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href="#Outras_equações"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Outras equações</span> </div> </a> <ul id="toc-Outras_equações-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Resolução_numérica_de_equações_diferenciais" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Resolução_numérica_de_equações_diferenciais"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Resolução numérica de equações diferenciais</span> </div> </a> <ul id="toc-Resolução_numérica_de_equações_diferenciais-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Ver_também" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Ver_também"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Ver também</span> </div> </a> <ul id="toc-Ver_também-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Referências" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Referências"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>Referências</span> </div> </a> <ul id="toc-Referências-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Conteúdo" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Alternar o índice" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Alternar o índice</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Trabalho (física)</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="Ir para um artigo noutra língua. Disponível em 102 línguas" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-102" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">102 línguas</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-af mw-list-item"><a href="https://af.wikipedia.org/wiki/Arbeid" title="Arbeid — africanês" lang="af" hreflang="af" data-title="Arbeid" data-language-autonym="Afrikaans" data-language-local-name="africanês" class="interlanguage-link-target"><span>Afrikaans</span></a></li><li class="interlanguage-link interwiki-als mw-list-item"><a href="https://als.wikipedia.org/wiki/Arbeit_(Physik)" title="Arbeit (Physik) — alemão suíço" lang="gsw" hreflang="gsw" data-title="Arbeit (Physik)" data-language-autonym="Alemannisch" data-language-local-name="alemão suíço" class="interlanguage-link-target"><span>Alemannisch</span></a></li><li class="interlanguage-link interwiki-am mw-list-item"><a href="https://am.wikipedia.org/wiki/%E1%88%A5%E1%88%AB" title="ሥራ — amárico" lang="am" hreflang="am" data-title="ሥራ" data-language-autonym="አማርኛ" data-language-local-name="amárico" class="interlanguage-link-target"><span>አማርኛ</span></a></li><li class="interlanguage-link interwiki-an mw-list-item"><a href="https://an.wikipedia.org/wiki/Treballo_mecanico" title="Treballo mecanico — aragonês" lang="an" hreflang="an" data-title="Treballo mecanico" data-language-autonym="Aragonés" data-language-local-name="aragonês" class="interlanguage-link-target"><span>Aragonés</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%B4%D8%BA%D9%84_(%D9%81%D9%8A%D8%B2%D9%8A%D8%A7%D8%A1)" title="شغل (فيزياء) — árabe" lang="ar" hreflang="ar" data-title="شغل (فيزياء)" data-language-autonym="العربية" data-language-local-name="árabe" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-as mw-list-item"><a href="https://as.wikipedia.org/wiki/%E0%A6%95%E0%A6%BE%E0%A7%B0%E0%A7%8D%E0%A6%AF%E0%A7%8D%E0%A6%AF" title="কাৰ্য্য — assamês" lang="as" hreflang="as" data-title="কাৰ্য্য" data-language-autonym="অসমীয়া" data-language-local-name="assamês" class="interlanguage-link-target"><span>অসমীয়া</span></a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/Trabayu_(f%C3%ADsica)" title="Trabayu (física) — asturiano" lang="ast" hreflang="ast" data-title="Trabayu (física)" data-language-autonym="Asturianu" data-language-local-name="asturiano" class="interlanguage-link-target"><span>Asturianu</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/Mexaniki_i%C5%9F" title="Mexaniki iş — azerbaijano" lang="az" hreflang="az" data-title="Mexaniki iş" data-language-autonym="Azərbaycanca" data-language-local-name="azerbaijano" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%9C%D0%B5%D1%85%D0%B0%D0%BD%D1%96%D1%87%D0%BD%D0%B0%D1%8F_%D1%80%D0%B0%D0%B1%D0%BE%D1%82%D0%B0" title="Механічная работа — bielorrusso" lang="be" hreflang="be" data-title="Механічная работа" data-language-autonym="Беларуская" data-language-local-name="bielorrusso" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-be-x-old mw-list-item"><a href="https://be-tarask.wikipedia.org/wiki/%D0%9C%D1%8D%D1%85%D0%B0%D0%BD%D1%96%D1%87%D0%BD%D0%B0%D1%8F_%D0%BF%D1%80%D0%B0%D1%86%D0%B0" title="Мэханічная праца — Belarusian (Taraškievica orthography)" lang="be-tarask" hreflang="be-tarask" data-title="Мэханічная праца" data-language-autonym="Беларуская (тарашкевіца)" data-language-local-name="Belarusian (Taraškievica orthography)" class="interlanguage-link-target"><span>Беларуская (тарашкевіца)</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%9C%D0%B5%D1%85%D0%B0%D0%BD%D0%B8%D1%87%D0%BD%D0%B0_%D1%80%D0%B0%D0%B1%D0%BE%D1%82%D0%B0" title="Механична работа — búlgaro" lang="bg" hreflang="bg" data-title="Механична работа" data-language-autonym="Български" data-language-local-name="búlgaro" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%95%E0%A6%BE%E0%A6%9C_(%E0%A6%AA%E0%A6%A6%E0%A6%BE%E0%A6%B0%E0%A7%8D%E0%A6%A5%E0%A6%AC%E0%A6%BF%E0%A6%9C%E0%A7%8D%E0%A6%9E%E0%A6%BE%E0%A6%A8)" title="কাজ (পদার্থবিজ্ঞান) — bengalês" lang="bn" hreflang="bn" data-title="কাজ (পদার্থবিজ্ঞান)" data-language-autonym="বাংলা" data-language-local-name="bengalês" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-br mw-list-item"><a href="https://br.wikipedia.org/wiki/Labour_un_nerzh" title="Labour un nerzh — bretão" lang="br" hreflang="br" data-title="Labour un nerzh" data-language-autonym="Brezhoneg" data-language-local-name="bretão" class="interlanguage-link-target"><span>Brezhoneg</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Rad_(fizika)" title="Rad (fizika) — bósnio" lang="bs" hreflang="bs" data-title="Rad (fizika)" data-language-autonym="Bosanski" data-language-local-name="bósnio" class="interlanguage-link-target"><span>Bosanski</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Treball_(f%C3%ADsica)" title="Treball (física) — catalão" lang="ca" hreflang="ca" data-title="Treball (física)" data-language-autonym="Català" data-language-local-name="catalão" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%DA%A9%D8%A7%D8%B1_(%D9%81%DB%8C%D8%B2%DB%8C%DA%A9)" title="کار (فیزیک) — curdo central" lang="ckb" hreflang="ckb" data-title="کار (فیزیک)" data-language-autonym="کوردی" data-language-local-name="curdo 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/Pr%C3%A1ce_(fyzika)" title="Práce (fyzika) — checo" lang="cs" hreflang="cs" data-title="Práce (fyzika)" data-language-autonym="Čeština" data-language-local-name="checo" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-csb mw-list-item"><a href="https://csb.wikipedia.org/wiki/Rob%C3%B2ta_(fizyka)" title="Robòta (fizyka) — kashubian" lang="csb" hreflang="csb" data-title="Robòta (fizyka)" data-language-autonym="Kaszëbsczi" data-language-local-name="kashubian" class="interlanguage-link-target"><span>Kaszëbsczi</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%9C%D0%B5%D1%85%D0%B0%D0%BD%D0%B8%D0%BA%C4%83%D0%BB%D0%BB%D0%B0_%C4%95%C3%A7" title="Механикăлла ĕç — chuvash" lang="cv" hreflang="cv" data-title="Механикăлла ĕç" data-language-autonym="Чӑвашла" data-language-local-name="chuvash" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Arbejde_(fysik)" title="Arbejde (fysik) — dinamarquês" lang="da" hreflang="da" data-title="Arbejde (fysik)" data-language-autonym="Dansk" data-language-local-name="dinamarquês" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Arbeit_(Physik)" title="Arbeit (Physik) — alemão" lang="de" hreflang="de" data-title="Arbeit (Physik)" data-language-autonym="Deutsch" data-language-local-name="alemão" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%88%CF%81%CE%B3%CE%BF_(%CF%86%CF%85%CF%83%CE%B9%CE%BA%CE%AE)" title="Έργο (φυσική) — grego" lang="el" hreflang="el" data-title="Έργο (φυσική)" data-language-autonym="Ελληνικά" data-language-local-name="grego" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Work_(physics)" title="Work (physics) — inglês" lang="en" hreflang="en" data-title="Work (physics)" data-language-autonym="English" data-language-local-name="inglês" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Laboro_(fiziko)" title="Laboro (fiziko) — esperanto" lang="eo" hreflang="eo" data-title="Laboro (fiziko)" data-language-autonym="Esperanto" data-language-local-name="esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Trabajo_(f%C3%ADsica)" title="Trabajo (física) — espanhol" lang="es" hreflang="es" data-title="Trabajo (física)" data-language-autonym="Español" data-language-local-name="espanhol" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Mehaaniline_t%C3%B6%C3%B6" title="Mehaaniline töö — estónio" lang="et" hreflang="et" data-title="Mehaaniline töö" data-language-autonym="Eesti" data-language-local-name="estónio" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Lan_(fisika)" title="Lan (fisika) — basco" lang="eu" hreflang="eu" data-title="Lan (fisika)" data-language-autonym="Euskara" data-language-local-name="basco" 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/%DA%A9%D8%A7%D8%B1_(%D9%81%DB%8C%D8%B2%DB%8C%DA%A9)" title="کار (فیزیک) — persa" lang="fa" hreflang="fa" data-title="کار (فیزیک)" 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/Ty%C3%B6_(fysiikka)" title="Työ (fysiikka) — finlandês" lang="fi" hreflang="fi" data-title="Työ (fysiikka)" data-language-autonym="Suomi" data-language-local-name="finlandês" 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/Travail_d%27une_force" title="Travail d'une force — francês" lang="fr" hreflang="fr" data-title="Travail d'une force" data-language-autonym="Français" data-language-local-name="francês" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-frr mw-list-item"><a href="https://frr.wikipedia.org/wiki/Werk" title="Werk — frísio setentrional" lang="frr" hreflang="frr" data-title="Werk" data-language-autonym="Nordfriisk" data-language-local-name="frísio setentrional" class="interlanguage-link-target"><span>Nordfriisk</span></a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/Obair_(fisic)" title="Obair (fisic) — irlandês" lang="ga" hreflang="ga" data-title="Obair (fisic)" data-language-autonym="Gaeilge" data-language-local-name="irlandês" class="interlanguage-link-target"><span>Gaeilge</span></a></li><li class="interlanguage-link interwiki-gd mw-list-item"><a href="https://gd.wikipedia.org/wiki/Obair_(fiosaigs)" title="Obair (fiosaigs) — gaélico escocês" lang="gd" hreflang="gd" data-title="Obair (fiosaigs)" data-language-autonym="Gàidhlig" data-language-local-name="gaélico escocês" class="interlanguage-link-target"><span>Gàidhlig</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Traballo_(f%C3%ADsica)" title="Traballo (física) — galego" lang="gl" hreflang="gl" data-title="Traballo (física)" data-language-autonym="Galego" data-language-local-name="galego" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-gv mw-list-item"><a href="https://gv.wikipedia.org/wiki/Obbyr_(fishig)" title="Obbyr (fishig) — manx" lang="gv" hreflang="gv" data-title="Obbyr (fishig)" data-language-autonym="Gaelg" data-language-local-name="manx" class="interlanguage-link-target"><span>Gaelg</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%A2%D7%91%D7%95%D7%93%D7%94_(%D7%A4%D7%99%D7%96%D7%99%D7%A7%D7%94)" title="עבודה (פיזיקה) — hebraico" lang="he" hreflang="he" data-title="עבודה (פיזיקה)" data-language-autonym="עברית" data-language-local-name="hebraico" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%95%E0%A4%BE%E0%A4%B0%E0%A5%8D%E0%A4%AF_(%E0%A4%AD%E0%A5%8C%E0%A4%A4%E0%A4%BF%E0%A4%95%E0%A5%80)" title="कार्य (भौतिकी) — hindi" lang="hi" hreflang="hi" data-title="कार्य (भौतिकी)" 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/Rad_(fizika)" title="Rad (fizika) — croata" lang="hr" hreflang="hr" data-title="Rad (fizika)" data-language-autonym="Hrvatski" data-language-local-name="croata" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-ht mw-list-item"><a href="https://ht.wikipedia.org/wiki/Travay_(fizik)" title="Travay (fizik) — haitiano" lang="ht" hreflang="ht" data-title="Travay (fizik)" data-language-autonym="Kreyòl ayisyen" data-language-local-name="haitiano" class="interlanguage-link-target"><span>Kreyòl ayisyen</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Mechanikai_munka" title="Mechanikai munka — húngaro" lang="hu" hreflang="hu" data-title="Mechanikai munka" data-language-autonym="Magyar" data-language-local-name="húngaro" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D4%B1%D5%B7%D5%AD%D5%A1%D5%BF%D5%A1%D5%B6%D6%84_(%D6%86%D5%AB%D5%A6%D5%AB%D5%AF%D5%A1)" title="Աշխատանք (ֆիզիկա) — arménio" lang="hy" hreflang="hy" data-title="Աշխատանք (ֆիզիկա)" data-language-autonym="Հայերեն" data-language-local-name="arménio" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Usaha_(fisika)" title="Usaha (fisika) — indonésio" lang="id" hreflang="id" data-title="Usaha (fisika)" data-language-autonym="Bahasa Indonesia" data-language-local-name="indonésio" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/Vinna_(e%C3%B0lisfr%C3%A6%C3%B0i)" title="Vinna (eðlisfræði) — islandês" lang="is" hreflang="is" data-title="Vinna (eðlisfræði)" data-language-autonym="Íslenska" data-language-local-name="islandês" class="interlanguage-link-target"><span>Íslenska</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Lavoro_(fisica)" title="Lavoro (fisica) — italiano" lang="it" hreflang="it" data-title="Lavoro (fisica)" data-language-autonym="Italiano" data-language-local-name="italiano" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E4%BB%95%E4%BA%8B_(%E7%89%A9%E7%90%86%E5%AD%A6)" title="仕事 (物理学) — japonês" lang="ja" hreflang="ja" data-title="仕事 (物理学)" data-language-autonym="日本語" data-language-local-name="japonês" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/%E1%83%9B%E1%83%A3%E1%83%A8%E1%83%90%E1%83%9D%E1%83%91%E1%83%90" title="მუშაობა — georgiano" lang="ka" hreflang="ka" data-title="მუშაობა" data-language-autonym="ქართული" data-language-local-name="georgiano" class="interlanguage-link-target"><span>ქართული</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%9C%D0%B5%D1%85%D0%B0%D0%BD%D0%B8%D0%BA%D0%B0%D0%BB%D1%8B%D2%9B_%D0%B6%D2%B1%D0%BC%D1%8B%D1%81" title="Механикалық жұмыс — cazaque" lang="kk" hreflang="kk" data-title="Механикалық жұмыс" data-language-autonym="Қазақша" data-language-local-name="cazaque" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-km mw-list-item"><a href="https://km.wikipedia.org/wiki/%E1%9E%80%E1%9E%98%E1%9F%92%E1%9E%98%E1%9E%93%E1%9F%92%E1%9E%8F_(%E1%9E%9A%E1%9E%BC%E1%9E%94%E1%9E%9C%E1%9E%B7%E1%9E%91%E1%9F%92%E1%9E%99%E1%9E%B6)" title="កម្មន្ត (រូបវិទ្យា) — khmer" lang="km" hreflang="km" data-title="កម្មន្ត (រូបវិទ្យា)" data-language-autonym="ភាសាខ្មែរ" data-language-local-name="khmer" class="interlanguage-link-target"><span>ភាសាខ្មែរ</span></a></li><li class="interlanguage-link interwiki-kn mw-list-item"><a href="https://kn.wikipedia.org/wiki/%E0%B2%95%E0%B3%86%E0%B2%B2%E0%B2%B8" title="ಕೆಲಸ — canarim" lang="kn" hreflang="kn" data-title="ಕೆಲಸ" data-language-autonym="ಕನ್ನಡ" data-language-local-name="canarim" class="interlanguage-link-target"><span>ಕನ್ನಡ</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%9D%BC_(%EB%AC%BC%EB%A6%AC%ED%95%99)" title="일 (물리학) — coreano" lang="ko" hreflang="ko" data-title="일 (물리학)" data-language-autonym="한국어" data-language-local-name="coreano" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-ku mw-list-item"><a href="https://ku.wikipedia.org/wiki/Kar_(fiz%C3%AEk)" title="Kar (fizîk) — curdo" lang="ku" hreflang="ku" data-title="Kar (fizîk)" data-language-autonym="Kurdî" data-language-local-name="curdo" class="interlanguage-link-target"><span>Kurdî</span></a></li><li class="interlanguage-link interwiki-ky mw-list-item"><a href="https://ky.wikipedia.org/wiki/%D0%9C%D0%B5%D1%85%D0%B0%D0%BD%D0%B8%D0%BA%D0%B0%D0%BB%D1%8B%D0%BA_%D0%B6%D1%83%D0%BC%D1%83%D1%88" title="Механикалык жумуш — quirguiz" lang="ky" hreflang="ky" data-title="Механикалык жумуш" data-language-autonym="Кыргызча" data-language-local-name="quirguiz" class="interlanguage-link-target"><span>Кыргызча</span></a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Labor_(physica)" title="Labor (physica) — latim" lang="la" hreflang="la" data-title="Labor (physica)" data-language-autonym="Latina" data-language-local-name="latim" class="interlanguage-link-target"><span>Latina</span></a></li><li class="interlanguage-link interwiki-lfn mw-list-item"><a href="https://lfn.wikipedia.org/wiki/Labora_(fisica)" title="Labora (fisica) — Lingua Franca Nova" lang="lfn" hreflang="lfn" data-title="Labora (fisica)" data-language-autonym="Lingua Franca Nova" data-language-local-name="Lingua Franca Nova" class="interlanguage-link-target"><span>Lingua Franca Nova</span></a></li><li class="interlanguage-link interwiki-lmo mw-list-item"><a href="https://lmo.wikipedia.org/wiki/La%C3%B4_(fisica)" title="Laô (fisica) — lombardo" lang="lmo" hreflang="lmo" data-title="Laô (fisica)" data-language-autonym="Lombard" data-language-local-name="lombardo" class="interlanguage-link-target"><span>Lombard</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Mechaninis_darbas" title="Mechaninis darbas — lituano" lang="lt" hreflang="lt" data-title="Mechaninis darbas" data-language-autonym="Lietuvių" data-language-local-name="lituano" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Darbs_(fizika)" title="Darbs (fizika) — letão" lang="lv" hreflang="lv" data-title="Darbs (fizika)" data-language-autonym="Latviešu" data-language-local-name="letão" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%A0%D0%B0%D0%B1%D0%BE%D1%82%D0%B0_(%D1%84%D0%B8%D0%B7%D0%B8%D0%BA%D0%B0)" title="Работа (физика) — macedónio" lang="mk" hreflang="mk" data-title="Работа (физика)" data-language-autonym="Македонски" data-language-local-name="macedónio" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%AA%E0%B5%8D%E0%B4%B0%E0%B4%B5%E0%B5%83%E0%B4%A4%E0%B5%8D%E0%B4%A4%E0%B4%BF" title="പ്രവൃത്തി — malaiala" lang="ml" hreflang="ml" data-title="പ്രവൃത്തി" data-language-autonym="മലയാളം" data-language-local-name="malaiala" class="interlanguage-link-target"><span>മലയാളം</span></a></li><li class="interlanguage-link interwiki-mn mw-list-item"><a href="https://mn.wikipedia.org/wiki/%D0%90%D0%B6%D0%B8%D0%BB_(%D1%84%D0%B8%D0%B7%D0%B8%D0%BA)" title="Ажил (физик) — mongol" lang="mn" hreflang="mn" data-title="Ажил (физик)" data-language-autonym="Монгол" data-language-local-name="mongol" class="interlanguage-link-target"><span>Монгол</span></a></li><li class="interlanguage-link interwiki-mr mw-list-item"><a href="https://mr.wikipedia.org/wiki/%E0%A4%95%E0%A4%BE%E0%A4%B0%E0%A5%8D%E0%A4%AF_(%E0%A4%AD%E0%A5%8C%E0%A4%A4%E0%A4%BF%E0%A4%95%E0%A4%B6%E0%A4%BE%E0%A4%B8%E0%A5%8D%E0%A4%A4%E0%A5%8D%E0%A4%B0)" title="कार्य (भौतिकशास्त्र) — marata" lang="mr" hreflang="mr" data-title="कार्य (भौतिकशास्त्र)" data-language-autonym="मराठी" data-language-local-name="marata" class="interlanguage-link-target"><span>मराठी</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Kerja_(fizik)" title="Kerja (fizik) — malaio" lang="ms" hreflang="ms" data-title="Kerja (fizik)" data-language-autonym="Bahasa Melayu" data-language-local-name="malaio" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-my mw-list-item"><a href="https://my.wikipedia.org/wiki/%E1%80%A1%E1%80%9C%E1%80%AF%E1%80%95%E1%80%BA_(%E1%80%9B%E1%80%B0%E1%80%95%E1%80%97%E1%80%B1%E1%80%92)" title="အလုပ် (ရူပဗေဒ) — birmanês" lang="my" hreflang="my" data-title="အလုပ် (ရူပဗေဒ)" data-language-autonym="မြန်မာဘာသာ" data-language-local-name="birmanês" class="interlanguage-link-target"><span>မြန်မာဘာသာ</span></a></li><li class="interlanguage-link interwiki-ne mw-list-item"><a href="https://ne.wikipedia.org/wiki/%E0%A4%95%E0%A4%BE%E0%A4%B0%E0%A5%8D%E0%A4%AF_(%E0%A4%AD%E0%A5%8C%E0%A4%A4%E0%A4%BF%E0%A4%95%E0%A4%B6%E0%A4%BE%E0%A4%B8%E0%A5%8D%E0%A4%A4%E0%A5%8D%E0%A4%B0)" title="कार्य (भौतिकशास्त्र) — nepalês" lang="ne" hreflang="ne" data-title="कार्य (भौतिकशास्त्र)" data-language-autonym="नेपाली" data-language-local-name="nepalês" class="interlanguage-link-target"><span>नेपाली</span></a></li><li class="interlanguage-link interwiki-new mw-list-item"><a href="https://new.wikipedia.org/wiki/%E0%A4%B5%E0%A5%87%E0%A4%B2%E0%A5%88_(%E0%A4%B8%E0%A4%A8%E0%A5%8D_%E0%A5%A7%E0%A5%AF%E0%A5%AF%E0%A5%AE%E0%A4%AF%E0%A4%BE_%E0%A4%B8%E0%A4%82%E0%A4%95%E0%A4%BF%E0%A4%AA%E0%A4%BE)" title="वेलै (सन् १९९८या संकिपा) — newari" lang="new" hreflang="new" data-title="वेलै (सन् १९९८या संकिपा)" data-language-autonym="नेपाल भाषा" data-language-local-name="newari" class="interlanguage-link-target"><span>नेपाल भाषा</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Arbeid_(natuurkunde)" title="Arbeid (natuurkunde) — neerlandês" lang="nl" hreflang="nl" data-title="Arbeid (natuurkunde)" data-language-autonym="Nederlands" data-language-local-name="neerlandês" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Arbeid_i_fysikk" title="Arbeid i fysikk — norueguês nynorsk" lang="nn" hreflang="nn" data-title="Arbeid i fysikk" data-language-autonym="Norsk nynorsk" data-language-local-name="norueguês nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Arbeid_(fysikk)" title="Arbeid (fysikk) — norueguês bokmål" lang="nb" hreflang="nb" data-title="Arbeid (fysikk)" data-language-autonym="Norsk bokmål" data-language-local-name="norueguês bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-oc mw-list-item"><a href="https://oc.wikipedia.org/wiki/Trabalh_(fisica)" title="Trabalh (fisica) — occitano" lang="oc" hreflang="oc" data-title="Trabalh (fisica)" data-language-autonym="Occitan" data-language-local-name="occitano" class="interlanguage-link-target"><span>Occitan</span></a></li><li class="interlanguage-link interwiki-om mw-list-item"><a href="https://om.wikipedia.org/wiki/Dalagaa" title="Dalagaa — oromo" lang="om" hreflang="om" data-title="Dalagaa" data-language-autonym="Oromoo" data-language-local-name="oromo" class="interlanguage-link-target"><span>Oromoo</span></a></li><li class="interlanguage-link interwiki-pa mw-list-item"><a href="https://pa.wikipedia.org/wiki/%E0%A8%95%E0%A9%B0%E0%A8%AE_(%E0%A8%AD%E0%A9%8C%E0%A8%A4%E0%A8%BF%E0%A8%95_%E0%A8%B5%E0%A8%BF%E0%A8%97%E0%A8%BF%E0%A8%86%E0%A8%A8)" title="ਕੰਮ (ਭੌਤਿਕ ਵਿਗਿਆਨ) — panjabi" lang="pa" hreflang="pa" data-title="ਕੰਮ (ਭੌਤਿਕ ਵਿਗਿਆਨ)" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="panjabi" class="interlanguage-link-target"><span>ਪੰਜਾਬੀ</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Praca_(fizyka)" title="Praca (fizyka) — polaco" lang="pl" hreflang="pl" data-title="Praca (fizyka)" data-language-autonym="Polski" data-language-local-name="polaco" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pms mw-list-item"><a href="https://pms.wikipedia.org/wiki/Travaj" title="Travaj — Piedmontese" lang="pms" hreflang="pms" data-title="Travaj" data-language-autonym="Piemontèis" data-language-local-name="Piedmontese" class="interlanguage-link-target"><span>Piemontèis</span></a></li><li class="interlanguage-link interwiki-qu mw-list-item"><a href="https://qu.wikipedia.org/wiki/Ruray" title="Ruray — quíchua" lang="qu" hreflang="qu" data-title="Ruray" data-language-autonym="Runa Simi" data-language-local-name="quíchua" class="interlanguage-link-target"><span>Runa Simi</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Lucru_mecanic" title="Lucru mecanic — romeno" lang="ro" hreflang="ro" data-title="Lucru mecanic" data-language-autonym="Română" data-language-local-name="romeno" 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%9C%D0%B5%D1%85%D0%B0%D0%BD%D0%B8%D1%87%D0%B5%D1%81%D0%BA%D0%B0%D1%8F_%D1%80%D0%B0%D0%B1%D0%BE%D1%82%D0%B0" title="Механическая работа — russo" lang="ru" hreflang="ru" data-title="Механическая работа" data-language-autonym="Русский" data-language-local-name="russo" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sat mw-list-item"><a href="https://sat.wikipedia.org/wiki/%E1%B1%A0%E1%B1%9F%E1%B1%B9%E1%B1%A2%E1%B1%A4" title="ᱠᱟᱹᱢᱤ — santali" lang="sat" hreflang="sat" data-title="ᱠᱟᱹᱢᱤ" data-language-autonym="ᱥᱟᱱᱛᱟᱲᱤ" data-language-local-name="santali" class="interlanguage-link-target"><span>ᱥᱟᱱᱛᱟᱲᱤ</span></a></li><li class="interlanguage-link interwiki-sd mw-list-item"><a href="https://sd.wikipedia.org/wiki/%DA%AA%D9%85_(%D8%B7%D8%A8%D8%B9%D9%8A%D8%A7%D8%AA)" title="ڪم (طبعيات) — sindi" lang="sd" hreflang="sd" data-title="ڪم (طبعيات)" data-language-autonym="سنڌي" data-language-local-name="sindi" class="interlanguage-link-target"><span>سنڌي</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Rad_(fizika)" title="Rad (fizika) — servo-croata" lang="sh" hreflang="sh" data-title="Rad (fizika)" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="servo-croata" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-si mw-list-item"><a href="https://si.wikipedia.org/wiki/%E0%B6%9A%E0%B7%8F%E0%B6%BB%E0%B7%8A%E0%B6%BA%E0%B6%BA_(%E0%B6%B7%E0%B7%9E%E0%B6%AD%E0%B7%92%E0%B6%9A_%E0%B7%80%E0%B7%92%E0%B6%AF%E0%B7%8A%E2%80%8D%E0%B6%BA%E0%B7%8F%E0%B7%80)" title="කාර්යය (භෞතික විද්‍යාව) — cingalês" lang="si" hreflang="si" data-title="කාර්යය (භෞතික විද්‍යාව)" data-language-autonym="සිංහල" data-language-local-name="cingalês" class="interlanguage-link-target"><span>සිංහල</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Work_(physics)" title="Work (physics) — Simple English" lang="en-simple" hreflang="en-simple" data-title="Work (physics)" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Pr%C3%A1ca_(fyzika)" title="Práca (fyzika) — eslovaco" lang="sk" hreflang="sk" data-title="Práca (fyzika)" data-language-autonym="Slovenčina" data-language-local-name="eslovaco" 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/Delo_(fizika)" title="Delo (fizika) — esloveno" lang="sl" hreflang="sl" data-title="Delo (fizika)" data-language-autonym="Slovenščina" data-language-local-name="esloveno" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sn mw-list-item"><a href="https://sn.wikipedia.org/wiki/Basa_(fundoyetsimba)" title="Basa (fundoyetsimba) — shona" lang="sn" hreflang="sn" data-title="Basa (fundoyetsimba)" data-language-autonym="ChiShona" data-language-local-name="shona" class="interlanguage-link-target"><span>ChiShona</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Puna_(fizik%C3%AB)" title="Puna (fizikë) — albanês" lang="sq" hreflang="sq" data-title="Puna (fizikë)" data-language-autonym="Shqip" data-language-local-name="albanês" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%9C%D0%B5%D1%85%D0%B0%D0%BD%D0%B8%D1%87%D0%BA%D0%B8_%D1%80%D0%B0%D0%B4" title="Механички рад — sérvio" lang="sr" hreflang="sr" data-title="Механички рад" data-language-autonym="Српски / srpski" data-language-local-name="sérvio" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-su mw-list-item"><a href="https://su.wikipedia.org/wiki/Usaha_m%C3%A9kanik" title="Usaha mékanik — sundanês" lang="su" hreflang="su" data-title="Usaha mékanik" data-language-autonym="Sunda" data-language-local-name="sundanês" class="interlanguage-link-target"><span>Sunda</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Arbete_(fysik)" title="Arbete (fysik) — sueco" lang="sv" hreflang="sv" data-title="Arbete (fysik)" data-language-autonym="Svenska" data-language-local-name="sueco" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%B5%E0%AF%87%E0%AE%B2%E0%AF%88_(%E0%AE%87%E0%AE%AF%E0%AE%B1%E0%AF%8D%E0%AE%AA%E0%AE%BF%E0%AE%AF%E0%AE%B2%E0%AF%8D)" title="வேலை (இயற்பியல்) — tâmil" lang="ta" hreflang="ta" data-title="வேலை (இயற்பியல்)" data-language-autonym="தமிழ்" data-language-local-name="tâmil" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-te mw-list-item"><a href="https://te.wikipedia.org/wiki/%E0%B0%AA%E0%B0%A8%E0%B0%BF" title="పని — telugu" lang="te" hreflang="te" data-title="పని" data-language-autonym="తెలుగు" data-language-local-name="telugu" class="interlanguage-link-target"><span>తెలుగు</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%87%E0%B8%B2%E0%B8%99_(%E0%B8%9F%E0%B8%B4%E0%B8%AA%E0%B8%B4%E0%B8%81%E0%B8%AA%E0%B9%8C)" title="งาน (ฟิสิกส์) — tailandês" lang="th" hreflang="th" data-title="งาน (ฟิสิกส์)" data-language-autonym="ไทย" data-language-local-name="tailandês" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/%C4%B0%C5%9F_(fizik)" title="İş (fizik) — turco" lang="tr" hreflang="tr" data-title="İş (fizik)" data-language-autonym="Türkçe" data-language-local-name="turco" 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%A0%D0%BE%D0%B1%D0%BE%D1%82%D0%B0_(%D1%84%D1%96%D0%B7%D0%B8%D0%BA%D0%B0)" title="Робота (фізика) — ucraniano" lang="uk" hreflang="uk" data-title="Робота (фізика)" data-language-autonym="Українська" data-language-local-name="ucraniano" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-ur mw-list-item"><a href="https://ur.wikipedia.org/wiki/%DA%A9%D8%A7%D9%85_(%D8%B7%D8%A8%DB%8C%D8%B9%DB%8C%D8%A7%D8%AA)" title="کام (طبیعیات) — urdu" lang="ur" hreflang="ur" data-title="کام (طبیعیات)" data-language-autonym="اردو" data-language-local-name="urdu" class="interlanguage-link-target"><span>اردو</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/C%C3%B4ng_(v%E1%BA%ADt_l%C3%BD_h%E1%BB%8Dc)" title="Công (vật lý học) — vietnamita" lang="vi" hreflang="vi" data-title="Công (vật lý học)" 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-wo mw-list-item"><a href="https://wo.wikipedia.org/wiki/Ligg%C3%A9ey_(j%C3%ABmm)" title="Liggéey (jëmm) — uólofe" lang="wo" hreflang="wo" data-title="Liggéey (jëmm)" data-language-autonym="Wolof" data-language-local-name="uólofe" class="interlanguage-link-target"><span>Wolof</span></a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E5%8A%9F" title="功 — wu" lang="wuu" hreflang="wuu" data-title="功" data-language-autonym="吴语" data-language-local-name="wu" class="interlanguage-link-target"><span>吴语</span></a></li><li class="interlanguage-link interwiki-yi mw-list-item"><a href="https://yi.wikipedia.org/wiki/%D7%90%D7%A8%D7%91%D7%A2%D7%98_(%D7%A4%D7%99%D7%96%D7%99%D7%A7)" title="ארבעט (פיזיק) — iídiche" lang="yi" hreflang="yi" data-title="ארבעט (פיזיק)" data-language-autonym="ייִדיש" data-language-local-name="iídiche" class="interlanguage-link-target"><span>ייִדיש</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E5%8A%9F" title="功 — chinês" lang="zh" hreflang="zh" data-title="功" data-language-autonym="中文" data-language-local-name="chinês" class="interlanguage-link-target"><span>中文</span></a></li><li class="interlanguage-link interwiki-zh-classical mw-list-item"><a href="https://zh-classical.wikipedia.org/wiki/%E5%8A%9F" title="功 — Literary Chinese" lang="lzh" hreflang="lzh" data-title="功" data-language-autonym="文言" data-language-local-name="Literary Chinese" class="interlanguage-link-target"><span>文言</span></a></li><li class="interlanguage-link interwiki-zh-min-nan mw-list-item"><a href="https://zh-min-nan.wikipedia.org/wiki/Kang_(bu%CC%8Dt-l%C3%AD-ha%CC%8Dk)" title="Kang (bu̍t-lí-ha̍k) — min nan" lang="nan" hreflang="nan" data-title="Kang (bu̍t-lí-ha̍k)" data-language-autonym="閩南語 / Bân-lâm-gú" data-language-local-name="min nan" class="interlanguage-link-target"><span>閩南語 / Bân-lâm-gú</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E5%8A%9F" title="功 — cantonês" lang="yue" hreflang="yue" data-title="功" data-language-autonym="粵語" data-language-local-name="cantonês" class="interlanguage-link-target"><span>粵語</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q42213#sitelinks-wikipedia" title="Editar hiperligações interlínguas" class="wbc-editpage">Editar hiperligações</a></span></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div class="vector-page-toolbar-container"> <div id="left-navigation"> <nav aria-label="Espaços nominais"> <div id="p-associated-pages" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-associated-pages" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-nstab-main" class="selected vector-tab-noicon mw-list-item"><a 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#b32424;background-color:#fee7e6}.mw-parser-output .tmbox-delete{border:2px solid #b32424}.mw-parser-output .tmbox-content{border:1px solid #c0c090}.mw-parser-output .tmbox-style{border:2px solid #fc3}.mw-parser-output .tmbox-move{border:2px solid #9932cc}.mw-parser-output .tmbox .mbox-text{border:none;padding:0.25em 0.9em;width:100%}.mw-parser-output .tmbox .mbox-image{border:none;padding:2px 0 2px 0.9em;text-align:center}.mw-parser-output .tmbox .mbox-imageright{border:none;padding:2px 0.9em 2px 0;text-align:center}.mw-parser-output .tmbox .mbox-empty-cell{border:none;padding:0;width:1px}.mw-parser-output .tmbox .mbox-invalid-type{text-align:center}@media(min-width:720px){.mw-parser-output .tmbox{margin:4px 10%}.mw-parser-output .tmbox.mbox-small{clear:right;float:right;margin:4px 0 4px 1em;width:238px}}@media screen{html.skin-theme-clientpref-night .mw-parser-output .tmbox{background-color:#2e2505}html.skin-theme-clientpref-night .mw-parser-output .tmbox-speedy{background-color:#310402}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .tmbox{background-color:#2e2505}html.skin-theme-clientpref-os .mw-parser-output .tmbox-speedy{background-color:#310402}}body.skin--responsive .mw-parser-output table.tmbox img{max-width:none!important}</style><table class="box-Mais_notas plainlinks metadata ambox ambox-content ambox-Refimprove" role="presentation"><tbody><tr><td class="mbox-image"><div style="width:52px"><span typeof="mw:File"><a href="/wiki/Ficheiro:Question_book-new.svg" class="mw-file-description"><img alt="Esta página cita fontes, mas não cobrem todo o conteúdo" src="//upload.wikimedia.org/wikipedia/commons/thumb/9/99/Question_book-new.svg/50px-Question_book-new.svg.png" decoding="async" width="50" height="39" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/99/Question_book-new.svg/75px-Question_book-new.svg.png 1.5x, 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href="https://www.google.com/search?as_eq=wikipedia&as_epq=Trabalho+%28f%C3%ADsica%29">Google</a> (<a rel="nofollow" class="external text" href="https://www.google.com/search?hl=pt&tbm=nws&q=Trabalho+%28f%C3%ADsica%29&oq=Trabalho+%28f%C3%ADsica%29">N</a> • <a rel="nofollow" class="external text" href="http://books.google.com/books?&as_brr=0&as_epq=Trabalho+%28f%C3%ADsica%29">L</a> • <a rel="nofollow" class="external text" href="https://scholar.google.com/scholar?hl=pt&q=Trabalho+%28f%C3%ADsica%29">A</a>)</span></small>).</span> <small class="date-container"><i>(<span class="date">Março de 2024</span>)</i></small></div></td></tr></tbody></table> <table class="vertical-navbox nowraplinks" style="float:right;clear:right;width:22.0em;margin:0.5em 0 0.5em 1em;background:var(--background-color-neutral-subtle, #f8f9fa);color:inherit;border:1px solid #aaa;padding:0.2em;border-spacing:0.4em 0;text-align:center;line-height:1.4em;font-size:88%;width:18.5em;"><tbody><tr><th style="padding:0.2em 0.4em 0.2em;font-size:145%;line-height:1.2em"><a href="/wiki/Mec%C3%A2nica_cl%C3%A1ssica" title="Mecânica clássica">Mecânica clássica</a></th></tr><tr><td style="padding:0.2em 0 0.4em"><span typeof="mw:File"><a href="/wiki/Ficheiro:Orbital_motion.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4e/Orbital_motion.gif/180px-Orbital_motion.gif" decoding="async" width="180" height="180" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4e/Orbital_motion.gif/270px-Orbital_motion.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/4/4e/Orbital_motion.gif 2x" data-file-width="300" data-file-height="300" /></a></span><div style="padding-top:0.2em;line-height:1.2em">Diagramas de movimento orbital de um satélite ao redor da Terra, mostrando a velocidade e aceleração.</div></td></tr><tr><td style="padding:0 0.1em 0.4em;padding-bottom:0.3em;"> <div class="NavFrame collapsed" style="border:none;padding:0"><div class="NavHead" style="font-size:105%;background:transparent;text-align:left;color:inherit">Cinemática</div><div class="NavContent hlist" style="font-size:105%;padding:0.2em 0 0.4em;text-align:center"> <ul><li><a href="/wiki/Deslocamento" title="Deslocamento">Deslocamento</a></li> <li><a href="/wiki/Velocidade" title="Velocidade">Velocidade</a></li> <li><a href="/wiki/Velocidade_escalar" class="mw-redirect" title="Velocidade escalar">Velocidade escalar</a></li> <li><a href="/wiki/Acelera%C3%A7%C3%A3o" title="Aceleração">Aceleração</a></li> <li><a href="/wiki/Acelera%C3%A7%C3%A3o_centr%C3%ADpeta" title="Aceleração centrípeta">Aceleração centrípeta</a></li> <li><a href="/wiki/Movimento_uniforme" class="mw-redirect" title="Movimento uniforme">Movimento uniforme</a></li> <li><a href="/wiki/Movimento_uniformemente_variado" title="Movimento uniformemente variado">Movimento uniformemente variado</a></li> <li><a href="/wiki/Movimento_retil%C3%ADneo" title="Movimento retilíneo">Movimento retilíneo</a></li> <li><a href="/wiki/Movimento_parab%C3%B3lico" title="Movimento parabólico">Movimento parabólico</a></li> <li><a href="/wiki/Movimento_circular" title="Movimento circular">Movimento circular</a></li> <li><a href="/wiki/Movimento_circular_uniforme" title="Movimento circular uniforme">Movimento circular uniforme</a></li> <li><a href="/wiki/Movimento_curvil%C3%ADneo" title="Movimento curvilíneo">Movimento curvilíneo</a></li> <li><a href="/wiki/Movimento_harm%C3%B4nico_simples" title="Movimento harmônico simples">Movimento harmônico simples</a></li> <li><a href="/wiki/Movimento_harm%C3%B4nico_complexo" title="Movimento harmônico complexo">Movimento harmônico complexo</a></li></ul></div></div></td> </tr><tr><td style="padding:0 0.1em 0.4em"> <div class="NavFrame collapsed" style="border:none;padding:0"><div class="NavHead" style="font-size:105%;background:transparent;text-align:left;color:inherit">Dinâmica</div><div class="NavContent hlist" style="font-size:105%;padding:0.2em 0 0.4em;text-align:center"> <ul><li><a href="/wiki/For%C3%A7a" title="Força">Força</a></li> <li><a href="/wiki/In%C3%A9rcia" title="Inércia">Inércia</a></li> <li><a href="/wiki/Produto_de_in%C3%A9rcia" title="Produto de inércia">Produto de inércia</a></li> <li><a href="/wiki/Leis_de_Newton" title="Leis de Newton">Leis de Newton</a></li> <li><a href="/wiki/Primeira_Lei_de_Newton" class="mw-redirect" title="Primeira Lei de Newton">Primeira Lei de Newton</a></li> <li><a href="/wiki/Segunda_Lei_de_Newton" class="mw-redirect" title="Segunda Lei de Newton">Segunda Lei de Newton</a></li> <li><a href="/wiki/Terceira_Lei_de_Newton" class="mw-redirect" title="Terceira Lei de Newton">Terceira Lei de Newton</a></li> <li><a href="/wiki/Equa%C3%A7%C3%B5es_de_movimento" title="Equações de movimento">Equações de movimento</a></li> <li><a href="/wiki/Resson%C3%A2ncia" title="Ressonância">Ressonância</a></li></ul></div></div></td> </tr><tr><td style="padding:0 0.1em 0.4em"> <div class="NavFrame