Kinetic energy - Wikipedia

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vector-toc-level-2"> <a class="vector-toc-link" href="#Kinetic_energy_of_rigid_bodies"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Kinetic energy of rigid bodies</span> </div> </a> <ul id="toc-Kinetic_energy_of_rigid_bodies-sublist" class="vector-toc-list"> <li id="toc-Derivation" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Derivation"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1.1</span> <span>Derivation</span> </div> </a> <ul id="toc-Derivation-sublist" class="vector-toc-list"> <li id="toc-Without_vector_calculus" class="vector-toc-list-item vector-toc-level-4"> <a class="vector-toc-link" href="#Without_vector_calculus"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1.1.1</span> <span>Without vector calculus</span> </div> </a> <ul id="toc-Without_vector_calculus-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-With_vector_calculus" class="vector-toc-list-item vector-toc-level-4"> <a class="vector-toc-link" href="#With_vector_calculus"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1.1.2</span> <span>With vector calculus</span> </div> </a> <ul id="toc-With_vector_calculus-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </li> <li id="toc-Rotating_bodies" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Rotating_bodies"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>Rotating bodies</span> </div> </a> <ul id="toc-Rotating_bodies-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Kinetic_energy_of_systems" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Kinetic_energy_of_systems"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.3</span> <span>Kinetic energy of systems</span> </div> </a> <ul id="toc-Kinetic_energy_of_systems-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Fluid_dynamics" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Fluid_dynamics"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.4</span> <span>Fluid dynamics</span> </div> </a> <ul id="toc-Fluid_dynamics-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Frame_of_reference" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Frame_of_reference"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.5</span> <span>Frame of reference</span> </div> </a> <ul id="toc-Frame_of_reference-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Rotation_in_systems" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Rotation_in_systems"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.6</span> <span>Rotation in systems</span> </div> </a> <ul id="toc-Rotation_in_systems-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Relativistic_kinetic_energy" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Relativistic_kinetic_energy"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Relativistic kinetic energy</span> </div> </a> <button aria-controls="toc-Relativistic_kinetic_energy-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Relativistic kinetic energy subsection</span> </button> <ul id="toc-Relativistic_kinetic_energy-sublist" class="vector-toc-list"> <li id="toc-Derivation_2" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Derivation_2"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>Derivation</span> </div> </a> <ul id="toc-Derivation_2-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Low_speed_limit" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Low_speed_limit"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.2</span> <span>Low speed limit</span> </div> </a> <ul id="toc-Low_speed_limit-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-General_relativity" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#General_relativity"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.3</span> <span>General relativity</span> </div> </a> <ul id="toc-General_relativity-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Kinetic_energy_in_quantum_mechanics" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Kinetic_energy_in_quantum_mechanics"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Kinetic energy in quantum mechanics</span> </div> </a> <ul id="toc-Kinetic_energy_in_quantum_mechanics-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Notes" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Notes</span> </div> </a> <ul id="toc-Notes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>External links</span> </div> </a> <ul id="toc-External_links-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="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" title="Table of Contents" > <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="Toggle the table of contents" > <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">Toggle the table of contents</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">Kinetic energy</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="Go to an article in another language. Available in 98 languages" > <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-98" 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">98 languages</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/Kinetiese_energie" title="Kinetiese energie – Afrikaans" lang="af" hreflang="af" data-title="Kinetiese energie" data-language-autonym="Afrikaans" data-language-local-name="Afrikaans" class="interlanguage-link-target"><span>Afrikaans</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%B7%D8%A7%D9%82%D8%A9_%D8%AD%D8%B1%D9%83%D9%8A%D8%A9" title="طاقة حركية – Arabic" lang="ar" hreflang="ar" data-title="طاقة حركية" data-language-autonym="العربية" data-language-local-name="Arabic" 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%97%E0%A6%A4%E0%A6%BF_%E0%A6%B6%E0%A6%95%E0%A7%8D%E0%A6%A4%E0%A6%BF" title="গতি শক্তি – Assamese" lang="as" hreflang="as" data-title="গতি শক্তি" data-language-autonym="অসমীয়া" data-language-local-name="Assamese" 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/Enerx%C3%ADa_cin%C3%A9tico" title="Enerxía cinético – Asturian" lang="ast" hreflang="ast" data-title="Enerxía cinético" data-language-autonym="Asturianu" data-language-local-name="Asturian" 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/Kinetik_enerji" title="Kinetik enerji – Azerbaijani" lang="az" hreflang="az" data-title="Kinetik enerji" data-language-autonym="Azərbaycanca" data-language-local-name="Azerbaijani" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%97%E0%A6%A4%E0%A6%BF%E0%A6%B6%E0%A6%95%E0%A7%8D%E0%A6%A4%E0%A6%BF" title="গতিশক্তি – Bangla" lang="bn" hreflang="bn" data-title="গতিশক্তি" data-language-autonym="বাংলা" data-language-local-name="Bangla" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%9A%D1%96%D0%BD%D0%B5%D1%82%D1%8B%D1%87%D0%BD%D0%B0%D1%8F_%D1%8D%D0%BD%D0%B5%D1%80%D0%B3%D1%96%D1%8F" title="Кінетычная энергія – Belarusian" lang="be" hreflang="be" data-title="Кінетычная энергія" data-language-autonym="Беларуская" data-language-local-name="Belarusian" 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%9A%D1%96%D0%BD%D1%8D%D1%82%D1%8B%D1%87%D0%BD%D0%B0%D1%8F_%D1%8D%D0%BD%D1%8D%D1%80%D0%B3%D1%96%D1%8F" title="Кінэтычная энэргія – Belarusian (Taraškievica orthography)" lang="be-tarask" hreflang="be-tarask" data-title="Кінэтычная энэргія" 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-bcl mw-list-item"><a href="https://bcl.wikipedia.org/wiki/Enerhiyang_kinetika" title="Enerhiyang kinetika – Central Bikol" lang="bcl" hreflang="bcl" data-title="Enerhiyang kinetika" data-language-autonym="Bikol Central" data-language-local-name="Central Bikol" class="interlanguage-link-target"><span>Bikol Central</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%9A%D0%B8%D0%BD%D0%B5%D1%82%D0%B8%D1%87%D0%BD%D0%B0_%D0%B5%D0%BD%D0%B5%D1%80%D0%B3%D0%B8%D1%8F" title="Кинетична енергия – Bulgarian" lang="bg" hreflang="bg" data-title="Кинетична енергия" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Kineti%C4%8Dka_energija" title="Kinetička energija – Bosnian" lang="bs" hreflang="bs" data-title="Kinetička energija" data-language-autonym="Bosanski" data-language-local-name="Bosnian" 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/Energia_cin%C3%A8tica" title="Energia cinètica – Catalan" lang="ca" hreflang="ca" data-title="Energia cinètica" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%9A%D0%B8%D0%BD%D0%B5%D1%82%D0%B8%D0%BA%C4%83%D0%BB%D0%BB%D0%B0_%D1%8D%D0%BD%D0%B5%D1%80%D0%B3%D0%B8" title="Кинетикăлла энерги – Chuvash" lang="cv" hreflang="cv" data-title="Кинетикăлла энерги" data-language-autonym="Чӑвашла" data-language-local-name="Chuvash" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Kinetick%C3%A1_energie" title="Kinetická energie – Czech" lang="cs" hreflang="cs" data-title="Kinetická energie" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-sn mw-list-item"><a href="https://sn.wikipedia.org/wiki/Simba_reNhekairo" title="Simba reNhekairo – Shona" lang="sn" hreflang="sn" data-title="Simba reNhekairo" data-language-autonym="ChiShona" data-language-local-name="Shona" class="interlanguage-link-target"><span>ChiShona</span></a></li><li class="interlanguage-link interwiki-cy mw-list-item"><a href="https://cy.wikipedia.org/wiki/Egni_cinetig" title="Egni cinetig – Welsh" lang="cy" hreflang="cy" data-title="Egni cinetig" data-language-autonym="Cymraeg" data-language-local-name="Welsh" class="interlanguage-link-target"><span>Cymraeg</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Kinetisk_energi" title="Kinetisk energi – Danish" lang="da" hreflang="da" data-title="Kinetisk energi" data-language-autonym="Dansk" data-language-local-name="Danish" 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/Kinetische_Energie" title="Kinetische Energie – German" lang="de" hreflang="de" data-title="Kinetische Energie" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Kineetiline_energia" title="Kineetiline energia – Estonian" lang="et" hreflang="et" data-title="Kineetiline energia" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%9A%CE%B9%CE%BD%CE%B7%CF%84%CE%B9%CE%BA%CE%AE_%CE%B5%CE%BD%CE%AD%CF%81%CE%B3%CE%B5%CE%B9%CE%B1" title="Κινητική ενέργεια – Greek" lang="el" hreflang="el" data-title="Κινητική ενέργεια" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Energ%C3%ADa_cin%C3%A9tica" title="Energía cinética – Spanish" lang="es" hreflang="es" data-title="Energía cinética" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Kineta_energio" title="Kineta energio – Esperanto" lang="eo" hreflang="eo" data-title="Kineta energio" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Energia_zinetiko" title="Energia zinetiko – Basque" lang="eu" hreflang="eu" data-title="Energia zinetiko" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%A7%D9%86%D8%B1%DA%98%DB%8C_%D8%AC%D9%86%D8%A8%D8%B4%DB%8C" title="انرژی جنبشی – Persian" lang="fa" hreflang="fa" data-title="انرژی جنبشی" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/%C3%89nergie_cin%C3%A9tique" title="Énergie cinétique – French" lang="fr" hreflang="fr" data-title="Énergie cinétique" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-fy mw-list-item"><a href="https://fy.wikipedia.org/wiki/Kinetyske_energy" title="Kinetyske energy – Western Frisian" lang="fy" hreflang="fy" data-title="Kinetyske energy" data-language-autonym="Frysk" data-language-local-name="Western Frisian" class="interlanguage-link-target"><span>Frysk</span></a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/Fuinneamh_cin%C3%A9iteach" title="Fuinneamh cinéiteach – Irish" lang="ga" hreflang="ga" data-title="Fuinneamh cinéiteach" data-language-autonym="Gaeilge" data-language-local-name="Irish" class="interlanguage-link-target"><span>Gaeilge</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Enerx%C3%ADa_cin%C3%A9tica" title="Enerxía cinética – Galician" lang="gl" hreflang="gl" data-title="Enerxía cinética" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%9A%B4%EB%8F%99_%EC%97%90%EB%84%88%EC%A7%80" title="운동 에너지 – Korean" lang="ko" hreflang="ko" data-title="운동 에너지" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D4%BF%D5%AB%D5%B6%D5%A5%D5%BF%D5%AB%D5%AF_%D5%A7%D5%B6%D5%A5%D6%80%D5%A3%D5%AB%D5%A1" title="Կինետիկ էներգիա – Armenian" lang="hy" hreflang="hy" data-title="Կինետիկ էներգիա" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%97%E0%A4%A4%E0%A4%BF%E0%A4%9C_%E0%A4%8A%E0%A4%B0%E0%A5%8D%E0%A4%9C%E0%A4%BE" title="गतिज ऊर्जा – Hindi" lang="hi" hreflang="hi" data-title="गतिज ऊर्जा" 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/Kineti%C4%8Dka_energija" title="Kinetička energija – Croatian" lang="hr" hreflang="hr" data-title="Kinetička energija" data-language-autonym="Hrvatski" data-language-local-name="Croatian" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-io mw-list-item"><a href="https://io.wikipedia.org/wiki/Cinetik_energio" title="Cinetik energio – Ido" lang="io" hreflang="io" data-title="Cinetik energio" data-language-autonym="Ido" data-language-local-name="Ido" class="interlanguage-link-target"><span>Ido</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Energi_kinetik" title="Energi kinetik – Indonesian" lang="id" hreflang="id" data-title="Energi kinetik" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-zu mw-list-item"><a href="https://zu.wikipedia.org/wiki/Impekumpeku" title="Impekumpeku – Zulu" lang="zu" hreflang="zu" data-title="Impekumpeku" data-language-autonym="IsiZulu" data-language-local-name="Zulu" class="interlanguage-link-target"><span>IsiZulu</span></a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/Hreyfiorka" title="Hreyfiorka – Icelandic" lang="is" hreflang="is" data-title="Hreyfiorka" data-language-autonym="Íslenska" data-language-local-name="Icelandic" 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/Energia_cinetica" title="Energia cinetica – Italian" lang="it" hreflang="it" data-title="Energia cinetica" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%90%D7%A0%D7%A8%D7%92%D7%99%D7%94_%D7%A7%D7%99%D7%A0%D7%98%D7%99%D7%AA" title="אנרגיה קינטית – Hebrew" lang="he" hreflang="he" data-title="אנרגיה קינטית" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-jv mw-list-item"><a href="https://jv.wikipedia.org/wiki/Energi_kinetis" title="Energi kinetis – Javanese" lang="jv" hreflang="jv" data-title="Energi kinetis" data-language-autonym="Jawa" data-language-local-name="Javanese" class="interlanguage-link-target"><span>Jawa</span></a></li><li class="interlanguage-link interwiki-kn mw-list-item"><a href="https://kn.wikipedia.org/wiki/%E0%B2%9A%E0%B2%B2%E0%B2%A8%E0%B2%B6%E0%B2%95%E0%B3%8D%E0%B2%A4%E0%B2%BF" title="ಚಲನಶಕ್ತಿ – Kannada" lang="kn" hreflang="kn" data-title="ಚಲನಶಕ್ತಿ" data-language-autonym="ಕನ್ನಡ" data-language-local-name="Kannada" class="interlanguage-link-target"><span>ಕನ್ನಡ</span></a></li><li class="interlanguage-link interwiki-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/%E1%83%99%E1%83%98%E1%83%9C%E1%83%94%E1%83%A2%E1%83%98%E1%83%99%E1%83%A3%E1%83%A0%E1%83%98_%E1%83%94%E1%83%9C%E1%83%94%E1%83%A0%E1%83%92%E1%83%98%E1%83%90" title="კინეტიკური ენერგია – Georgian" lang="ka" hreflang="ka" data-title="კინეტიკური ენერგია" data-language-autonym="ქართული" data-language-local-name="Georgian" 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%9A%D0%B8%D0%BD%D0%B5%D1%82%D0%B8%D0%BA%D0%B0%D0%BB%D1%8B%D2%9B_%D1%8D%D0%BD%D0%B5%D1%80%D0%B3%D0%B8%D1%8F" title="Кинетикалық энергия – Kazakh" lang="kk" hreflang="kk" data-title="Кинетикалық энергия" data-language-autonym="Қазақша" data-language-local-name="Kazakh" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-ht mw-list-item"><a href="https://ht.wikipedia.org/wiki/En%C3%A8ji_sinetik" title="Enèji sinetik – Haitian Creole" lang="ht" hreflang="ht" data-title="Enèji sinetik" data-language-autonym="Kreyòl ayisyen" data-language-local-name="Haitian Creole" class="interlanguage-link-target"><span>Kreyòl ayisyen</span></a></li><li class="interlanguage-link interwiki-gcr mw-list-item"><a href="https://gcr.wikipedia.org/wiki/%C3%89nerji_sin%C3%A9tik" title="Énerji sinétik – Guianan Creole" lang="gcr" hreflang="gcr" data-title="Énerji sinétik" data-language-autonym="Kriyòl gwiyannen" data-language-local-name="Guianan Creole" class="interlanguage-link-target"><span>Kriyòl gwiyannen</span></a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Energia_cinetica" title="Energia cinetica – Latin" lang="la" hreflang="la" data-title="Energia cinetica" data-language-autonym="Latina" data-language-local-name="Latin" class="interlanguage-link-target"><span>Latina</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Kin%C4%93tisk%C4%81_ener%C4%A3ija" title="Kinētiskā enerģija – Latvian" lang="lv" hreflang="lv" data-title="Kinētiskā enerģija" data-language-autonym="Latviešu" data-language-local-name="Latvian" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Kinetin%C4%97_energija" title="Kinetinė energija – Lithuanian" lang="lt" hreflang="lt" data-title="Kinetinė energija" data-language-autonym="Lietuvių" data-language-local-name="Lithuanian" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-lfn mw-list-item"><a href="https://lfn.wikipedia.org/wiki/Enerjia_cinetica" title="Enerjia cinetica – Lingua Franca Nova" lang="lfn" hreflang="lfn" data-title="Enerjia cinetica" 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-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Mozg%C3%A1si_energia" title="Mozgási energia – Hungarian" lang="hu" hreflang="hu" data-title="Mozgási energia" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%9A%D0%B8%D0%BD%D0%B5%D1%82%D0%B8%D1%87%D0%BA%D0%B0_%D0%B5%D0%BD%D0%B5%D1%80%D0%B3%D0%B8%D1%98%D0%B0" title="Кинетичка енергија – Macedonian" lang="mk" hreflang="mk" data-title="Кинетичка енергија" data-language-autonym="Македонски" data-language-local-name="Macedonian" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-mg mw-list-item"><a href="https://mg.wikipedia.org/wiki/Angovon-kalaky" title="Angovon-kalaky – Malagasy" lang="mg" hreflang="mg" data-title="Angovon-kalaky" data-language-autonym="Malagasy" data-language-local-name="Malagasy" class="interlanguage-link-target"><span>Malagasy</span></a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%97%E0%B4%A4%E0%B4%BF%E0%B4%95%E0%B5%8B%E0%B5%BC%E0%B4%9C%E0%B5%8D%E0%B4%9C%E0%B4%82" title="ഗതികോർജ്ജം – Malayalam" lang="ml" hreflang="ml" data-title="ഗതികോർജ്ജം" data-language-autonym="മലയാളം" data-language-local-name="Malayalam" class="interlanguage-link-target"><span>മലയാളം</span></a></li><li class="interlanguage-link interwiki-mr mw-list-item"><a href="https://mr.wikipedia.org/wiki/%E0%A4%97%E0%A4%A4%E0%A4%BF%E0%A4%9C_%E0%A4%8A%E0%A4%B0%E0%A5%8D%E0%A4%9C%E0%A4%BE" title="गतिज ऊर्जा – Marathi" lang="mr" hreflang="mr" data-title="गतिज ऊर्जा" data-language-autonym="मराठी" data-language-local-name="Marathi" 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/Tenaga_kinetik" title="Tenaga kinetik – Malay" lang="ms" hreflang="ms" data-title="Tenaga kinetik" data-language-autonym="Bahasa Melayu" data-language-local-name="Malay" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-my mw-list-item"><a