור</a> הקצר ביותר המחבר בין נקודת <a href="/wiki/%D7%9E%D7%93%D7%99%D7%93%D7%94" title="מדידה">מדידה</a> התחלתית לנקודת מדידה סופית של מיקומו של גוף. כיוונו של וקטור ההעתק הוא ככיוון <a href="/wiki/%D7%99%D7%A9%D7%A8" title="ישר">ישר</a> דמיוני המחבר את שתי נקודות המדידה, וגודלו כאורך הישר. מתמטית, העתק מוגדר כ<a href="/wiki/%D7%94%D7%A4%D7%A8%D7%A9" class="mw-redirect" title="הפרש">הפרש</a> בין וקטור מיקום הגוף בסוף התנועה לווקטור מיקומו בתחילת ה<a href="/wiki/%D7%AA%D7%A0%D7%95%D7%A2%D7%94_(%D7%A4%D7%99%D7%96%D7%99%D7%A7%D7%94)" title="תנועה (פיזיקה)">תנועה</a>. מהגדרה זו ניתן להסיק כי וקטור ההעתק אינו מושפע מהדרך אותה הגוף עבר בין נקודות המדידה, אלא רק ממיקומו הסופי וההתחלתי. כלומר אם מיקום הגוף בזמן <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_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cb0768c0bd659f2f84fb5ef9f4b74f336123d915" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.894ex; height:2.343ex;" alt="{\displaystyle t_{1}}"></span> היה <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 {\vec {x_{1}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {x_{1}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/84ddc66e7b753ee96992397191d8bf10e2632a06" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.384ex; height:3.343ex;" alt="{\displaystyle {\vec {x_{1}}}}"></span> וב- <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_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/749fee708b41e7079eabd50d61c8bf3e965db16f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.894ex; height:2.343ex;" alt="{\displaystyle t_{2}}"></span> היה <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 {\vec {x_{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {x_{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/325239622396b04745fc7106902500ad96b1d5ca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.384ex; height:3.343ex;" alt="{\displaystyle {\vec {x_{2}}}}"></span> הרי שווקטור ההעתק <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 {\vec {x}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta {\vec {x}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/28bc37487416d3059535e86be27b48fd5d6b211c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.266ex; height:2.343ex;" alt="{\displaystyle \Delta {\vec {x}}}"></span> הוא <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 {\vec {x}}\;{\overset {\underset {\mathrm {def} }{}}{=}}\;{\vec {x_{2}}}-{\vec {x_{1}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mover> <mo>=</mo> <munder> <mrow /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">f</mi> </mrow> </munder> </mover> </mrow> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta {\vec {x}}\;{\overset {\underset {\mathrm {def} }{}}{=}}\;{\vec {x_{2}}}-{\vec {x_{1}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c0a041097bb093d5602d427ceca0cda734de115e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.995ex; height:3.676ex;" alt="{\displaystyle \Delta {\vec {x}}\;{\overset {\underset {\mathrm {def} }{}}{=}}\;{\vec {x_{2}}}-{\vec {x_{1}}}}"></span>. בהקשר זה האות היוונית <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 }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32769037c408874e1890f77554c65f39c523ebe2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.936ex; height:2.176ex;" alt="{\displaystyle \Delta }"></span> (<a href="/wiki/%D7%93%D7%9C%D7%AA%D7%90_(%D7%90%D7%95%D7%AA)" title="דלתא (אות)">דלתא</a>) משמשת לציון <a href="/wiki/%D7%94%D7%A4%D7%A8%D7%A9" class="mw-redirect" title="הפרש">הפרש</a>, שימוש נפוץ בה. