Moviment rectilini - Viquipèdia, l'enciclopèdia lliure

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[n]" accesskey="n"><span>Discussió per aquest IP</span></a></li> </ul> </div> </div> </div> </div> </nav> </div> </header> </div> <div class="mw-page-container"> <div class="mw-page-container-inner"> <div class="vector-sitenotice-container"> <div id="siteNotice"></div> </div> <div class="vector-column-start"> <div class="vector-main-menu-container"> <div id="mw-navigation"> <nav id="mw-panel" class="vector-main-menu-landmark" aria-label="Lloc"> <div id="vector-main-menu-pinned-container" class="vector-pinned-container"> </div> </nav> </div> </div> <div class="vector-sticky-pinned-container"> <nav id="mw-panel-toc" aria-label="Contingut" data-event-name="ui.sidebar-toc" class="mw-table-of-contents-container vector-toc-landmark"> <div id="vector-toc-pinned-container" class="vector-pinned-container"> <div id="vector-toc" class="vector-toc vector-pinnable-element"> <div class="vector-pinnable-header vector-toc-pinnable-header vector-pinnable-header-pinned" data-feature-name="toc-pinned" data-pinnable-element-id="vector-toc" > <h2 class="vector-pinnable-header-label">Contingut</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">mou a la barra lateral</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">amaga</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">Inici</div> </a> </li> <li id="toc-Moviment_rectilini_uniforme_(MRU)" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Moviment_rectilini_uniforme_(MRU)"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Moviment rectilini uniforme (MRU)</span> </div> </a> <ul id="toc-Moviment_rectilini_uniforme_(MRU)-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Moviment_Rectilini_Uniformement_Accelerat_(MRUA)" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Moviment_Rectilini_Uniformement_Accelerat_(MRUA)"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Moviment Rectilini Uniformement Accelerat (MRUA)</span> </div> </a> <button aria-controls="toc-Moviment_Rectilini_Uniformement_Accelerat_(MRUA)-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Commuta la subsecció Moviment Rectilini Uniformement Accelerat (MRUA)</span> </button> <ul id="toc-Moviment_Rectilini_Uniformement_Accelerat_(MRUA)-sublist" class="vector-toc-list"> <li id="toc-Equació_de_la_velocitat" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Equació_de_la_velocitat"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Equació de la velocitat</span> </div> </a> <ul id="toc-Equació_de_la_velocitat-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Equació_de_la_posició" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Equació_de_la_posició"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Equació de la posició</span> </div> </a> <ul id="toc-Equació_de_la_posició-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Equacions_no_horàries" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Equacions_no_horàries"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3</span> <span>Equacions no horàries</span> </div> </a> <ul id="toc-Equacions_no_horàries-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Cas_particular_de_la_caiguda_lliure" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Cas_particular_de_la_caiguda_lliure"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.4</span> <span>Cas particular de la caiguda lliure</span> </div> </a> <ul id="toc-Cas_particular_de_la_caiguda_lliure-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Acceleració" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Acceleració"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Acceleració</span> </div> </a> <ul id="toc-Acceleració-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Referències" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Referències"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Referències</span> </div> </a> <ul id="toc-Referències-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> 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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">Moviment rectilini</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="Vés a un article en una altra llengua. 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href="https://be.wikipedia.org/wiki/%D0%9F%D1%80%D0%B0%D0%BC%D0%B0%D0%BB%D1%96%D0%BD%D0%B5%D0%B9%D0%BD%D1%8B_%D1%80%D1%83%D1%85" title="Прамалінейны рух - belarús" lang="be" hreflang="be" data-title="Прамалінейны рух" data-language-autonym="Беларуская" data-language-local-name="belarús" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%B0%E0%A7%88%E0%A6%96%E0%A6%BF%E0%A6%95_%E0%A6%97%E0%A6%A4%E0%A6%BF" title="রৈখিক গতি - bengalí" lang="bn" hreflang="bn" data-title="রৈখিক গতি" data-language-autonym="বাংলা" data-language-local-name="bengalí" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Jednoliko_kretanje_po_pravcu" title="Jednoliko kretanje po pravcu - bosnià" lang="bs" hreflang="bs" data-title="Jednoliko kretanje po pravcu" data-language-autonym="Bosanski" data-language-local-name="bosnià" class="interlanguage-link-target"><span>Bosanski</span></a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%D8%AC%D9%88%D9%88%DA%B5%DB%95%DB%8C_%DA%95%D8%A7%D8%B3%D8%AA%DB%95%D9%88%DA%95%D8%A7%D8%B3%D8%AA" title="جووڵەی ڕاستەوڕاست - kurd central" lang="ckb" hreflang="ckb" data-title="جووڵەی ڕاستەوڕاست" data-language-autonym="کوردی" data-language-local-name="kurd central" class="interlanguage-link-target"><span>کوردی</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/P%C5%99%C3%ADmo%C4%8Dar%C3%BD_pohyb" title="Přímočarý pohyb - txec" lang="cs" hreflang="cs" data-title="Přímočarý pohyb" data-language-autonym="Čeština" data-language-local-name="txec" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%A2%D3%B3%D1%80%C4%95_%D0%BA%D1%83%C3%A7%C4%83%D0%BC" title="Тӳрĕ куçăм - txuvaix" lang="cv" hreflang="cv" data-title="Тӳрĕ куçăм" data-language-autonym="Чӑвашла" data-language-local-name="txuvaix" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Linear_motion" title="Linear motion - anglès" lang="en" hreflang="en" data-title="Linear motion" data-language-autonym="English" data-language-local-name="anglès" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Movimiento_rectil%C3%ADneo" title="Movimiento rectilíneo - espanyol" lang="es" hreflang="es" data-title="Movimiento rectilíneo" data-language-autonym="Español" data-language-local-name="espanyol" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Higidura_zuzen" title="Higidura zuzen - basc" lang="eu" hreflang="eu" data-title="Higidura zuzen" data-language-autonym="Euskara" data-language-local-name="basc" 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%AD%D8%B1%DA%A9%D8%AA_%D8%AE%D8%B7%DB%8C" title="حرکت خطی - persa" lang="fa" hreflang="fa" data-title="حرکت خطی" data-language-autonym="فارسی" data-language-local-name="persa" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Suoraviivainen_liike" title="Suoraviivainen liike - finès" lang="fi" hreflang="fi" data-title="Suoraviivainen liike" data-language-autonym="Suomi" data-language-local-name="finès" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Mouvement_rectiligne" title="Mouvement rectiligne - francès" lang="fr" hreflang="fr" data-title="Mouvement rectiligne" data-language-autonym="Français" data-language-local-name="francès" 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%B0%E0%A5%88%E0%A4%96%E0%A4%BF%E0%A4%95_%E0%A4%97%E0%A4%A4%E0%A4%BF" title="रैखिक गति - hindi" lang="hi" hreflang="hi" data-title="रैखिक गति" data-language-autonym="हिन्दी" data-language-local-name="hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Translacija" title="Translacija - croat" lang="hr" hreflang="hr" data-title="Translacija" data-language-autonym="Hrvatski" data-language-local-name="croat" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Gerak_lurus" title="Gerak lurus - indonesi" lang="id" hreflang="id" data-title="Gerak lurus" data-language-autonym="Bahasa Indonesia" data-language-local-name="indonesi" 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/Moto_rettilineo" title="Moto rettilineo - italià" lang="it" hreflang="it" data-title="Moto rettilineo" data-language-autonym="Italiano" data-language-local-name="italià" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-pa mw-list-item"><a href="https://pa.wikipedia.org/wiki/%E0%A8%B0%E0%A9%87%E0%A8%96%E0%A8%BF%E0%A8%95_%E0%A8%97%E0%A8%A4%E0%A9%80" title="ਰੇਖਿਕ ਗਤੀ - panjabi" lang="pa" hreflang="pa" data-title="ਰੇਖਿਕ ਗਤੀ" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="panjabi" class="interlanguage-link-target"><span>ਪੰਜਾਬੀ</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Movimento_retil%C3%ADneo" title="Movimento retilíneo - portuguès" lang="pt" hreflang="pt" data-title="Movimento retilíneo" data-language-autonym="Português" data-language-local-name="portuguès" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Mi%C8%99care_rectilinie" title="Mișcare rectilinie - romanès" lang="ro" hreflang="ro" data-title="Mișcare rectilinie" data-language-autonym="Română" data-language-local-name="romanès" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9F%D1%80%D1%8F%D0%BC%D0%BE%D0%BB%D0%B8%D0%BD%D0%B5%D0%B9%D0%BD%D0%BE%D0%B5_%D0%B4%D0%B2%D0%B8%D0%B6%D0%B5%D0%BD%D0%B8%D0%B5" title="Прямолинейное движение - rus" lang="ru" hreflang="ru" data-title="Прямолинейное движение" data-language-autonym="Русский" data-language-local-name="rus" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sc mw-list-item"><a href="https://sc.wikipedia.org/wiki/Movimentu_in_caminu_deretu" title="Movimentu in caminu deretu - sard" lang="sc" hreflang="sc" data-title="Movimentu in caminu deretu" data-language-autonym="Sardu" data-language-local-name="sard" class="interlanguage-link-target"><span>Sardu</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Pravolinijsko_kretanje" title="Pravolinijsko kretanje - serbocroat" lang="sh" hreflang="sh" data-title="Pravolinijsko kretanje" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="serbocroat" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Priamo%C4%8Diary_pohyb" title="Priamočiary pohyb - eslovac" lang="sk" hreflang="sk" data-title="Priamočiary pohyb" data-language-autonym="Slovenčina" data-language-local-name="eslovac" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Premo_gibanje" title="Premo