href="https://my.wikipedia.org/wiki/%E1%80%A1%E1%80%9B%E1%80%BD%E1%80%B1%E1%80%B7%E1%80%85%E1%80%BD%E1%80%99%E1%80%BA%E1%80%B8%E1%80%A1%E1%80%84%E1%80%BA" title="အရွေ့စွမ်းအင် – Burmese" lang="my" hreflang="my" data-title="အရွေ့စွမ်းအင်" data-language-autonym="မြန်မာဘာသာ" data-language-local-name="Burmese" class="interlanguage-link-target"><span>မြန်မာဘာသာ</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Kinetische_energie" title="Kinetische energie – Dutch" lang="nl" hreflang="nl" data-title="Kinetische energie" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ne mw-list-item"><a href="https://ne.wikipedia.org/wiki/%E0%A4%9A%E0%A4%BE%E0%A4%B2_%E0%A4%B6%E0%A4%95%E0%A5%8D%E0%A4%A4%E0%A4%BF" title="चाल शक्ति – Nepali" lang="ne" hreflang="ne" data-title="चाल शक्ति" data-language-autonym="नेपाली" data-language-local-name="Nepali" class="interlanguage-link-target"><span>नेपाली</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E9%81%8B%E5%8B%95%E3%82%A8%E3%83%8D%E3%83%AB%E3%82%AE%E3%83%BC" title="運動エネルギー – Japanese" lang="ja" hreflang="ja" data-title="運動エネルギー" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Kinetisk_energi" title="Kinetisk energi – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Kinetisk energi" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Kinetisk_energi" title="Kinetisk energi – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Kinetisk energi" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegian Nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-oc mw-list-item"><a href="https://oc.wikipedia.org/wiki/Energia_cinetica" title="Energia cinetica – Occitan" lang="oc" hreflang="oc" data-title="Energia cinetica" data-language-autonym="Occitan" data-language-local-name="Occitan" 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/Anniisaa_sochii" title="Anniisaa sochii – Oromo" lang="om" hreflang="om" data-title="Anniisaa sochii" data-language-autonym="Oromoo" data-language-local-name="Oromo" class="interlanguage-link-target"><span>Oromoo</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Kinetik_energiya" title="Kinetik energiya – Uzbek" lang="uz" hreflang="uz" data-title="Kinetik energiya" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="Uzbek" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></a></li><li class="interlanguage-link interwiki-pa mw-list-item"><a href="https://pa.wikipedia.org/wiki/%E0%A8%97%E0%A8%A4%E0%A8%BF%E0%A8%9C_%E0%A8%8A%E0%A8%B0%E0%A8%9C%E0%A8%BE" title="ਗਤਿਜ ਊਰਜਾ – Punjabi" lang="pa" hreflang="pa" data-title="ਗਤਿਜ ਊਰਜਾ" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="Punjabi" class="interlanguage-link-target"><span>ਪੰਜਾਬੀ</span></a></li><li class="interlanguage-link interwiki-pnb mw-list-item"><a href="https://pnb.wikipedia.org/wiki/%DA%A9%D8%A7%D8%A6%DB%8C%D9%86%DB%8C%D9%B9%DA%A9_%D8%AA%D9%88%D8%A7%D9%86%D8%A7%D8%A6%DB%8C" title="کائینیٹک توانائی – Western Punjabi" lang="pnb" hreflang="pnb" data-title="کائینیٹک توانائی" data-language-autonym="پنجابی" data-language-local-name="Western Punjabi" class="interlanguage-link-target"><span>پنجابی</span></a></li><li class="interlanguage-link interwiki-jam mw-list-item"><a href="https://jam.wikipedia.org/wiki/Kinetik_enaji" title="Kinetik enaji – Jamaican Creole English" lang="jam" hreflang="jam" data-title="Kinetik enaji" data-language-autonym="Patois" data-language-local-name="Jamaican Creole English" class="interlanguage-link-target"><span>Patois</span></a></li><li class="interlanguage-link interwiki-pms mw-list-item"><a href="https://pms.wikipedia.org/wiki/Energ%C3%ACa_cin%C3%A9tica" title="Energìa cinética – Piedmontese" lang="pms" hreflang="pms" data-title="Energìa cinética" data-language-autonym="Piemontèis" data-language-local-name="Piedmontese" class="interlanguage-link-target"><span>Piemontèis</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Energia_kinetyczna" title="Energia kinetyczna – Polish" lang="pl" hreflang="pl" data-title="Energia kinetyczna" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Energia_cin%C3%A9tica" title="Energia cinética – Portuguese" lang="pt" hreflang="pt" data-title="Energia cinética" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Energie_cinetic%C4%83" title="Energie cinetică – Romanian" lang="ro" hreflang="ro" data-title="Energie cinetică" data-language-autonym="Română" data-language-local-name="Romanian" 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%9A%D0%B8%D0%BD%D0%B5%D1%82%D0%B8%D1%87%D0%B5%D1%81%D0%BA%D0%B0%D1%8F_%D1%8D%D0%BD%D0%B5%D1%80%D0%B3%D0%B8%D1%8F" title="Кинетическая энергия – Russian" lang="ru" hreflang="ru" data-title="Кинетическая энергия" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sco mw-list-item"><a href="https://sco.wikipedia.org/wiki/Kinetic_energy" title="Kinetic energy – Scots" lang="sco" hreflang="sco" data-title="Kinetic energy" data-language-autonym="Scots" data-language-local-name="Scots" class="interlanguage-link-target"><span>Scots</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Energjia_kinetike" title="Energjia kinetike – Albanian" lang="sq" hreflang="sq" data-title="Energjia kinetike" data-language-autonym="Shqip" data-language-local-name="Albanian" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-si mw-list-item"><a href="https://si.wikipedia.org/wiki/%E0%B6%A0%E0%B7%8F%E0%B6%BD%E0%B6%9A_%E0%B7%81%E0%B6%9A%E0%B7%8A%E0%B6%AD%E0%B7%92%E0%B6%BA" title="චාලක ශක්තිය – Sinhala" lang="si" hreflang="si" data-title="චාලක ශක්තිය" data-language-autonym="සිංහල" data-language-local-name="Sinhala" 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/Kinetic_energy" title="Kinetic energy – Simple English" lang="en-simple" hreflang="en-simple" data-title="Kinetic energy" 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-sd mw-list-item"><a href="https://sd.wikipedia.org/wiki/%D8%AD%D8%B1%DA%AA%D9%8A_%D8%AA%D9%88%D8%A7%D9%86%D8%A7%D8%A6%D9%8A" title="حرڪي توانائي – Sindhi" lang="sd" hreflang="sd" data-title="حرڪي توانائي" data-language-autonym="سنڌي" data-language-local-name="Sindhi" class="interlanguage-link-target"><span>سنڌي</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Kinetick%C3%A1_energia" title="Kinetická energia – Slovak" lang="sk" hreflang="sk" data-title="Kinetická energia" data-language-autonym="Slovenčina" data-language-local-name="Slovak" 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/Kineti%C4%8Dna_energija" title="Kinetična energija – Slovenian" lang="sl" hreflang="sl" data-title="Kinetična energija" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-so mw-list-item"><a href="https://so.wikipedia.org/wiki/Awood_Socota" title="Awood Socota – Somali" lang="so" hreflang="so" data-title="Awood Socota" data-language-autonym="Soomaaliga" data-language-local-name="Somali" class="interlanguage-link-target"><span>Soomaaliga</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%9A%D0%B8%D0%BD%D0%B5%D1%82%D0%B8%D1%87%D0%BA%D0%B0_%D0%B5%D0%BD%D0%B5%D1%80%D0%B3%D0%B8%D1%98%D0%B0" title="Кинетичка енергија – Serbian" lang="sr" hreflang="sr" data-title="Кинетичка енергија" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Kineti%C4%8Dka_energija" title="Kinetička energija – Serbo-Croatian" lang="sh" hreflang="sh" data-title="Kinetička energija" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="Serbo-Croatian" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Liike-energia" title="Liike-energia – Finnish" lang="fi" hreflang="fi" data-title="Liike-energia" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Kinetisk_energi" title="Kinetisk energi – Swedish" lang="sv" hreflang="sv" data-title="Kinetisk energi" data-language-autonym="Svenska" data-language-local-name="Swedish" 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%87%E0%AE%AF%E0%AE%95%E0%AF%8D%E0%AE%95_%E0%AE%86%E0%AE%B1%E0%AF%8D%E0%AE%B1%E0%AE%B2%E0%AF%8D" title="இயக்க ஆற்றல் – Tamil" lang="ta" hreflang="ta" data-title="இயக்க ஆற்றல்" data-language-autonym="தமிழ்" data-language-local-name="Tamil" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-tt mw-list-item"><a href="https://tt.wikipedia.org/wiki/Kinetik_energi%C3%A4" title="Kinetik energiä – Tatar" lang="tt" hreflang="tt" data-title="Kinetik energiä" data-language-autonym="Татарча / tatarça" data-language-local-name="Tatar" class="interlanguage-link-target"><span>Татарча / 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.infobox-full-data:not(.notheme)>div:not(.notheme)[style]{background:#1f1f23!important;color:#f8f9fa}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .infobox-full-data:not(.notheme) div:not(.notheme){background:#1f1f23!important;color:#f8f9fa}}@media(min-width:640px){body.skin--responsive .mw-parser-output .infobox-table{display:table!important}body.skin--responsive .mw-parser-output .infobox-table>caption{display:table-caption!important}body.skin--responsive .mw-parser-output .infobox-table>tbody{display:table-row-group}body.skin--responsive .mw-parser-output .infobox-table tr{display:table-row!important}body.skin--responsive .mw-parser-output .infobox-table th,body.skin--responsive .mw-parser-output .infobox-table td{padding-left:inherit;padding-right:inherit}}</style><table class="infobox"><tbody><tr><th colspan="2" class="infobox-above">Kinetic energy</th></tr><tr><td colspan="2" class="infobox-image"><span typeof="mw:File"><a href="/wiki/File:Wooden_roller_coaster_txgi.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/30/Wooden_roller_coaster_txgi.jpg/220px-Wooden_roller_coaster_txgi.jpg" decoding="async" width="220" height="271" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/30/Wooden_roller_coaster_txgi.jpg/330px-Wooden_roller_coaster_txgi.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/30/Wooden_roller_coaster_txgi.jpg/440px-Wooden_roller_coaster_txgi.jpg 2x" data-file-width="800" data-file-height="987" /></a></span><div class="infobox-caption">The cars of a <a href="/wiki/Roller_coaster" title="Roller coaster">roller coaster</a> reach their maximum kinetic energy when at the bottom of the path. When they start rising, the kinetic energy begins to be converted to gravitational <a href="/wiki/Potential_energy" title="Potential energy">potential energy</a>. The sum of kinetic and potential energy in the system remains constant, ignoring losses to <a href="/wiki/Friction" title="Friction">friction</a>.</div></td></tr><tr><th scope="row" class="infobox-label"><div style="display: inline-block; line-height: 1.2em; padding: .1em 0;">Common symbols</div></th><td class="infobox-data">KE, <i>E</i><sub>k</sub>, <i>K</i> or T</td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/SI_unit" class="mw-redirect" title="SI unit">SI unit</a></th><td class="infobox-data"><a href="/wiki/Joule" title="Joule">joule</a> (J)</td></tr><tr><th scope="row" class="infobox-label"><div style="display: inline-block; line-height: 1.2em; padding: .1em 0;">Derivations from<br />other quantities</div></th><td class="infobox-data"><i>E</i><sub>k</sub> = <style data-mw-deduplicate="TemplateStyles:r1214402035">.mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}</style><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span><i><a href="/wiki/Mass" title="Mass">m</a><a href="/wiki/Speed" title="Speed">v</a></i><span style="padding-left:0.12em;"><sup>2</sup></span> <br /> <i>E</i><sub>k</sub> = <i>E</i><sub>t</sub> + <i>E</i><sub>r</sub></td></tr></tbody></table> <style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist 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rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><table class="sidebar sidebar-collapse nomobile nowraplinks"><tbody><tr><td class="sidebar-pretitle">Part of a series on</td></tr><tr><th class="sidebar-title-with-pretitle" style="padding-left:0.9em;padding-right:0.9em;"><a href="/wiki/Classical_mechanics" title="Classical mechanics">Classical mechanics</a></th></tr><tr><td class="sidebar-image"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\textbf {F}}={\frac {d\mathbf {p} }{dt}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext mathvariant="bold">F</mtext> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\textbf {F}}={\frac {d\mathbf {p} }{dt}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c2ad0a6d6780c3abc5247abd82bd8a2249d56ff3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:8.318ex; height:5.509ex;" alt="{\displaystyle {\textbf {F}}={\frac {d\mathbf {p} }{dt}}}"></span><div class="sidebar-caption" style="font-size:90%;padding:0.6em 0;font-style:italic;"><a href="/wiki/Second_law_of_motion" class="mw-redirect" title="Second law of motion">Second law of motion</a></div></td></tr><tr><th class="sidebar-heading" style="font-weight: bold; display:block;margin-bottom:1.0em;"> <div class="hlist"> <ul><li><a href="/wiki/History_of_classical_mechanics" title="History of classical mechanics">History</a></li> <li><a href="/wiki/Timeline_of_classical_mechanics" title="Timeline of classical mechanics">Timeline</a></li> <li><a href="/wiki/List_of_textbooks_on_classical_mechanics_and_quantum_mechanics" title="List of textbooks on classical mechanics and quantum mechanics">Textbooks</a></li></ul> </div></th></tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="border-bottom: 1px solid black;text-align:center;;color: var(--color-base)">Branches</div><div class="sidebar-list-content mw-collapsible-content plainlist" style="padding-top:0.35em;"><div class="hlist"> <ul><li><a href="/wiki/Applied_mechanics" title="Applied mechanics">Applied</a></li> <li><a href="/wiki/Celestial_mechanics" title="Celestial mechanics">Celestial</a></li> <li><a href="/wiki/Continuum_mechanics" title="Continuum mechanics">Continuum</a></li> <li><a href="/wiki/Analytical_dynamics" class="mw-redirect" title="Analytical dynamics">Dynamics</a></li> <li><a href="/wiki/Classical_field_theory" title="Classical field theory">Field theory</a></li> <li><a href="/wiki/Kinematics" title="Kinematics">Kinematics</a></li> <li><a href="/wiki/Kinetics_(physics)" title="Kinetics (physics)">Kinetics</a></li> <li><a href="/wiki/Statics" title="Statics">Statics</a></li> <li><a href="/wiki/Statistical_mechanics" title="Statistical mechanics">Statistical mechanics</a></li></ul> </div></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="border-bottom: 1px solid black;text-align:center;;color: var(--color-base)">Fundamentals</div><div class="sidebar-list-content mw-collapsible-content plainlist" style="padding-top:0.35em;"><div class="hlist"> <ul><li><a href="/wiki/Acceleration" title="Acceleration">Acceleration</a></li> <li><a href="/wiki/Angular_momentum" title="Angular momentum">Angular momentum</a></li> <li><a href="/wiki/Couple_(mechanics)" title="Couple (mechanics)">Couple</a></li> <li><a href="/wiki/D%27Alembert%27s_principle" title="D'Alembert's principle">D'Alembert's principle</a></li> <li><a href="/wiki/Energy" title="Energy">Energy</a> <ul><li><a class="mw-selflink-fragment" href="#Newtonian_kinetic_energy">kinetic</a></li> <li><a href="/wiki/Potential_energy" title="Potential energy">potential</a></li></ul></li> <li><a href="/wiki/Force" title="Force">Force</a></li> <li><a href="/wiki/Frame_of_reference" title="Frame of reference">Frame of reference</a></li> <li><a href="/wiki/Inertial_frame_of_reference" title="Inertial frame of reference">Inertial frame of reference</a></li> <li><a href="/wiki/Impulse_(physics)" title="Impulse (physics)">Impulse</a></li> <li><span class="nowrap"><a href="/wiki/Inertia" title="Inertia">Inertia</a> / <a href="/wiki/Moment_of_inertia" title="Moment of inertia">Moment of inertia</a></span></li> <li><a href="/wiki/Mass" title="Mass">Mass</a></li> <li><br /><a href="/wiki/Mechanical_power_(physics)" class="mw-redirect" title="Mechanical power (physics)">Mechanical power</a></li> <li><a href="/wiki/Work_(physics)" title="Work (physics)">Mechanical work</a></li> <li><br /><a href="/wiki/Moment_(physics)" title="Moment (physics)">Moment</a></li> <li><a href="/wiki/Momentum" title="Momentum">Momentum</a></li> <li><a href="/wiki/Space" title="Space">Space</a></li> <li><a href="/wiki/Speed" title="Speed">Speed</a></li> <li><a href="/wiki/Time" title="Time">Time</a></li> <li><a href="/wiki/Torque" title="Torque">Torque</a></li> <li><a href="/wiki/Velocity" title="Velocity">Velocity</a></li> <li><a href="/wiki/Virtual_work" title="Virtual work">Virtual work</a></li></ul> </div></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="border-bottom: 1px solid black;text-align:center;;color: var(--color-base)">Formulations</div><div class="sidebar-list-content mw-collapsible-content plainlist" style="padding-top:0.35em;"> <ul><li><div style="display: inline-block; line-height: 1.2em; padding: .1em 0;"><b><a href="/wiki/Newton%27s_laws_of_motion" title="Newton's laws of motion">Newton's laws of motion</a></b></div></li> <li><div style="display: inline-block; line-height: 1.2em; padding: .1em 0;"><b><a href="/wiki/Analytical_mechanics" title="Analytical mechanics">Analytical mechanics</a></b> <div class="plainlist"><ul><li><a href="/wiki/Lagrangian_mechanics" title="Lagrangian mechanics">Lagrangian mechanics</a></li><li><a href="/wiki/Hamiltonian_mechanics" title="Hamiltonian mechanics">Hamiltonian mechanics</a></li><li><a href="/wiki/Routhian_mechanics" title="Routhian mechanics">Routhian mechanics</a></li><li><a href="/wiki/Hamilton%E2%80%93Jacobi_equation" title="Hamilton–Jacobi equation">Hamilton–Jacobi equation</a></li><li><a href="/wiki/Appell%27s_equation_of_motion" title="Appell's equation of motion">Appell's equation of motion</a></li><li><a href="/wiki/Koopman%E2%80%93von_Neumann_classical_mechanics" title="Koopman–von Neumann classical mechanics">Koopman–von Neumann mechanics</a></li></ul></div></div></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="border-bottom: 1px solid black;text-align:center;;color: var(--color-base)">Core topics</div><div class="sidebar-list-content mw-collapsible-content plainlist" style="padding-top:0.35em;"><div class="hlist"> <ul><li><a href="/wiki/Damping" title="Damping">Damping</a></li> <li><a href="/wiki/Displacement_(geometry)" title="Displacement (geometry)">Displacement</a></li> <li><a href="/wiki/Equations_of_motion" title="Equations of motion">Equations of motion</a></li> <li><a href="/wiki/Euler%27s_laws_of_motion" title="Euler's laws of motion"><span class="wrap">Euler's laws of motion</span></a></li> <li><a href="/wiki/Fictitious_force" title="Fictitious force">Fictitious force</a></li> <li><a href="/wiki/Friction" title="Friction">Friction</a></li> <li><a href="/wiki/Harmonic_oscillator" title="Harmonic oscillator">Harmonic oscillator</a></li></ul> </div> <ul><li><span class="nowrap"><a href="/wiki/Inertial_frame_of_reference" title="Inertial frame of reference">Inertial</a> / <a href="/wiki/Non-inertial_reference_frame" title="Non-inertial reference frame">Non-inertial reference frame</a></span></li></ul> <div class="hlist"> <ul><li><a href="/wiki/Motion" title="Motion">Motion</a> (<a href="/wiki/Linear_motion" title="Linear motion">linear</a>)</li> <li><a href="/wiki/Newton%27s_law_of_universal_gravitation" title="Newton's law of universal gravitation"><span class="wrap">Newton's law of universal gravitation</span></a></li> <li><a href="/wiki/Newton%27s_laws_of_motion" title="Newton's laws of motion">Newton's laws of motion</a></li> <li><a href="/wiki/Relative_velocity" title="Relative