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="מהירות_ותאוצה"><span id=".D7.9E.D7.94.D7.99.D7.A8.D7.95.D7.AA_.D7.95.D7.AA.D7.90.D7.95.D7.A6.D7.94"></span>מהירות ותאוצה</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D7%94%D7%A2%D7%AA%D7%A7_(%D7%A4%D7%99%D7%96%D7%99%D7%A7%D7%94)&action=edit&section=1" title="עריכת קוד המקור של הפרק: מהירות ותאוצה"><span>עריכת קוד מקור</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D7%94%D7%A2%D7%AA%D7%A7_(%D7%A4%D7%99%D7%96%D7%99%D7%A7%D7%94)&veaction=edit&section=1" title="עריכת פסקה: "מהירות ותאוצה"" class="mw-editsection-visualeditor"><span>עריכה</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>על בסיס ההעתק ניתן להגדיר <a href="/wiki/%D7%9E%D7%94%D7%99%D7%A8%D7%95%D7%AA" title="מהירות">מהירות</a> של גוף - אפשר למדוד את מיקום הגוף בשתי נקודות סמוכת בזמן לרגע המבוקש, ולחשב את וקטור ההעתק על סמך שתי מדידות אלו. הממוצע ליחידת זמן של וקטור ההעתק יוגדר כווקטור ה<a href="/wiki/%D7%9E%D7%94%D7%99%D7%A8%D7%95%D7%AA_%D7%9E%D7%9E%D7%95%D7%A6%D7%A2%D7%AA" class="mw-redirect" title="מהירות ממוצעת">מהירות הממוצעת</a> על המקטע המחבר את שתי הנקודות <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 {\vec {v}}}={\frac {\Delta {\vec {x}}}{\Delta t}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mrow> <mrow> <mi mathvariant="normal">Δ</mi> <mi>t</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bar {\vec {v}}}={\frac {\Delta {\vec {x}}}{\Delta t}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c2b4bc6c57dd01d62ca24921b99b9ffcad025755" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:8.41ex; height:5.509ex;" alt="{\displaystyle {\bar {\vec {v}}}={\frac {\Delta {\vec {x}}}{\Delta t}}}"></span>. בהנחה ותנועת הגוף רציפה, אזי ככל שהמדידות יהיו סמוכות יותר, כך גודל זה יתקרב למספר מסוים. מספר זה, אשר מבטא את ה<a href="/wiki/%D7%92%D7%91%D7%95%D7%9C_%D7%A9%D7%9C_%D7%A1%D7%93%D7%A8%D7%94" title="גבול של סדרה">גבול</a> כאשר <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> שואף לאפס מוגדר כווקטור המהירות הרגעית בנקודה (Velocity), וגודלו הוא המהירות (Speed) בנקודה. </p><p>דרך נפוצה להתייחסות למיקום גוף במרחב היא באמצעות <a href="/wiki/%D7%A4%D7%95%D7%A0%D7%A7%D7%A6%D7%99%D7%94" title="פונקציה">פונקציה</a> וקטורית <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 {\vec {x}}(t)}"> <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 stretchy="false">→</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {x}}(t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6d1a21f93f4898468d3be62d985d996a11b8f561" 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 {\vec {x}}(t)}"></span> המחזירה עבור כל נקודה בזמן את וקטור מיקום הגוף. במקרה זה ניתן להציב בפונקציה שתי נקודות זמן <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_{i+1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t_{i+1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6c3e7533fd04354c4a859a8d42b4ca2e81465c8d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.74ex; height:2.343ex;" alt="{\displaystyle t_{i+1}}"></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_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b61e3d4d909be4a19c9a554a301684232f59e5a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.639ex; height:2.343ex;" alt="{\displaystyle t_{i}}"></span> אשר ההפרש בין שני הערכים המוחזרים ייתן את וקטור ההעתק. באופן דומה לחישוב לעיל, ההעתק ה<a href="/wiki/%D7%9E%D7%9E%D7%95%D7%A6%D7%A2" title="ממוצע">ממוצע</a> ליחידת זמן יוגדר כווקטור המהירות הממוצעת: <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 {\vec {v}}}\;{\overset {\underset {\mathrm {def} }{}}{=}}\;{\frac {{\vec {x}}(t_{i+1})-{\vec {x}}(t_{i})}{t_{i+1}-t_{i}}}={\frac {\Delta {\vec {x}}}{\Delta t}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mover> <mo>=</mo> <munder> <mrow /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">f</mi> </mrow> </munder> </mover> </mrow> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mrow> <mrow> <mi mathvariant="normal">Δ</mi> <mi>t</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,{\bar {\vec {v}}}\;{\overset {\underset {\mathrm {def} }{}}{=}}\;{\frac {{\vec {x}}(t_{i+1})-{\vec {x}}(t_{i})}{t_{i+1}-t_{i}}}={\frac {\Delta {\vec {x}}}{\Delta t}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7e8b89d9566f94324910ce35576e379c44ab59" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:27.252ex; height:6.176ex;" alt="{\displaystyle \,{\bar {\vec {v}}}\;{\overset {\underset {\mathrm {def} }{}}{=}}\;{\frac {{\vec {x}}(t_{i+1})-{\vec {x}}(t_{i})}{t_{i+1}-t_{i}}}={\frac {\Delta {\vec {x}}}{\Delta t}}}"></span>. ופונקציית המהירות תוגדר באמצעות הגבול בו <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> שואף ל-<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 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 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 {\vec {v}}(t){\overset {\underset {\mathrm {def} }{}}{=}}\lim _{\Delta t\to 0}{\frac {\Delta {\vec {x}}}{\Delta t}}=\lim _{\Delta t\to 0}{\frac {{\vec {x}}(t+\Delta t)-{\vec {x}}(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>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mo>=</mo> <munder> <mrow /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">f</mi> </mrow> </munder> </mover> </mrow> <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 mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mrow> <mrow> <mi mathvariant="normal">Δ</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <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> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo>+</mo> <mi mathvariant="normal">Δ</mi> <mi>t</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <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 {\vec {v}}(t){\overset {\underset {\mathrm {def} }{}}{=}}\lim _{\Delta t\to 0}{\frac {\Delta {\vec {x}}}{\Delta t}}=\lim _{\Delta t\to 0}{\frac {{\vec {x}}(t+\Delta t)-{\vec {x}}(t)}{\Delta t}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/88fbd72623a88b6a19555aac671d10e61cd3f93a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:40.122ex; height:6.009ex;" alt="{\displaystyle {\vec {v}}(t){\overset {\underset {\mathrm {def} }{}}{=}}\lim _{\Delta t\to 0}{\frac {\Delta {\vec {x}}}{\Delta t}}=\lim _{\Delta t\to 0}{\frac {{\vec {x}}(t+\Delta t)-{\vec {x}}(t)}{\Delta t}}}"></span>, כלומר פונקציית המהירות היא ה<a href="/wiki/%D7%A0%D7%92%D7%96%D7%A8%D7%AA" title="נגזרת">נגזרת</a> של פונקציית וקטור המיקום. </p><p>באופן דומה וקטור <a href="/wiki/%D7%AA%D7%90%D7%95%D7%A6%D7%94" title="תאוצה">התאוצה</a> מוגדר כווקטור שינוי המהירות הרגעית של הגוף ביחס לזמן.<span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {a}}(t)=\lim _{\Delta t\to 0}{\frac {{\vec {v}}(t+\Delta t)-{\vec {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 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> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo>+</mo> <mi mathvariant="normal">Δ</mi> <mi>t</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <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 {\vec {a}}(t)=\lim _{\Delta t\to 0}{\frac {{\vec {v}}(t+\Delta t)-{\vec {v}}(t)}{\Delta t}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c60be8b7d27b8ed87e21b2b66a19555e81d34011" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:28.733ex; height:6.009ex;" alt="{\displaystyle {\vec {a}}(t)=\lim _{\Delta t\to 0}{\frac {{\vec {v}}(t+\Delta t)-{\vec {v}}(t)}{\Delta t}}}"></span> </p><p>בתהליך ההופכי כדי למצוא את וקטור המהירות מבצעים <a href="/wiki/%D7%90%D7%99%D7%A0%D7%98%D7%92%D7%A8%D7%9C" title="אינטגרל">סכימה</a> (אינטגרציה) על וקטור התאוצה, וכדי למצוא את ההעתק מבצעים סכימה על וקטור המהירות. </p><p><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 \int {\vec {a}}dt={\vec {v}}\;\;;\int {\vec {v}}dt={\vec {\Delta x}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mi>d</mi> <mi>t</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mo>;</mo> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mi>d</mi> <mi>t</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi mathvariant="normal">Δ</mi> <mi>x</mi> </mrow> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int {\vec {a}}dt={\vec {v}}\;\;;\int {\vec {v}}dt={\vec {\Delta x}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c725193cd7ae85d709ea8ebf429909a50112eb74" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:24.64ex; height:5.676ex;" alt="{\displaystyle \int {\vec {a}}dt={\vec {v}}\;\;;\int {\vec {v}}dt={\vec {\Delta x}}}"></span> </p> <div class="mw-heading mw-heading2"><h2 id="הגדרת_אורך_עקום_התנועה"><span id=".D7.94.D7.92.D7.93.D7.A8.D7.AA_.D7.90.D7.95.D7.A8.D7.9A_.D7.A2.D7.A7.D7.95.D7.9D_.D7.94.D7.AA.D7.A0.D7.95.D7.A2.D7.94"></span>הגדרת אורך עקום התנועה</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D7%94%D7%A2%D7%AA%D7%A7_(%D7%A4%D7%99%D7%96%D7%99%D7%A7%D7%94)&action=edit&section=2" title="עריכת קוד המקור של הפרק: הגדרת אורך עקום התנועה"><span>עריכת קוד מקור</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D7%94%D7%A2%D7%AA%D7%A7_(%D7%A4%D7%99%D7%96%D7%99%D7%A7%D7%94)&veaction=edit&section=2" title="עריכת פסקה: "הגדרת אורך עקום התנועה"" class="mw-editsection-visualeditor"><span>עריכה</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>הגדרת וקטור ההעתק בין שתי נקודות אינה מאפשרת לחשב את אורך הדרך שעבר גוף בין שתי נקודות במקרה בו הגוף לא נע בצורה ישרה בין שתי הנקודות. לפיכך, כאשר רוצים לחשב את אורך הדרך אותה עבר הגוף, יש לחלק את תנועתו למקטעים קטנים ככל שניתן, ולחבר את אורכי ההעתקים המחושבים על סמך מדידות בתחילת ובסוף כל קטע. </p> <figure typeof="mw:File/Thumb"><a href="/wiki/%D7%A7%D7%95%D7%91%D7%A5:Arclength.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/11/Arclength.png/354px-Arclength.png" decoding="async" width="354" height="91" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/11/Arclength.png/531px-Arclength.png 1.5x, //upload.wikimedia.org/wikipedia/commons/1/11/Arclength.png 2x" data-file-width="653" data-file-height="168" /></a><figcaption>קירוב גודל עקומה S באמצעות חלוקה למקטעים</figcaption></figure> <p><br /> <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=\lVert \Delta {\vec {x}}_{1}\rVert +\lVert \Delta {\vec {x}}_{2}\rVert \cdots \lVert \Delta {\vec {x}}_{n-1}\rVert +\lVert \Delta {\vec {x}}_{n}\rVert =\sum _{i=1}^{n}\lVert \Delta {{\vec {x}}_{i}}\rVert }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo>=</mo> <mo fence="false" stretchy="false">‖</mo> <mi mathvariant="normal">Δ</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo fence="false" stretchy="false">‖</mo> <mo>+</mo> <mo fence="false" stretchy="false">‖</mo> <mi mathvariant="normal">Δ</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo fence="false" stretchy="false">‖</mo> <mo>⋯</mo> <mo fence="false" stretchy="false">‖</mo> <mi mathvariant="normal">Δ</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo fence="false" stretchy="false">‖</mo> <mo>+</mo> <mo fence="false" stretchy="false">‖</mo> <mi mathvariant="normal">Δ</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo fence="false" stretchy="false">‖</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mo fence="false" stretchy="false">‖</mo> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mo fence="false" stretchy="false">‖</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S=\lVert \Delta {\vec {x}}_{1}\rVert +\lVert \Delta {\vec {x}}_{2}\rVert \cdots \lVert \Delta {\vec {x}}_{n-1}\rVert +\lVert \Delta {\vec {x}}_{n}\rVert =\sum _{i=1}^{n}\lVert \Delta {{\vec {x}}_{i}}\rVert }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/08cc9adb5d007f992788d8289797650dc599ffba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:55.627ex; height:6.843ex;" alt="{\displaystyle S=\lVert \Delta {\vec {x}}_{1}\rVert +\lVert \Delta {\vec {x}}_{2}\rVert \cdots \lVert \Delta {\vec {x}}_{n-1}\rVert +\lVert \Delta {\vec {x}}_{n}\rVert =\sum _{i=1}^{n}\lVert \Delta {{\vec {x}}_{i}}\rVert }"></span> </p><p>בהנחה והתנועה של הגוף רציפה, ככל שהחלוקה תהיה מעודנת יותר, כלומר תכיל מקטעים קצרים יותר, כך כל מקטע יתקרב לקו ישר וחישוב אורך הדרך יהיה מדויק יותר. </p><p>באופן דומה, כאשר נתונה פונקציה <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 {\vec {x}}(t)}"> <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 stretchy="false">→</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {x}}(t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6d1a21f93f4898468d3be62d985d996a11b8f561" 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 {\vec {x}}(t)}"></span> המקבלת ערכים בטווח של <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_{1}<t<t_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo><</mo> <mi>t</mi> <mo><</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t_{1}<t<t_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b990a6be17cb6ce90aa7df197d07845186610d45" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.824ex; height:2.343ex;" alt="{\displaystyle t_{1}<t<t_{2}}"></span> ומחזירה עבור כל נקודה בזמן את ווקטור המיקום של גוף הנע על מסלול <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>, ניתן לחלק את מסלול הגוף <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> למספר סופי <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 n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> של מקטעי תנועה המוגדרים על ידי נקודת התחלה בזמן ונקודת סוף בזמן <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_{i},t_{i+1}]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [t_{i},t_{i+1}]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5847784c78dde5ff1000d6a2885311bb3bc4a95" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.707ex; height:2.843ex;" alt="{\displaystyle [t_{i},t_{i+1}]}"></span>. בכל מקטע זמן שכזה, ניתן לבחור בנקודה <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_{i}^{*}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t_{i}^{*}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6344d9a81d0b06a4f3f9903d659a850c9e3af620" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:1.894ex; height:2.843ex;" alt="{\displaystyle t_{i}^{*}}"></span> השייכת לקטע, לחשב את גודל וקטור המהירות הרגעית באותה הנקודה <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\Vert v(t_{i}^{*})\right\Vert }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo symmetric="true">‖</mo> <mrow> <mi>v</mi> <mo stretchy="false">(</mo> <msubsup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> <mo stretchy="false">)</mo> </mrow> <mo symmetric="true">‖</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left\Vert v(t_{i}^{*})\right\Vert }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/55ed5778a18dba36534d357d3fc7a028e8cea2e5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.156ex; height:3.176ex;" alt="{\displaystyle \left\Vert v(t_{i}^{*})\right\Vert }"></span>, ולהכפיל בהפרש הזמנים <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_{i}=t_{i+1}-t_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta t_{i}=t_{i+1}-t_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2aa47f1a9b88c5bf4946bedc4440dfe59d7d6b1d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.893ex; height:2.509ex;" alt="{\displaystyle \Delta t_{i}=t_{i+1}-t_{i}}"></span>. כלומר לחשב את הביטוי <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\Vert {{\vec {v}}(t_{i}^{*})}\right\Vert \Delta t_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo symmetric="true">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <msubsup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> <mo stretchy="false">)</mo> </mrow> <mo symmetric="true">‖</mo> </mrow> <mi