gibanje - eslovè" lang="sl" hreflang="sl" data-title="Premo gibanje" data-language-autonym="Slovenščina" data-language-local-name="eslovè" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/L%C3%ABvizja_drejtvizore_e_nj%C3%ABtrajtshme" title="Lëvizja drejtvizore e njëtrajtshme - albanès" lang="sq" hreflang="sq" data-title="Lëvizja drejtvizore e njëtrajtshme" data-language-autonym="Shqip" data-language-local-name="albanès" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Do%C4%9Frusal_hareket" title="Doğrusal hareket - turc" lang="tr" hreflang="tr" data-title="Doğrusal hareket" data-language-autonym="Türkçe" data-language-local-name="turc" 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%D1%80%D1%8F%D0%BC%D0%BE%D0%BB%D1%96%D0%BD%D1%96%D0%B9%D0%BD%D0%B8%D0%B9_%D1%80%D1%83%D1%85" title="Прямолінійний рух - ucraïnès" lang="uk" hreflang="uk" data-title="Прямолінійний рух" data-language-autonym="Українська" data-language-local-name="ucraïnès" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/To%27g%27ri_chiziqli_harakat" title="To'g'ri chiziqli harakat - uzbek" lang="uz" hreflang="uz" data-title="To'g'ri chiziqli harakat" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="uzbek" 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/Chuy%E1%BB%83n_%C4%91%E1%BB%99ng_th%E1%BA%B3ng" title="Chuyển động thẳng - vietnamita" lang="vi" hreflang="vi" data-title="Chuyển động thẳng" data-language-autonym="Tiếng Việt" data-language-local-name="vietnamita" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E7%9B%B4%E7%B7%9A%E9%81%8B%E5%8B%95" title="直線運動 - xinès" lang="zh" 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class="vector-column-end"> <div class="vector-sticky-pinned-container"> <nav class="vector-page-tools-landmark" aria-label="Eines de la pàgina"> <div id="vector-page-tools-pinned-container" class="vector-pinned-container"> </div> </nav> <nav class="vector-appearance-landmark" aria-label="Aparença"> <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">Aparença</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">mou a la barra lateral</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">amaga</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">De la Viquipèdia, l'enciclopèdia lliure</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="ca" dir="ltr"><p>Un <b>moviment rectilini</b> és aquell <a href="/wiki/Moviment" title="Moviment">moviment</a> on el cos que es mou té una <a href="/wiki/Traject%C3%B2ria_(cinem%C3%A0tica)" title="Trajectòria (cinemàtica)">trajectòria</a> <a href="/wiki/Recta" title="Recta">rectilínia</a>. Per tant, la seva <a href="/wiki/Velocitat" title="Velocitat">velocitat</a> es mantindrà constant en <a href="/wiki/Direcci%C3%B3_(geometria)" title="Direcció (geometria)">direcció</a> i <a href="/wiki/Sentit" class="mw-redirect" title="Sentit">sentit</a> - però no necessàriament en <a href="/wiki/M%C3%B2dul" title="Mòdul">mòdul</a> -, i l'<a href="/wiki/Acceleraci%C3%B3" title="Acceleració">acceleració</a> normal — a causa de les variacions de direcció de la velocitat — serà nul·la.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p><p>El moviment rectilini és el més simple de tots els <a href="/wiki/Moviment" title="Moviment">moviments</a>. Segons la primera <a href="/wiki/Lleis_de_Newton" title="Lleis de Newton">llei de moviment</a> de <a href="/wiki/Isaac_Newton" title="Isaac Newton">Newton</a>,<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> els objectes que no experimenten cap <a href="/wiki/For%C3%A7a" title="Força">força</a> neta continuaran movent-se en línia recta amb una <a href="/wiki/Velocitat" title="Velocitat">velocitat</a> constant fins que estiguin subjectes a una força neta. En circumstàncies quotidianes, les forces externes com la <a href="/wiki/Gravetat" title="Gravetat">gravetat</a> i la <a href="/wiki/Fricci%C3%B3" title="Fricció">fricció</a> poden fer que un objecte canviï la direcció del seu moviment, de manera que el seu moviment deixa de ser lineal. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Moviment_rectilini_uniforme_(MRU)"><span id="Moviment_rectilini_uniforme_.28MRU.29"></span>Moviment rectilini uniforme (MRU)</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Moviment_rectilini&action=edit&section=1" title="Modifica la secció: Moviment rectilini uniforme (MRU)"><span>modifica</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r30997230">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}</style><div role="note" class="hatnote navigation-not-searchable">Article principal: <a href="/wiki/Moviment_rectilini_uniforme" title="Moviment rectilini uniforme">Moviment rectilini uniforme</a></div> <p>És aquell moviment rectilini que es produeix quan el cos es mou amb <a href="/wiki/Velocitat" title="Velocitat">velocitat</a> constant. Tindrà, com qualsevol altre moviment rectilini, l'acceleració normal nul·la; per altra banda, també serà nul·la l'<a href="/w/index.php?title=Acceleraci%C3%B3_tangencial&action=edit&redlink=1" class="new" title="Acceleració tangencial (encara no existeix)">acceleració tangencial</a>, ja que la velocitat es manté constant.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p><p>En la vida quotidiana és molt difícil, trobar objectes que es moguin amb velocitat constant, ja que sempre hi ha un <a href="/wiki/Fregament" class="mw-redirect" title="Fregament">fregament</a> que fa que la velocitat disminueixi. Però és molt útil com a aproximació estudiar el moviment amb velocitat constant, ja que és senzill i ens aporta molta informació. </p><p>Com que la velocitat és una magnitud vectorial, direm que es manté constant quan no canvia: </p> <ul><li>El <b>mòdul</b>, és a dir, el valor numèric.</li> <li>La <b>direcció</b>, que és la recta que conté.</li> <li>El <b>sentit</b>, que es determina per la punta de la fletxa.</li></ul> <p>Per tant, perquè no canviï la direcció, la trajectòria ha de ser una línia recta i el mòbil sempre ha d'anar en el mateix sentit i amb el mateix valor de la velocitat. </p><p>Aquest moviment s'anomena <a href="/wiki/Moviment_rectilini_uniforme" title="Moviment rectilini uniforme">moviment rectilini uniforme</a> (m.r.u.) </p><p>Es pot modelitzar el comportament d'un mòbil en MRU a partir d'una única funció o equació,<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> que ens dona la posició en cada instant en funció del temps: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x(t)=x_{0}+v\left(t-t_{0}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <mi>v</mi> <mrow> <mo>(</mo> <mrow> <mi>t</mi> <mo>−</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x(t)=x_{0}+v\left(t-t_{0}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e6c56825c3d41951ad3c397fe44874f01b9f334" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.199ex; height:2.843ex;" alt="{\displaystyle x(t)=x_{0}+v\left(t-t_{0}\right)}"></span></dd></dl> <p>On: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4bf17264a35330beeb310c35f9676cf9837482e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.91ex; height:1.676ex;" alt="{\displaystyle x\ }"></span> = Posició del mòbil respecte al <a href="/wiki/Sistema_de_refer%C3%A8ncia" title="Sistema de referència">sistema de referència</a> triat</dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{0}\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{0}\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d98cdbdb39511d1a53c0c6b72772bc13c55cfd54" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.965ex; height:2.009ex;" alt="{\displaystyle x_{0}\ }"></span> = Posició inicial (quan <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t=t_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> <mo>=</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t=t_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/be6d7492e2d48bf34fdd5dffa189b188c140820c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.832ex; height:2.343ex;" alt="{\displaystyle t=t_{0}}"></span>)</dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/84c8c79f007e986ade705cc35653c07ac717bc31" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.708ex; height:1.676ex;" alt="{\displaystyle v\ }"></span> = velocitat, que es manté constant</dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/37f32f091511cddfa156e1660810e5c638f386b1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.42ex; height:2.009ex;" alt="{\displaystyle t\ }"></span> = temps</dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t_{0}\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t_{0}\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3aa4da2116d5f5c360fe989968525b44a819b14e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.475ex; height:2.343ex;" alt="{\displaystyle t_{0}\ }"></span> = temps inicial (quan <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=x_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=x_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/04e899fc6eba0b387b91f070adc7bc4fe5a706cb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.812ex; height:2.009ex;" alt="{\displaystyle x=x_{0}}"></span>)</dd></dl> <div class="NavFrame" style="background-color: transparent; width:100%;margin-bottom:0px;float:left; border-radius:4px"> <div class="NavPic" style="display: none;"></div> <div class="NavHead" align="center" style="background-color: transparent;border-radius:4px;">Obtenció de l'equació general d'un MRU</div> <div class="NavContent" align="left" style="padding:7px;"><div align="left"> <p>Partint de la definició de la velocitat com a derivada de la posició respecte del temps: </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 v={\frac {dx}{dt}}\Rightarrow \int {dx}=\int {v\cdot dt}\Rightarrow x=\int {v\cdot dt}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>x</mi> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo stretchy="false">⇒</mo> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> <mi>x</mi> </mrow> <mo>=</mo> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>v</mi> <mo>⋅</mo> <mi>d</mi> <mi>t</mi> </mrow> <mo stretchy="false">⇒</mo> <mi>x</mi> <mo>=</mo> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>v</mi> <mo>⋅</mo> <mi>d</mi> <mi>t</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v={\frac {dx}{dt}}\Rightarrow \int {dx}=\int {v\cdot dt}\Rightarrow x=\int {v\cdot