velocity">Relative velocity</a></li> <li><a href="/wiki/Rigid_body" title="Rigid body">Rigid body</a> <ul><li><a href="/wiki/Rigid_body_dynamics" title="Rigid body dynamics">dynamics</a></li> <li><a href="/wiki/Euler%27s_equations_(rigid_body_dynamics)" title="Euler's equations (rigid body dynamics)">Euler's equations</a></li></ul></li> <li><a href="/wiki/Simple_harmonic_motion" title="Simple harmonic motion">Simple harmonic motion</a></li> <li><a href="/wiki/Vibration" title="Vibration">Vibration</a></li></ul> </div></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="border-bottom: 1px solid black;text-align:center;;color: var(--color-base)"><a href="/wiki/Rotation_around_a_fixed_axis" title="Rotation around a fixed axis">Rotation</a></div><div class="sidebar-list-content mw-collapsible-content plainlist" style="padding-top:0.35em;"><div class="hlist"> <ul><li><a href="/wiki/Circular_motion" title="Circular motion">Circular motion</a></li> <li><a href="/wiki/Rotating_reference_frame" title="Rotating reference frame">Rotating reference frame</a></li> <li><a href="/wiki/Centripetal_force" title="Centripetal force">Centripetal force</a></li> <li><a href="/wiki/Centrifugal_force" title="Centrifugal force">Centrifugal force</a> <ul><li><a href="/wiki/Reactive_centrifugal_force" title="Reactive centrifugal force">reactive</a></li></ul></li> <li><a href="/wiki/Coriolis_force" title="Coriolis force">Coriolis force</a></li> <li><a href="/wiki/Pendulum_(mechanics)" title="Pendulum (mechanics)">Pendulum</a></li> <li><a href="/wiki/Tangential_speed" title="Tangential speed">Tangential speed</a></li> <li><a href="/wiki/Rotational_frequency" title="Rotational frequency">Rotational frequency</a></li></ul> </div> <ul><li><a href="/wiki/Angular_acceleration" title="Angular acceleration">Angular acceleration</a> / <a href="/wiki/Angular_displacement" title="Angular displacement">displacement</a> / <a href="/wiki/Angular_frequency" title="Angular frequency">frequency</a> / <a href="/wiki/Angular_velocity" title="Angular velocity">velocity</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="border-bottom: 1px solid black;text-align:center;;color: var(--color-base)">Scientists</div><div class="sidebar-list-content mw-collapsible-content plainlist" style="padding-top:0.35em;"><div class="hlist"> <ul><li><a href="/wiki/Johannes_Kepler" title="Johannes Kepler">Kepler</a></li> <li><a href="/wiki/Galileo_Galilei" title="Galileo Galilei">Galileo</a></li> <li><a href="/wiki/Christiaan_Huygens" title="Christiaan Huygens">Huygens</a></li> <li><a href="/wiki/Isaac_Newton" title="Isaac Newton">Newton</a></li> <li><a href="/wiki/Jeremiah_Horrocks" title="Jeremiah Horrocks">Horrocks</a></li> <li><a href="/wiki/Edmond_Halley" title="Edmond Halley">Halley</a></li> <li><a href="/wiki/Pierre_Louis_Maupertuis" title="Pierre Louis Maupertuis">Maupertuis</a></li> <li><a href="/wiki/Daniel_Bernoulli" title="Daniel Bernoulli">Daniel Bernoulli</a></li> <li><a href="/wiki/Johann_Bernoulli" title="Johann Bernoulli">Johann Bernoulli</a></li> <li><a href="/wiki/Leonhard_Euler" title="Leonhard Euler">Euler</a></li> <li><a href="/wiki/Jean_le_Rond_d%27Alembert" title="Jean le Rond d'Alembert">d'Alembert</a></li> <li><a href="/wiki/Alexis_Clairaut" title="Alexis Clairaut">Clairaut</a></li> <li><a href="/wiki/Joseph-Louis_Lagrange" title="Joseph-Louis Lagrange">Lagrange</a></li> <li><a href="/wiki/Pierre-Simon_Laplace" title="Pierre-Simon Laplace">Laplace</a></li> <li><a href="/wiki/Sim%C3%A9on_Denis_Poisson" title="Siméon Denis Poisson">Poisson</a></li> <li><a href="/wiki/William_Rowan_Hamilton" title="William Rowan Hamilton">Hamilton</a></li> <li><a href="/wiki/Carl_Gustav_Jacob_Jacobi" 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href="/wiki/Template:Classical_mechanics" title="Template:Classical mechanics"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Classical_mechanics" title="Template talk:Classical mechanics"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Classical_mechanics" title="Special:EditPage/Template:Classical mechanics"><abbr title="Edit this template">e</abbr></a></li></ul></div></td></tr></tbody></table> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Emilie_Chatelet_portrait_by_Latour.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/ba/Emilie_Chatelet_portrait_by_Latour.jpg/220px-Emilie_Chatelet_portrait_by_Latour.jpg" decoding="async" width="220" height="263" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/ba/Emilie_Chatelet_portrait_by_Latour.jpg/330px-Emilie_Chatelet_portrait_by_Latour.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/ba/Emilie_Chatelet_portrait_by_Latour.jpg/440px-Emilie_Chatelet_portrait_by_Latour.jpg 2x" data-file-width="866" data-file-height="1037" /></a><figcaption><a href="/wiki/%C3%89milie_du_Ch%C3%A2telet" title="Émilie du Châtelet">Émilie du Châtelet</a> (1706–1749) was the first to publish the relation for kinetic energy <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{\text{kin}}\propto mv^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>kin</mtext> </mrow> </msub> <mo>∝</mo> <mi>m</mi> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{\text{kin}}\propto mv^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/167288255099338e01843213778f1f697d1a3a38" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.508ex; height:3.009ex;" alt="{\displaystyle E_{\text{kin}}\propto mv^{2}}"></span>. This means that an object with twice the speed hits four times harder. (Portrait by <a href="/wiki/Maurice_Quentin_de_La_Tour" title="Maurice Quentin de La Tour">Maurice Quentin de La Tour</a>.)</figcaption></figure> <p>In <a href="/wiki/Physics" title="Physics">physics</a>, the <b>kinetic energy</b> of an object is the form of <a href="/wiki/Energy" title="Energy">energy</a> that it possesses due to its <a href="/wiki/Motion_(physics)" class="mw-redirect" title="Motion (physics)">motion</a>.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p><p>In <a href="/wiki/Classical_mechanics" title="Classical mechanics">classical mechanics</a>, the kinetic energy of a non-rotating object of <a href="/wiki/Mass" title="Mass">mass</a> <i>m</i> traveling at a <a href="/wiki/Speed" title="Speed">speed</a> <i>v</i> is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\frac {1}{2}}mv^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>m</mi> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle {\frac {1}{2}}mv^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/689701aed83c55c0c88edc47453833d1abeab74e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:5.88ex; height:3.509ex;" alt="{\textstyle {\frac {1}{2}}mv^{2}}"></span>.<sup id="cite_ref-R&H_2-0" class="reference"><a href="#cite_note-R&H-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p><p>The kinetic energy of an object is equal to the <a href="/wiki/Work_(physics)" title="Work (physics)">work</a>, or force (<a href="/wiki/Force" title="Force">F</a>) in the direction of motion times its displacement (<a href="/wiki/Displacement_(geometry)" title="Displacement (geometry)">s</a>), needed to accelerate the mass from <a href="/wiki/Rest_(physics)" class="mw-redirect" title="Rest (physics)">rest</a> to its stated <a href="/wiki/Velocity" title="Velocity">velocity</a>. Having gained this energy during its <a href="/wiki/Acceleration" title="Acceleration">acceleration</a>, it maintains this kinetic energy unless its speed changes. The same amount of work is done by the object when decelerating from its current <a href="/wiki/Speed" title="Speed">speed</a> to a state of rest.<sup id="cite_ref-R&H_2-1" class="reference"><a href="#cite_note-R&H-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p><p>The <a href="/wiki/International_System_of_Units" title="International System of Units">SI unit</a> of kinetic energy is the <a href="/wiki/Joule" title="Joule">joule</a>, while the <a href="/wiki/English_Engineering_Units" title="English Engineering Units">English unit</a> of kinetic energy is the <a href="/wiki/Foot-pound" class="mw-redirect" title="Foot-pound">foot-pound</a>. </p><p>In <a href="/wiki/Special_relativity" title="Special relativity">relativistic mechanics</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="{\textstyle {\frac {1}{2}}mv^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>m</mi> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle {\frac {1}{2}}mv^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/689701aed83c55c0c88edc47453833d1abeab74e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:5.88ex; height:3.509ex;" alt="{\textstyle {\frac {1}{2}}mv^{2}}"></span> is a good <a href="/wiki/Approximation" title="Approximation">approximation</a> of kinetic energy only when <i>v</i> is much less than the <a href="/wiki/Speed_of_light" title="Speed of light">speed of light</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="History_and_etymology">History and etymology</h2></div> <p>The adjective <i>kinetic</i> has its roots in the <a href="/wiki/Ancient_Greek" title="Ancient Greek">Greek</a> word κίνησις <i>kinesis</i>, meaning "motion". The dichotomy between kinetic energy and <a href="/wiki/Potential_energy" title="Potential energy">potential energy</a> can be traced back to <a href="/wiki/Aristotle" title="Aristotle">Aristotle</a>'s concepts of <a href="/wiki/Actuality_and_potentiality" class="mw-redirect" title="Actuality and potentiality">actuality and potentiality</a>.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p><p>The principle of <a href="/wiki/Classical_mechanics" title="Classical mechanics">classical mechanics</a> that <i>E</i> ∝ <i>mv</i><sup>2</sup> is conserved was first developed by <a href="/wiki/Gottfried_Leibniz" class="mw-redirect" title="Gottfried Leibniz">Gottfried Leibniz</a> and <a href="/wiki/Johann_Bernoulli" title="Johann Bernoulli">Johann Bernoulli</a>, who described kinetic energy as the <i>living force</i> or <i><a href="/wiki/Vis_viva" title="Vis viva">vis viva</a></i>.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 227">: 227 </span></sup> <a href="/wiki/Willem_%27s_Gravesande" title="Willem 's Gravesande">Willem 's Gravesande</a> of the Netherlands provided experimental evidence of this relationship in 1722. By dropping weights from different heights into a block of clay, Gravesande determined that their penetration depth was proportional to the square of their impact speed. <a href="/wiki/%C3%89milie_du_Ch%C3%A2telet" title="Émilie du Châtelet">Émilie du Châtelet</a> recognized the implications of the experiment and published an explanation.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p><p>The terms <i>kinetic energy</i> and <i>work</i> in their present scientific meanings date back to the mid-19th century. Early understandings of these ideas can be attributed to <a href="/wiki/Thomas_Young_(scientist)" title="Thomas Young (scientist)">Thomas Young</a>, who in his 1802 lecture to the Royal Society, was the first to use the term <i>energy</i> to refer to kinetic energy in its modern sense, instead of <i>vis viva</i>. <a href="/wiki/Gaspard-Gustave_Coriolis" class="mw-redirect" title="Gaspard-Gustave Coriolis">Gaspard-Gustave Coriolis</a> published in 1829 the paper titled <i>Du Calcul de l'Effet des Machines</i> outlining the mathematics of kinetic energy. <a href="/wiki/William_Thomson,_1st_Baron_Kelvin" class="mw-redirect" title="William Thomson, 1st Baron Kelvin">William Thomson</a>, later Lord Kelvin, is given the credit for coining the term "kinetic energy" c. 1849–1851.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> <a href="/wiki/William_Rankine" title="William Rankine">William Rankine</a>, who had introduced the term "potential energy" in 1853, and the phrase "actual energy" to complement it,<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> later cites <a href="/wiki/William_Thomson,_1st_Baron_Kelvin" class="mw-redirect" title="William Thomson, 1st Baron Kelvin">William Thomson</a> and <a href="/wiki/Peter_Tait_(physicist)" class="mw-redirect" title="Peter Tait (physicist)">Peter Tait</a> as substituting the word "kinetic" for "actual".<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Overview">Overview</h2></div> <p><a href="/wiki/Energy" title="Energy">Energy</a> occurs in many forms, including <a href="/wiki/Chemical_energy" title="Chemical energy">chemical energy</a>, <a href="/wiki/Thermal_energy" title="Thermal energy">thermal energy</a>, <a href="/wiki/Electromagnetic_radiation" title="Electromagnetic radiation">electromagnetic radiation</a>, <a href="/wiki/Gravitational_energy" title="Gravitational energy">gravitational energy</a>, <a href="/wiki/Electric_energy" class="mw-redirect" title="Electric energy">electric energy</a>, <a href="/wiki/Elastic_energy" title="Elastic energy">elastic energy</a>, <a href="/wiki/Nuclear_binding_energy" title="Nuclear binding energy">nuclear energy</a>, and <a href="/wiki/Rest_energy" class="mw-redirect" title="Rest energy">rest energy</a>. These can be categorized in two main classes: <a href="/wiki/Potential_energy" title="Potential energy">potential energy</a> and kinetic energy. Kinetic energy is the movement energy of an object. Kinetic energy can be transferred between objects and transformed into other kinds of energy.<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> </p><p>Kinetic energy may be best understood by examples that demonstrate how it is transformed to and from other forms of energy. For example, a <a href="/wiki/Cyclist" class="mw-redirect" title="Cyclist">cyclist</a> transfers <a href="/wiki/Food_energy" title="Food energy">chemical energy provided by food</a> to the bicycle and cyclist's store of kinetic energy as they increase their speed. On a level surface, this speed can be maintained without further work, except to overcome <a href="/wiki/Drag_(physics)" title="Drag (physics)">air resistance</a> and <a href="/wiki/Friction" title="Friction">friction</a>. The chemical energy has been converted into kinetic energy, the energy of motion, but the process is not completely efficient and produces thermal energy within the cyclist. </p><p>The kinetic energy in the moving cyclist and the bicycle can be converted to other forms. For example, the cyclist could encounter a hill just high enough to coast up, so that the bicycle comes to a complete halt at the top. The kinetic energy has now largely been converted to gravitational potential energy that can be released by freewheeling down the other side of the hill. Since the bicycle lost some of its energy to friction, it never regains all of its speed without additional pedaling. The energy is not destroyed; it has only been converted to another form by friction. Alternatively, the cyclist could connect a <a href="/wiki/Bottle_dynamo" title="Bottle dynamo">dynamo</a> to one of the wheels and generate some electrical energy on the descent. The bicycle would be traveling slower at the bottom of the hill than without the generator because some of the energy has been diverted into electrical energy. Another possibility would be for the cyclist to apply the brakes, in which case the kinetic energy would be dissipated through friction as <a href="/wiki/Heat" title="Heat">heat</a>. </p><p>Like any physical quantity that is a function of velocity, the kinetic energy of an object depends on the relationship between the object and the observer's <a href="/wiki/Frame_of_reference" title="Frame of reference">frame of reference</a>. Thus, the kinetic energy of an object is not <a href="/wiki/Galilean_invariance" title="Galilean invariance">invariant</a>. </p><p><a href="/wiki/Spacecraft" title="Spacecraft">Spacecraft</a> use chemical energy to launch and gain considerable kinetic energy to reach <a href="/wiki/Orbital_speed" title="Orbital speed">orbital velocity</a>. In an entirely circular orbit, this kinetic energy remains constant because there is almost no friction in near-earth space. However, it becomes apparent at re-entry when some of the kinetic energy is converted to heat. If the orbit is <a href="/wiki/Elliptic_orbit" title="Elliptic orbit">elliptical</a> or <a href="/wiki/Hyperbolic_trajectory" title="Hyperbolic trajectory">hyperbolic</a>, then throughout the orbit kinetic and <a href="/wiki/Potential_energy" title="Potential energy">potential energy</a> are exchanged; kinetic energy is greatest and potential energy lowest at closest approach to the earth or other massive body, while potential energy is greatest and kinetic energy the lowest at maximum distance. Disregarding loss or gain however, the sum of the kinetic and potential energy remains constant. </p><p>Kinetic energy can be passed from one object to another. In the game of <a href="/wiki/Billiards" class="mw-redirect" title="Billiards">billiards</a>, the player imposes kinetic energy on the cue ball by striking it with the cue stick. If the cue ball collides with another ball, it slows down dramatically, and the ball it hit accelerates as the kinetic energy is passed on to it. <a href="/wiki/Collisions" class="mw-redirect" title="Collisions">Collisions</a> in billiards are effectively <a href="/wiki/Elastic_collision" title="Elastic collision">elastic collisions</a>, in which kinetic energy is preserved. In <a href="/wiki/Inelastic_collision" title="Inelastic collision">inelastic collisions</a>, kinetic energy is dissipated in various forms of energy, such as heat, sound and binding energy (breaking bound structures). </p><p><a href="/wiki/Flywheel" title="Flywheel">Flywheels</a> have been developed as a method of <a href="/wiki/Flywheel_energy_storage" title="Flywheel energy storage">energy storage</a>. This illustrates that kinetic energy is also stored in rotational motion. </p><p>Several mathematical descriptions of kinetic energy exist that describe it in the appropriate physical situation. For objects and processes in common human experience, the formula <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span>mv<sup>2</sup> given by <a href="/wiki/Classical_mechanics" title="Classical mechanics">classical mechanics</a> is suitable. However, if the speed of the object is comparable to the speed of light, <a href="/wiki/Special_relativity" title="Special relativity">relativistic effects</a> become significant and the relativistic formula is used. If the object is on the atomic or <a href="/wiki/Sub-atomic_scale" class="mw-redirect" title="Sub-atomic scale">sub-atomic scale</a>, <a href="/wiki/Quantum_mechanical" class="mw-redirect" title="Quantum mechanical">quantum mechanical</a> effects are significant, and a quantum mechanical model must be employed. </p> <div class="mw-heading mw-heading2"><h2 id="Kinetic_energy_for_non-relativistic_velocity">Kinetic energy for non-relativistic velocity</h2></div> <p>Treatments of kinetic energy depend upon the relative velocity of objects compared to the fixed <a href="/wiki/Speed_of_light" title="Speed of light">speed of light</a>. Speeds experienced directly by humans are <b>non-relativisitic</b>; higher speeds require the <a href="/wiki/Theory_of_relativity" title="Theory of relativity">theory of