collapsed" style="border:none;padding:0"><div class="NavHead" style="font-size:105%;background:transparent;text-align:left;color:inherit">História</div><div class="NavContent hlist" style="font-size:105%;padding:0.2em 0 0.4em;text-align:center"> <ul><li><a href="/wiki/Hist%C3%B3ria_da_f%C3%ADsica" title="História da física">História da física</a></li></ul></div></div></td> </tr><tr><td style="padding:0 0.1em 0.4em"> <div class="NavFrame" style="border:none;padding:0"><div class="NavHead" style="font-size:105%;background:transparent;text-align:left;color:inherit">Trabalho e Mecânica</div><div class="NavContent hlist" style="font-size:105%;padding:0.2em 0 0.4em;text-align:center"> <ul><li><a href="/wiki/Energia_cin%C3%A9tica" title="Energia cinética">Energia cinética</a></li> <li><a href="/wiki/Energia_potencial" title="Energia potencial">Energia potencial</a></li> <li><a class="mw-selflink selflink">Trabalho</a></li> <li><a href="/wiki/Lei_da_conserva%C3%A7%C3%A3o_da_energia" title="Lei da conservação da energia">Conservação da energia</a></li> <li><a href="/wiki/For%C3%A7a_conservativa" title="Força conservativa">Força conservativa</a></li> <li><a href="/wiki/For%C3%A7a_de_contato" title="Força de contato">Força de contato</a></li> <li><a href="/wiki/Fun%C3%A7%C3%A3o_de_Lagrange" title="Função de Lagrange">Função de Lagrange</a></li> <li><a href="/wiki/Pot%C3%AAncia" title="Potência">Potência</a></li> <li><a href="/wiki/Retropropuls%C3%A3o" title="Retropropulsão">Retropropulsão</a></li> <li><a href="/wiki/Princ%C3%ADpio_de_Hamilton" title="Princípio de Hamilton">Princípio de Hamilton</a></li></ul></div></div></td> </tr><tr><td style="padding:0 0.1em 0.4em"> <div class="NavFrame collapsed" style="border:none;padding:0"><div class="NavHead" style="font-size:105%;background:transparent;text-align:left;color:inherit">Sistema de partículas</div><div class="NavContent hlist" style="font-size:105%;padding:0.2em 0 0.4em;text-align:center"> <ul><li><a href="/wiki/Centro_de_massas" class="mw-redirect" title="Centro de massas">Centro de massa</a></li> <li><a href="/wiki/Corpo_r%C3%ADgido" title="Corpo rígido">Corpo rígido</a></li> <li><a href="/wiki/Momento_linear" title="Momento linear">Momento linear</a></li> <li><a href="/wiki/Conserva%C3%A7%C3%A3o_do_momento_linear" title="Conservação do momento linear">Conservação do momento linear</a></li> <li><a href="/wiki/Equil%C3%ADbrio_din%C3%A2mico" title="Equilíbrio dinâmico">Equilíbrio dinâmico</a></li> <li><a href="/wiki/Princ%C3%ADpio_de_d%27Alembert" title="Princípio de d'Alembert">Princípio de d'Alembert</a></li> <li><a href="/wiki/Sistema_massa-mola" title="Sistema massa-mola">Sistema massa-mola</a></li></ul></div></div></td> </tr><tr><td style="padding:0 0.1em 0.4em"> <div class="NavFrame collapsed" style="border:none;padding:0"><div class="NavHead" style="font-size:105%;background:transparent;text-align:left;color:inherit">Colisões</div><div class="NavContent hlist" style="font-size:105%;padding:0.2em 0 0.4em;text-align:center"> <ul><li><a href="/wiki/Impulso" title="Impulso">Impulso</a></li> <li><a href="/wiki/Colis%C3%A3o_el%C3%A1stica" title="Colisão elástica">Colisão elástica</a></li> <li><a href="/wiki/Colis%C3%A3o_inel%C3%A1stica" title="Colisão inelástica">Colisão inelástica</a></li></ul></div></div></td> </tr><tr><td style="padding:0 0.1em 0.4em"> <div class="NavFrame collapsed" style="border:none;padding:0"><div class="NavHead" style="font-size:105%;background:transparent;text-align:left;color:inherit">Movimento rotacional</div><div class="NavContent hlist" style="font-size:105%;padding:0.2em 0 0.4em;text-align:center"> <ul><li><a href="/wiki/Posi%C3%A7%C3%A3o_angular" title="Posição angular">Posição angular</a></li> <li><a href="/wiki/Deslocamento_angular" class="mw-redirect" title="Deslocamento angular">Deslocamento angular</a></li> <li><a href="/wiki/Velocidade_angular" title="Velocidade angular">Velocidade angular</a></li> <li><a href="/wiki/Acelera%C3%A7%C3%A3o_angular" title="Aceleração angular">Aceleração angular</a></li> <li><a href="/wiki/Momento_de_in%C3%A9rcia" title="Momento de inércia">Momento de inércia</a></li> <li><a href="/wiki/Torque" title="Torque">Torque</a></li> <li><a href="/wiki/Momento_angular" title="Momento angular">Momento angular</a></li></ul></div></div></td> </tr><tr><td style="padding:0 0.1em 0.4em"> <div class="NavFrame collapsed" style="border:none;padding:0"><div class="NavHead" style="font-size:105%;background:transparent;text-align:left;color:inherit">Sistemas Clássicos</div><div class="NavContent hlist" style="font-size:105%;padding:0.2em 0 0.4em;text-align:center"> <ul><li><a href="/wiki/Sistema_din%C3%A2mico" title="Sistema dinâmico">Sistema dinâmico</a></li> <li><a href="/wiki/Sistema_de_equa%C3%A7%C3%B5es_lineares" title="Sistema de equações lineares">Sistema linear</a></li> <li><a href="/wiki/Sistema_din%C3%A2mico_n%C3%A3o_linear" title="Sistema dinâmico não linear">Sistema não linear</a></li> <li><a href="/wiki/Sistema_hamiltoniano" title="Sistema hamiltoniano">Sistema hamiltoniano</a></li> <li><a href="/wiki/Teoria_do_caos" title="Teoria do caos">Sistema caótico</a></li></ul></div></div></td> </tr><tr><td style="padding:0 0.1em 0.4em"> <div class="NavFrame collapsed" style="border:none;padding:0"><div class="NavHead" style="font-size:105%;background:transparent;text-align:left;color:inherit">Formulações</div><div class="NavContent hlist" style="font-size:105%;padding:0.2em 0 0.4em;text-align:center"> <ul><li><a href="/wiki/Mec%C3%A2nica_newtoniana" class="mw-redirect" title="Mecânica newtoniana">Mecânica newtoniana</a></li> <li><a href="/wiki/Mec%C3%A2nica_hamiltoniana" title="Mecânica hamiltoniana">Mecânica hamiltoniana</a></li> <li><a href="/wiki/Mec%C3%A2nica_de_Lagrange" title="Mecânica de Lagrange">Mecânica lagrangiana</a></li> <li><a href="/wiki/Mec%C3%A2nica_cl%C3%A1ssica_de_Koopman-von_Neumann" title="Mecânica clássica de Koopman-von Neumann">Mecânica KvN</a></li> <li><a href="/wiki/Equa%C3%A7%C3%A3o_de_Udwadia-Kalaba" title="Equação de Udwadia-Kalaba">Equação de Udwadia-Kalaba</a></li> <li><a href="/wiki/Mec%C3%A2nica_de_Routhian" title="Mecânica de Routhian">Mecânica de Routhian</a></li></ul></div></div></td> </tr><tr><td style="padding:0 0.1em 0.4em"> <div class="NavFrame collapsed" style="border:none;padding:0"><div class="NavHead" style="font-size:105%;background:transparent;text-align:left;color:inherit">Gravitação</div><div class="NavContent hlist" style="font-size:105%;padding:0.2em 0 0.4em;text-align:center"> <ul><li><a href="/wiki/Lei_da_gravita%C3%A7%C3%A3o_universal" title="Lei da gravitação universal">Lei da gravitação universal</a></li> <li><a href="/wiki/Princ%C3%ADpio_da_superposi%C3%A7%C3%A3o" title="Princípio da superposição">Princípio da superposição</a></li> <li><a href="/wiki/Constante_gravitacional_universal" title="Constante gravitacional universal">Constante gravitacional</a></li> <li><a href="/wiki/Velocidade_de_escape" title="Velocidade de escape">Velocidade de escape</a></li> <li><a href="/wiki/Leis_de_Kepler" title="Leis de Kepler">Leis de Kepler</a></li> <li><a href="/wiki/Princ%C3%ADpio_da_equival%C3%AAncia" title="Princípio da equivalência">Princípio da equivalência</a></li></ul></div></div></td> </tr><tr><td style="padding:0 0.1em 0.4em"> <div class="NavFrame collapsed" style="border:none;padding:0"><div class="NavHead" style="font-size:105%;background:transparent;text-align:left;color:inherit">Físicos</div><div class="NavContent hlist" style="font-size:105%;padding:0.2em 0 0.4em;text-align:center"> <ul><li><a href="/wiki/Isaac_Newton" title="Isaac Newton">Isaac Newton</a></li> <li><a href="/wiki/Leonhard_Euler" title="Leonhard Euler">Leonhard Euler</a></li> <li><a href="/wiki/Joseph-Louis_Lagrange" title="Joseph-Louis Lagrange">Joseph-Louis Lagrange</a></li> <li><a href="/wiki/Pierre-Simon_Laplace" class="mw-redirect" title="Pierre-Simon Laplace">Pierre-Simon Laplace</a></li> <li><a href="/wiki/Galileu_Galilei" title="Galileu Galilei">Galileu Galilei</a></li> <li><a href="/wiki/William_Rowan_Hamilton" title="William Rowan Hamilton">William Rowan Hamilton</a></li> <li><a href="/wiki/Johannes_Kepler" title="Johannes Kepler">Johannes Kepler</a></li></ul></div></div></td> </tr><tr><td style="text-align:right;font-size:115%;padding-top: 0.6em;"><div class="plainlinks hlist navbar mini"><ul><li class="nv-ver"><a href="/wiki/Predefini%C3%A7%C3%A3o:Mec%C3%A2nica_Cl%C3%A1ssica" title="Predefinição:Mecânica Clássica"><abbr title="Ver esta predefinição">v</abbr></a></li><li class="nv-discutir"><a href="/wiki/Predefini%C3%A7%C3%A3o_Discuss%C3%A3o:Mec%C3%A2nica_Cl%C3%A1ssica" title="Predefinição Discussão:Mecânica Clássica"><abbr title="Discutir esta predefinição">d</abbr></a></li><li class="nv-editar"><a class="external text" href="https://pt.wikipedia.org/w/index.php?title=Predefini%C3%A7%C3%A3o:Mec%C3%A2nica_Cl%C3%A1ssica&action=edit"><abbr title="Editar esta predefinição">e</abbr></a></li></ul></div></td></tr></tbody></table> <p>Em <a href="/wiki/F%C3%ADsica" title="Física">física</a>, <b>trabalho</b> (normalmente representado por <i>W</i>, do inglês <i>work</i>, ou pela letra grega <i><a href="/wiki/%CE%A4" title="Τ"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>τ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tau }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/38a7dcde9730ef0853809fefc18d88771f95206c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.202ex; height:1.676ex;" alt="{\displaystyle \tau }"></span></a></i>) é uma medida da <a href="/wiki/Energia" title="Energia">energia</a> transferida pela aplicação de uma <a href="/wiki/For%C3%A7a" title="Força">força</a> ao longo de um <a href="/wiki/Deslocamento" title="Deslocamento">deslocamento</a>. </p><p>O trabalho de uma força <b>F</b> aplicada ao longo de um caminho <i>C</i> pode ser calculado de forma geral através da seguinte <a href="/wiki/Integral_de_linha" title="Integral de linha">integral de linha</a>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {W} _{c}=\int _{c}\mathbf {F} \cdot d\mathbf {r} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">W</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> <mo>=</mo> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mo>⋅</mo> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {W} _{c}=\int _{c}\mathbf {F} \cdot d\mathbf {r} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1e47eb6f519b69c0d781da93dd0f47052dd15d1e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:14.735ex; height:5.676ex;" alt="{\displaystyle \operatorname {W} _{c}=\int _{c}\mathbf {F} \cdot d\mathbf {r} }"></span></dd></dl> <dl><dd>onde:</dd> <dd><b>F</b> é o vector <a href="/wiki/For%C3%A7a" title="Força">força</a></dd> <dd><b>r</b> é o vector <a href="/wiki/Deslocamento" title="Deslocamento">deslocamento</a>.</dd></dl> <p>O trabalho é um <a href="/wiki/N%C3%BAmero_real" title="Número real">número real</a>, que pode ser positivo ou negativo. Quando a força atua no sentido do deslocamento, o trabalho é positivo, isto é, existe energia sendo acrescentada ao corpo ou sistema. O contrário também é verdadeiro, uma força no sentido oposto ao deslocamento retira energia do corpo ou sistema. Qual tipo de energia, se <a href="/wiki/Energia_cin%C3%A9tica" title="Energia cinética">energia cinética</a> ou <a href="/wiki/Energia_potencial" title="Energia potencial">energia potencial</a>, depende do sistema em consideração. </p><p>Como mostra a equação acima, a existência de uma força não é sinônimo de realização de trabalho. Para que tal aconteça, é necessário que haja deslocamento do ponto de aplicação da força e que haja uma componente não nula da força na direcção do deslocamento. É por esta razão que aparece um <a href="/wiki/Produto_interno" title="Produto interno">produto interno</a> entre <b>F</b> e <b>r</b>. Por exemplo, um corpo em <a href="/wiki/Movimento_circular_uniforme" title="Movimento circular uniforme">movimento circular uniforme</a> (velocidade angular constante) está sujeito a uma <a href="/wiki/For%C3%A7a_centr%C3%ADpeta" title="Força centrípeta">força centrípeta</a>. No entanto, esta força não realiza trabalho, visto que é perpendicular à trajectória. </p><p>Portanto há duas condições para que uma força realize trabalho: </p> <ol><li>Que haja deslocamento;</li> <li>Que haja força ou componente da força na direção do deslocamento.</li></ol> <p>Esta definição é válida para qualquer tipo de força, independentemente da sua origem. Assim, pode tratar-se de uma força de atrito, gravítica (gravitacional), eléctrica, magnética, etc. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Tipos_de_trabalho">Tipos de trabalho</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Trabalho_(f%C3%ADsica)&veaction=edit&section=1" title="Editar secção: Tipos de trabalho" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Trabalho_(f%C3%ADsica)&action=edit&section=1" title="Editar código-fonte da secção: Tipos de trabalho"><span>editar código-fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Trabalho nulo, quando o trabalho é igual a zero;</li> <li>Trabalho potente/motor, quando a força e o deslocamento estão no mesmo sentido;</li> <li>Trabalho resistente, quando a força e deslocamento possuem sentidos contrários (geralmente representado por T= -F.d).