dt}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b6dade7f043c208ba1eb4d82d50ad92adb6fbda5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:42.375ex; height:5.843ex;" alt="{\displaystyle v={\frac {dx}{dt}}\Rightarrow \int {dx}=\int {v\cdot dt}\Rightarrow x=\int {v\cdot dt}}"></span> </p><p>Resolent la integral indefinida per v constant: </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 x=vt+C\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mi>v</mi> <mi>t</mi> <mo>+</mo> <mi>C</mi> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=vt+C\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e688c70bb02d26ee7d0a25bc11c8cc8b11c8cba8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:11.583ex; height:2.343ex;" alt="{\displaystyle x=vt+C\ }"></span> </p><p>Aplicant condicions inicials, <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=x_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=x_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/04e899fc6eba0b387b91f070adc7bc4fe5a706cb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.812ex; height:2.009ex;" alt="{\displaystyle x=x_{0}}"></span> quan <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t=t_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> <mo>=</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t=t_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/be6d7492e2d48bf34fdd5dffa189b188c140820c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.832ex; height:2.343ex;" alt="{\displaystyle t=t_{0}}"></span>, podem aïllar C: </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 x_{0}=vt_{0}+C\Rightarrow C=x_{0}-vt_{0}\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mi>v</mi> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <mi>C</mi> <mo stretchy="false">⇒</mo> <mi>C</mi> <mo>=</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>−</mo> <mi>v</mi> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{0}=vt_{0}+C\Rightarrow C=x_{0}-vt_{0}\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/38f0a3802a0496236d63dce3aee761cba6fb2b5b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:30.416ex; height:2.509ex;" alt="{\displaystyle x_{0}=vt_{0}+C\Rightarrow C=x_{0}-vt_{0}\ }"></span> </p><p>Substituint C en l'equació anterior: </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 x=vt+x_{0}-vt_{0}\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mi>v</mi> <mi>t</mi> <mo>+</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>−</mo> <mi>v</mi> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=vt+x_{0}-vt_{0}\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e78b97df95b148ba51027fbc293dc64f79e6fc87" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:18.062ex; height:2.343ex;" alt="{\displaystyle x=vt+x_{0}-vt_{0}\ }"></span> </p><p>I finalment, traient v factor comú, obtenim l'equació general de l'MRU: </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 x=x_{0}+v\left(t-t_{0}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <mi>v</mi> <mrow> <mo>(</mo> <mrow> <mi>t</mi> <mo>−</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=x_{0}+v\left(t-t_{0}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a9a55dda08657318edcccd1a1f922396e1a81283" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.55ex; height:2.843ex;" alt="{\displaystyle x=x_{0}+v\left(t-t_{0}\right)}"></span> </p> </div></div> <div style="clear:both;"></div> </div> <div class="mw-heading mw-heading2"><h2 id="Moviment_Rectilini_Uniformement_Accelerat_(MRUA)"><span id="Moviment_Rectilini_Uniformement_Accelerat_.28MRUA.29"></span>Moviment Rectilini Uniformement Accelerat (MRUA)</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Moviment_rectilini&action=edit&section=2" title="Modifica la secció: Moviment Rectilini Uniformement Accelerat (MRUA)"><span>modifica</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>És aquell moviment rectilini on la velocitat varia de manera constant, o sigui, existeix una acceleració tangencial constant i diferent de 0. Com en el cas anterior l'acceleració normal serà nul·la. </p><p>Aquest moviment es pot modelitzar amb dues equacions bàsiques que es poden ajuntar en una tercera que no depèn del temps. En aquest apartat estudiarem alguns tipus de moviments en què la velocitat canvia; són els moviments accelerats. Limitarem l'estudi als casos en què l'acceleració és constant, és a dir, als moviments rectilinis uniformement accelerats. </p><p>El moviment rectilini uniformement accelerat(m.r.u.a) és un moviment en què la trajectòria és una línia recta i l'acceleració és constant. </p> <div class="mw-heading mw-heading3"><h3 id="Equació_de_la_velocitat"><span id="Equaci.C3.B3_de_la_velocitat"></span>Equació de la velocitat</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Moviment_rectilini&action=edit&section=3" title="Modifica la secció: Equació de la velocitat"><span>modifica</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>En aquest cas, com que la trajectòria és rectilínia, no canvia ni la direcció ni el sentit, només el mòdul de la velocitat: per això només hi ha acceleració tangencial. </p><p>Aquesta equació/funció relaciona la velocitat del mòbil amb el temps. </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v(t)=v_{0}+a\left(t-t_{0}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <mi>a</mi> <mrow> <mo>(</mo> <mrow> <mi>t</mi> <mo>−</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v(t)=v_{0}+a\left(t-t_{0}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/70bed2d15f2c0e90dd1ac32592729e4d51e5045b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.897ex; height:2.843ex;" alt="{\displaystyle v(t)=v_{0}+a\left(t-t_{0}\right)}"></span></dd></dl> <p>On: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/84c8c79f007e986ade705cc35653c07ac717bc31" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.708ex; height:1.676ex;" alt="{\displaystyle v\ }"></span> = velocitat del mòbil en funció del temps t</dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v_{0}\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v_{0}\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/64f4a31e2c1d8a0bf253408da4e1bdb15de5cf33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.763ex; height:2.009ex;" alt="{\displaystyle v_{0}\ }"></span> = velocitat inicial (quan <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t=t_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> <mo>=</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t=t_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/be6d7492e2d48bf34fdd5dffa189b188c140820c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.832ex; height:2.343ex;" alt="{\displaystyle t=t_{0}}"></span>)</dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cab11b679bd92051611e727cc8cc095c5c883745" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.81ex; height:1.676ex;" alt="{\displaystyle a\ }"></span> = acceleració, que es manté constant</dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/37f32f091511cddfa156e1660810e5c638f386b1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.42ex; height:2.009ex;" alt="{\displaystyle t\ }"></span> = temps</dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t_{0}\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t_{0}\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3aa4da2116d5f5c360fe989968525b44a819b14e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.475ex; height:2.343ex;" alt="{\displaystyle t_{0}\ }"></span> = temps inicial (quan <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=x_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=x_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/04e899fc6eba0b387b91f070adc7bc4fe5a706cb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.812ex; height:2.009ex;" alt="{\displaystyle x=x_{0}}"></span>)</dd></dl> <div class="NavFrame" style="background-color: transparent; width:100%;margin-bottom:0px;float:left; border-radius:4px"> <div class="NavPic" style="display: none;"></div> <div class="NavHead" align="center" style="background-color: transparent;border-radius:4px;">Obtenció de l'equació de la velocitat</div> <div class="NavContent" align="left" style="padding:7px;"><div align="left"> <p>Partim de la definició de l'acceleració com a la derivada de la velocitat: </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 a={\frac {dv}{dt}}\Rightarrow \int {dv}=\int {a\cdot dt}\Rightarrow v=\int {a\cdot dt}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>v</mi> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo stretchy="false">⇒</mo> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> <mi>v</mi> </mrow> <mo>=</mo> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mo>⋅</mo> <mi>d</mi> <mi>t</mi> </mrow> <mo stretchy="false">⇒</mo> <mi>v</mi> <mo>=</mo> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mo>⋅</mo> <mi>d</mi> <mi>t</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a={\frac {dv}{dt}}\Rightarrow \int {dv}=\int {a\cdot dt}\Rightarrow v=\int {a\cdot dt}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7bdd5efe76960699f1da7675a268f1f89737bb4e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:42.075ex; height:5.843ex;" alt="{\displaystyle a={\frac {dv}{dt}}\Rightarrow \int {dv}=\int {a\cdot dt}\Rightarrow v=\int {a\cdot dt}}"></span> </p><p>Resolent la integral indefinida per <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span> constant: </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 v=at+C\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> <mo>=</mo> <mi>a</mi> <mi>t</mi> <mo>+</mo> <mi>C</mi> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v=at+C\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/67997d7c3338628edfeb46ffeaecf56ba6c5bbe7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:11.483ex; height:2.343ex;" alt="{\displaystyle v=at+C\ }"></span> </p><p>Aplicant condicions inicials, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v=v_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> <mo>=</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v=v_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/600f1fced4041372dcb68ef2f69706559e45c024" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.408ex; height:2.009ex;" alt="{\displaystyle v=v_{0}}"></span> quan <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t=t_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> <mo>=</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t=t_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/be6d7492e2d48bf34fdd5dffa189b188c140820c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.832ex; height:2.343ex;" alt="{\displaystyle