relativity</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Kinetic_energy_of_rigid_bodies">Kinetic energy of rigid bodies</h3></div> <p>In <a href="/wiki/Classical_mechanics" title="Classical mechanics">classical mechanics</a>, the kinetic energy of a <i>point object</i> (an object so small that its mass can be assumed to exist at one point), or a non-rotating <a href="/wiki/Rigid_body" title="Rigid body">rigid body</a> depends on the <a href="/wiki/Mass" title="Mass">mass</a> of the body as well as its <a href="/wiki/Speed" title="Speed">speed</a>. The kinetic energy is equal to half the <a href="/wiki/Multiplication" title="Multiplication">product</a> of the mass and the square of the speed. In formula form: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{\text{k}}={\frac {1}{2}}mv^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>k</mtext> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>m</mi> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{\text{k}}={\frac {1}{2}}mv^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3fe4395503647681d6282df4937207bf30bf5dd9" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:12.135ex; height:5.176ex;" alt="{\displaystyle E_{\text{k}}={\frac {1}{2}}mv^{2}}"></span> </p><p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}"></span> is the mass and <span class="mwe-math-element"><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> is the speed (magnitude of the velocity) of the body. In <a href="/wiki/SI" class="mw-redirect" title="SI">SI</a> units, mass is measured in <a href="/wiki/Kilogram" title="Kilogram">kilograms</a>, speed in <a href="/wiki/Metres_per_second" class="mw-redirect" title="Metres per second">metres per second</a>, and the resulting kinetic energy is in <a href="/wiki/Joule" title="Joule">joules</a>. </p><p>For example, one would calculate the kinetic energy of an 80 kg mass (about 180 lbs) traveling at 18 metres per second (about 40 mph, or 65 km/h) as </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{\text{k}}={\frac {1}{2}}\cdot 80\,{\text{kg}}\cdot \left(18\,{\text{m/s}}\right)^{2}=12,960\,{\text{J}}=12.96\,{\text{kJ}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>k</mtext> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>⋅</mo> <mn>80</mn> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>kg</mtext> </mrow> <mo>⋅</mo> <msup> <mrow> <mo>(</mo> <mrow> <mn>18</mn> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>m/s</mtext> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mn>12</mn> <mo>,</mo> <mn>960</mn> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>J</mtext> </mrow> <mo>=</mo> <mn>12.96</mn> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>kJ</mtext> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{\text{k}}={\frac {1}{2}}\cdot 80\,{\text{kg}}\cdot \left(18\,{\text{m/s}}\right)^{2}=12,960\,{\text{J}}=12.96\,{\text{kJ}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1d3ea23e54bf7dde8e38eaef21876f7296091acb" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:48.694ex; height:5.176ex;" alt="{\displaystyle E_{\text{k}}={\frac {1}{2}}\cdot 80\,{\text{kg}}\cdot \left(18\,{\text{m/s}}\right)^{2}=12,960\,{\text{J}}=12.96\,{\text{kJ}}}"></span> </p><p>When a person throws a ball, the person does <a href="/wiki/Work_(physics)" title="Work (physics)">work</a> on it to give it speed as it leaves the hand. The moving ball can then hit something and push it, doing work on what it hits. The kinetic energy of a moving object is equal to the work required to bring it from rest to that speed, or the work the object can do while being brought to rest: <b>net force × displacement = kinetic energy</b>, i.e., </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Fs={\frac {1}{2}}mv^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mi>s</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>m</mi> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Fs={\frac {1}{2}}mv^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/69c392c47fce36ea6b3e831d61f125285e560f38" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:12.151ex; height:5.176ex;" alt="{\displaystyle Fs={\frac {1}{2}}mv^{2}}"></span> </p><p>Since the kinetic energy increases with the square of the speed, an object doubling its speed has four times as much kinetic energy. For example, a car traveling twice as fast as another requires four times as much distance to stop, assuming a constant braking force. As a consequence of this quadrupling, it takes four times the work to double the speed. </p><p>The kinetic energy of an object is related to its <a href="/wiki/Momentum" title="Momentum">momentum</a> by the equation: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{\text{k}}={\frac {p^{2}}{2m}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>k</mtext> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mn>2</mn> <mi>m</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{\text{k}}={\frac {p^{2}}{2m}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/558732496342cbd68af16cafdeff4e390e9b56a2" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:9.953ex; height:5.676ex;" alt="{\displaystyle E_{\text{k}}={\frac {p^{2}}{2m}}}"></span> </p><p>where: </p> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span> is momentum</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}"></span> is mass of the body</li></ul> <p>For the <i>translational kinetic energy,</i> that is the kinetic energy associated with <a href="/wiki/Rectilinear_motion" class="mw-redirect" title="Rectilinear motion">rectilinear motion</a>, of a <a href="/wiki/Rigid_body" title="Rigid body">rigid body</a> with constant <a href="/wiki/Mass" title="Mass">mass</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 m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}"></span>, whose <a href="/wiki/Center_of_mass" title="Center of mass">center of mass</a> is moving in a straight line with speed <span class="mwe-math-element"><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>, as seen above is equal to </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{\text{t}}={\frac {1}{2}}mv^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>t</mtext> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>m</mi> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{\text{t}}={\frac {1}{2}}mv^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/67698b94624b9afd114a9ffca4c55353d9fe09a7" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:11.906ex; height:5.176ex;" alt="{\displaystyle E_{\text{t}}={\frac {1}{2}}mv^{2}}"></span> </p><p>where: </p> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}"></span> is the mass of the body</li> <li><span class="mwe-math-element"><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> is the speed of the <a href="/wiki/Center_of_mass" title="Center of mass">center of mass</a> of the body.</li></ul> <p>The kinetic energy of any entity depends on the reference frame in which it is measured. However, the total energy of an isolated system, i.e. one in which energy can neither enter nor leave, does not change over time in the reference frame in which it is measured. Thus, the chemical energy converted to kinetic energy by a rocket engine is divided differently between the rocket ship and its exhaust stream depending upon the chosen reference frame. This is called the <a href="/wiki/Oberth_effect" title="Oberth effect">Oberth effect</a>. But the total energy of the system, including kinetic energy, fuel chemical energy, heat, etc., is conserved over time, regardless of the choice of reference frame. Different observers moving with different reference frames would however disagree on the value of this conserved energy. </p><p>The kinetic energy of such systems depends on the choice of reference frame: the reference frame that gives the minimum value of that energy is the <a href="/wiki/Center_of_momentum" class="mw-redirect" title="Center of momentum">center of momentum</a> frame, i.e. the reference frame in which the total momentum of the system is zero. This minimum kinetic energy contributes to the <a href="/wiki/Invariant_mass" title="Invariant mass">invariant mass</a> of the system as a whole. </p> <div class="mw-heading mw-heading4"><h4 id="Derivation">Derivation</h4></div> <div class="mw-heading mw-heading5"><h5 id="Without_vector_calculus">Without vector calculus</h5></div> <p>The work W done by a force <i>F</i> on an object over a distance <i>s</i> parallel to <i>F</i> equals </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W=F\cdot s.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>W</mi> <mo>=</mo> <mi>F</mi> <mo>⋅</mo> <mi>s</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W=F\cdot s.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a970e400f46c95636a393346b1ae6f97f47d385c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.691ex; height:2.176ex;" alt="{\displaystyle W=F\cdot s.}"></span> </p><p>Using <a href="/wiki/Newton%27s_Second_Law" class="mw-redirect" title="Newton's Second Law">Newton's Second Law</a> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F=ma}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo>=</mo> <mi>m</mi> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F=ma}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1ca4e42b7d6d66f52294364928cb5f7c590f514c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.109ex; height:2.176ex;" alt="{\displaystyle F=ma}"></span> </p><p>with <i>m</i> the mass and <i>a</i> the <a href="/wiki/Acceleration#Uniform_acceleration" title="Acceleration">acceleration</a> of the object and </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s={\frac {at^{2}}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>a</mi> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s={\frac {at^{2}}{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bc86011c2cedf6a5e2886d6e9dfd02a645eb7f33" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:8.149ex; height:5.676ex;" alt="{\displaystyle s={\frac {at^{2}}{2}}}"></span> </p><p>the distance traveled by the accelerated object in time <i>t</i>, we find with <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v=at}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> <mo>=</mo> <mi>a</mi> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v=at}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ab8eed79a675337b2155d9be4004717b9e9a4c9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.296ex; height:2.009ex;" alt="{\displaystyle v=at}"></span> for the velocity <i>v</i> of the object </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W=ma{\frac {at^{2}}{2}}={\frac {m(at)^{2}}{2}}={\frac {mv^{2}}{2}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>W</mi> <mo>=</mo> <mi>m</mi> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>a</mi> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>m</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mi>t</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>m</mi> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W=ma{\frac {at^{2}}{2}}={\frac {m(at)^{2}}{2}}={\frac {mv^{2}}{2}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/42b5e9ba4acc0734e1d53b862e2f5fa808b39780" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:32.475ex; height:5.843ex;" alt="{\displaystyle W=ma{\frac {at^{2}}{2}}={\frac {m(at)^{2}}{2}}={\frac {mv^{2}}{2}}.}"></span> </p> <div class="mw-heading mw-heading5"><h5 id="With_vector_calculus">With vector calculus</h5></div> <p>The work done in accelerating a particle with mass <i>m</i> during the infinitesimal time interval <i>dt</i> is given by the dot product of <i>force</i> <b>F</b> and the infinitesimal <i>displacement</i> <i>d</i><b>x</b> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {F} \cdot d\mathbf {x} =\mathbf {F} \cdot \mathbf {v} dt={\frac {d\mathbf {p} }{dt}}\cdot \mathbf {v} dt=\mathbf {v} \cdot d\mathbf {p} =\mathbf {v} \cdot d(m\mathbf {v} )\,,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mo>⋅</mo> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mi>d</mi> <mi>t</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mi>d</mi> <mi>t</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>⋅</mo> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>⋅</mo> <mi>d</mi> <mo stretchy="false">(</mo> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {F} \cdot d\mathbf {x} =\mathbf {F} \cdot \mathbf {v} dt={\frac {d\mathbf {p} }{dt}}\cdot \mathbf {v} dt=\mathbf {v} \cdot d\mathbf {p} =\mathbf {v} \cdot d(m\mathbf {v} )\,,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1e70c2eed00bb6c845e2237918874127228a2953" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:50.285ex; height:5.509ex;" alt="{\displaystyle \mathbf {F} \cdot d\mathbf {x} =\mathbf {F} \cdot \mathbf {v} dt={\frac {d\mathbf {p} }{dt}}\cdot \mathbf {v} dt=\mathbf {v} \cdot d\mathbf {p} =\mathbf {v} \cdot d(m\mathbf {v} )\,,}"></span> </p><p>where we have assumed the relationship <b>p</b> = <i>m</i> <b>v</b> and the validity of <a href="/wiki/Newton%27s_Second_Law" class="mw-redirect" title="Newton's Second Law">Newton's Second Law</a>. (However, also see the special relativistic derivation <a href="#Relativistic_kinetic_energy_of_rigid_bodies">below</a>.) </p><p>Applying the <a href="/wiki/Product_rule" title="Product rule">product rule</a> we see that: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d(\mathbf {v} \cdot \mathbf {v} )=(d\mathbf {v} )\cdot \mathbf {v} +\mathbf {v} \cdot (d\mathbf {v} )=2(\mathbf {v} \cdot d\mathbf {v} ).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo stretchy="false">)</mo> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>⋅</mo> <mo stretchy="false">(</mo> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mn>2</mn> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>⋅</mo> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d(\mathbf {v} \cdot \mathbf {v} )=(d\mathbf {v} )\cdot \mathbf {v} +\mathbf {v} \cdot (d\mathbf {v} )=2(\mathbf {v} \cdot d\mathbf {v} ).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/02f09e02c1df7f87732a08473cf08c19c22308c9" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:40.952ex; height:2.843ex;" alt="{\displaystyle d(\mathbf {v} \cdot \mathbf {v} )=(d\mathbf {v} )\cdot \mathbf {v} +\mathbf {v} \cdot (d\mathbf {v} )=2(\mathbf {v} \cdot d\mathbf {v} ).}"></span> </p><p>Therefore (assuming constant mass so that <i>dm</i> = 0), we have </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {v} \cdot d(m\mathbf {v} )={\frac {m}{2}}d(\mathbf {v} \cdot \mathbf {v} )={\frac {m}{2}}dv^{2}=d\left({\frac {mv^{2}}{2}}\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>⋅</mo> <mi>d</mi> <mo stretchy="false">(</mo> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>m</mi> <mn>2</mn> </mfrac> </mrow> <mi>d</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>m</mi> <mn>2</mn> </mfrac> </mrow> <mi>d</mi> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mi>d</mi> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>m</mi> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {v} \cdot d(m\mathbf {v} )={\frac {m}{2}}d(\mathbf {v} \cdot \mathbf {v} )={\frac {m}{2}}dv^{2}=d\left({\frac {mv^{2}}{2}}\right).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8d91ba24815a4413f3d574735ea1194deb78b95c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:46.655ex; height:6.343ex;" alt="{\displaystyle \mathbf {v} \cdot d(m\mathbf {v} )={\frac {m}{2}}d(\mathbf {v} \cdot \mathbf {v} )={\frac {m}{2}}dv^{2}=d\left({\frac {mv^{2}}{2}}\right).}"></span> </p><p>Since this is a <a href="/wiki/Total_differential" class="mw-redirect" title="Total differential">total differential</a> (that is, it only depends on the final state, not how the particle got there), we can integrate it and call the result kinetic energy: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{\text{k}}=\int _{v_{1}}^{v_{2}}\mathbf {p} d\mathbf {v} =\int _{v_{1}}^{v_{2}}m\mathbf {v} d\mathbf {v} ={mv^{2} \over 2}{\bigg \vert }_{v_{1}}^{v_{2}}={1 \over 2}m(v_{2}^{2}-v_{1}^{2}).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>k</mtext> </mrow> </msub> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </msubsup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>m</mi> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mn>2</mn> </mfrac> </mrow> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">|</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </msubsup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>m</mi> <mo stretchy="false">(</mo> <msubsup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>−</mo> <msubsup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{\text{k}}=\int _{v_{1}}^{v_{2}}\mathbf {p} d\mathbf {v} =\int _{v_{1}}^{v_{2}}m\mathbf {v} d\mathbf {v} ={mv^{2} \over 2}{\bigg \vert }_{v_{1}}^{v_{2}}={1 \over 2}m(v_{2}^{2}-v_{1}^{2}).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f508a80ff377567de0e55d22106f0af8bd9a2a29" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:56.09ex; height:6.509ex;" alt="{\displaystyle E_{\text{k}}=\int _{v_{1}}^{v_{2}}\mathbf {p} d\mathbf {v} =\int _{v_{1}}^{v_{2}}m\mathbf {v} d\mathbf {v} ={mv^{2} \over 2}{\bigg \vert }_{v_{1}}^{v_{2}}={1 \over 2}m(v_{2}^{2}-v_{1}^{2}).