</li></ul> <div class="mw-heading mw-heading2"><h2 id="Trabalho_e_energia">Trabalho e energia</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Trabalho_(f%C3%ADsica)&veaction=edit&section=2" title="Editar secção: Trabalho e energia" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Trabalho_(f%C3%ADsica)&action=edit&section=2" title="Editar código-fonte da secção: Trabalho e energia"><span>editar código-fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Se uma força <b>F</b> é aplicada num corpo que realiza um deslocamento <b>dr</b>, o trabalho realizado pela força é uma grandeza escalar de valor: </p> <dl><dd><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 d{\operatorname {W} }={\mathbf {F} }\cdot d{\mathbf {r} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">W</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> </mrow> <mo>⋅</mo> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d{\operatorname {W} }={\mathbf {F} }\cdot d{\mathbf {r} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aaf26f93252b862c5cabb05ad93fab1e48c06f20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:12.383ex; height:2.176ex;" alt="{\displaystyle d{\operatorname {W} }={\mathbf {F} }\cdot d{\mathbf {r} }}"></span></dd></dl></dd></dl> <p>Se a massa do corpo for suposta constante, e obtivermos <i>dW</i><sub>total</sub> como o trabalho total realizado sobre o corpo (obtido pela soma do trabalho realizado por cada uma das forças que atua sobre o mesmo), então, aplicando a segunda lei de Newton pode-se demonstrar que: </p> <dl><dd><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 d\operatorname {W} _{total}=d\operatorname {E_{c}} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <msub> <mi mathvariant="normal">W</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> <mi>o</mi> <mi>t</mi> <mi>a</mi> <mi>l</mi> </mrow> </msub> <mo>=</mo> <mi>d</mi> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <msub> <mi mathvariant="normal">E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">c</mi> </mrow> </msub> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d\operatorname {W} _{total}=d\operatorname {E_{c}} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c1a28aa5f27cd691703b4a10454ec50d8c57ba8c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.815ex; height:2.509ex;" alt="{\displaystyle d\operatorname {W} _{total}=d\operatorname {E_{c}} }"></span></dd></dl></dd></dl> <p>onde <i>E</i><sub>c</sub> é a <a href="/wiki/Energia_cin%C3%A9tica" title="Energia cinética">energia cinética</a>. Para um <a href="/wiki/Ponto_material" title="Ponto material">ponto material</a>, <i>E</i><sub>c</sub> é definida como: </p> <dl><dd><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 \operatorname {E_{c}} ={\frac {\operatorname {m} \operatorname {v^{2}} }{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <msub> <mi mathvariant="normal">E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">c</mi> </mrow> </msub> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">m</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <msup> <mi mathvariant="normal">v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mrow> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {E_{c}} ={\frac {\operatorname {m} \operatorname {v^{2}} }{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c4258c46ae2b5812be2a4ee9a3c2aae4eee57869" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:11.084ex; height:5.676ex;" alt="{\displaystyle \operatorname {E_{c}} ={\frac {\operatorname {m} \operatorname {v^{2}} }{2}}}"></span></dd></dl></dd></dl> <p>Para objectos extensos compostos por diversos pontos, a energia cinética é a soma das energias cinéticas das partículas que constituem um tipo especial de forças, conhecidas como forças conservativas, pode ser expresso como o <a href="/wiki/Gradiente" title="Gradiente">gradiente</a> de uma função escalar, a <a href="/wiki/Energia_potencial" title="Energia potencial">energia potencial</a>, V: </p> <dl><dd><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathbf {F} }=-grad{\operatorname {(V)} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> </mrow> <mo>=</mo> <mo>−</mo> <mi>g</mi> <mi>r</mi> <mi>a</mi> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mo stretchy="false">(</mo> <mi mathvariant="normal">V</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathbf {F} }=-grad{\operatorname {(V)} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f0fa2b59316b706c37976818176c1e386571aeb6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.752ex; height:2.843ex;" alt="{\displaystyle {\mathbf {F} }=-grad{\operatorname {(V)} }}"></span></dd></dl></dd></dl> <p>Se supusermos que todas as forças que atuam sobre um corpo são conservativas, e V é a energia potencial do sistema (obtida pela soma das energias potenciais de cada ponto, devidas a cada força), então: </p> <dl><dd><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathbf {F} }\cdot d{\mathbf {r} }=-grad{\operatorname {(V)} }\cdot d{\mathbf {r} }=-d\operatorname {V} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> </mrow> <mo>⋅</mo> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> </mrow> <mo>=</mo> <mo>−</mo> <mi>g</mi> <mi>r</mi> <mi>a</mi> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mo stretchy="false">(</mo> <mi mathvariant="normal">V</mi> <mo stretchy="false">)</mo> </mrow> </mrow> <mo>⋅</mo> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> </mrow> <mo>=</mo> <mo>−</mo> <mi>d</mi> <mi mathvariant="normal">V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathbf {F} }\cdot d{\mathbf {r} }=-grad{\operatorname {(V)} }\cdot d{\mathbf {r} }=-d\operatorname {V} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a558b5b973437bb67843a107bb9e773856ac53d6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.999ex; height:2.843ex;" alt="{\displaystyle {\mathbf {F} }\cdot d{\mathbf {r} }=-grad{\operatorname {(V)} }\cdot d{\mathbf {r} }=-d\operatorname {V} }"></span></dd></dl></dd></dl> <p>logo, </p> <dl><dd><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 -d\operatorname {V} =d{\operatorname {E_{c}} }\Rightarrow d{(\operatorname {E_{c}+V} )}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mi>d</mi> <mi mathvariant="normal">V</mi> <mo>=</mo> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <msub> <mi mathvariant="normal">E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">c</mi> </mrow> </msub> </mrow> </mrow> <mo stretchy="false">⇒</mo> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <msub> <mi mathvariant="normal">E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">c</mi> </mrow> </msub> <mo>+</mo> <mi mathvariant="normal">V</mi> </mrow> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -d\operatorname {V} =d{\operatorname {E_{c}} }\Rightarrow d{(\operatorname {E_{c}+V} )}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/46eb246b4143b7ddfbc62285e305dec9c0457c74" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.042ex; height:2.843ex;" alt="{\displaystyle -d\operatorname {V} =d{\operatorname {E_{c}} }\Rightarrow d{(\operatorname {E_{c}+V} )}=0}"></span></dd></dl></dd></dl> <p>Este resultado é conhecido como a <a href="/wiki/Lei_de_conserva%C3%A7%C3%A3o_da_energia" class="mw-redirect" title="Lei de conservação da energia">lei de conservação da energia</a>, indicando que a energia total <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {E_{t}} =\operatorname {E_{c}+V} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <msub> <mi mathvariant="normal">E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">t</mi> </mrow> </msub> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <msub> <mi mathvariant="normal">E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">c</mi> </mrow> </msub> <mo>+</mo> <mi mathvariant="normal">V</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {E_{t}} =\operatorname {E_{c}+V} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01e5f6210dbda60f755649eb51a17f9488a2ec91" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.682ex; height:2.509ex;" alt="{\displaystyle \operatorname {E_{t}} =\operatorname {E_{c}+V} }"></span> é constante (não é função do tempo). </p> <div class="mw-heading mw-heading2"><h2 id="Conceito">Conceito</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Trabalho_(f%C3%ADsica)&veaction=edit&section=3" title="Editar secção: Conceito" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Trabalho_(f%C3%ADsica)&action=edit&section=3" title="Editar código-fonte da secção: Conceito"><span>editar código-fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Os princípios do conceito de trabalho remontam às <a href="/wiki/Transforma%C3%A7%C3%A3o_de_Galileu" title="Transformação de Galileu">equações de Galileu</a> do movimento retilíneo uniformemente variado (<a href="/wiki/MRUV" class="mw-redirect" title="MRUV">MRUV</a>). Temos que o deslocamento <span class="mwe-math-element"><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 s}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mi>s</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta s}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dd7783a6d29d2ca4d9f1e0f501c4c6483fc058bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.026ex; height:2.176ex;" alt="{\displaystyle \Delta s}"></span> (positivo para uma direção da reta e negativo para a outra) equivale a </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta s={\frac {v^{2}-v_{0}^{2}}{2a}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mi>s</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> <msubsup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mrow> <mrow> <mn>2</mn> <mi>a</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta s={\frac {v^{2}-v_{0}^{2}}{2a}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2895d3498827e3aa172a2a65fbc8c26124e4c48a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:14.165ex; height:6.009ex;" alt="{\displaystyle \Delta s={\frac {v^{2}-v_{0}^{2}}{2a}}}"></span></dd></dl> <p>O que nos dá uma relação entre o deslocamento e a mudança de velocidade (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e07b00e7fc0847fbd16391c778d65bc25c452597" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle v}"></span> é a velocidade correspondente ao final do deslocamento e <span class="mwe-math-element"><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> é a velocidade correspondente ao seu início). </p><p>Essa equação é o primeiro passo para um tratamento da mecânica que seja independente do tempo envolvido. Mas ainda há em si um fator que remete para o tempo: a aceleração. De forma qualitativa, essa equação diz-nos que quanto maior for o módulo da aceleração que leva um corpo de velocidade <span class="mwe-math-element"><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> à velocidade <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e07b00e7fc0847fbd16391c778d65bc25c452597" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle v}"></span>, menor é o espaço percorrido durante essa transformação. De modo simples: se a mudança de velocidade demora mais, então sobra mais tempo para que o corpo se mova enquanto isso. Para eliminar esse fator que é tão dependente da maneira como se deu a mudança de velocidades (o que é contraditório com um tratamento atemporal), devemos multiplicar ambos os membros da equação por <span class="mwe-math-element"><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> e passar a pensar em <span class="mwe-math-element"><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\Delta s}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mi mathvariant="normal">Δ</mi> <mi>s</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\Delta s}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f0eaa6dadcc7326bc5d322a7926bd58e22eadd1d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.256ex; height:2.176ex;" alt="{\displaystyle a\Delta s}"></span> como uma entidade única, relacionada apenas com a variação absoluta do quadrado da velocidade dividido por dois: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\Delta s={\frac {v^{2}}{2}}-{\frac {v_{0}^{2}}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mi mathvariant="normal">Δ</mi> <mi>s</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mn>2</mn> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msubsup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\Delta s={\frac {v^{2}}{2}}-{\frac {v_{0}^{2}}{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/16f78381fb614a21e74371f4fbf3ff21fe97f723" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:16.231ex; height:6.009ex;" alt="{\displaystyle a\Delta s={\frac {v^{2}}{2}}-{\frac {v_{0}^{2}}{2}}}"></span></dd></dl> <p>Independentemente de como foi realizada a transformação, o <span class="mwe-math-element"><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 {v^{2}}{2}}-{\frac {v_{0}^{2}}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mn>2</mn> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msubsup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {v^{2}}{2}}-{\frac {v_{0}^{2}}{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/03c7544a434f123178a9252a59718d3fc8d0a969" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:8.876ex; height:6.009ex;" alt="{\displaystyle {\frac {v^{2}}{2}}-{\frac {v_{0}^{2}}{2}}}"></span> será sempre igual à entidade <span class="mwe-math-element"><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\Delta s}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mi mathvariant="normal">Δ</mi> <mi>s</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\Delta s}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f0eaa6dadcc7326bc5d322a7926bd58e22eadd1d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.256ex; height:2.176ex;" alt="{\displaystyle a\Delta s}"></span>, de modo que finalmente temos um tratamento atemporal no movimento uniformemente variado. </p><p>Entretanto, queremos estender isso ao movimento geral. Para isso, primeiro temos que estabelecer uma relação entre o movimento retilíneo e o movimento curvilíneo, a fim de estender os nossos conceitos de um para o outro. Para tal, recordamos as relações entre os vetores velocidade, posição e aceleração: a aceleração é a <a href="/wiki/Derivada" title="Derivada">derivada</a> temporal da velocidade e a velocidade é a derivada temporal da posição. Agora pensemos em qualquer "deslocamento infinitesimal" <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d{\vec {r}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d{\vec {r}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d18ac21ce60897ec4d2f48d720d513bf74e94301" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.439ex; height:2.343ex;" alt="{\displaystyle d{\vec {r}}}"></span>. Temos que: </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 d{\vec {r}}={\frac {d{\vec {r}}}{dt}}.dt={\vec {v}}dt}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>.</mo> <mi>d</mi> <mi>t</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mi>d</mi> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d{\vec {r}}={\frac {d{\vec {r}}}{dt}}.dt={\vec {v}}dt}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/afa0c241760398101a3006571cdd0eb3169164f4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:18.231ex; height:5.509ex;" alt="{\displaystyle d{\vec {r}}={\frac {d{\vec {r}}}{dt}}.dt={\vec {v}}dt}"></span></dd></dl> <p>Ou seja, qualquer deslocamento infinitesimal dá-se na direção da velocidade instantânea (desde que a posição seja descrita por uma <a href="/wiki/Fun%C3%A7%C3%A3o_vetorial" class="mw-redirect" title="Função vetorial">função vetorial</a> <a href="/wiki/Fun%C3%A7%C3%A3o_cont%C3%ADnua" title="Função contínua">contínua</a>). Como a direção da velocidade instantânea é uma só, então cada deslocamento infinitesimal é retilíneo. </p><p>Agora, devemos descobrir o quanto a nossa entidade <span class="mwe-math-element"><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 {v^{2}}{2}}-{\frac {v_{0}^{2}}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mn>2</mn> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msubsup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {v^{2}}{2}}-{\frac {v_{0}^{2}}{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/03c7544a434f123178a9252a59718d3fc8d0a969" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:8.876ex; height:6.009ex;" alt="{\displaystyle {\frac {v^{2}}{2}}-{\frac {v_{0}^{2}}{2}}}"></span> varia nesse intervalo infinitesimal de tempo em que os deslocamentos são retilíneos. Para isso, derivamos a entidade em relação ao tempo: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {d}{dt}}\left[{\frac {v^{2}}{2}}-{\frac {v_{0}^{2}}{2}}\right]=v{\frac {dv}{dt}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mrow> <mo>[</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mn>2</mn> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msubsup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mn>2</mn> </mfrac> </mrow> </mrow> <mo>]</mo> </mrow> <mo>=</mo> <mi>v</mi> <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 {\frac {d}{dt}}\left[{\frac {v^{2}}{2}}-{\frac {v_{0}^{2}}{2}}\right]=v{\frac {dv}{dt}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b036198afcd2d10423b1b661f15ac77eab64adc9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:22.271ex; height:7.509ex;" alt="{\displaystyle {\frac {d}{dt}}\left[{\frac {v^{2}}{2}}-{\frac {v_{0}^{2}}{2}}\right]=v{\frac {dv}{dt}}}"></span></dd></dl> <p>Note que a derivada <span class="mwe-math-element"><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 {dv}{dt}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <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 {\frac {dv}{dt}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e20b178c1895c793f8e906575cd7f2f537b17cc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:3.18ex; height:5.509ex;" alt="{\displaystyle {\frac {dv}{dt}}}"></span> NÃO corresponde ao <a href="/wiki/Vetor_(matem%C3%A1tica)" title="Vetor (matemática)">vetor</a> aceleração, como mostraremos logo. </p><p>Antes disso, voltemos por um instante à nossa entidade <span class="mwe-math-element"><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\Delta s={\frac {v^{2}}{2}}-{\frac {v_{0}^{2}}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mi mathvariant="normal">Δ</mi> <mi>s</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mn>2</mn> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msubsup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\Delta s={\frac {v^{2}}{2}}-{\frac {v_{0}^{2}}{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/16f78381fb614a21e74371f4fbf3ff21fe97f723" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:16.231ex; height:6.009ex;" alt="{\displaystyle a\Delta s={\frac {v^{2}}{2}}-{\frac {v_{0}^{2}}{2}}}"></span> (que só é válida para o MRUV). Claramente, se considerarmos o deslocamento como sendo sempre positivo, então uma aceleração negativa (no sentido oposto ao do movimento) implica uma diminuição da magnitude da velocidade, enquanto que uma aceleração positiva (no mesmo sentido do movimento) aumenta a magnitude da velocidade. </p><p>E quanto a uma aceleração que não se dá na mesma direção do deslocamento? Vejamos a seguinte relação: </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 {\vec {v}}=v{\hat {v}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {v}}=v{\hat {v}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3857abe2a274b37bc12106fa3d7aa0e1a67f35f5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.628ex; height:2.343ex;" alt="{\displaystyle {\vec {v}}=v{\hat {v}}}"></span></dd></dl> <p>Onde <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e07b00e7fc0847fbd16391c778d65bc25c452597" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle v}"></span> é a magnitude da velocidade e <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {v}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {v}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2790b6cafab4cc0f98a9ae8beb550947e60062f1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.227ex; height:2.176ex;" alt="{\displaystyle {\hat {v}}}"></span> é o <a href="/wiki/Vetor_unit%C3%A1rio" title="Vetor unitário">vetor unitário</a> que indica a direção da velocidade. Sendo assim, para obter a aceleração derivamos a expressão <span class="mwe-math-element"><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{\hat {v}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v{\hat {v}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fbbec65c244d8fa3496b18e1bcd16944bf9b10b4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.355ex; height:2.176ex;" alt="{\displaystyle v{\hat {v}}}"></span>, usando a <a href="/wiki/Regra_da_cadeia" title="Regra da cadeia">regra da cadeia</a>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {a}}={\frac {dv}{dt}}{\hat {v}}+v{\frac {d{\hat {v}}}{dt}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>v</mi> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>+</mo> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {a}}={\frac {dv}{dt}}{\hat {v}}+v{\frac {d{\hat {v}}}{dt}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/37a1e8bfd9c1d7b29f6782d7c7f087f367130e01" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:15.982ex; height:5.509ex;" alt="{\displaystyle {\vec {a}}={\frac {dv}{dt}}{\hat {v}}+v{\frac {d{\hat {v}}}{dt}}}"></span></dd></dl> <p>Onde vemos que um componente da aceleração (na mesma direção da velocidade), muda a magnitude da velocidade <span class="mwe-math-element"><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({\frac {dv}{dt}}{\hat {v}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>v</mi> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left({\frac {dv}{dt}}{\hat {v}}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0dc3d9bcf03cd89bdbc508ec4263eade7f56430b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:7.828ex; height:6.176ex;" alt="{\displaystyle \left({\frac {dv}{dt}}{\hat {v}}\right)}"></span>, enquanto o outro componente muda apenas a direção da velocidade <span class="mwe-math-element"><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(v{\frac {d{\hat {v}}}{dt}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mrow> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(v{\frac {d{\hat {v}}}{dt}}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/769db069db8b93485597b85871450f986948fa57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:7.828ex; height:6.176ex;" alt="{\displaystyle \left(v{\frac {d{\hat {v}}}{dt}}\right)}"></span>, lembrando que a derivada de um vetor unitário é sempre na direção <a href="/wiki/Perpendicular" class="mw-redirect" title="Perpendicular">perpendicular</a> a esse vetor unitário). Ou seja, como destacámos acima, a derivada <span class="mwe-math-element"><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 {dv}{dt}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <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 {\frac {dv}{dt}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e20b178c1895c793f8e906575cd7f2f537b17cc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:3.18ex; height:5.509ex;" alt="{\displaystyle {\frac {dv}{dt}}}"></span> corresponde apenas a um componente da aceleração: o componente que se dá na direção da velocidade. </p><p>Esse componente equivale a: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {dv}{dt}}={\frac {{\vec {a}}\cdot {\vec {v}}}{v}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <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>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mrow> <mi>v</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {dv}{dt}}={\frac {{\vec {a}}\cdot {\vec {v}}}{v}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/89afd9e1d7b4819ab9f9157d579cc723a01bd7c5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:11.198ex; height:5.509ex;" alt="{\displaystyle {\frac {dv}{dt}}={\frac {{\vec {a}}\cdot {\vec {v}}}{v}}}"></span></dd></dl> <p>Note que quando esse <a href="/wiki/Produto_escalar" title="Produto escalar">produto escalar</a> é negativo, a componente da aceleração que está na direção do deslocamento tem o sentido oposto ao seu. Isso implica uma diminuição da magnitude da velocidade, em concordância com a situação encontrada no MRUV. </p><p>Agora, a mudança infinitesimal na nossa entidade <span class="mwe-math-element"><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 {v^{2}}{2}}-{\frac {v_{0}^{2}}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mn>2</mn> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msubsup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {v^{2}}{2}}-{\frac {v_{0}^{2}}{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/03c7544a434f123178a9252a59718d3fc8d0a969" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:8.876ex; height:6.009ex;" alt="{\displaystyle {\frac {v^{2}}{2}}-{\frac {v_{0}^{2}}{2}}}"></span> fica: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {d}{dt}}\left[{\frac {v^{2}}{2}}-{\frac {v_{0}^{2}}{2}}\right]=v{\frac {dv}{dt}}={\vec {a}}\cdot {\vec {v}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mrow> <mo>[</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mn>2</mn> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msubsup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mn>2</mn> </mfrac> </mrow> </mrow> <mo>]</mo> </mrow> <mo>=</mo> <mi>v</mi> <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>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {d}{dt}}\left[{\frac {v^{2}}{2}}-{\frac {v_{0}^{2}}{2}}\right]=v{\frac {dv}{dt}}={\vec {a}}\cdot {\vec {v}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/590b02ef9e5355f998fea8a0d54db64fd06ce200" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:29.454ex; height:7.509ex;" alt="{\displaystyle {\frac {d}{dt}}\left[{\frac {v^{2}}{2}}-{\frac {v_{0}^{2}}{2}}\right]=v{\frac {dv}{dt}}={\vec {a}}\cdot {\vec {v}}}"></span></dd></dl> <p>Mas queremos saber essa mudança em um intervalo de tempo qualquer. Então integramos com relação ao tempo: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {v^{2}}{2}}-{\frac {v_{0}^{2}}{2}}=\int {\vec {a}}\cdot {\vec {v}}dt}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mn>2</mn> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msubsup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mn>2</mn> </mfrac> </mrow> <mo>=</mo> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mi>d</mi> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {v^{2}}{2}}-{\frac {v_{0}^{2}}{2}}=\int {\vec {a}}\cdot {\vec {v}}dt}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e58e15ae54a591616fbe7f86a4ba1799263ec209" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:20.695ex; height:6.509ex;" alt="{\displaystyle {\frac {v^{2}}{2}}-{\frac {v_{0}^{2}}{2}}=\int {\vec {a}}\cdot {\vec {v}}dt}"></span></dd></dl> <p>Finalmente encontramos a nossa entidade. No entanto, em analogia ao que aconteceu no MRUV, o que temos aqui é uma <a href="/wiki/Integral" title="Integral">integral</a> dependente do tempo, o que não condiz com do que estamos desde o início à procura: um tratamento atemporal. Assim, fazemos simplesmente: </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 \int {\vec {a}}\cdot {\vec {v}}dt=\int {\vec {a}}\cdot {\frac {d{\vec {r}}}{dt}}dt}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mi>d</mi> <mi>t</mi> <mo>=</mo> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mi>d</mi> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int {\vec {a}}\cdot {\vec {v}}dt=\int {\vec {a}}\cdot {\frac {d{\vec {r}}}{dt}}dt}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/63b15edb8be7384fe9da3018c1cd624f09050391" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:22.639ex; height:5.843ex;" alt="{\displaystyle \int {\vec {a}}\cdot {\vec {v}}dt=\int {\vec {a}}\cdot {\frac {d{\vec {r}}}{dt}}dt}"></span></dd></dl> <p>O que constitui uma <a href="/wiki/Integral_de_linha" title="Integral de linha">integral de linha</a>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{C}{\vec {a}}\cdot d{\vec {r}}={\frac {v^{2}}{2}}-{\frac {v_{0}^{2}}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>⋅</mo> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mn>2</mn> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msubsup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{C}{\vec {a}}\cdot d{\vec {r}}={\frac {v^{2}}{2}}-{\frac {v_{0}^{2}}{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3909baae1b6a0dd3024f7161c08118c4b98251b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:20.484ex; height:6.509ex;" alt="{\displaystyle \int _{C}{\vec {a}}\cdot d{\vec {r}}={\frac {v^{2}}{2}}-{\frac {v_{0}^{2}}{2}}}"></span></dd></dl> <p>Com os limites de integração, obviamente, correspondendo aos pontos inicial e final da trajetória. </p><p>O nosso *trabalho* está quase pronto. Só precisamos de multiplicar essa entidade que encontramos pela <a href="/wiki/Massa" title="Massa">massa</a>. Isso tem inúmeras vantagens, mas aqui daremos apenas uma razão conceitual: a aceleração é um conceito secundário em comparação com a importância da força. Trocar na equação acima a aceleração pela força implica trazer essa entidade para mais perto do mundo físico. Isso também se deve à ligação do trabalho com o conceito de <a href="/wiki/Energia" title="Energia">energia</a>, que é uma quantidade que se conserva e que está ligada à massa. </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 \int _{C}m{\vec {a}}\cdot d{\vec {r}}=\int _{C}{\vec {F}}\cdot d{\vec {r}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> </msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>⋅</mo> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>F</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>⋅</mo> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{C}m{\vec {a}}\cdot d{\vec {r}}=\int _{C}{\vec {F}}\cdot d{\vec {r}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/863cee55c57f6e796d1a02ce168223d3c685b3ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:22.697ex; height:5.676ex;" alt="{\displaystyle \int _{C}m{\vec {a}}\cdot d{\vec {r}}=\int _{C}{\vec {F}}\cdot d{\vec {r}}}"></span></dd></dl> <p>Assim, obtemos o trabalho integral realizado sobre uma partícula: </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 W=\int _{C}{\vec {F}}\cdot d{\vec {r}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>W</mi> <mo>=</mo> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>F</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>⋅</mo> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W=\int _{C}{\vec {F}}\cdot d{\vec {r}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/09d3c927ea8d8f744ee78fbebe132e15bd0374c1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:14.584ex; height:5.676ex;" alt="{\displaystyle W=\int _{C}{\vec {F}}\cdot d{\vec {r}}}"></span></dd></dl> <p>Onde <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {F}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>F</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {F}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ef40edff397a115ecdce7d3518001dfcc7f37d9e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.771ex; height:2.843ex;" alt="{\displaystyle {\vec {F}}}"></span> é a força resultante. O trabalho realizado por uma outra força qualquer é análogo, trocando-se a força total pela força qualquer. Note que a componente do trabalho de uma força qualquer que contribui para a componente força resultante na direção do deslocamento é, justamente, o produto escalar entre a força qualquer e a direção do deslocamento, o que justifica essa similaridade. </p> <div class="mw-heading mw-heading2"><h2 id="Unidades">Unidades</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Trabalho_(f%C3%ADsica)&veaction=edit&section=4" title="Editar secção: Unidades" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Trabalho_(f%C3%ADsica)&action=edit&section=4" title="Editar código-fonte da secção: Unidades"><span>editar código-fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A unidade <a href="/wiki/Sistema_Internacional_de_Unidades" title="Sistema Internacional de Unidades">SI</a> de trabalho é o <a href="/wiki/Joule" title="Joule">joule</a> (J), que se define como o trabalho realizado por uma força de um <a href="/wiki/Newton" class="mw-redirect" title="Newton">newton</a> (N) atuando ao longo de 1 metro (m) na direção do deslocamento. O trabalho pode igualmente exprimir-se em N.m, como se depreende desta definição. Estas são as unidades mais correntes; no entanto, na medida em que o trabalho é uma forma de <a href="/wiki/Energia" title="Energia">energia</a>, outras unidades são por vezes empregues. </p> <div class="mw-heading mw-heading2"><h2 id="Outras_unidades">Outras unidades</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Trabalho_(f%C3%ADsica)&veaction=edit&section=5" title="Editar secção: Outras unidades" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Trabalho_(f%C3%ADsica)&action=edit&section=5" title="Editar código-fonte da secção: Outras unidades"><span>editar código-fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>O quilojoule, equivalente a <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 10^{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 10^{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9dbd60b35ef8ca5859f59d14662a63139ffe0a50" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.379ex; height:2.676ex;" alt="{\displaystyle 10^{3}}"></span> joules e o erg, que equivale a: 1 joule = <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 10^{3}*10^{2}*10^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>∗</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>∗</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 10^{3}*10^{2}*10^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/248a645c56263c125b04cbec89ceeec9b7b0e6ea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:14.527ex; height:2.676ex;" alt="{\displaystyle 10^{3}*10^{2}*10^{2}}"></span> erg = <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 10^{7}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>7</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 10^{7}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c8013a64c98fba31f457676460cc7752ddd4d491" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.379ex; height:2.676ex;" alt="{\displaystyle 10^{7}}"></span> erg. </p> <div class="mw-heading mw-heading2"><h2 id="Outras_equações"><span id="Outras_equa.C3.A7.C3.B5es"></span>Outras equações</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Trabalho_(f%C3%ADsica)&veaction=edit&section=6" title="Editar secção: Outras equações" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Trabalho_(f%C3%ADsica)&action=edit&section=6" title="Editar código-fonte da secção: Outras equações"><span>editar código-fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Para o caso simples em que o corpo se desloca em movimento retilíneo e a força é paralela à direção do movimento, o trabalho é dado pela equação: </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 \operatorname {W} =\operatorname {Fr} \;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">W</mi> <mo>=</mo> <mi>Fr</mi> <mspace width="thickmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {W} =\operatorname {Fr} \;}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ed1c5d186b61cb70482d091615e26c5b6de64370" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.562ex; height:2.176ex;" alt="{\displaystyle \operatorname {W} =\operatorname {Fr} \;}"></span></dd></dl> <p>onde <i>F</i> é apenas a magnitude da força e <i>r</i> é a distância percorrida pelo corpo. Caso a força se oponha ao movimento, o trabalho é negativo. De forma mais geral, a força e o deslocamento podem ser tomados como grandezas vectoriais e combinados através do <a href="/wiki/Produto_interno" title="Produto interno">produto interno</a>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {W} =\mathbf {F} \cdot \mathbf {r} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">W</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {W} =\mathbf {F} \cdot \mathbf {r} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01e3ab6fc5264e5cb81c0e974349aeca8dfa759b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.951ex; height:2.176ex;" alt="{\displaystyle \operatorname {W} =\mathbf {F} \cdot \mathbf {r} }"></span></dd></dl> <p>Esta fórmula é válida para situações em que a força forma um ângulo com a direção do movimento, desde que a magnitude da força e direcção do deslocamento sejam constantes. A generalização desta equação para situações em que a força e a direção variam ao longo da trajetória (ou do tempo) pode ser feita recorrendo ao uso de diferenciais. O trabalho infinitesimal <i>dW</i> realizado pela força <i><b>F</b></i> ao longo do deslocamento infinitesimal <i><b>dr</b></i> é então dado por: </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 d\operatorname {W} =\mathbf {F} \cdot d{\mathbf {r} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mi mathvariant="normal">W</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mo>⋅</mo> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d\operatorname {W} =\mathbf {F} \cdot d{\mathbf {r} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d7c63352882a718527898fdc42903f58afdde444" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:12.77ex; height:2.176ex;" alt="{\displaystyle d\operatorname {W} =\mathbf {F} \cdot d{\mathbf {r} }}"></span></dd></dl> <p>A <a href="/wiki/Integra%C3%A7%C3%A3o" class="mw-redirect" title="Integração">integração</a> de ambos os lados desta equação ao longo da trajetória resulta na equação geral inicialmente apresentada. </p> <div class="mw-heading mw-heading2"><h2 id="Resolução_numérica_de_equações_diferenciais"><span id="Resolu.C3.A7.C3.A3o_num.C3.A9rica_de_equa.C3.A7.C3.B5es_diferenciais"></span>Resolução numérica de equações diferenciais</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Trabalho_(f%C3%ADsica)&veaction=edit&section=7" title="Editar secção: Resolução numérica de equações diferenciais" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Trabalho_(f%C3%ADsica)&action=edit&section=7" title="Editar código-fonte da secção: Resolução numérica de equações diferenciais"><span>editar código-fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>O movimento dos projéteis, em sendo considerado uma única força — o <a href="/wiki/Peso" title="Peso">peso</a> -, seria o movimento real se o movimento fosse realizado no vácuo. Uma solução mais realista obtém-se tendo em conta também