t=t_{0}}"></span>, podem aïllar C: </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 v_{0}=at_{0}+C\Rightarrow C=v_{0}-at_{0}\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mi>a</mi> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <mi>C</mi> <mo stretchy="false">⇒</mo> <mi>C</mi> <mo>=</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>−</mo> <mi>a</mi> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v_{0}=at_{0}+C\Rightarrow C=v_{0}-at_{0}\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/29b0f4660f8fe62bef5b36fc05ea48fb6863128e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:30.216ex; height:2.509ex;" alt="{\displaystyle v_{0}=at_{0}+C\Rightarrow C=v_{0}-at_{0}\ }"></span> </p><p>Substituint C en l'equació anterior: </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 v=at+v_{0}-at_{0}\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> <mo>=</mo> <mi>a</mi> <mi>t</mi> <mo>+</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>−</mo> <mi>a</mi> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v=at+v_{0}-at_{0}\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81b1849f6acdb6e8515f08983047c60c67da65d5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:17.862ex; height:2.343ex;" alt="{\displaystyle v=at+v_{0}-at_{0}\ }"></span> </p><p>I finalment, traient <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span> factor comú, obtenim l'equació de la velocitat de l'MRUA: </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 v=v_{0}+a\left(t-t_{0}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> <mo>=</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <mi>a</mi> <mrow> <mo>(</mo> <mrow> <mi>t</mi> <mo>−</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v=v_{0}+a\left(t-t_{0}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/60710b8a1895213a200b760dfe439bfa8a477f9c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.248ex; height:2.843ex;" alt="{\displaystyle v=v_{0}+a\left(t-t_{0}\right)}"></span> </p> </div></div> <div style="clear:both;"></div> </div> <div class="mw-heading mw-heading3"><h3 id="Equació_de_la_posició"><span id="Equaci.C3.B3_de_la_posici.C3.B3"></span>Equació de la posició</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Moviment_rectilini&action=edit&section=4" title="Modifica la secció: Equació de la posició"><span>modifica</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Aquesta funció o equació ens dona la posició d'un mòbil en MRUA en cada instant, en funció del temps: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=x_{0}+v_{0}\left(t-t_{0}\right)+{\frac {1}{2}}a\left(t-t_{0}\right)^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <mi>t</mi> <mo>−</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>a</mi> <msup> <mrow> <mo>(</mo> <mrow> <mi>t</mi> <mo>−</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=x_{0}+v_{0}\left(t-t_{0}\right)+{\frac {1}{2}}a\left(t-t_{0}\right)^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cd4f0643fb96fcdd69502e554d0a0ed25ffd1313" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:34.111ex; height:5.176ex;" alt="{\displaystyle x=x_{0}+v_{0}\left(t-t_{0}\right)+{\frac {1}{2}}a\left(t-t_{0}\right)^{2}}"></span></dd></dl> <p>On: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4bf17264a35330beeb310c35f9676cf9837482e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.91ex; height:1.676ex;" alt="{\displaystyle x\ }"></span> = Posició del mòbil respecte al <a href="/wiki/Sistema_de_refer%C3%A8ncia" title="Sistema de referència">sistema de referència</a> triat</dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{0}\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{0}\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d98cdbdb39511d1a53c0c6b72772bc13c55cfd54" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.965ex; height:2.009ex;" alt="{\displaystyle x_{0}\ }"></span> = Posició inicial (quan <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t=t_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> <mo>=</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t=t_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/be6d7492e2d48bf34fdd5dffa189b188c140820c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.832ex; height:2.343ex;" alt="{\displaystyle t=t_{0}}"></span>)</dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v_{0}\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v_{0}\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/64f4a31e2c1d8a0bf253408da4e1bdb15de5cf33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.763ex; height:2.009ex;" alt="{\displaystyle v_{0}\ }"></span> = velocitat inicial del mòbil (quan <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t=t_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> <mo>=</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t=t_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/be6d7492e2d48bf34fdd5dffa189b188c140820c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.832ex; height:2.343ex;" alt="{\displaystyle t=t_{0}}"></span>)</dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cab11b679bd92051611e727cc8cc095c5c883745" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.81ex; height:1.676ex;" alt="{\displaystyle a\ }"></span> = acceleració, que es manté constant</dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/37f32f091511cddfa156e1660810e5c638f386b1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.42ex; height:2.009ex;" alt="{\displaystyle t\ }"></span> = temps</dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t_{0}\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t_{0}\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3aa4da2116d5f5c360fe989968525b44a819b14e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.475ex; height:2.343ex;" alt="{\displaystyle t_{0}\ }"></span> = temps inicial (quan <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=x_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=x_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/04e899fc6eba0b387b91f070adc7bc4fe5a706cb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.812ex; height:2.009ex;" alt="{\displaystyle x=x_{0}}"></span> i <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v=v_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> <mo>=</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v=v_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/600f1fced4041372dcb68ef2f69706559e45c024" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.408ex; height:2.009ex;" alt="{\displaystyle v=v_{0}}"></span>)</dd></dl> <div class="NavFrame" style="background-color: transparent; width:100%;margin-bottom:0px;float:left; border-radius:4px"> <div class="NavPic" style="display: none;"></div> <div class="NavHead" align="center" style="background-color: transparent;border-radius:4px;">Obtenció de l'equació de la posició</div> <div class="NavContent" align="left" style="padding:7px;"><div align="left"> <p>Partim, igual que per a la demostració de l'equació del MRU, de la definició de la velocitat com a la derivada de la posició respecte del temps: </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 v={\frac {dx}{dt}}\Rightarrow \int {dx}=\int {v\cdot dt}\Rightarrow x=\int {v\cdot dt}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>x</mi> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo stretchy="false">⇒</mo> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> <mi>x</mi> </mrow> <mo>=</mo> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>v</mi> <mo>⋅</mo> <mi>d</mi> <mi>t</mi> </mrow> <mo stretchy="false">⇒</mo> <mi>x</mi> <mo>=</mo> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>v</mi> <mo>⋅</mo> <mi>d</mi> <mi>t</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v={\frac {dx}{dt}}\Rightarrow \int {dx}=\int {v\cdot dt}\Rightarrow x=\int {v\cdot dt}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b6dade7f043c208ba1eb4d82d50ad92adb6fbda5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:42.375ex; height:5.843ex;" alt="{\displaystyle v={\frac {dx}{dt}}\Rightarrow \int {dx}=\int {v\cdot dt}\Rightarrow x=\int {v\cdot dt}}"></span> </p><p>Com que v no és constant, cal aplicar l'equació de la velocitat <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v=v_{0}+a\left(t-t_{0}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> <mo>=</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <mi>a</mi> <mrow> <mo>(</mo> <mrow> <mi>t</mi> <mo>−</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v=v_{0}+a\left(t-t_{0}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/60710b8a1895213a200b760dfe439bfa8a477f9c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.248ex; height:2.843ex;" alt="{\displaystyle v=v_{0}+a\left(t-t_{0}\right)}"></span> per resoldre la integral: </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 x=\int {\left[v_{0}+a\left(t-t_{0}\right)\right]\cdot dt}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mrow> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <mi>a</mi> <mrow> <mo>(</mo> <mrow> <mi>t</mi> <mo>−</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> </mrow> <mo>]</mo> </mrow> <mo>⋅</mo> <mi>d</mi> <mi>t</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=\int {\left[v_{0}+a\left(t-t_{0}\right)\right]\cdot dt}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5b9e3449b5617f570f7af17b0e432c2c146c340d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:26.06ex; height:5.676ex;" alt="{\displaystyle x=\int {\left[v_{0}+a\left(t-t_{0}\right)\right]\cdot dt}}"></span> </p><p>Desenvolupem els parèntesis, per poder separar els sumands que formen la integral en diferents integrals separades: </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 x=\int {\left[v_{0}+at-at_{0}\right]\cdot dt}=\int {v_{0}\cdot dt}+\int {at\cdot dt}-\int {at_{0}}\cdot dt}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mrow> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <mi>a</mi> <mi>t</mi> <mo>−</mo> <mi>a</mi> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> <mo>]</mo> </mrow> <mo>⋅</mo> <mi>d</mi> <mi>t</mi> </mrow> <mo>=</mo> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>⋅</mo> <mi>d</mi> <mi>t</mi> </mrow> <mo>+</mo> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mi>t</mi> <mo>⋅</mo> <mi>d</mi> <mi>t</mi> </mrow> <mo>−</mo> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> <mo>⋅</mo> <mi>d</mi> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=\int {\left[v_{0}+at-at_{0}\right]\cdot dt}=\int {v_{0}\cdot dt}+\int {at\cdot dt}-\int {at_{0}}\cdot dt}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4465afd8012738879be9641bb67d8a1ebeb20a37" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:60.193ex; height:5.676ex;" alt="{\displaystyle x=\int {\left[v_{0}+at-at_{0}\right]\cdot dt}=\int {v_{0}\cdot dt}+\int {at\cdot dt}-\int {at_{0}}\cdot dt}"></span> </p><p>Resolem les diferents integrals, tenint en compte que <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/60faad24775635f4722ccc438093dbbfe05f34ae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.182ex; height:2.009ex;" alt="{\displaystyle v_{0}}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span> i <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/02d3006c4190b1939b04d9b9bb21006fb4e6fa4a" 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_{0}}"></span> són constants: </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 x=v_{0}t+{\frac {1}{2}}at^{2}-at_{0}t+C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mi>t</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>a</mi> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mi>a</mi> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mi>t</mi> <mo>+</mo> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=v_{0}t+{\frac {1}{2}}at^{2}-at_{0}t+C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5656a35299058c517052482e628dad979b2b1905" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:26.823ex; height:5.176ex;" alt="{\displaystyle x=v_{0}t+{\frac {1}{2}}at^{2}-at_{0}t+C}"></span> </p><p>Apliquem condicions inicials, <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=x_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=x_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/04e899fc6eba0b387b91f070adc7bc4fe5a706cb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.812ex; height:2.009ex;" alt="{\displaystyle x=x_{0}}"></span> quan <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t=t_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> <mo>=</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t=t_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/be6d7492e2d48bf34fdd5dffa189b188c140820c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.832ex; height:2.343ex;" alt="{\displaystyle t=t_{0}}"></span> i operem: </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 x_{0}=v_{0}t_{0}+{\frac {1}{2}}a{t_{0}}^{2}-at_{0}t_{0}+C=v_{0}t_{0}+{\frac {1}{2}}a{t_{0}}^{2}-a{t_{0}}^{2}+C=v_{0}t_{0}+a{t_{0}}^{2}\left({\frac {1}{2}}-1\right)+C=v_{0}t_{0}-{\frac {1}{2}}a{t_{0}}^{2}+C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>a</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mi>a</mi> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <mi>C</mi> <mo>=</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>a</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mi>a</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>C</mi> <mo>=</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <mi>a</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>−</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mi>C</mi> <mo>=</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>a</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{0}=v_{0}t_{0}+{\frac {1}{2}}a{t_{0}}^{2}-at_{0}t_{0}+C=v_{0}t_{0}+{\frac {1}{2}}a{t_{0}}^{2}-a{t_{0}}^{2}+C=v_{0}t_{0}+a{t_{0}}^{2}\left({\frac {1}{2}}-1\right)+C=v_{0}t_{0}-{\frac {1}{2}}a{t_{0}}^{2}+C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ad3523b509823667e2447c6e6586377d34a1d77b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:108.262ex; height:6.176ex;" alt="{\displaystyle x_{0}=v_{0}t_{0}+{\frac {1}{2}}a{t_{0}}^{2}-at_{0}t_{0}+C=v_{0}t_{0}+{\frac {1}{2}}a{t_{0}}^{2}-a{t_{0}}^{2}+C=v_{0}t_{0}+a{t_{0}}^{2}\left({\frac {1}{2}}-1\right)+C=v_{0}t_{0}-{\frac {1}{2}}a{t_{0}}^{2}+C}"></span> </p><p>Aïllem C: </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 C=x_{0}-v_{0}t_{0}+{\frac {1}{2}}a{t_{0}}^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> <mo>=</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>a</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C=x_{0}-v_{0}t_{0}+{\frac {1}{2}}a{t_{0}}^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4e187e1304524295ada587b7a1a8f51c95265121" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:23.182ex; height:5.176ex;" alt="{\displaystyle C=x_{0}-v_{0}t_{0}+{\frac {1}{2}}a{t_{0}}^{2}}"></span> </p><p>Substitüim en l'equació obtinguda en resoldre la integral: </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 x=v_{0}t+{\frac {1}{2}}at^{2}-at_{0}t+x_{0}-v_{0}t_{0}+{\frac {1}{2}}a{t_{0}}^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mi>t</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>a</mi> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mi>a</mi> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mi>t</mi> <mo>+</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>a</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=v_{0}t+{\frac {1}{2}}at^{2}-at_{0}t+x_{0}-v_{0}t_{0}+{\frac {1}{2}}a{t_{0}}^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b03733a2a25a8915619c2b85d0cccaf6c3c23842" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:43.373ex; height:5.176ex;" alt="{\displaystyle x=v_{0}t+{\frac {1}{2}}at^{2}-at_{0}t+x_{0}-v_{0}t_{0}+{\frac {1}{2}}a{t_{0}}^{2}}"></span> </p><p>Traiem factor comú <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/60faad24775635f4722ccc438093dbbfe05f34ae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.182ex; height:2.009ex;" alt="{\displaystyle v_{0}}"></span> i reduïm a denominador comú els termes que contenen <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span>: </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 x=x_{0}+v_{0}\left(t-t_{0}\right)+{\frac {at^{2}}{2}}-{\frac {2at_{0}t}{2}}+{\frac {a{t_{0}}^{2}}{2}}=x_{0}+v_{0}\left(t-t_{0}\right)+{\frac {at^{2}-2at_{0}t+a{t_{0}}^{2}}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <mi>t</mi> <mo>−</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>a</mi> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>a</mi> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mi>t</mi> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>a</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>=</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <mi>t</mi> <mo>−</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>a</mi> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>2</mn> <mi>a</mi> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mi>t</mi> <mo>+</mo> <mi>a</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=x_{0}+v_{0}\left(t-t_{0}\right)+{\frac {at^{2}}{2}}-{\frac {2at_{0}t}{2}}+{\frac {a{t_{0}}^{2}}{2}}=x_{0}+v_{0}\left(t-t_{0}\right)+{\frac {at^{2}-2at_{0}t+a{t_{0}}^{2}}{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/858446f5e92558cd03b5c09917db65e880dd80c0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:83.121ex; height:5.676ex;" alt="{\displaystyle x=x_{0}+v_{0}\left(t-t_{0}\right)+{\frac {at^{2}}{2}}-{\frac {2at_{0}t}{2}}+{\frac {a{t_{0}}^{2}}{2}}=x_{0}+v_{0}\left(t-t_{0}\right)+{\frac {at^{2}-2at_{0}t+a{t_{0}}^{2}}{2}}}"></span> </p><p>Traiem factor comú <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{2}}a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{2}}a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bdf63eb594535b8b789f5c53d0a1ea168661404e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:3.228ex; height:5.176ex;" alt="{\displaystyle {\frac {1}{2}}a}"></span> </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 x=x_{0}+v_{0}\left(t-t_{0}\right)+{\frac {1}{2}}a\left(t^{2}-2t_{0}t+{t_{0}}^{2}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <mi>t</mi> <mo>−</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>a</mi> <mrow> <mo>(</mo> <mrow> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>2</mn> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mi>t</mi> <mo>+</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=x_{0}+v_{0}\left(t-t_{0}\right)+{\frac {1}{2}}a\left(t^{2}-2t_{0}t+{t_{0}}^{2}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c0b22deed0345b2938db72e1dbc69388c20a0b2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:42.609ex; height:5.176ex;" alt="{\displaystyle x=x_{0}+v_{0}\left(t-t_{0}\right)+{\frac {1}{2}}a\left(t^{2}-2t_{0}t+{t_{0}}^{2}\right)}"></span> </p><p>Podem simplificar els termes de l'interior dels últims parèntesis, ja que són el resultat d'un producte notable. D'aquesta manera, aconseguim l'equació de la posició per al MRUA: </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 x=x_{0}+v_{0}\left(t-t_{0}\right)+{\frac {1}{2}}a\left(t-t_{0}\right)^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <mi>t</mi> <mo>−</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>a</mi> <msup> <mrow> <mo>(</mo> <mrow> <mi>t</mi> <mo>−</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=x_{0}+v_{0}\left(t-t_{0}\right)+{\frac {1}{2}}a\left(t-t_{0}\right)^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cd4f0643fb96fcdd69502e554d0a0ed25ffd1313" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:34.111ex; height:5.176ex;" alt="{\displaystyle x=x_{0}+v_{0}\left(t-t_{0}\right)+{\frac {1}{2}}a\left(t-t_{0}\right)^{2}}"></span> </p> </div></div> <div style="clear:both;"></div> </div> <div class="mw-heading mw-heading3"><h3 id="Equacions_no_horàries"><span id="Equacions_no_hor.C3.A0ries"></span>Equacions no horàries</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Moviment_rectilini&action=edit&section=5" title="Modifica la secció: Equacions no horàries"><span>modifica</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Tot i que no són necessàries per a la caracterització i l'estudi d'un MRUA, a l'hora de resoldre problemes poden resultar molt útils, ja que no incorporen el temps en la seva expressió (d'aquí el seu nom). S'obtenen a partir d'un sistema format per les dues equacions anteriors. </p><p><i>Equació no horària