}"></span> </p><p>This equation states that the kinetic energy (<i>E</i><sub>k</sub>) is equal to the <a href="/wiki/Integral" title="Integral">integral</a> of the <a href="/wiki/Dot_product" title="Dot product">dot product</a> of the <a href="/wiki/Momentum" title="Momentum">momentum</a> (<b>p</b>) of a body and the <a href="/wiki/Infinitesimal" title="Infinitesimal">infinitesimal</a> change of the <a href="/wiki/Velocity" title="Velocity">velocity</a> (<b>v</b>) of the body. It is assumed that the body starts with no kinetic energy when it is at rest (motionless). </p> <div class="mw-heading mw-heading3"><h3 id="Rotating_bodies">Rotating bodies</h3></div> <p>If a rigid body Q is rotating about any line through the center of mass then it has <a href="/wiki/Rotational_energy" title="Rotational energy"><i>rotational kinetic energy</i></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 E_{\text{r}}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>r</mtext> </mrow> </msub> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{\text{r}}\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b569dd02f795addac571047a77bce131ad9dd5c2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.979ex; height:2.509ex;" alt="{\displaystyle E_{\text{r}}\,}"></span>) which is simply the sum of the kinetic energies of its moving parts, and is thus given by: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{\text{r}}=\int _{Q}{\frac {v^{2}dm}{2}}=\int _{Q}{\frac {(r\omega )^{2}dm}{2}}={\frac {\omega ^{2}}{2}}\int _{Q}{r^{2}}dm={\frac {\omega ^{2}}{2}}I={\frac {1}{2}}I\omega ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>r</mtext> </mrow> </msub> <mo>=</mo> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>Q</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>d</mi> <mi>m</mi> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>=</mo> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>Q</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mi>r</mi> <mi>ω</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>d</mi> <mi>m</mi> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mn>2</mn> </mfrac> </mrow> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>Q</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mi>d</mi> <mi>m</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mn>2</mn> </mfrac> </mrow> <mi>I</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>I</mi> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{\text{r}}=\int _{Q}{\frac {v^{2}dm}{2}}=\int _{Q}{\frac {(r\omega )^{2}dm}{2}}={\frac {\omega ^{2}}{2}}\int _{Q}{r^{2}}dm={\frac {\omega ^{2}}{2}}I={\frac {1}{2}}I\omega ^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0753180c8a737bd014d265c52cd4140b4d2f2c22" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:62.705ex; height:6.676ex;" alt="{\displaystyle E_{\text{r}}=\int _{Q}{\frac {v^{2}dm}{2}}=\int _{Q}{\frac {(r\omega )^{2}dm}{2}}={\frac {\omega ^{2}}{2}}\int _{Q}{r^{2}}dm={\frac {\omega ^{2}}{2}}I={\frac {1}{2}}I\omega ^{2}}"></span> </p><p>where: </p> <ul><li>ω is the body's <a href="/wiki/Angular_velocity" title="Angular velocity">angular velocity</a></li> <li><i>r</i> is the distance of any mass <i>dm</i> from that line</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/535ea7fc4134a31cbe2251d9d3511374bc41be9f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.172ex; height:2.176ex;" alt="{\displaystyle I}"></span> is the body's <a href="/wiki/Moment_of_inertia" title="Moment of inertia">moment of inertia</a>, equal to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \int _{Q}{r^{2}}dm}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>Q</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mi>d</mi> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \int _{Q}{r^{2}}dm}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9c18d096a930d40c61569b1e6b1ec8af92808992" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:8.376ex; height:3.509ex;" alt="{\textstyle \int _{Q}{r^{2}}dm}"></span>.</li></ul> <p>(In this equation the moment of <a href="/wiki/Inertia" title="Inertia">inertia</a> must be taken about an axis through the center of mass and the rotation measured by ω must be around that axis; more general equations exist for systems where the object is subject to wobble due to its eccentric shape). </p> <div class="mw-heading mw-heading3"><h3 id="Kinetic_energy_of_systems">Kinetic energy of systems</h3></div> <p>A system of bodies may have internal kinetic energy due to the relative motion of the bodies in the system. For example, in the <a href="/wiki/Solar_System" title="Solar System">Solar System</a> the planets and planetoids are orbiting the Sun. In a tank of gas, the molecules are moving in all directions. The kinetic energy of the system is the sum of the kinetic energies of the bodies it contains. </p><p>A macroscopic body that is stationary (i.e. a reference frame has been chosen to correspond to the body's <a href="/wiki/Center_of_momentum" class="mw-redirect" title="Center of momentum">center of momentum</a>) may have various kinds of <a href="/wiki/Internal_energy" title="Internal energy">internal energy</a> at the molecular or atomic level, which may be regarded as kinetic energy, due to molecular translation, rotation, and vibration, electron translation and spin, and nuclear spin. These all contribute to the body's mass, as provided by the special theory of relativity. When discussing movements of a macroscopic body, the kinetic energy referred to is usually that of the macroscopic movement only. However, all internal energies of all types contribute to a body's mass, inertia, and total energy. </p> <div class="mw-heading mw-heading3"><h3 id="Fluid_dynamics">Fluid dynamics</h3></div> <p>In <a href="/wiki/Fluid_dynamics" title="Fluid dynamics">fluid dynamics</a>, the kinetic energy per unit volume at each point in an incompressible fluid flow field is called the <a href="/wiki/Dynamic_pressure" title="Dynamic pressure">dynamic pressure</a> at that point.<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{\text{k}}={\frac {1}{2}}mv^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>k</mtext> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>m</mi> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{\text{k}}={\frac {1}{2}}mv^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3fe4395503647681d6282df4937207bf30bf5dd9" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:12.135ex; height:5.176ex;" alt="{\displaystyle E_{\text{k}}={\frac {1}{2}}mv^{2}}"></span> </p><p>Dividing by V, the unit of volume: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}{\frac {E_{\text{k}}}{V}}&={\frac {1}{2}}{\frac {m}{V}}v^{2}\\q&={\frac {1}{2}}\rho v^{2}\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>k</mtext> </mrow> </msub> <mi>V</mi> </mfrac> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>m</mi> <mi>V</mi> </mfrac> </mrow> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mtd> </mtr> <mtr> <mtd> <mi>q</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>ρ</mi> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\frac {E_{\text{k}}}{V}}&={\frac {1}{2}}{\frac {m}{V}}v^{2}\\q&={\frac {1}{2}}\rho v^{2}\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f763ebdcdc434c2088d6fdb0fc813ce7ccce1483" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.546ex; margin-bottom: -0.292ex; width:14.558ex; height:10.843ex;" alt="{\displaystyle {\begin{aligned}{\frac {E_{\text{k}}}{V}}&={\frac {1}{2}}{\frac {m}{V}}v^{2}\\q&={\frac {1}{2}}\rho v^{2}\end{aligned}}}"></span> </p><p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06809d64fa7c817ffc7e323f85997f783dbdf71d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}"></span> is the dynamic pressure, and ρ is the density of the incompressible fluid. </p> <div class="mw-heading mw-heading3"><h3 id="Frame_of_reference">Frame of reference</h3></div> <p>The speed, and thus the kinetic energy of a single object is frame-dependent (relative): it can take any non-negative value, by choosing a suitable <a href="/wiki/Inertial_frame_of_reference" title="Inertial frame of reference">inertial frame of reference</a>. For example, a bullet passing an observer has kinetic energy in the reference frame of this observer. The same bullet is stationary to an observer moving with the same velocity as the bullet, and so has zero kinetic energy.<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> By contrast, the total kinetic energy of a system of objects cannot be reduced to zero by a suitable choice of the inertial reference frame, unless all the objects have the same velocity. In any other case, the total kinetic energy has a non-zero minimum, as no inertial reference frame can be chosen in which all the objects are stationary. This minimum kinetic energy contributes to the system's <a href="/wiki/Invariant_mass" title="Invariant mass">invariant mass</a>, which is independent of the reference frame. </p><p>The total kinetic energy of a system depends on the <a href="/wiki/Inertial_frame_of_reference" title="Inertial frame of reference">inertial frame of reference</a>: it is the sum of the total kinetic energy in a <a href="/wiki/Center_of_momentum_frame" class="mw-redirect" title="Center of momentum frame">center of momentum frame</a> and the kinetic energy the total mass would have if it were concentrated in the <a href="/wiki/Center_of_mass" title="Center of mass">center of mass</a>. </p><p>This may be simply shown: let <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle \mathbf {V} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">V</mi> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \textstyle \mathbf {V} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2fce5593a61408934c8eda76759f756d4fe0a25c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.019ex; height:2.176ex;" alt="{\displaystyle \textstyle \mathbf {V} }"></span> be the relative velocity of the center of mass frame <i>i</i> in the frame <i>k</i>. Since </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v^{2}=\left(v_{i}+V\right)^{2}=\left(\mathbf {v} _{i}+\mathbf {V} \right)\cdot \left(\mathbf {v} _{i}+\mathbf {V} \right)=\mathbf {v} _{i}\cdot \mathbf {v} _{i}+2\mathbf {v} _{i}\cdot \mathbf {V} +\mathbf {V} \cdot \mathbf {V} =v_{i}^{2}+2\mathbf {v} _{i}\cdot \mathbf {V} +V^{2},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msup> <mrow> <mo>(</mo> <mrow> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>+</mo> <mi>V</mi> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">V</mi> </mrow> </mrow> <mo>)</mo> </mrow> <mo>⋅</mo> <mrow> <mo>(</mo> <mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">V</mi> </mrow> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>⋅</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>+</mo> <mn>2</mn> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">V</mi> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">V</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">V</mi> </mrow> <mo>=</mo> <msubsup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <mn>2</mn> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">V</mi> </mrow> <mo>+</mo> <msup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v^{2}=\left(v_{i}+V\right)^{2}=\left(\mathbf {v} _{i}+\mathbf {V} \right)\cdot \left(\mathbf {v} _{i}+\mathbf {V} \right)=\mathbf {v} _{i}\cdot \mathbf {v} _{i}+2\mathbf {v} _{i}\cdot \mathbf {V} +\mathbf {V} \cdot \mathbf {V} =v_{i}^{2}+2\mathbf {v} _{i}\cdot \mathbf {V} +V^{2},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e5ddcb085de8b874425b03fab72413dde1d3742f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:86.556ex; height:3.509ex;" alt="{\displaystyle v^{2}=\left(v_{i}+V\right)^{2}=\left(\mathbf {v} _{i}+\mathbf {V} \right)\cdot \left(\mathbf {v} _{i}+\mathbf {V} \right)=\mathbf {v} _{i}\cdot \mathbf {v} _{i}+2\mathbf {v} _{i}\cdot \mathbf {V} +\mathbf {V} \cdot \mathbf {V} =v_{i}^{2}+2\mathbf {v} _{i}\cdot \mathbf {V} +V^{2},}"></span> </p><p>Then, </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{\text{k}}=\int {\frac {v^{2}}{2}}dm=\int {\frac {v_{i}^{2}}{2}}dm+\mathbf {V} \cdot \int \mathbf {v} _{i}dm+{\frac {V^{2}}{2}}\int dm.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>k</mtext> </mrow> </msub> <mo>=</mo> <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> <mi>d</mi> <mi>m</mi> <mo>=</mo> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msubsup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mn>2</mn> </mfrac> </mrow> <mi>d</mi> <mi>m</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">V</mi> </mrow> <mo>⋅</mo> <mo>∫</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mi>d</mi> <mi>m</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> <mi>d</mi> <mi>m</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{\text{k}}=\int {\frac {v^{2}}{2}}dm=\int {\frac {v_{i}^{2}}{2}}dm+\mathbf {V} \cdot \int \mathbf {v} _{i}dm+{\frac {V^{2}}{2}}\int dm.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/13f2e14b365ca3eb167ef01b35069f49d17678b9" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:54.828ex; height:6.343ex;" alt="{\displaystyle E_{\text{k}}=\int {\frac {v^{2}}{2}}dm=\int {\frac {v_{i}^{2}}{2}}dm+\mathbf {V} \cdot \int \mathbf {v} _{i}dm+{\frac {V^{2}}{2}}\int dm.}"></span> </p><p>However, let <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \int {\frac {v_{i}^{2}}{2}}dm=E_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msubsup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mn>2</mn> </mfrac> </mrow> <mi>d</mi> <mi>m</mi> <mo>=</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \int {\frac {v_{i}^{2}}{2}}dm=E_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75f0e76e8f9043ea5f8a1523f6d57bbdb5fcb84a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:13.14ex; height:4.509ex;" alt="{\textstyle \int {\frac {v_{i}^{2}}{2}}dm=E_{i}}"></span> the kinetic energy in the center of mass frame, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \int \mathbf {v} _{i}dm}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mo>∫</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mi>d</mi> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \int \mathbf {v} _{i}dm}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bbd4ac48bc9617d1cb390495b92f42f235d6bb78" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.272ex; height:3.176ex;" alt="{\textstyle \int \mathbf {v} _{i}dm}"></span> would be simply the total momentum that is by definition zero in the center of mass frame, and let the total mass: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \int dm=M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mo>∫</mo> <mi>d</mi> <mi>m</mi> <mo>=</mo> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \int dm=M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/96d04cc988fd6a5a14ddbebe108de01211f960bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.602ex; height:3.176ex;" alt="{\textstyle \int dm=M}"></span>. Substituting, we get:<sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{\text{k}}=E_{i}+{\frac {MV^{2}}{2}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>k</mtext> </mrow> </msub> <mo>=</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>M</mi> <msup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{\text{k}}=E_{i}+{\frac {MV^{2}}{2}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cbf9570dcaba8fc06e0edd436a22caee1633d3ca" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:18.166ex; height:5.676ex;" alt="{\displaystyle E_{\text{k}}=E_{i}+{\frac {MV^{2}}{2}}.}"></span> </p><p>Thus the kinetic energy of a system is lowest to center of momentum reference frames, i.e., frames of reference in which the center of mass is stationary (either the <a href="/wiki/Center_of_mass_frame" class="mw-redirect" title="Center of mass frame">center of mass frame</a> or any other <a href="/wiki/Center_of_momentum_frame" class="mw-redirect" title="Center of momentum frame">center of momentum frame</a>). In any different frame of reference, there is additional kinetic energy corresponding to the total mass moving at the speed of the center of mass. The kinetic energy of the system in the <a href="/wiki/Center_of_momentum_frame" class="mw-redirect" title="Center of momentum frame">center of momentum frame</a> is a quantity that is invariant (all observers see it to be the same). </p> <div class="mw-heading mw-heading3"><h3 id="Rotation_in_systems">Rotation in systems</h3></div> <p>It sometimes is convenient to split the total kinetic energy of a body into the sum of the body's center-of-mass translational kinetic energy and the energy of rotation around the center of mass (<a href="/wiki/Rotational_energy" title="Rotational energy">rotational energy</a>): </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{\text{k}}=E_{\text{t}}+E_{\text{r}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>k</mtext> </mrow> </msub> <mo>=</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>t</mtext> </mrow> </msub> <mo>+</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>r</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{\text{k}}=E_{\text{t}}+E_{\text{r}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e95bdc6a6590681b52c13baaf88ceec68fbf3216" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.934ex; height:2.509ex;" alt="{\displaystyle E_{\text{k}}=E_{\text{t}}+E_{\text{r}}}"></span> </p><p>where: </p> <ul><li><i>E</i><sub>k</sub> is the total kinetic energy</li> <li><i>E</i><sub>t</sub> is the translational kinetic energy</li> <li><i>E</i><sub>r</sub> is the <i>rotational energy</i> or <i>angular kinetic energy</i> in the rest frame</li></ul> <p>Thus the kinetic energy of a tennis ball in flight is the kinetic energy due to its rotation, plus the kinetic energy due to its translation. </p> <div class="mw-heading mw-heading2"><h2 id="Relativistic_kinetic_energy">Relativistic kinetic energy</h2></div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Mass_in_special_relativity" title="Mass in special relativity">Mass in special relativity</a> and <a href="/wiki/Tests_of_relativistic_energy_and_momentum" title="Tests of relativistic energy and momentum">Tests of relativistic energy and momentum</a></div> <p>If a body's speed is a significant fraction of the <a href="/wiki/Speed_of_light" title="Speed of light">speed of light</a>, it is necessary to use relativistic mechanics to calculate its kinetic energy. In relativity, the total energy is given by the <a href="/wiki/Energy-momentum_relation" class="mw-redirect" title="Energy-momentum relation">energy-momentum relation</a>: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E^{2}=(p{\textrm {c}})^{2}+\left(m_{0}{\textrm {c}}^{2}\right)^{2}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mo stretchy="false">(</mo> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext>c</mtext> </mrow> </mrow> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mrow> <mo>(</mo> <mrow> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext>c</mtext> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E^{2}=(p{\textrm {c}})^{2}+\left(m_{0}{\textrm {c}}^{2}\right)^{2}\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c30e6a9271d66617b81202ea6f615de13e4fd3b2" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:22.605ex; height:3.843ex;" alt="{\displaystyle E^{2}=(p{\textrm {c}})^{2}+\left(m_{0}{\textrm {c}}^{2}\right)^{2}\,}"></span> </p><p>Here we use the relativistic expression for linear momentum: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p=m\gamma v}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>=</mo> <mi>m</mi> <mi>γ</mi> <mi>v</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p=m\gamma v}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7b296776d9f129fec70e3696fc4fbc1255c3e287" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:8.788ex; height:2.176ex;" alt="{\displaystyle p=m\gamma v}"></span>, where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \gamma =1/{\sqrt {1-v^{2}/c^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>γ</mi> <mo>=</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \gamma =1/{\sqrt {1-v^{2}/c^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/16a28ea842c016e78a603843b6bbbb136e0418e7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:18.418ex; height:3.343ex;" alt="{\textstyle \gamma =1/{\sqrt {1-v^{2}/c^{2}}}}"></span>. with <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}"></span> being an object's <a href="/wiki/Invariant_mass" title="Invariant mass">(rest) mass</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 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> speed, and <i>c</i> the speed of light in vacuum. Then kinetic energy is the <a href="/wiki/Energy%E2%80%93momentum_relation#Connection_to_E_=_mc2" title="Energy–momentum relation"> total relativistic energy minus the rest energy</a>: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{K}=E-m_{0}c^{2}={\sqrt {(p{\textrm {c}})^{2}+\left(m_{0}{\textrm {c}}^{2}\right)^{2}}}-m_{0}c^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>K</mi> </mrow> </msub> <mo>=</mo> <mi>E</mi> <mo>−</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo stretchy="false">(</mo> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext>c</mtext> </mrow> </mrow> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mrow> <mo>(</mo> <mrow> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext>c</mtext> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mo>−</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{K}=E-m_{0}c^{2}={\sqrt {(p{\textrm {c}})^{2}+\left(m_{0}{\textrm {c}}^{2}\right)^{2}}}-m_{0}c^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8ba96348953891bd329a87ac6f752001dcc452a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.505ex; width:45.968ex; height:4.843ex;" alt="{\displaystyle E_{K}=E-m_{0}c^{2}={\sqrt {(p{\textrm {c}})^{2}+\left(m_{0}{\textrm {c}}^{2}\right)^{2}}}-m_{0}c^{2}}"></span> </p><p>At low speeds, the square root can be expanded and the rest energy drops out, giving the Newtonian kinetic energy. </p> <figure class="mw-halign-center" typeof="mw:File/Thumb"><a href="/wiki/File:KEvsMOMENTUM.