mathvariant="normal">Δ</mi> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left\Vert {{\vec {v}}(t_{i}^{*})}\right\Vert \Delta t_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/19bf419de5ea0d987691ab949e0d15a11c644ca8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.166ex; height:3.176ex;" alt="{\displaystyle \left\Vert {{\vec {v}}(t_{i}^{*})}\right\Vert \Delta t_{i}}"></span> השווה בקירוב לווקטור ההעתק המוגדר על המקטע. </p><p>כאשר <a href="/wiki/%D7%A1%D7%9B%D7%95%D7%9D" title="סכום">נסכום</a> את כל המקטעים, נקבל את הסכום הבא: <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=\sum _{i=1}^{n}\left\Vert {{\vec {v}}(t_{i}^{*})}\right\Vert \Delta t_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mrow> <mo symmetric="true">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <msubsup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> <mo stretchy="false">)</mo> </mrow> <mo symmetric="true">‖</mo> </mrow> <mi mathvariant="normal">Δ</mi> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S=\sum _{i=1}^{n}\left\Vert {{\vec {v}}(t_{i}^{*})}\right\Vert \Delta t_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8fdc53ec0b2fa57b2f430667561ad54c2dbc0c7e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:19.505ex; height:6.843ex;" alt="{\displaystyle S=\sum _{i=1}^{n}\left\Vert {{\vec {v}}(t_{i}^{*})}\right\Vert \Delta t_{i}}"></span>, אם קיים לפונקציה גבול כאשר אורך המקטעים <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> שואף ל-<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 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}"></span> אז נגדיר גבול זה כאורך הדרך או העקומה אותה עבר הגוף. על מנת לחשב את גודל הסכום, נזהה כי סכום זה מהווה <a href="/wiki/%D7%A1%D7%9B%D7%95%D7%9D_%D7%A8%D7%99%D7%9E%D7%9F" class="mw-redirect" title="סכום רימן">סכום רימן</a>. תכונתו של סכום רימן היא שאם הוא מתכנס כאשר אורך המקטעים <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> שואף ל-<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 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}"></span>, אזי הוא הופך ל<a href="/wiki/%D7%90%D7%99%D7%A0%D7%98%D7%92%D7%A8%D7%9C" title="אינטגרל">אינטגרל</a>, ובפרט במקרה זה הוא מקבל את הביטוי: <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 \int _{t1}^{t2}{\left\Vert {{\vec {v}}(t)}\right\Vert dt}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> <mn>2</mn> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo symmetric="true">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mo symmetric="true">‖</mo> </mrow> <mi>d</mi> <mi>t</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{t1}^{t2}{\left\Vert {{\vec {v}}(t)}\right\Vert dt}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/816b42f3842f3dd59f6a614bca2358f35809fd18" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:13.091ex; height:6.176ex;" alt="{\displaystyle \int _{t1}^{t2}{\left\Vert {{\vec {v}}(t)}\right\Vert dt}}"></span> . </p><p>כאשר פותחים את פונקציית המהירות הווקטורית לרכיבי <a href="/wiki/%D7%95%D7%A7%D7%98%D7%95%D7%A8_%D7%99%D7%97%D7%99%D7%93%D7%94" title="וקטור יחידה">ווקטורי היחידה</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 x,y,z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x,y,z}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bbeca34b28f569a407ef74a955d041df9f360268" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.641ex; height:2.009ex;" alt="{\displaystyle x,y,z}"></span> מקבלים את הביטוי הבא: <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=\sum _{i=1}^{n}\left\Vert {{\vec {v}}(t_{i}^{*})}\right\Vert \Delta t_{i}=\int _{t1}^{t2}{\sqrt {v_{x}^{2}{\hat {x}}+v_{y}^{2}{\hat {y}}+v_{z}^{2}{\hat {z}}}}dt=\int _{t1}^{t2}{\sqrt {{\dot {x}}_{x}^{2}(t){\hat {x}}+{\dot {x}}_{y}^{2}(t){\hat {y}}+{\dot {x}}_{z}^{2}(t){\hat {z}}}}dt}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mrow> <mo