completa:</i> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v^{2}={v_{0}}^{2}+2a\left(x-x_{0}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>2</mn> <mi>a</mi> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v^{2}={v_{0}}^{2}+2a\left(x-x_{0}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/493a96cf7225cd4f3de23f18bc33987fb63be5ac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.499ex; height:3.176ex;" alt="{\displaystyle v^{2}={v_{0}}^{2}+2a\left(x-x_{0}\right)}"></span></dd></dl> <p><i>Equació no horària simplificada:</i> aconseguida considerant <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v_{0}=0\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mn>0</mn> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v_{0}=0\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/445abf2c39ecde1a2303b13dce4beef7347a1034" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.023ex; height:2.509ex;" alt="{\displaystyle v_{0}=0\ }"></span> i <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_{0}=0\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mn>0</mn> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{0}=0\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4ea6d371505d0e52c79509a705e511dd0a9d95ad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.225ex; height:2.509ex;" alt="{\displaystyle x_{0}=0\ }"></span>. </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v={\sqrt {2ax}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> <mi>a</mi> <mi>x</mi> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v={\sqrt {2ax}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3057b13a38bb9b69af4af21a77990022f1b0a66" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.884ex; height:3.009ex;" alt="{\displaystyle v={\sqrt {2ax}}}"></span></dd></dl> <div class="NavFrame" style="background-color: transparent; width:100%;margin-bottom:0px;float:left; border-radius:4px"> <div class="NavPic" style="display: none;"></div> <div class="NavHead" align="center" style="background-color: transparent;border-radius:4px;">Obtenció a partir de les dues equacions anteriors</div> <div class="NavContent" align="left" style="padding:7px;"><div align="left"> <p>Per obtenir l'equació no horària completa partim d'un sistema format per l'equació de la posició i la de la velocitat el qual, en resoldre'l, ens portarà a l'equació desitjada. </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 \left\{{\begin{aligned}&x=x_{0}+v_{0}\left(t-t_{0}\right)+{\frac {1}{2}}a\left(t-t_{0}\right)^{2}\\&v=v_{0}+a\left(t-t_{0}\right)\\\end{aligned}}\right.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>{</mo> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd /> <mtd> <mi>x</mi> <mo>=</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <mi>t</mi> <mo>−</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>a</mi> <msup> <mrow> <mo>(</mo> <mrow> <mi>t</mi> <mo>−</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi>v</mi> <mo>=</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <mi>a</mi> <mrow> <mo>(</mo> <mrow> <mi>t</mi> <mo>−</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> </mtable> </mrow> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left\{{\begin{aligned}&x=x_{0}+v_{0}\left(t-t_{0}\right)+{\frac {1}{2}}a\left(t-t_{0}\right)^{2}\\&v=v_{0}+a\left(t-t_{0}\right)\\\end{aligned}}\right.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2a0144c7b4d1e1400db12b8556869f1811856574" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.347ex; margin-bottom: -0.325ex; width:36.928ex; height:8.509ex;" alt="{\displaystyle \left\{{\begin{aligned}&x=x_{0}+v_{0}\left(t-t_{0}\right)+{\frac {1}{2}}a\left(t-t_{0}\right)^{2}\\&v=v_{0}+a\left(t-t_{0}\right)\\\end{aligned}}\right.}"></span> </p><p>Movem <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_{0}\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{0}\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d98cdbdb39511d1a53c0c6b72772bc13c55cfd54" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.965ex; height:2.009ex;" alt="{\displaystyle x_{0}\ }"></span> a l'esquerra de la igualtat: </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 \left\{{\begin{aligned}&x-x_{0}=v_{0}\left(t-t_{0}\right)+{\frac {1}{2}}a\left(t-t_{0}\right)^{2}\\&v=v_{0}+a\left(t-t_{0}\right)\\\end{aligned}}\right.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>{</mo> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd /> <mtd> <mi>x</mi> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <mi>t</mi> <mo>−</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>a</mi> <msup> <mrow> <mo>(</mo> <mrow> <mi>t</mi> <mo>−</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi>v</mi> <mo>=</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <mi>a</mi> <mrow> <mo>(</mo> <mrow> <mi>t</mi> <mo>−</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> </mtable> </mrow> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left\{{\begin{aligned}&x-x_{0}=v_{0}\left(t-t_{0}\right)+{\frac {1}{2}}a\left(t-t_{0}\right)^{2}\\&v=v_{0}+a\left(t-t_{0}\right)\\\end{aligned}}\right.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2ddcd7752ecb3c3d11a10122f54027c1f548635d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.347ex; margin-bottom: -0.325ex; width:36.928ex; height:8.509ex;" alt="{\displaystyle \left\{{\begin{aligned}&x-x_{0}=v_{0}\left(t-t_{0}\right)+{\frac {1}{2}}a\left(t-t_{0}\right)^{2}\\&v=v_{0}+a\left(t-t_{0}\right)\\\end{aligned}}\right.}"></span> </p><p>Definim <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 x=x-x_{0}\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mi>x</mi> <mo>=</mo> <mi>x</mi> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta x=x-x_{0}\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e087f11294bff077347f3fcba7dfd112c88e3f3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.499ex; height:2.509ex;" alt="{\displaystyle \Delta x=x-x_{0}\ }"></span> i <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=t-t_{0}\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mi>t</mi> <mo>=</mo> <mi>t</mi> <mo>−</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta t=t-t_{0}\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0389a48c4e8b5d3da0293843b08ed46a253419da" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.028ex; height:2.509ex;" alt="{\displaystyle \Delta t=t-t_{0}\ }"></span>, i ho substituïm en les dues equacions: </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 \left\{{\begin{aligned}&\Delta x=v_{0}\Delta t+{\frac {1}{2}}a\Delta t^{2}\\&v=v_{0}+a\Delta t\\\end{aligned}}\right.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>{</mo> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd /> <mtd> <mi mathvariant="normal">Δ</mi> <mi>x</mi> <mo>=</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mi mathvariant="normal">Δ</mi> <mi>t</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>a</mi> <mi mathvariant="normal">Δ</mi> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi>v</mi> <mo>=</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <mi>a</mi> <mi mathvariant="normal">Δ</mi> <mi>t</mi> </mtd> </mtr> </mtable> </mrow> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left\{{\begin{aligned}&\Delta x=v_{0}\Delta t+{\frac {1}{2}}a\Delta t^{2}\\&v=v_{0}+a\Delta t\\\end{aligned}}\right.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dc280c9615bd551371512ed280c4e1e29f6f9413" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:24.037ex; height:8.176ex;" alt="{\displaystyle \left\{{\begin{aligned}&\Delta x=v_{0}\Delta t+{\frac {1}{2}}a\Delta t^{2}\\&v=v_{0}+a\Delta t\\\end{aligned}}\right.}"></span> </p><p>Aïllem <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> <mtext> </mtext> </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/57a3491cf87244c3f1b810ef17807ad72e833a01" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.356ex; height:2.176ex;" alt="{\displaystyle \Delta t\ }"></span> en la segona equació: </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 \Delta t={\frac {v-v_{0}}{a}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mi>t</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>v</mi> <mo>−</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> <mi>a</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta t={\frac {v-v_{0}}{a}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4976215f003b392ae217521596be982d3f33d71" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:12.86ex; height:5.009ex;" alt="{\displaystyle \Delta t={\frac {v-v_{0}}{a}}}"></span> </p><p>Substituïm <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> <mtext> </mtext> </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/57a3491cf87244c3f1b810ef17807ad72e833a01" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.356ex; height:2.176ex;" alt="{\displaystyle \Delta t\ }"></span> en la primera equació per eliminar el temps: </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 \Delta x=v_{0}{\frac {v-v_{0}}{a}}+{\frac {1}{2}}a\left({\frac {v-v_{0}}{a}}\right)^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mi>x</mi> <mo>=</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>v</mi> <mo>−</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> <mi>a</mi> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>a</mi> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>v</mi> <mo>−</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> <mi>a</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta x=v_{0}{\frac {v-v_{0}}{a}}+{\frac {1}{2}}a\left({\frac {v-v_{0}}{a}}\right)^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8788cc06c91f5d911941452fb3b1ba75318c9188" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:33.062ex; height:6.509ex;" alt="{\displaystyle \Delta x=v_{0}{\frac {v-v_{0}}{a}}+{\frac {1}{2}}a\left({\frac {v-v_{0}}{a}}\right)^{2}}"></span> </p><p>Operem: </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 {\begin{aligned}&\Delta x=v_{0}{\frac {v-v_{0}}{a}}+{\frac {1}{2}}a\left({\frac {v-v_{0}}{a}}\right)^{2}={\frac {v_{0}v-{v_{0}}^{2}}{a}}+{\frac {1}{2}}a{\frac {\left(v-v_{0}\right)^{2}}{a^{2}}}={\frac {v_{0}v-{v_{0}}^{2}}{a}}+{\frac {\left(v-v_{0}\right)^{2}}{2a}}=\\&{\frac {2v_{0}v-2{v_{0}}^{2}}{2a}}+{\frac {v^{2}-2vv_{0}+{v_{0}}^{2}}{2a}}={\frac {2v_{0}v-2{v_{0}}^{2}+v^{2}-2vv_{0}+{v_{0}}^{2}}{2a}}={\frac {v^{2}-{v_{0}}^{2}}{2a}}\\\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd /> <mtd> <mi