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ea/KEvsMOMENTUM.png/450px-KEvsMOMENTUM.png" decoding="async" width="450" height="424" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/e/ea/KEvsMOMENTUM.png 1.5x" data-file-width="495" data-file-height="466" /></a><figcaption> Log of relativistic kinetic energy versus log relativistic momentum, for many objects of vastly different scales. The intersections of the object lines with the bottom axis approaches the rest energy. At low kinetic energy the slope of the object lines reflect Newtonian mechanics. As the lines approach <span class="mwe-math-element"><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/86a67b81c2de995bd608d5b2df50cd8cd7d92455" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.007ex; height:1.676ex;" alt="{\displaystyle c}"></span> the slope bends at the lightspeed barrier.</figcaption></figure> <div class="mw-heading mw-heading3"><h3 id="Derivation_2">Derivation</h3></div> <p>Start with the expression for linear momentum <span class="mwe-math-element"><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 {p} =m\gamma \mathbf {v} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> <mo>=</mo> <mi>m</mi> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {p} =m\gamma \mathbf {v} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/212dc9b9ce048465636309e9442073e1c1ce0b27" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.297ex; height:2.176ex;" alt="{\displaystyle \mathbf {p} =m\gamma \mathbf {v} }"></span>, where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \gamma =1/{\sqrt {1-v^{2}/c^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>γ</mi> <mo>=</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \gamma =1/{\sqrt {1-v^{2}/c^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/16a28ea842c016e78a603843b6bbbb136e0418e7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:18.418ex; height:3.343ex;" alt="{\textstyle \gamma =1/{\sqrt {1-v^{2}/c^{2}}}}"></span>. <a href="/wiki/Integration_by_parts" title="Integration by parts">Integrating by parts</a> yields </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{\text{k}}=\int \mathbf {v} \cdot d\mathbf {p} =\int \mathbf {v} \cdot d(m\gamma \mathbf {v} )=m\gamma \mathbf {v} \cdot \mathbf {v} -\int m\gamma \mathbf {v} \cdot d\mathbf {v} =m\gamma v^{2}-{\frac {m}{2}}\int \gamma d\left(v^{2}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>k</mtext> </mrow> </msub> <mo>=</mo> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>⋅</mo> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> <mo>=</mo> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>⋅</mo> <mi>d</mi> <mo stretchy="false">(</mo> <mi>m</mi> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mi>m</mi> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>−</mo> <mo>∫</mo> <mi>m</mi> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>⋅</mo> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>=</mo> <mi>m</mi> <mi>γ</mi> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>m</mi> <mn>2</mn> </mfrac> </mrow> <mo>∫</mo> <mi>γ</mi> <mi>d</mi> <mrow> <mo>(</mo> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{\text{k}}=\int \mathbf {v} \cdot d\mathbf {p} =\int \mathbf {v} \cdot d(m\gamma \mathbf {v} )=m\gamma \mathbf {v} \cdot \mathbf {v} -\int m\gamma \mathbf {v} \cdot d\mathbf {v} =m\gamma v^{2}-{\frac {m}{2}}\int \gamma d\left(v^{2}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fb1b0f44c3102c1d6609c561de57ca517487c796" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:80.582ex; height:5.676ex;" alt="{\displaystyle E_{\text{k}}=\int \mathbf {v} \cdot d\mathbf {p} =\int \mathbf {v} \cdot d(m\gamma \mathbf {v} )=m\gamma \mathbf {v} \cdot \mathbf {v} -\int m\gamma \mathbf {v} \cdot d\mathbf {v} =m\gamma v^{2}-{\frac {m}{2}}\int \gamma d\left(v^{2}\right)}"></span> </p><p>Since <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma =\left(1-v^{2}/c^{2}\right)^{-{\frac {1}{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>γ</mi> <mo>=</mo> <msup> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma =\left(1-v^{2}/c^{2}\right)^{-{\frac {1}{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af70a9fa55fa80a28cefd2814a73060b7b51dfa1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:18.913ex; height:4.509ex;" alt="{\displaystyle \gamma =\left(1-v^{2}/c^{2}\right)^{-{\frac {1}{2}}}}"></span>, </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}E_{\text{k}}&=m\gamma v^{2}-{\frac {-mc^{2}}{2}}\int \gamma d\left(1-{\frac {v^{2}}{c^{2}}}\right)\\&=m\gamma v^{2}+mc^{2}\left(1-{\frac {v^{2}}{c^{2}}}\right)^{\frac {1}{2}}-E_{0}\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>k</mtext> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>m</mi> <mi>γ</mi> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>−</mo> <mi>m</mi> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>∫</mo> <mi>γ</mi> <mi>d</mi> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mi>m</mi> <mi>γ</mi> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>m</mi> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </msup> <mo>−</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}E_{\text{k}}&=m\gamma v^{2}-{\frac {-mc^{2}}{2}}\int \gamma d\left(1-{\frac {v^{2}}{c^{2}}}\right)\\&=m\gamma v^{2}+mc^{2}\left(1-{\frac {v^{2}}{c^{2}}}\right)^{\frac {1}{2}}-E_{0}\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c2d84a4c9ece3283cc41da63cffb6ca1d379da74" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.338ex; width:38.011ex; height:13.843ex;" alt="{\displaystyle {\begin{aligned}E_{\text{k}}&=m\gamma v^{2}-{\frac {-mc^{2}}{2}}\int \gamma d\left(1-{\frac {v^{2}}{c^{2}}}\right)\\&=m\gamma v^{2}+mc^{2}\left(1-{\frac {v^{2}}{c^{2}}}\right)^{\frac {1}{2}}-E_{0}\end{aligned}}}"></span> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/411d268de7b1cf300d7481e3fe59f3b20887e0d0" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.769ex; height:2.509ex;" alt="{\displaystyle E_{0}}"></span> is a <a href="/wiki/Constant_of_integration" title="Constant of integration">constant of integration</a> for the <a href="/wiki/Indefinite_integral" class="mw-redirect" title="Indefinite integral">indefinite integral</a>. </p><p>Simplifying the expression we obtain </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}E_{\text{k}}&=m\gamma \left(v^{2}+c^{2}\left(1-{\frac {v^{2}}{c^{2}}}\right)\right)-E_{0}\\&=m\gamma \left(v^{2}+c^{2}-v^{2}\right)-E_{0}\\&=m\gamma c^{2}-E_{0}\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>k</mtext> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>m</mi> <mi>γ</mi> <mrow> <mo>(</mo> <mrow> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mrow> <mo>)</mo> </mrow> <mo>−</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mi>m</mi> <mi>γ</mi> <mrow> <mo>(</mo> <mrow> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mo>−</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mi>m</mi> <mi>γ</mi> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}E_{\text{k}}&=m\gamma \left(v^{2}+c^{2}\left(1-{\frac {v^{2}}{c^{2}}}\right)\right)-E_{0}\\&=m\gamma \left(v^{2}+c^{2}-v^{2}\right)-E_{0}\\&=m\gamma c^{2}-E_{0}\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d015c23055c4b638a89a527da19fa172e49d1d61" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.005ex; width:37.299ex; height:13.176ex;" alt="{\displaystyle {\begin{aligned}E_{\text{k}}&=m\gamma \left(v^{2}+c^{2}\left(1-{\frac {v^{2}}{c^{2}}}\right)\right)-E_{0}\\&=m\gamma \left(v^{2}+c^{2}-v^{2}\right)-E_{0}\\&=m\gamma c^{2}-E_{0}\end{aligned}}}"></span> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/411d268de7b1cf300d7481e3fe59f3b20887e0d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.769ex; height:2.509ex;" alt="{\displaystyle E_{0}}"></span> is found by observing that when <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {v} =0,\ \gamma =1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mtext> </mtext> <mi>γ</mi> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {v} =0,\ \gamma =1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/24c3438a80f8af95fdabc87f6b3ac9adc9cf7a8e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.81ex; height:2.676ex;" alt="{\displaystyle \mathbf {v} =0,\ \gamma =1}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{\text{k}}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>k</mtext> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{\text{k}}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7d6184134abeacd1bc9a6e9d78c23e61f3fb63e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.076ex; height:2.509ex;" alt="{\displaystyle E_{\text{k}}=0}"></span>, giving </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{0}=mc^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mi>m</mi> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{0}=mc^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c817d727d88d16ad580951c6c13b40834a634746" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.969ex; height:3.009ex;" alt="{\displaystyle E_{0}=mc^{2}}"></span> </p><p>resulting in the formula </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{\text{k}}=m\gamma c^{2}-mc^{2}={\frac {mc^{2}}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}-mc^{2}=(\gamma -1)mc^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>k</mtext> </mrow> </msub> <mo>=</mo> <mi>m</mi> <mi>γ</mi> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mi>m</mi> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>m</mi> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <msqrt> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> </msqrt> </mfrac> </mrow> <mo>−</mo> <mi>m</mi> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mo stretchy="false">(</mo> <mi>γ</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mi>m</mi> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{\text{k}}=m\gamma c^{2}-mc^{2}={\frac {mc^{2}}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}-mc^{2}=(\gamma -1)mc^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee27d816c3d93fa560f4ae20689e98cb11e52408" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.671ex; width:52.162ex; height:8.509ex;" alt="{\displaystyle E_{\text{k}}=m\gamma c^{2}-mc^{2}={\frac {mc^{2}}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}-mc^{2}=(\gamma -1)mc^{2}}"></span> </p><p>This formula shows that the work expended accelerating an object from rest approaches infinity as the velocity approaches the speed of light. Thus it is impossible to accelerate an object across this boundary. </p> <div class="mw-heading mw-heading3"><h3 id="Low_speed_limit">Low speed limit</h3></div> <p>The mathematical by-product of this calculation is the <a href="/wiki/Mass%E2%80%93energy_equivalence" title="Mass–energy equivalence">mass–energy equivalence</a> formula, that mass and energy are essentially the same thing:<sup id="cite_ref-Einstein1922_14-0" class="reference"><a href="#cite_note-Einstein1922-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 51">: 51 </span></sup><sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 121">: 121 </span></sup> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{\text{rest}}=E_{0}=mc^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>rest</mtext> </mrow> </msub> <mo>=</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mi>m</mi> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{\text{rest}}=E_{0}=mc^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/93c46d5ba8fed0dc3da10ee98fcb485c1dbb8343" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:17.678ex; height:3.009ex;" alt="{\displaystyle E_{\text{rest}}=E_{0}=mc^{2}}"></span> </p><p>At a low speed (<i>v</i> ≪ <i>c</i>), the relativistic kinetic energy is approximated well by the classical kinetic energy. To see this, apply the <a href="/wiki/Binomial_approximation" title="Binomial approximation">binomial approximation</a> or take the first two terms of the <a href="/wiki/Taylor_expansion" class="mw-redirect" title="Taylor expansion">Taylor expansion</a> in powers of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4faa98a21ac8133ab466999288849492be28b3d1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.182ex; height:2.676ex;" alt="{\displaystyle v^{2}}"></span> for the reciprocal square root:<sup id="cite_ref-Einstein1922_14-1" class="reference"><a href="#cite_note-Einstein1922-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 51">: 51 </span></sup> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{\text{k}}\approx mc^{2}\left(1+{\frac {1}{2}}{\frac {v^{2}}{c^{2}}}\right)-mc^{2}={\frac {1}{2}}mv^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>k</mtext> </mrow> </msub> <mo>≈</mo> <mi>m</mi> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>−</mo> <mi>m</mi> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>m</mi> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{\text{k}}\approx mc^{2}\left(1+{\frac {1}{2}}{\frac {v^{2}}{c^{2}}}\right)-mc^{2}={\frac {1}{2}}mv^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7c395e82a0ee909de937f8fa787d7503edcc67a4" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:39.104ex; height:6.343ex;" alt="{\displaystyle E_{\text{k}}\approx mc^{2}\left(1+{\frac {1}{2}}{\frac {v^{2}}{c^{2}}}\right)-mc^{2}={\frac {1}{2}}mv^{2}}"></span> </p><p>So, the total energy <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7587849b44d775263271e89499f4327eeac5dc81" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.804ex; height:2.509ex;" alt="{\displaystyle E_{k}}"></span> can be partitioned into the rest mass energy plus the non-relativistic kinetic energy at low speeds. </p><p>When objects move at a speed much slower than light (e.g. in everyday phenomena on Earth), the first two terms of the series predominate. The next term in the Taylor series approximation </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{\text{k}}\approx mc^{2}\left(1+{\frac {1}{2}}{\frac {v^{2}}{c^{2}}}+{\frac {3}{8}}{\frac {v^{4}}{c^{4}}}\right)-mc^{2}={\frac {1}{2}}mv^{2}+{\frac {3}{8}}m{\frac {v^{4}}{c^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>k</mtext> </mrow> </msub> <mo>≈</mo> <mi>m</mi> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>8</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>−</mo> <mi>m</mi> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>m</mi> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>8</mn> </mfrac> </mrow> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{\text{k}}\approx mc^{2}\left(1+{\frac {1}{2}}{\frac {v^{2}}{c^{2}}}+{\frac {3}{8}}{\frac {v^{4}}{c^{4}}}\right)-mc^{2}={\frac {1}{2}}mv^{2}+{\frac {3}{8}}m{\frac {v^{4}}{c^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b0eb384d3b48e0d9ed727f0e77afb8545feb9b73" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:56.859ex; height:6.343ex;" alt="{\displaystyle E_{\text{k}}\approx mc^{2}\left(1+{\frac {1}{2}}{\frac {v^{2}}{c^{2}}}+{\frac {3}{8}}{\frac {v^{4}}{c^{4}}}\right)-mc^{2}={\frac {1}{2}}mv^{2}+{\frac {3}{8}}m{\frac {v^{4}}{c^{2}}}}"></span> </p><p>is small for low speeds. For example, for a speed of 10 km/s (22,000 mph) the correction to the non-relativistic kinetic energy is 0.0417 J/kg (on a non-relativistic kinetic energy of 50 MJ/kg) and for a speed of 100 km/s it is 417 J/kg (on a non-relativistic kinetic energy of 5 GJ/kg). </p><p>The relativistic relation between kinetic energy and momentum is given by </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{\text{k}}={\sqrt {p^{2}c^{2}+m^{2}c^{4}}}-mc^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>k</mtext> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </msqrt> </mrow> <mo>−</mo> <mi>m</mi> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{\text{k}}={\sqrt {p^{2}c^{2}+m^{2}c^{4}}}-mc^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bd11e7bc315624e7d27ed2258de4afddcb4a44f8" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:27.46ex; height:4.843ex;" alt="{\displaystyle E_{\text{k}}={\sqrt {p^{2}c^{2}+m^{2}c^{4}}}-mc^{2}}"></span> </p><p>This can also be expanded as a <a href="/wiki/Taylor_series" title="Taylor series">Taylor series</a>, the first term of which is the simple expression from Newtonian mechanics:<sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{\text{k}}\approx {\frac {p^{2}}{2m}}-{\frac {p^{4}}{8m^{3}c^{2}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>k</mtext> </mrow> </msub> <mo>≈</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mn>2</mn> <mi>m</mi> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mrow> <mn>8</mn> <msup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{\text{k}}\approx {\frac {p^{2}}{2m}}-{\frac {p^{4}}{8m^{3}c^{2}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bcd1ecd7a0f9a3ee5ff1280d474dc946edd70ee2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:20.594ex; height:6.009ex;" alt="{\displaystyle E_{\text{k}}\approx {\frac {p^{2}}{2m}}-{\frac {p^{4}}{8m^{3}c^{2}}}.