symmetric="true">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <msubsup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> <mo stretchy="false">)</mo> </mrow> <mo symmetric="true">‖</mo> </mrow> <mi mathvariant="normal">Δ</mi> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> <mn>2</mn> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msubsup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>+</mo> <msubsup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>y</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>+</mo> <msubsup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>z</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </msqrt> </mrow> <mi>d</mi> <mi>t</mi> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> <mn>2</mn> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>+</mo> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>y</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>+</mo> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>z</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </msqrt> </mrow> <mi>d</mi> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S=\sum _{i=1}^{n}\left\Vert {{\vec {v}}(t_{i}^{*})}\right\Vert \Delta t_{i}=\int _{t1}^{t2}{\sqrt {v_{x}^{2}{\hat {x}}+v_{y}^{2}{\hat {y}}+v_{z}^{2}{\hat {z}}}}dt=\int _{t1}^{t2}{\sqrt {{\dot {x}}_{x}^{2}(t){\hat {x}}+{\dot {x}}_{y}^{2}(t){\hat {y}}+{\dot {x}}_{z}^{2}(t){\hat {z}}}}dt}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e4e590bf4b7e2011eb2a1c2ec7fd4124b108e8ff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:84.558ex; height:6.843ex;" alt="{\displaystyle S=\sum _{i=1}^{n}\left\Vert {{\vec {v}}(t_{i}^{*})}\right\Vert \Delta t_{i}=\int _{t1}^{t2}{\sqrt {v_{x}^{2}{\hat {x}}+v_{y}^{2}{\hat {y}}+v_{z}^{2}{\hat {z}}}}dt=\int _{t1}^{t2}{\sqrt {{\dot {x}}_{x}^{2}(t){\hat {x}}+{\dot {x}}_{y}^{2}(t){\hat {y}}+{\dot {x}}_{z}^{2}(t){\hat {z}}}}dt}"></span>. </p> <div class="mw-heading mw-heading2"><h2 id="גוף_קשיח"><span id=".D7.92.D7.95.D7.A3_.D7.A7.D7.A9.D7.99.D7.97"></span>גוף קשיח</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D7%94%D7%A2%D7%AA%D7%A7_(%D7%A4%D7%99%D7%96%D7%99%D7%A7%D7%94)&action=edit&section=3" title="עריכת קוד המקור של הפרק: גוף קשיח"><span>עריכת קוד מקור</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D7%94%D7%A2%D7%AA%D7%A7_(%D7%A4%D7%99%D7%96%D7%99%D7%A7%D7%94)&veaction=edit&section=3" title="עריכת פסקה: "גוף קשיח"" class="mw-editsection-visualeditor"><span>עריכה</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>כאשר מנתחים תנועה של <a href="/wiki/%D7%9E%D7%9B%D7%A0%D7%99%D7%A7%D7%94_%D7%A9%D7%9C_%D7%92%D7%95%D7%A3_%D7%A7%D7%A9%D7%99%D7%97" title="מכניקה של גוף קשיח">גוף קשיח</a> וקטור ההעתק צריך להכיל גם את התנועה הסיבובית אותה עבר הגוף, ולכן במקרה כזה ההעתק לאורך הדרך יקרא ההעתק הליניארי של הגוף, ההעתק הסיבובי (המתאר את הזווית אותה עבר הגוף) יקרא העתק זוויתי וההעתק הכולל יהיה <a href="/wiki/%D7%A6%D7%99%D7%A8%D7%95%D7%A3_%D7%9C%D7%99%D7%A0%D7%99%D7%90%D7%A8%D7%99" title="צירוף ליניארי">הצירוף הליניארי</a> של שניהם. </p> <div class="mw-heading mw-heading2"><h2 id="מערכות_ייחוס"><span id=".D7.9E.D7.A2.D7.A8.D7.9B.D7.95.D7.AA_.D7.99.D7.99.D7.97.D7.95.D7.A1"></span>מערכות ייחוס</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D7%94%D7%A2%D7%AA%D7%A7_(%D7%A4%D7%99%D7%96%D7%99%D7%A7%D7%94)&action=edit&section=4" title="עריכת קוד המקור של הפרק: מערכות ייחוס"><span>עריכת קוד מקור</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D7%94%D7%A2%D7%AA%D7%A7_(%D7%A4%D7%99%D7%96%D7%99%D7%A7%D7%94)&veaction=edit&section=4" title="עריכת פסקה: "מערכות ייחוס"" class="mw-editsection-visualeditor"><span>עריכה</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>עקרון היחסות של גלילאו קובע שחוקי ה<a href="/wiki/%D7%A4%D7%99%D7%96%D7%99%D7%A7%D7%94" title="פיזיקה">פיזיקה</a> הם זהים בכל שתי <a