mathvariant="normal">Δ</mi> <mi>x</mi> <mo>=</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>v</mi> <mo>−</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> <mi>a</mi> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>a</mi> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>v</mi> <mo>−</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> <mi>a</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mi>v</mi> <mo>−</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mi>a</mi> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mrow> <mo>(</mo> <mrow> <mi>v</mi> <mo>−</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mi>v</mi> <mo>−</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mi>a</mi> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mrow> <mo>(</mo> <mrow> <mi>v</mi> <mo>−</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mn>2</mn> <mi>a</mi> </mrow> </mfrac> </mrow> <mo>=</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mi>v</mi> <mo>−</mo> <mn>2</mn> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mn>2</mn> <mi>a</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>2</mn> <mi>v</mi> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mn>2</mn> <mi>a</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mi>v</mi> <mo>−</mo> <mn>2</mn> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>2</mn> <mi>v</mi> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mn>2</mn> <mi>a</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mn>2</mn> <mi>a</mi> </mrow> </mfrac> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}&\Delta x=v_{0}{\frac {v-v_{0}}{a}}+{\frac {1}{2}}a\left({\frac {v-v_{0}}{a}}\right)^{2}={\frac {v_{0}v-{v_{0}}^{2}}{a}}+{\frac {1}{2}}a{\frac {\left(v-v_{0}\right)^{2}}{a^{2}}}={\frac {v_{0}v-{v_{0}}^{2}}{a}}+{\frac {\left(v-v_{0}\right)^{2}}{2a}}=\\&{\frac {2v_{0}v-2{v_{0}}^{2}}{2a}}+{\frac {v^{2}-2vv_{0}+{v_{0}}^{2}}{2a}}={\frac {2v_{0}v-2{v_{0}}^{2}+v^{2}-2vv_{0}+{v_{0}}^{2}}{2a}}={\frac {v^{2}-{v_{0}}^{2}}{2a}}\\\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c85c4aa43567bcbc4ef4aa68a46c523984ba6ac8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.671ex; width:91.516ex; height:12.509ex;" alt="{\displaystyle {\begin{aligned}&\Delta x=v_{0}{\frac {v-v_{0}}{a}}+{\frac {1}{2}}a\left({\frac {v-v_{0}}{a}}\right)^{2}={\frac {v_{0}v-{v_{0}}^{2}}{a}}+{\frac {1}{2}}a{\frac {\left(v-v_{0}\right)^{2}}{a^{2}}}={\frac {v_{0}v-{v_{0}}^{2}}{a}}+{\frac {\left(v-v_{0}\right)^{2}}{2a}}=\\&{\frac {2v_{0}v-2{v_{0}}^{2}}{2a}}+{\frac {v^{2}-2vv_{0}+{v_{0}}^{2}}{2a}}={\frac {2v_{0}v-2{v_{0}}^{2}+v^{2}-2vv_{0}+{v_{0}}^{2}}{2a}}={\frac {v^{2}-{v_{0}}^{2}}{2a}}\\\end{aligned}}}"></span> </p><p>Aïllem <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v^{2}\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v^{2}\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af573c08b15190ca0739a5904481a6097bdbfb27" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.763ex; height:2.676ex;" alt="{\displaystyle v^{2}\ }"></span>: </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 \Delta x={\frac {v^{2}-{v_{0}}^{2}}{2a}}\Rightarrow 2a\Delta x=v^{2}-{v_{0}}^{2}\Rightarrow v^{2}={v_{0}}^{2}+2a\Delta x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mi>x</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mn>2</mn> <mi>a</mi> </mrow> </mfrac> </mrow> <mo stretchy="false">⇒</mo> <mn>2</mn> <mi>a</mi> <mi mathvariant="normal">Δ</mi> <mi>x</mi> <mo>=</mo> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">⇒</mo> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>2</mn> <mi>a</mi> <mi mathvariant="normal">Δ</mi> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta x={\frac {v^{2}-{v_{0}}^{2}}{2a}}\Rightarrow 2a\Delta x=v^{2}-{v_{0}}^{2}\Rightarrow v^{2}={v_{0}}^{2}+2a\Delta x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d2b5aa4ee3998088279c673cf2a48eb19ea8cd16" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:56.716ex; height:5.676ex;" alt="{\displaystyle \Delta x={\frac {v^{2}-{v_{0}}^{2}}{2a}}\Rightarrow 2a\Delta x=v^{2}-{v_{0}}^{2}\Rightarrow v^{2}={v_{0}}^{2}+2a\Delta x}"></span> </p><p>I, finalment, substituïm altra vegada <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-x_{0}=\Delta x\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mi mathvariant="normal">Δ</mi> <mi>x</mi> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x-x_{0}=\Delta x\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3609c6078ed81fa8dfdcf9949633f1d6d9df9871" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.499ex; height:2.509ex;" alt="{\displaystyle x-x_{0}=\Delta x\ }"></span> per obtenir l'equació no horària completa: </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 v^{2}={v_{0}}^{2}+2a\left(x-x_{0}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>2</mn> <mi>a</mi> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v^{2}={v_{0}}^{2}+2a\left(x-x_{0}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/493a96cf7225cd4f3de23f18bc33987fb63be5ac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.499ex; height:3.176ex;" alt="{\displaystyle v^{2}={v_{0}}^{2}+2a\left(x-x_{0}\right)}"></span> </p><p>A partir d'aquí podem considerar <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v_{0}=0\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mn>0</mn> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v_{0}=0\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/445abf2c39ecde1a2303b13dce4beef7347a1034" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.023ex; height:2.509ex;" alt="{\displaystyle v_{0}=0\ }"></span> i <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_{0}=0\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mn>0</mn> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{0}=0\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4ea6d371505d0e52c79509a705e511dd0a9d95ad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.225ex; height:2.509ex;" alt="{\displaystyle x_{0}=0\ }"></span> i operar per aconseguir l'equació no horària simple: </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 v^{2}=0^{2}+2a\left(x-0\right)=2ax\Rightarrow v={\sqrt {2ax}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msup> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>2</mn> <mi>a</mi> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>−</mo> <mn>0</mn> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mn>2</mn> <mi>a</mi> <mi>x</mi> <mo stretchy="false">⇒</mo> <mi>v</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> <mi>a</mi> <mi>x</mi> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v^{2}=0^{2}+2a\left(x-0\right)=2ax\Rightarrow v={\sqrt {2ax}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c8780591a3cc8c76ffbbd7e077cd9d153e682171" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:40.577ex; height:3.176ex;" alt="{\displaystyle v^{2}=0^{2}+2a\left(x-0\right)=2ax\Rightarrow v={\sqrt {2ax}}}"></span> </p> </div></div> <div style="clear:both;"></div> </div> <div class="mw-heading mw-heading3"><h3 id="Cas_particular_de_la_caiguda_lliure">Cas particular de la caiguda lliure</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Moviment_rectilini&action=edit&section=6" title="Modifica la secció: Cas particular de la caiguda lliure"><span>modifica</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Es considera caiguda lliure al model de moviment on un cos cau a causa de l'acció de la <a href="/wiki/Gravetat" title="Gravetat">gravetat</a>. L'acceleració en aquest cas és l'acceleració de la gravetat, en la <a href="/wiki/Superf%C3%ADcie" class="mw-redirect" title="Superfície">superfície</a> de la Terra g=-9.81 m/s². No es té en compte la <a href="/wiki/Fricci%C3%B3" title="Fricció">fricció</a> de l'<a href="/wiki/Aire" title="Aire">aire</a> ni les variacions que sofreix la gravetat en la superfície de la Terra. S'acostuma a agafar com a <a href="/wiki/Sistema_de_refer%C3%A8ncia" title="Sistema de referència">sistema de referència</a> aquell que consisteix en una línia vertical on cap amunt es considera positiu, i cap avall negatiu. D'aquesta manera un cos que baixa té un desplaçament negatiu, i un que puja té un desplaçament positiu. En aquest sistema és important agafar l'acceleració amb valor negatiu. </p><p>Si prenem h com alçada, les fórmules resulten: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h=h_{0}+v_{0}(t-t_{0})+{\frac {1}{2}}g(t-t_{0})^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> <mo>=</mo> <msub> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo>−</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>g</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo>−</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h=h_{0}+v_{0}(t-t_{0})+{\frac {1}{2}}g(t-t_{0})^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/98cefa795177b19a1a5dcf0774f3749d3886d1a1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:33.628ex; height:5.176ex;" alt="{\displaystyle h=h_{0}+v_{0}(t-t_{0})+{\frac {1}{2}}g(t-t_{0})^{2}}"></span></dd></dl> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v=v_{0}+g(t-t_{0})\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> <mo>=</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo>−</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v=v_{0}+g(t-t_{0})\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/18c5c962c7600b822fb7ff492d0e6b37fa4b3dcf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.328ex; height:2.843ex;" alt="{\displaystyle v=v_{0}+g(t-t_{0})\ }"></span></dd></dl> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g=-9.81m/s^{2}\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo>=</mo> <mo>−</mo> <mn>9.81</mn> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g=-9.81m/s^{2}\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c8f5f9e9d8138365b8c9613764ebc10ffd1c85c0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.085ex; height:3.176ex;" alt="{\displaystyle g=-9.81m/s^{2}\ }"></span></dd></dl> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v^{2}={v_{0}}^{2}+2g(h-h_{0})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>2</mn> <mi>g</mi> <mo stretchy="false">(</mo> <mi>h</mi> <mo>−</mo> <msub> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v^{2}={v_{0}}^{2}+2g(h-h_{0})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f4c9b3b5cce6dabf7e07eb95b068b77c7f3918c4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.017ex; height:3.176ex;" alt="{\displaystyle v^{2}={v_{0}}^{2}+2g(h-h_{0})}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Acceleració"><span