}"></span></dd></dl> <p>This suggests that the formulae for energy and momentum are not special and axiomatic, but concepts emerging from the equivalence of mass and energy and the principles of relativity. </p> <div class="mw-heading mw-heading3"><h3 id="General_relativity">General relativity</h3></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Schwarzschild_geodesics" title="Schwarzschild geodesics">Schwarzschild geodesics</a></div> <p>Using the convention that </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g_{\alpha \beta }\,u^{\alpha }\,u^{\beta }\,=\,-c^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mi>β</mi> </mrow> </msub> <mspace width="thinmathspace" /> <msup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msup> <mspace width="thinmathspace" /> <msup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msup> <mspace width="thinmathspace" /> <mo>=</mo> <mspace width="thinmathspace" /> <mo>−</mo> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g_{\alpha \beta }\,u^{\alpha }\,u^{\beta }\,=\,-c^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b80f85569fd2cc8427a40acc059cbc6a642ade1a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:16.969ex; height:3.343ex;" alt="{\displaystyle g_{\alpha \beta }\,u^{\alpha }\,u^{\beta }\,=\,-c^{2}}"></span> </p><p>where the <a href="/wiki/Four-velocity" title="Four-velocity">four-velocity</a> of a particle is </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u^{\alpha }\,=\,{\frac {dx^{\alpha }}{d\tau }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msup> <mspace width="thinmathspace" /> <mo>=</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msup> </mrow> <mrow> <mi>d</mi> <mi>τ</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u^{\alpha }\,=\,{\frac {dx^{\alpha }}{d\tau }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/590a832fa927b18c99c655612e1ed772fc8bb3cf" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:11.152ex; height:5.509ex;" alt="{\displaystyle u^{\alpha }\,=\,{\frac {dx^{\alpha }}{d\tau }}}"></span> </p><p>and <span class="mwe-math-element"><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> is the <a href="/wiki/Proper_time" title="Proper time">proper time</a> of the particle, there is also an expression for the kinetic energy of the particle in <a href="/wiki/General_relativity" title="General relativity">general relativity</a>. </p><p>If the particle has momentum </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{\beta }\,=\,m\,g_{\beta \alpha }\,u^{\alpha }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msub> <mspace width="thinmathspace" /> <mo>=</mo> <mspace width="thinmathspace" /> <mi>m</mi> <mspace width="thinmathspace" /> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> <mi>α</mi> </mrow> </msub> <mspace width="thinmathspace" /> <msup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{\beta }\,=\,m\,g_{\beta \alpha }\,u^{\alpha }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fac56f4780a74b03e504e3632ff8531a06ef108f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; margin-left: -0.089ex; width:15.069ex; height:3.009ex;" alt="{\displaystyle p_{\beta }\,=\,m\,g_{\beta \alpha }\,u^{\alpha }}"></span> </p><p>as it passes by an observer with four-velocity <i>u</i><sub>obs</sub>, then the expression for total energy of the particle as observed (measured in a local inertial frame) is </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E\,=\,-\,p_{\beta }\,u_{\text{obs}}^{\beta }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mspace width="thinmathspace" /> <mo>=</mo> <mspace width="thinmathspace" /> <mo>−</mo> <mspace width="thinmathspace" /> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msub> <mspace width="thinmathspace" /> <msubsup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>obs</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E\,=\,-\,p_{\beta }\,u_{\text{obs}}^{\beta }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bb06940c5dea9257a446738f54eae5025a9c43eb" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:14.52ex; height:3.509ex;" alt="{\displaystyle E\,=\,-\,p_{\beta }\,u_{\text{obs}}^{\beta }}"></span> </p><p>and the kinetic energy can be expressed as the total energy minus the rest energy: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{k}\,=\,-\,p_{\beta }\,u_{\text{obs}}^{\beta }\,-\,m\,c^{2}\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mspace width="thinmathspace" /> <mo>=</mo> <mspace width="thinmathspace" /> <mo>−</mo> <mspace width="thinmathspace" /> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msub> <mspace width="thinmathspace" /> <msubsup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>obs</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msubsup> <mspace width="thinmathspace" /> <mo>−</mo> <mspace width="thinmathspace" /> <mi>m</mi> <mspace width="thinmathspace" /> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{k}\,=\,-\,p_{\beta }\,u_{\text{obs}}^{\beta }\,-\,m\,c^{2}\,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ac20132d0de0d2f0081d627665ef7a1854eb04b6" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:24.685ex; height:3.509ex;" alt="{\displaystyle E_{k}\,=\,-\,p_{\beta }\,u_{\text{obs}}^{\beta }\,-\,m\,c^{2}\,.}"></span> </p><p>Consider the case of a metric that is diagonal and spatially isotropic (<i>g</i><sub><i>tt</i></sub>, <i>g</i><sub><i>ss</i></sub>, <i>g</i><sub><i>ss</i></sub>, <i>g</i><sub><i>ss</i></sub>). Since </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u^{\alpha }={\frac {dx^{\alpha }}{dt}}{\frac {dt}{d\tau }}=v^{\alpha }u^{t}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msup> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>t</mi> </mrow> <mrow> <mi>d</mi> <mi>τ</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msup> <msup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u^{\alpha }={\frac {dx^{\alpha }}{dt}}{\frac {dt}{d\tau }}=v^{\alpha }u^{t}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/58234444527ae29e2400ca87884f55959319e93d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:21.298ex; height:5.509ex;" alt="{\displaystyle u^{\alpha }={\frac {dx^{\alpha }}{dt}}{\frac {dt}{d\tau }}=v^{\alpha }u^{t}}"></span> </p><p>where <i>v</i><sup>α</sup> is the ordinary velocity measured w.r.t. the coordinate system, we get </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -c^{2}=g_{\alpha \beta }u^{\alpha }u^{\beta }=g_{tt}\left(u^{t}\right)^{2}+g_{ss}v^{2}\left(u^{t}\right)^{2}\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mi>β</mi> </mrow> </msub> <msup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msup> <msup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msup> <mo>=</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> <mi>t</mi> </mrow> </msub> <msup> <mrow> <mo>(</mo> <msup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msup> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> <mi>s</mi> </mrow> </msub> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mrow> <mo>(</mo> <msup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msup> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -c^{2}=g_{\alpha \beta }u^{\alpha }u^{\beta }=g_{tt}\left(u^{t}\right)^{2}+g_{ss}v^{2}\left(u^{t}\right)^{2}\,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/378a7fc27fcec901aea79e82941f34fe3b4097df" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:40.667ex; height:3.676ex;" alt="{\displaystyle -c^{2}=g_{\alpha \beta }u^{\alpha }u^{\beta }=g_{tt}\left(u^{t}\right)^{2}+g_{ss}v^{2}\left(u^{t}\right)^{2}\,.}"></span> </p><p>Solving for <i>u</i><sup>t</sup> gives </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u^{t}=c{\sqrt {\frac {-1}{g_{tt}+g_{ss}v^{2}}}}\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msup> <mo>=</mo> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mrow> <mo>−</mo> <mn>1</mn> </mrow> <mrow> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> <mi>t</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> <mi>s</mi> </mrow> </msub> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </msqrt> </mrow> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u^{t}=c{\sqrt {\frac {-1}{g_{tt}+g_{ss}v^{2}}}}\,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/84da52ec1237ff6751e671764fa2c4dc2a532d6f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:20.889ex; height:7.509ex;" alt="{\displaystyle u^{t}=c{\sqrt {\frac {-1}{g_{tt}+g_{ss}v^{2}}}}\,.}"></span> </p><p>Thus for a stationary observer (<i>v</i> = 0) </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u_{\text{obs}}^{t}=c{\sqrt {\frac {-1}{g_{tt}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>obs</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msubsup> <mo>=</mo> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mrow> <mo>−</mo> <mn>1</mn> </mrow> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> <mi>t</mi> </mrow> </msub> </mfrac> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u_{\text{obs}}^{t}=c{\sqrt {\frac {-1}{g_{tt}}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e71ae55733c0f90e9a0b886ba99a7e08938bd375" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:14.182ex; height:7.676ex;" alt="{\displaystyle u_{\text{obs}}^{t}=c{\sqrt {\frac {-1}{g_{tt}}}}}"></span> </p><p>and thus the kinetic energy takes the form </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{\text{k}}=-mg_{tt}u^{t}u_{\text{obs}}^{t}-mc^{2}=mc^{2}{\sqrt {\frac {g_{tt}}{g_{tt}+g_{ss}v^{2}}}}-mc^{2}\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>k</mtext> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mi>m</mi> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> <mi>t</mi> </mrow> </msub> <msup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msup> <msubsup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>obs</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msubsup> <mo>−</mo> <mi>m</mi> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mi>m</mi> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> <mi>t</mi> </mrow> </msub> <mrow> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> <mi>t</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> <mi>s</mi> </mrow> </msub> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </msqrt> </mrow> <mo>−</mo> <mi>m</mi> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{\text{k}}=-mg_{tt}u^{t}u_{\text{obs}}^{t}-mc^{2}=mc^{2}{\sqrt {\frac {g_{tt}}{g_{tt}+g_{ss}v^{2}}}}-mc^{2}\,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/45db3f8527096ca42e44be7c33d9f0abc27429db" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:54.104ex; height:6.176ex;" alt="{\displaystyle E_{\text{k}}=-mg_{tt}u^{t}u_{\text{obs}}^{t}-mc^{2}=mc^{2}{\sqrt {\frac {g_{tt}}{g_{tt}+g_{ss}v^{2}}}}-mc^{2}\,.}"></span> </p><p>Factoring out the rest energy gives: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{\text{k}}=mc^{2}\left({\sqrt {\frac {g_{tt}}{g_{tt}+g_{ss}v^{2}}}}-1\right)\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>k</mtext> </mrow> </msub> <mo>=</mo> <mi>m</mi> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> <mi>t</mi> </mrow> </msub> <mrow> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> <mi>t</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> <mi>s</mi> </mrow> </msub> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </msqrt> </mrow> <mo>−</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{\text{k}}=mc^{2}\left({\sqrt {\frac {g_{tt}}{g_{tt}+g_{ss}v^{2}}}}-1\right)\,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9608aecac2a83aff9e45831be8594de3a12da048" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:33.102ex; height:7.509ex;" alt="{\displaystyle E_{\text{k}}=mc^{2}\left({\sqrt {\frac {g_{tt}}{g_{tt}+g_{ss}v^{2}}}}-1\right)\,.}"></span> </p><p>This expression reduces to the special relativistic case for the flat-space metric where </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}g_{tt}&=-c^{2}\\g_{ss}&=1\,.\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> <mi>t</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo>−</mo> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mtd> </mtr> <mtr> <mtd> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> <mi>s</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>1</mn> <mspace width="thinmathspace" /> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}g_{tt}&=-c^{2}\\g_{ss}&=1\,.\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9c9120c9858564101689d216ac5b9caf5b722d67" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:10.603ex; height:6.176ex;" alt="{\displaystyle {\begin{aligned}g_{tt}&=-c^{2}\\g_{ss}&=1\,.\end{aligned}}}"></span> </p><p>In the Newtonian approximation to general relativity </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}g_{tt}&=-\left(c^{2}+2\Phi \right)\\g_{ss}&=1-{\frac {2\Phi }{c^{2}}}\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> <mi>t</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo>−</mo> <mrow> <mo>(</mo> <mrow> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>2</mn> <mi mathvariant="normal">Φ</mi> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> <mi>s</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi mathvariant="normal">Φ</mi> </mrow> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}g_{tt}&=-\left(c^{2}+2\Phi \right)\\g_{ss}&=1-{\frac {2\Phi }{c^{2}}}\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/135968cca6b455dfaf7a92d27386ae842ce36c19" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:18.8ex; height:9.176ex;" alt="{\displaystyle {\begin{aligned}g_{tt}&=-\left(c^{2}+2\Phi \right)\\g_{ss}&=1-{\frac {2\Phi }{c^{2}}}\end{aligned}}}"></span> </p><p>where Φ is the Newtonian <a href="/wiki/Gravitational_potential" title="Gravitational potential">gravitational potential</a>. This means clocks run slower and measuring rods are shorter near massive bodies. </p> <div class="mw-heading mw-heading2"><h2 id="Kinetic_energy_in_quantum_mechanics">Kinetic energy in quantum mechanics</h2></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Further information: <a href="/wiki/Hamiltonian_(quantum_mechanics)" title="Hamiltonian (quantum mechanics)">Hamiltonian (quantum mechanics)</a></div> <p>In <a href="/wiki/Quantum_mechanics" title="Quantum mechanics">quantum mechanics</a>, observables like kinetic energy are represented as <a href="/wiki/Operator_(physics)" title="Operator (physics)">operators</a>. For one particle of mass <i>m</i>, the kinetic energy operator appears as a term in the <a href="/wiki/Hamiltonian_(quantum_mechanics)" title="Hamiltonian (quantum mechanics)">Hamiltonian</a> and is defined in terms of the more fundamental momentum operator <span class="mwe-math-element"><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 {p}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>p</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {p}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8bd4c026f1b3413adc58b9b65e89e62bce92c85a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.449ex; height:2.509ex;" alt="{\displaystyle {\hat {p}}}"></span>. The kinetic energy operator in the <a href="/wiki/Relativistic_quantum_mechanics#Non-relativistic_and_relativistic_Hamiltonians" title="Relativistic quantum mechanics">non-relativistic</a> case can be written as </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {T}}={\frac {{\hat {p}}^{2}}{2m}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>T</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>p</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mn>2</mn> <mi>m</mi> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {T}}={\frac {{\hat {p}}^{2}}{2m}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c8b54fcc1e23ef7bdf7e0ef90ce2e39ed830ac7" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:9.558ex; height:5.843ex;" alt="{\displaystyle {\hat {T}}={\frac {{\hat {p}}^{2}}{2m}}.}"></span> </p><p>Notice that this can be obtained by replacing <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span> by <span class="mwe-math-element"><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 {p}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>p</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {p}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8bd4c026f1b3413adc58b9b65e89e62bce92c85a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.449ex; height:2.509ex;" alt="{\displaystyle {\hat {p}}}"></span> in the classical expression for kinetic energy in terms of <a href="/wiki/Momentum" title="Momentum">momentum</a>, </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{\text{k}}={\frac {p^{2}}{2m}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>k</mtext> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mn>2</mn> <mi>m</mi> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{\text{k}}={\frac {p^{2}}{2m}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7041db2558d1e6e149508a392076a3eeea7fc1be" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:10.6ex; height:5.676ex;" alt="{\displaystyle E_{\text{k}}={\frac {p^{2}}{2m}}.}"></span> </p><p>In the <a href="/wiki/Schr%C3%B6dinger_picture" title="Schrödinger picture">Schrödinger picture</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 {\hat {p}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>p</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {p}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8bd4c026f1b3413adc58b9b65e89e62bce92c85a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.449ex; height:2.509ex;" alt="{\displaystyle {\hat {p}}}"></span> takes the form <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -i\hbar \nabla }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mi>i</mi> <mi class="MJX-variant">ℏ</mi> <mi mathvariant="normal">∇</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -i\hbar \nabla }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7282d9d33646ecea0e5441adc5b48e535a703db6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.853ex; height:2.343ex;" alt="{\displaystyle -i\hbar \nabla }"></span> where the derivative is taken with respect to position coordinates and hence </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {T}}=-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>T</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi class="MJX-variant">ℏ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mn>2</mn> <mi>m</mi> </mrow> </mfrac> </mrow> <msup> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {T}}=-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/147dd6a27d8746092989005194552ba14cfa8077" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:14.357ex; height:5.676ex;" alt="{\displaystyle {\hat {T}}=-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}.