a força de resistência do ar. A <a href="/wiki/Acelera%C3%A7%C3%A3o" title="Aceleração">aceleração</a> do <a href="/wiki/Proj%C3%A9til" title="Projétil">projétil</a> será então </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 {\vec {a}}={\vec {g}}+{\dfrac {{\vec {F}}_{\mathrm {r} }}{m}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>g</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mfrac> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>F</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">r</mi> </mrow> </mrow> </msub> <mi>m</mi> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {a}}={\vec {g}}+{\dfrac {{\vec {F}}_{\mathrm {r} }}{m}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4a8537d83f46d3adde5ad8a431bd801d6af8b877" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:11.827ex; height:6.009ex;" alt="{\displaystyle {\vec {a}}={\vec {g}}+{\dfrac {{\vec {F}}_{\mathrm {r} }}{m}}}"></span></dd></dl> <p>em 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 {\vec {g}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>g</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {g}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8aad928c73fda5199478a151663f0ce3a57a8027" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.174ex; height:2.676ex;" alt="{\displaystyle {\vec {g}}}"></span> é a aceleração da gravidade e <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {F}}_{\mathrm {r} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>F</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">r</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {F}}_{\mathrm {r} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/38228840f0827624ae1d71a5bfab73598954034c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.648ex; height:3.176ex;" alt="{\displaystyle {\vec {F}}_{\mathrm {r} }}"></span> é a força de resistência do ar, força essa que depende da velocidade, da massa volúmica do ar, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f7d439671d1289b6a816e6af7a304be40608d64" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.202ex; height:2.176ex;" alt="{\displaystyle \rho }"></span>, e da forma e do tamanho do projétil. </p><p>Se o projétil for uma esfera de raio <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}"></span> a expressão para a força de resistência do ar será, a partir da equação ... </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 F_{\mathrm {r} }=6\,\pi \,\eta \,r\,v+{\frac {1}{4}}\,\pi \,\rho \,r^{2}\,v^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">r</mi> </mrow> </mrow> </msub> <mo>=</mo> <mn>6</mn> <mspace width="thinmathspace" /> <mi>π</mi> <mspace width="thinmathspace" /> <mi>η</mi> <mspace width="thinmathspace" /> <mi>r</mi> <mspace width="thinmathspace" /> <mi>v</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mrow> <mspace width="thinmathspace" /> <mi>π</mi> <mspace width="thinmathspace" /> <mi>ρ</mi> <mspace width="thinmathspace" /> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="thinmathspace" /> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{\mathrm {r} }=6\,\pi \,\eta \,r\,v+{\frac {1}{4}}\,\pi \,\rho \,r^{2}\,v^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/666a504a4090a0784c98d45de50ff94e92e73ac1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:26.065ex; height:5.176ex;" alt="{\displaystyle F_{\mathrm {r} }=6\,\pi \,\eta \,r\,v+{\frac {1}{4}}\,\pi \,\rho \,r^{2}\,v^{2}}"></span></dd></dl> <p>para uma esfera, a força de resistência do ar pode escrever-se na forma vetorial,<sup id="cite_ref-Villate_1-0" class="reference"><a href="#cite_note-Villate-1"><span>[</span>1<span>]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {F}}_{\mathrm {r} }=-{\dfrac {1}{4}}\,\pi \,\rho \,R^{2}\,v\,{\vec {v}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>F</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">r</mi> </mrow> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mstyle> </mrow> <mspace width="thinmathspace" /> <mi>π</mi> <mspace width="thinmathspace" /> <mi>ρ</mi> <mspace width="thinmathspace" /> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="thinmathspace" /> <mi>v</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {F}}_{\mathrm {r} }=-{\dfrac {1}{4}}\,\pi \,\rho \,R^{2}\,v\,{\vec {v}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4bf0c605a932501a441ca8dc8a7af6552a907f83" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:19.144ex; height:5.176ex;" alt="{\displaystyle {\vec {F}}_{\mathrm {r} }=-{\dfrac {1}{4}}\,\pi \,\rho \,R^{2}\,v\,{\vec {v}}}"></span></dd></dl> <p>Escolhendo um sistema de eixos em que a gravidade aponta no sentido negativo do eixo dos <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}"></span> e a velocidade <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {v_{0}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {v_{0}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bfc08345b567f582c88c76e59d8486a7e544de8c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.324ex; height:3.343ex;" alt="{\displaystyle {\vec {v_{0}}}}"></span> com que é lançado o projétil está no plano <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle xy}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle xy}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c72eb345e496513fb8b2fa4aa8c4d89b855f9a01" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.485ex; height:2.009ex;" alt="{\displaystyle xy}"></span>, o peso e a força de resistência do ar atuarão sempre no plano <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle xy}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle xy}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c72eb345e496513fb8b2fa4aa8c4d89b855f9a01" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.485ex; height:2.009ex;" alt="{\displaystyle xy}"></span> e o movimento do projétil estará limitado a esse plano.<sup id="cite_ref-Villate_1-1" class="reference"><a href="#cite_note-Villate-1"><span>[</span>1<span>]</span></a></sup> </p><p>Assim sendo, a velocidade e a aceleração têm duas componentes, segundo <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> e <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}"></span>, e combinando as duas equações anteriores, as derivadas das componentes da velocidade são, </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 {\dfrac {\mathrm {d} \,v_{x}}{\mathrm {d} \,t}}=-C\,v_{x}\,{\sqrt {v_{x}^{2}+v_{y}^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mspace width="thinmathspace" /> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mspace width="thinmathspace" /> <mi>t</mi> </mrow> </mfrac> </mstyle> </mrow> <mo>=</mo> <mo>−</mo> <mi>C</mi> <mspace width="thinmathspace" /> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msubsup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\dfrac {\mathrm {d} \,v_{x}}{\mathrm {d} \,t}}=-C\,v_{x}\,{\sqrt {v_{x}^{2}+v_{y}^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/98f56ded8e922235550dcf8544e908b8f977bfe6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:24.209ex; height:5.509ex;" alt="{\displaystyle {\dfrac {\mathrm {d} \,v_{x}}{\mathrm {d} \,t}}=-C\,v_{x}\,{\sqrt {v_{x}^{2}+v_{y}^{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 {\dfrac {\mathrm {d} \,v_{y}}{\mathrm {d} \,t}}=-g-k\,v_{y}\,{\sqrt {v_{x}^{2}+v_{y}^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mspace width="thinmathspace" /> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mspace width="thinmathspace" /> <mi>t</mi> </mrow> </mfrac> </mstyle> </mrow> <mo>=</mo> <mo>−</mo> <mi>g</mi> <mo>−</mo> <mi>k</mi> <mspace width="thinmathspace" /> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msubsup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\dfrac {\mathrm {d} \,v_{y}}{\mathrm {d} \,t}}=-g-k\,v_{y}\,{\sqrt {v_{x}^{2}+v_{y}^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a4723f807ab710ceba184ce85ba823d1de59161e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:27.364ex; height:5.843ex;" alt="{\displaystyle {\dfrac {\mathrm {d} \,v_{y}}{\mathrm {d} \,t}}=-g-k\,v_{y}\,{\sqrt {v_{x}^{2}+v_{y}^{2}}}}"></span></dd></dl> <p>em que a constante positiva <span class="mwe-math-element"><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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}"></span> é igual a, </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C={\dfrac {\pi \,\rho \,R^{2}}{4\,m}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mfrac> <mrow> <mi>π</mi> <mspace width="thinmathspace" /> <mi>ρ</mi> <mspace width="thinmathspace" /> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mn>4</mn> <mspace width="thinmathspace" /> <mi>m</mi> </mrow> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C={\dfrac {\pi \,\rho \,R^{2}}{4\,m}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/403c50cd03eab421633b75b8dfc46792e1753db1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:11.827ex; height:5.676ex;" alt="{\displaystyle C={\dfrac {\pi \,\rho \,R^{2}}{4\,m}}}"></span></dd></dl> <p>A introdução do efeito da resistência do ar complica muito o problema, porque estas equações não são equações de variáveis separáveis e deverão ser resolvidas em simultâneo, já que as duas componentes <span class="mwe-math-element"><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_{x}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v_{x}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/704b7ad1ece77840fde455daa6d2e51e64282b5e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.3ex; height:2.009ex;" alt="{\displaystyle v_{x}}"></span> e <span class="mwe-math-element"><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_{y}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v_{y}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/922aa64dad09633a401e14be9b4389795835cd8c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.177ex; height:2.343ex;" alt="{\displaystyle v_{y}}"></span> aparecem nas duas equações. </p><p>Um caso particular é o caso da queda livre vertical, em que a velocidade inicial é zero; nesse caso, a força de resistência do ar atua unicamente na vertical e em sentido oposto ao peso, o movimento é unicamente ao longo do eixo dos <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}"></span> e a equação diferencial para a componente vertical da velocidade é, </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 {\dfrac {\mathrm {d} \,v_{y}}{\mathrm {d} \,t}}=-g+C\,v_{y}^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mspace width="thinmathspace" /> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mspace width="thinmathspace" /> <mi>t</mi> </mrow> </mfrac> </mstyle> </mrow> <mo>=</mo> <mo>−</mo> <mi>g</mi> <mo>+</mo> <mi>C</mi> <mspace width="thinmathspace" /> <msubsup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\dfrac {\mathrm {d} \,v_{y}}{\mathrm {d} \,t}}=-g+C\,v_{y}^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eaf74059bba9ceb57362fff9d974aab09575a5b8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:17.891ex; height:5.843ex;" alt="{\displaystyle {\dfrac {\mathrm {d} \,v_{y}}{\mathrm {d} \,t}}=-g+C\,v_{y}^{2}}"></span></dd></dl> <p>que sim é uma equação de variáveis separáveis e pode ser resolvida facilmente. </p><p>Usando o resultado desse problema, tendo em conta que a constante <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}"></span> é <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {C/g}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>g</mi> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {C/g}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a293577b655b49f7ade9550ba38ce8ff0b9aa84e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:6.369ex; height:4.843ex;" alt="{\displaystyle {\sqrt {C/g}}}"></span> e o valor da velocidade é <span class="mwe-math-element"><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_{y}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -v_{y}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a840284cdcdad9ac6df64fb2dd14a6bc7d46ca20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.985ex; height:2.676ex;" alt="{\displaystyle -v_{y}}"></span>, a solução dessa equação diferencial é: </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_{y}=-{\sqrt {\dfrac {g}{C}}}\,\tanh \left({\sqrt {g\,C}}\,t\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mstyle displaystyle="true" scriptlevel="0"> <mfrac> <mi>g</mi> <mi>C</mi> </mfrac> </mstyle> </msqrt> </mrow> <mspace width="thinmathspace" /> <mi>tanh</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>g</mi> <mspace width="thinmathspace" /> <mi>C</mi> </msqrt> </mrow> <mspace width="thinmathspace" /> <mi>t</mi> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v_{y}=-{\sqrt {\dfrac {g}{C}}}\,\tanh \left({\sqrt {g\,C}}\,t\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4eae05eead8521b4d7b670391ec90dd864d2a9f4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; 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