id="Acceleraci.C3.B3"></span>Acceleració</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Moviment_rectilini&action=edit&section=7" title="Modifica la secció: Acceleració"><span>modifica</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Acceleraci%C3%B3" title="Acceleració">L'acceleració</a> es defineix com la variació de la <a href="/wiki/Velocitat" title="Velocitat">velocitat</a> amb el <a href="/wiki/Temps" title="Temps">temps</a>. L'acceleració instantània és la segona derivada del desplaçament, és a dir, l'acceleració instantània es pot trobar diferenciant la posició respecte al temps dues vegades o diferenciant la velocitat respecte al temps una vegada, que es calcula de la següent manera: </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 a={\frac {dv}{dt}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>v</mi> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a={\frac {dv}{dt}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fdc68e21cfa9e1bbb7f366143c8588aea3025d3a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:7.508ex; height:5.509ex;" alt="{\displaystyle a={\frac {dv}{dt}}}"></span> o el que és el mateix, </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 a={\frac {d^{2}x}{dt^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>x</mi> </mrow> <mrow> <mi>d</mi> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a={\frac {d^{2}x}{dt^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5c8af19f5fc980fc4db2b42c5e896b9877ba29a0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:8.766ex; height:6.009ex;" alt="{\displaystyle a={\frac {d^{2}x}{dt^{2}}}}"></span> </p><p>Les unitats de l'acceleració en el <a href="/wiki/Sistema_Internacional_d%27Unitats" title="Sistema Internacional d'Unitats">SI</a> són <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m\cdot s^{-2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo>⋅</mo> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m\cdot s^{-2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6bdc23df54ac24cae2c991dfab4471da5ff1608d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.143ex; height:2.676ex;" alt="{\displaystyle m\cdot s^{-2}}"></span> o metres per segon al quadrat. </p><p>Si <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{m}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{m}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e579a0eee7d28a69a7e8b666784aeed3baa8d617" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.905ex; height:2.009ex;" alt="{\displaystyle a_{m}}"></span> és l'acceleració mitjana i <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 v=v_{2}-v_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mi>v</mi> <mo>=</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta v=v_{2}-v_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ef32ef576dac0f0264c386686c6520f03bfe93b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.366ex; height:2.509ex;" alt="{\displaystyle \Delta v=v_{2}-v_{1}}"></span> és l'increment de la velocitat durant l'interval de temps <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> llavors </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 a_{m}={\frac {\Delta v}{\Delta t}}={\frac {v_{2}-v_{1}}{t_{2}-t_{1}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">Δ</mi> <mi>v</mi> </mrow> <mrow> <mi mathvariant="normal">Δ</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mrow> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{m}={\frac {\Delta v}{\Delta t}}={\frac {v_{2}-v_{1}}{t_{2}-t_{1}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/03173fd0771985fb81178f6db7c3b34d3a830216" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:21.042ex; height:5.676ex;" alt="{\displaystyle a_{m}={\frac {\Delta v}{\Delta t}}={\frac {v_{2}-v_{1}}{t_{2}-t_{1}}}}"></span> </p> <div class="mw-heading mw-heading2"><h2 id="Referències"><span id="Refer.C3.A8ncies"></span>Referències</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Moviment_rectilini&action=edit&section=8" title="Modifica la secció: Referències"><span>modifica</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="reflist {{#if: | references-column-count references-column-count-{{{col}}}" style="list-style-type: decimal;"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><a href="#cite_ref-1">↑</a></span> <span class="reference-text"><span class="citation book" style="font-style:normal" id="CITEREFWalker2011"><span style="font-variant: small-caps;">Walker</span>, Jearl. «Capítol 3». A: <a rel="nofollow" class="external text" href="http://archive.org/details/hallidayresnickp0000walk_r2u2"><i>Halliday & Resnick principles of physics</i></a>.  Hoboken, N.J. : Wiley, 2011. <span style="font-size:90%; white-space:nowrap;"><a href="/wiki/Especial:Fonts_bibliogr%C3%A0fiques/978-0-470-56837-8" title="Especial:Fonts bibliogràfiques/978-0-470-56837-8">ISBN 978-0-470-56837-8</a></span>.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Halliday+%26+Resnick+principles+of+physics&rft.atitle=Cap%C3%ADtol+3&rft.aulast=Walker&rft.aufirst=Jearl&rft.date=2011&rft.pub=Hoboken%2C+N.J.+%3A+Wiley&rft.isbn=978-0-470-56837-8&rft_id=http%3A%2F%2Farchive.org%2Fdetails%2Fhallidayresnickp0000walk_r2u2"><span style="display: none;"> </span></span></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><a href="#cite_ref-2">↑</a></span> <span class="reference-text"><span class="citation" style="font-style:normal">«<a rel="nofollow" class="external text" href="https://www.enciclopedia.cat/gran-enciclopedia-catalana/lleis-del-moviment-de-newton">lleis del moviment de Newton</a>».  Gran Enciclopèdia Catalana. [Consulta: 22 octubre 2022].</span></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><a href="#cite_ref-3">↑</a></span> <span class="reference-text"><span class="citation book" style="font-style:normal"> <a rel="nofollow" class="external text" href="https://books.google.cat/books?id=XpgVXRsXeXsC&pg=PA81&dq=movimiento+rectilineo+uniforme&hl=ca&sa=X&ved=2ahUKEwj15NLYl_b6AhW5g84BHRh3A0sQ6AF6BAgKEAI#v=onepage&q=movimiento%20rectilineo%20uniforme&f=false"><i>Física mecánica conceptos básicos y problemas</i></a> (en castellà).  ITM, 2008, p. 81. <span style="font-size:90%; white-space:nowrap;"><a href="/wiki/Especial:Fonts_bibliogr%C3%A0fiques/978-958-8351-47-6" title="Especial:Fonts bibliogràfiques/978-958-8351-47-6">ISBN 978-958-8351-47-6</a></span>.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=F%C3%ADsica+mec%C3%A1nica+conceptos+b%C3%A1sicos+y+problemas&rft.date=2008&rft.pub=ITM&rft.pages=81&rft.isbn=978-958-8351-47-6&rft_id=https%3A%2F%2Fbooks.google.cat%2Fbooks%3Fid%3DXpgVXRsXeXsC%26pg%3DPA81%26dq%3Dmovimiento%2Brectilineo%2Buniforme%26hl%3Dca%26sa%3DX%26ved%3D2ahUKEwj15NLYl_b6AhW5g84BHRh3A0sQ6AF6BAgKEAI%23v%3Donepage%26q%3Dmovimiento%2520rectilineo%2520uniforme%26f%3Dfalse"><span style="display: none;"> </span></span></span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><a href="#cite_ref-4">↑</a></span> <span class="reference-text"><span class="citation" style="font-style:normal">«<a rel="nofollow" class="external text" href="https://www.iau.org/publications/proceedings_rules/units/">SI Units</a>».  International Astronomical Union. [Consulta: 23 octubre 2022].</span></span> </li> </ol></div></div> <style data-mw-deduplicate="TemplateStyles:r33663753">.mw-parser-output .side-box{margin:4px 0;box-sizing:border-box;border:1px solid #aaa;font-size:88%;line-height:1.25em;background-color:#f9f9f9;display:flow-root}.mw-parser-output .side-box-abovebelow,.mw-parser-output .side-box-text{padding:0.25em 0.9em}.mw-parser-output .side-box-image{padding:2px 0 2px 0.9em;text-align:center}.mw-parser-output .side-box-imageright{padding:2px 0.9em 2px 0;text-align:center}@media(min-width:500px){.mw-parser-output .side-box-flex{display:flex;align-items:center}.mw-parser-output .side-box-text{flex:1;min-width:0}}@media(min-width:720px){.mw-parser-output .side-box{width:238px}.mw-parser-output .side-box-right{clear:right;float:right;margin-left:1em}.mw-parser-output .side-box-left{margin-right:1em}.mw-parser-output .side-box-center{clear:both;margin:auto}}</style><div class="side-box metadata side-box-right plainlinks"> <div class="side-box-flex"> <div class="side-box-image"><span typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/30px-Commons-logo.svg.png" decoding="async" width="30" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/45px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/59px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /></span></span></div> <div class="side-box-text plainlist">A <span class="plainlinks"><a class="external text" href="https://commons.wikimedia.org/wiki/P%C3%A0gina_principal?uselang=ca">Wikimedia Commons</a></span> hi ha contingut multimèdia relatiu a: <i><b><a href="https://commons.wikimedia.org/wiki/Category:Linear_movement" class="extiw" title="commons:Category:Linear movement">Moviment rectilini</a></b></i></div></div> </div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1&useformat=desktop" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Obtingut de «<a dir="ltr" href="https://ca.wikipedia.org/w/index.php?title=Moviment_rectilini&oldid=34124832">https://ca.wikipedia.org/w/index.php?title=Moviment_rectilini&oldid=34124832</a>»</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Especial:Categorias" title="Especial:Categorias">Categoria</a>: <ul><li><a href="/wiki/Categoria:Cinem%C3%A0tica" title="Categoria:Cinemàtica">Cinemàtica</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Categoria oculta: <ul><li><a href="/wiki/Categoria:P%C3%A0gines_amb_enlla%C3%A7_commonscat_des_de_Wikidata" title="Categoria:Pàgines amb enllaç commonscat des de Wikidata">Pàgines amb enllaç commonscat des de Wikidata</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> La pàgina va ser modificada per darrera vegada el 18 oct 2024 a les 15:56.</li> <li id="footer-info-copyright">El text està disponible sota la <a href="/wiki/Viquip%C3%A8dia:Text_de_la_llic%C3%A8ncia_de_Creative_Commons_Reconeixement-Compartir_Igual_4.0_No_adaptada" title="Viquipèdia:Text de la llicència de Creative Commons Reconeixement-Compartir Igual 4.0 No adaptada"> Llicència de Creative Commons Reconeixement i Compartir-Igual</a>; es poden aplicar termes addicionals. 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Moviment rectilini - Viquipèdia, l'enciclopèdia lliure