href="/wiki/%D7%9E%D7%A2%D7%A8%D7%9B%D7%AA_%D7%99%D7%99%D7%97%D7%95%D7%A1" title="מערכת ייחוס">מערכות ייחוס</a> הזזות במהירות קבועה זו יחסית לזו, ולכן אין אפשרות להבדיל בין מערכות הייחוס מי מהן נעה ומי מהן ניצבת, אלא ניתן לדבר רק על הפרש המהירויות בין שתי המערכות. מכאן שאין שום משמעות למיקום ללא מערכת ייחוס, ובהתאמה אין משמעות להעתק ללא מערכת ייחוס. לדוגמה: רכבת נעה במהירות <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="{\textstyle {\vec {u}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle {\vec {u}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/67f564b03b889c20668364cd7910c0d35e1cbfb9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:2.343ex;" alt="{\textstyle {\vec {u}}}"></span> ונוסע הולך בתוך הרכבת במהירות <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="{\textstyle {\vec {v}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle {\vec {v}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e0172ab9c0689e9c1b0cfd84e54fdceec8719b09" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.175ex; height:2.343ex;" alt="{\textstyle {\vec {v}}}"></span>, אם מערכת הייחוס היא הרכבת עצמה, ההעתקו של האדם יבוטא לפי סכימה של מהירות הליכתו כלומר, <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="{\textstyle .\int ({\vec {v}})dt={\vec {v}}t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mo>.</mo> <mo>∫</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <mi>d</mi> <mi>t</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle .\int ({\vec {v}})dt={\vec {v}}t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/242374f71838130e6b6de757c4036fc892ae5db1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.605ex; height:3.176ex;" alt="{\textstyle .\int ({\vec {v}})dt={\vec {v}}t}"></span> אך אם מערכת הייחוס היא מערכת אינרציאלית יחסית לרכבת אזי וקטור העתקו יבוטא על ידי סכימה של מהירות הרכבת בנוסף למהירות הנוסע כלומר, <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="{\textstyle .\int ({\vec {u}}+{\vec {v}})dt={\vec {u}}t+{\vec {v}}t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mo>.</mo> <mo>∫</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <mi>d</mi> <mi>t</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mi>t</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle .\int ({\vec {u}}+{\vec {v}})dt={\vec {u}}t+{\vec {v}}t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aaf8a9abeaf5f7e3dc12d9dc19947e9706fa5819" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:21.785ex; height:3.176ex;" alt="{\textstyle .\int ({\vec {u}}+{\vec {v}})dt={\vec {u}}t+{\vec {v}}t}"></span> מכאן שבמערכות ייחוס שונות יהיו העתקים שונים לאותו הגוף שנע, ושיש חשיבות רבה לבחירת מערכת הייחוס ממנה מודדים את המיקום וההעתק של הגוף. באופן דומה ניתן להראות זאת תוך שימוש בעקרונות <a href="/wiki/%D7%AA%D7%95%D7%A8%D7%AA_%D7%94%D7%99%D7%97%D7%A1%D7%95%D7%AA_%D7%94%D7%A4%D7%A8%D7%98%D7%99%D7%AA" title="תורת היחסות הפרטית">היחסות הפרטית</a> <a href="/wiki/%D7%94%D7%9B%D7%9C%D7%9C%D7%94_(%D7%9E%D7%AA%D7%9E%D7%98%D7%99%D7%A7%D7%94)" title="הכללה (מתמטיקה)">המכליל</a> את עקרון היחסות של גלילי למהירויות הקרובות ל<a href="/wiki/%D7%9E%D7%94%D7%99%D7%A8%D7%95%D7%AA_%D7%94%D7%90%D7%95%D7%A8" title="מהירות האור">מהירות האור</a>. </p> <div class="mw-heading mw-heading2"><h2 id="קישורים_חיצוניים"><span id=".D7.A7.D7.99.D7.A9.D7.95.D7.A8.D7.99.D7.9D_.D7.97.D7.99.D7.A6.D7.95.D7.A0.D7.99.D7.99.D7.9D"></span>קישורים חיצוניים</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D7%94%D7%A2%D7%AA%D7%A7_(%D7%A4%D7%99%D7%96%D7%99%D7%A7%D7%94)&action=edit&section=5" title="עריכת קוד המקור של הפרק: קישורים חיצוניים"><span>עריכת קוד מקור</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D7%94%D7%A2%D7%AA%D7%A7_(%D7%A4%D7%99%D7%96%D7%99%D7%A7%D7%94)&veaction=edit&section=5" title="עריכת פסקה: "קישורים 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