}"></span> </p><p>The expectation value of the electron kinetic energy, <span class="mwe-math-element"><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\langle {\hat {T}}\right\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>⟨</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>T</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>⟩</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left\langle {\hat {T}}\right\rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/03d71a29760cb8580d1d9e2eede254da1534f744" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:4.615ex; height:4.843ex;" alt="{\displaystyle \left\langle {\hat {T}}\right\rangle }"></span>, for a system of <i>N</i> electrons described by the <a href="/wiki/Wave_function" title="Wave function">wavefunction</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 \vert \psi \rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">|</mo> <mi>ψ</mi> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \vert \psi \rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8addc466341daafbbf224730defba59505d1e760" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.065ex; height:2.843ex;" alt="{\displaystyle \vert \psi \rangle }"></span> is a sum of 1-electron operator expectation values: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left\langle {\hat {T}}\right\rangle =\left\langle \psi \left\vert \sum _{i=1}^{N}{\frac {-\hbar ^{2}}{2m_{\text{e}}}}\nabla _{i}^{2}\right\vert \psi \right\rangle =-{\frac {\hbar ^{2}}{2m_{\text{e}}}}\sum _{i=1}^{N}\left\langle \psi \left\vert \nabla _{i}^{2}\right\vert \psi \right\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>⟨</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>T</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>⟩</mo> </mrow> <mo>=</mo> <mrow> <mo>⟨</mo> <mrow> <mi>ψ</mi> <mrow> <mo>|</mo> <mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>−</mo> <msup> <mi class="MJX-variant">ℏ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mn>2</mn> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>e</mtext> </mrow> </msub> </mrow> </mfrac> </mrow> <msubsup> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mrow> <mo>|</mo> </mrow> <mi>ψ</mi> </mrow> <mo>⟩</mo> </mrow> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi class="MJX-variant">ℏ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mn>2</mn> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>e</mtext> </mrow> </msub> </mrow> </mfrac> </mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </munderover> <mrow> <mo>⟨</mo> <mrow> <mi>ψ</mi> <mrow> <mo>|</mo> <msubsup> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>|</mo> </mrow> <mi>ψ</mi> </mrow> <mo>⟩</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left\langle {\hat {T}}\right\rangle =\left\langle \psi \left\vert \sum _{i=1}^{N}{\frac {-\hbar ^{2}}{2m_{\text{e}}}}\nabla _{i}^{2}\right\vert \psi \right\rangle =-{\frac {\hbar ^{2}}{2m_{\text{e}}}}\sum _{i=1}^{N}\left\langle \psi \left\vert \nabla _{i}^{2}\right\vert \psi \right\rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aa88e2c6bca46a10409fcf958dd865603028d9cb" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:52.622ex; height:7.509ex;" alt="{\displaystyle \left\langle {\hat {T}}\right\rangle =\left\langle \psi \left\vert \sum _{i=1}^{N}{\frac {-\hbar ^{2}}{2m_{\text{e}}}}\nabla _{i}^{2}\right\vert \psi \right\rangle =-{\frac {\hbar ^{2}}{2m_{\text{e}}}}\sum _{i=1}^{N}\left\langle \psi \left\vert \nabla _{i}^{2}\right\vert \psi \right\rangle }"></span> </p><p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m_{\text{e}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>e</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m_{\text{e}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c45a5eb082ea4dafd3cb43ca39e033989e4a52eb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.003ex; height:2.009ex;" alt="{\displaystyle m_{\text{e}}}"></span> is the mass of the electron and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nabla _{i}^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nabla _{i}^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/036dc2fbf8e2560ea75b0b67841cb36e83dfab1c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.99ex; height:3.176ex;" alt="{\displaystyle \nabla _{i}^{2}}"></span> is the <a href="/wiki/Laplacian" class="mw-redirect" title="Laplacian">Laplacian</a> operator acting upon the coordinates of the <i>i</i><sup>th</sup> electron and the summation runs over all electrons. </p><p>The <a href="/wiki/Density_functional_theory" title="Density functional theory">density functional</a> formalism of quantum mechanics requires knowledge of the electron density <i>only</i>, i.e., it formally does not require knowledge of the wavefunction. Given an electron density <span class="mwe-math-element"><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 (\mathbf {r} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho (\mathbf {r} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/77f477411625125978c0a18946bdfae2c1f13bcb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.113ex; height:2.843ex;" alt="{\displaystyle \rho (\mathbf {r} )}"></span>, the exact N-electron kinetic energy functional is unknown; however, for the specific case of a 1-electron system, the kinetic energy can be written as </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T[\rho ]={\frac {1}{8}}\int {\frac {\nabla \rho (\mathbf {r} )\cdot \nabla \rho (\mathbf {r} )}{\rho (\mathbf {r} )}}d^{3}r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo stretchy="false">[</mo> <mi>ρ</mi> <mo stretchy="false">]</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>8</mn> </mfrac> </mrow> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∇</mi> <mi>ρ</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">)</mo> <mo>⋅</mo> <mi mathvariant="normal">∇</mi> <mi>ρ</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>ρ</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T[\rho ]={\frac {1}{8}}\int {\frac {\nabla \rho (\mathbf {r} )\cdot \nabla \rho (\mathbf {r} )}{\rho (\mathbf {r} )}}d^{3}r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/25c6547c29b24c193b6dc8a0276e1e820cfb8da6" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:30.131ex; height:6.509ex;" alt="{\displaystyle T[\rho ]={\frac {1}{8}}\int {\frac {\nabla \rho (\mathbf {r} )\cdot \nabla \rho (\mathbf {r} )}{\rho (\mathbf {r} )}}d^{3}r}"></span> </p><p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T[\rho ]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo stretchy="false">[</mo> <mi>ρ</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T[\rho ]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9f33f6be9ea929173b30036aae6d57718be7b9c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.132ex; height:2.843ex;" alt="{\displaystyle T[\rho ]}"></span> is known as the <a href="/wiki/Carl_Friedrich_von_Weizs%C3%A4cker" title="Carl Friedrich von Weizsäcker">von Weizsäcker</a> kinetic energy functional. </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2></div> <style data-mw-deduplicate="TemplateStyles:r1266661725">.mw-parser-output .portalbox{padding:0;margin:0.5em 0;display:table;box-sizing:border-box;max-width:175px;list-style:none}.mw-parser-output .portalborder{border:1px solid var(--border-color-base,#a2a9b1);padding:0.1em;background:var(--background-color-neutral-subtle,#f8f9fa)}.mw-parser-output .portalbox-entry{display:table-row;font-size:85%;line-height:110%;height:1.9em;font-style:italic;font-weight:bold}.mw-parser-output .portalbox-image{display:table-cell;padding:0.2em;vertical-align:middle;text-align:center}.mw-parser-output .portalbox-link{display:table-cell;padding:0.2em 0.2em 0.2em 0.3em;vertical-align:middle}@media(min-width:720px){.mw-parser-output .portalleft{margin:0.5em 1em 0.5em 0}.mw-parser-output .portalright{clear:right;float:right;margin:0.5em 0 0.5em 1em}}</style><ul role="navigation" aria-label="Portals" class="noprint portalbox portalborder portalright"> <li class="portalbox-entry"><span class="portalbox-image"><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Crystal_energy.svg" class="mw-file-description"><img alt="icon" src="//upload.wikimedia.org/wikipedia/commons/thumb/1/14/Crystal_energy.svg/29px-Crystal_energy.svg.png" decoding="async" width="29" height="28" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/14/Crystal_energy.svg/44px-Crystal_energy.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/14/Crystal_energy.svg/59px-Crystal_energy.svg.png 2x" data-file-width="130" data-file-height="124" /></a></span></span><span class="portalbox-link"><a href="/wiki/Portal:Energy" title="Portal:Energy">Energy portal</a></span></li></ul> <ul><li><a href="/wiki/Escape_velocity" title="Escape velocity">Escape velocity</a></li> <li><a href="/wiki/Foot-pound" class="mw-redirect" title="Foot-pound">Foot-pound</a></li> <li><a href="/wiki/Joule" title="Joule">Joule</a></li> <li><a href="/wiki/Kinetic_energy_penetrator" title="Kinetic energy penetrator">Kinetic energy penetrator</a></li> <li><a href="/wiki/Projectile#Typical_projectile_speeds" title="Projectile">Kinetic energy per unit mass of projectiles</a></li> <li><a href="/wiki/Projectile#Kinetic_projectiles" title="Projectile">Kinetic projectile</a></li> <li><a href="/wiki/Parallel_axis_theorem" title="Parallel axis theorem">Parallel axis theorem</a></li> <li><a href="/wiki/Potential_energy" title="Potential energy">Potential energy</a></li> <li><a href="/wiki/Recoil" title="Recoil">Recoil</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFJain2009" class="citation book cs1">Jain, Mahesh C. (2009). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=wKeDYbTuiPAC"><i>Textbook of Engineering Physics (Part I)</i></a>. PHI Learning Pvt. p. 9. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-81-203-3862-3" title="Special:BookSources/978-81-203-3862-3"><bdi>978-81-203-3862-3</bdi></a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20200804012822/https://books.google.com/books?id=wKeDYbTuiPAC">Archived</a> from the original on 2020-08-04<span class="reference-accessdate">. Retrieved <span class="nowrap">2018-06-21</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Textbook+of+Engineering+Physics+%28Part+I%29&rft.pages=9&rft.pub=PHI+Learning+Pvt.&rft.date=2009&rft.isbn=978-81-203-3862-3&rft.aulast=Jain&rft.aufirst=Mahesh+C.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DwKeDYbTuiPAC&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKinetic+energy" class="Z3988"></span>, <a rel="nofollow" class="external text" href="https://books.google.com/books?id=wKeDYbTuiPAC&q=kinetic&pg=PA9">Chapter 1, p. 9</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20200804012414/https://books.google.com/books?id=wKeDYbTuiPAC&pg=PA9#v=snippet&q=kinetic&f=false">Archived</a> 2020-08-04 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a></span> </li> <li id="cite_note-R&H-2"><span class="mw-cite-backlink">^ <a href="#cite_ref-R&H_2-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-R&H_2-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text">Resnick, Robert and Halliday, David (1960) <i>Physics</i>, Section 7-5, Wiley International Edition</span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBrenner2008" class="citation book cs1">Brenner, Joseph (2008). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=Jnj5E6C9UwsC"><i>Logic in Reality</i></a> (illustrated ed.). Springer Science & Business Media. p. 93. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4020-8375-4" title="Special:BookSources/978-1-4020-8375-4"><bdi>978-1-4020-8375-4</bdi></a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20200125133150/https://books.google.com/books?id=Jnj5E6C9UwsC">Archived</a> from the original on 2020-01-25<span class="reference-accessdate">. Retrieved <span class="nowrap">2016-02-01</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Logic+in+Reality&rft.pages=93&rft.edition=illustrated&rft.pub=Springer+Science+%26+Business+Media&rft.date=2008&rft.isbn=978-1-4020-8375-4&rft.aulast=Brenner&rft.aufirst=Joseph&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DJnj5E6C9UwsC&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKinetic+energy" class="Z3988"></span> <a rel="nofollow" class="external text" href="https://books.google.com/books?id=Jnj5E6C9UwsC&pg=PA93">Extract of page 93</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20200804010734/https://books.google.com/books?id=Jnj5E6C9UwsC&pg=PA93">Archived</a> 2020-08-04 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a></span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFeather1959" class="citation book cs1">Feather, Norman (1959). <i>An Introduction to the Physics of Mass Length and Time – Hardcover</i>. Edinburgh University Press.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=An+Introduction+to+the+Physics+of+Mass+Length+and+Time+%E2%80%93+Hardcover&rft.pub=Edinburgh+University+Press&rft.date=1959&rft.aulast=Feather&rft.aufirst=Norman&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKinetic+energy" class="Z3988"></span></span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJudith_P._Zinsser2007" class="citation book cs1">Judith P. Zinsser (2007). <i>Emilie du Chatelet: Daring Genius of the Enlightenment</i>. Penguin. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-14-311268-6" title="Special:BookSources/978-0-14-311268-6"><bdi>978-0-14-311268-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Emilie+du+Chatelet%3A+Daring+Genius+of+the+Enlightenment&rft.pub=Penguin&rft.date=2007&rft.isbn=978-0-14-311268-6&rft.au=Judith+P.+Zinsser&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKinetic+energy" class="Z3988"></span></span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCrosbie_Smith,_M._Norton_Wise1989" class="citation book cs1">Crosbie Smith, M. Norton Wise (1989-10-26). <i>Energy and Empire: A Biographical Study of Lord Kelvin</i>. Cambridge University Press. p. 866. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-521-26173-2" title="Special:BookSources/0-521-26173-2"><bdi>0-521-26173-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Energy+and+Empire%3A+A+Biographical+Study+of+Lord+Kelvin&rft.pages=866&rft.pub=Cambridge+University+Press&rft.date=1989-10-26&rft.isbn=0-521-26173-2&rft.au=Crosbie+Smith%2C+M.+Norton+Wise&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKinetic+energy" class="Z3988"></span></span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJohn_Theodore_Merz1912" class="citation book cs1">John Theodore Merz (1912). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/historyofeuropea00merz_1/page/139"><i>A History of European Thought in the Nineteenth Century</i></a></span>. Blackwood. p. <a rel="nofollow" class="external text" href="https://archive.org/details/historyofeuropea00merz_1/page/139">139</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-8446-2579-5" title="Special:BookSources/0-8446-2579-5"><bdi>0-8446-2579-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+History+of+European+Thought+in+the+Nineteenth+Century&rft.pages=139&rft.pub=Blackwood&rft.date=1912&rft.isbn=0-8446-2579-5&rft.au=John+Theodore+Merz&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fhistoryofeuropea00merz_1%2Fpage%2F139&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKinetic+energy" class="Z3988"></span></span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWilliam_John_Macquorn_Rankine1853" class="citation journal cs1">William John Macquorn Rankine (1853). <a rel="nofollow" class="external text" href="https://archive.org/details/miscellaneoussci00rank/page/202/mode/2up">"On the general law of the transformation of energy"</a>. <i>Proceedings of the Philosophical Society of Glasgow</i>. <b>3</b> (5).</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Proceedings+of+the+Philosophical+Society+of+Glasgow&rft.atitle=On+the+general+law+of+the+transformation+of+energy&rft.volume=3&rft.issue=5&rft.date=1853&rft.au=William+John+Macquorn+Rankine&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fmiscellaneoussci00rank%2Fpage%2F202%2Fmode%2F2up&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKinetic+energy" class="Z3988"></span></span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text">"... what remained to be done, was to qualify the noun 'energy' by appropriate adjectives, so as to distinguish between energy of activity and energy of configuration. The well-known pair of antithetical adjectives, 'actual' and 'potential,' seemed exactly suited for that purpose. ... Sir William Thomson and Professor Tait have lately substituted the word 'kinetic' for 'actual.<span style="padding-right:.15em;">'</span>" <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWilliam_John_Macquorn_Rankine1867" class="citation journal cs1"><a href="/wiki/William_John_Macquorn_Rankine" class="mw-redirect" title="William John Macquorn Rankine">William John Macquorn Rankine</a> (1867). <a rel="nofollow" class="external text" href="https://archive.org/details/miscellaneoussci00rank/page/230/mode/2up">"On the Phrase "Potential Energy," and on the Definitions of Physical Quantities"</a>. <i>Proceedings of the Philosophical Society of Glasgow</i>. <b>VI</b> (III).</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Proceedings+of+the+Philosophical+Society+of+Glasgow&rft.atitle=On+the+Phrase+%22Potential+Energy%2C%22+and+on+the+Definitions+of+Physical+Quantities&rft.volume=VI&rft.issue=III&rft.date=1867&rft.au=William+John+Macquorn+Rankine&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fmiscellaneoussci00rank%2Fpage%2F230%2Fmode%2F2up&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKinetic+energy" class="Z3988"></span></span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGoel2007" class="citation book cs1">Goel, V. K. (2007). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=2PaYugROudwC"><i>Fundamentals Of Physics Xi</i></a> (illustrated ed.). Tata McGraw-Hill Education. p. 12.30. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-07-062060-5" title="Special:BookSources/978-0-07-062060-5"><bdi>978-0-07-062060-5</bdi></a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20200803032820/https://books.google.com/books?id=2PaYugROudwC">Archived</a> from the original on 2020-08-03<span class="reference-accessdate">. Retrieved <span class="nowrap">2020-07-07</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Fundamentals+Of+Physics+Xi&rft.pages=12.30&rft.edition=illustrated&rft.pub=Tata+McGraw-Hill+Education&rft.date=2007&rft.isbn=978-0-07-062060-5&rft.aulast=Goel&rft.aufirst=V.+K.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D2PaYugROudwC&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKinetic+energy" class="Z3988"></span> <a rel="nofollow" class="external text" href="https://books.google.com/books?id=2PaYugROudwC&pg=RA11-PA30">Extract of page 12.30</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20200707183114/https://books.google.be/books?id=2PaYugROudwC&pg=RA11-PA30">Archived</a> 2020-07-07 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a></span> </li> <li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text">A. M. Kuethe and J. D. Schetzer (1959). <i>Foundations of Aerodynamics</i>, 2nd edition, p.53. John Wiley & Sons <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-471-50952-3" title="Special:BookSources/0-471-50952-3">0-471-50952-3</a></span> </li> <li id="cite_note-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-12">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSearsBrehme1968" class="citation book cs1">Sears, Francis Weston; Brehme, Robert W. (1968). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/introductiontoth0000sear_c2m9"><i>Introduction to the theory of relativity</i></a></span>. Addison-Wesley. p. <a rel="nofollow" class="external text" href="https://archive.org/details/introductiontoth0000sear_c2m9/page/127">127</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+the+theory+of+relativity&rft.pages=127&rft.pub=Addison-Wesley&rft.date=1968&rft.aulast=Sears&rft.aufirst=Francis+Weston&rft.au=Brehme%2C+Robert+W.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fintroductiontoth0000sear_c2m9&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKinetic+energy" class="Z3988"></span>, <a rel="nofollow" class="external text" href="https://books.google.com/books?id=cpzvAAAAMAAJ&q=%22in+its+own+rest+frame%22">Snippet view of page 127</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20200804041331/https://books.google.com/books?id=cpzvAAAAMAAJ&dq=%22in+its+own+rest+frame%22+%22kinetic+energy%22&q=%22in+its+own+rest+frame%22">Archived</a> 2020-08-04 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a></span> </li> <li id="cite_note-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-13">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="http://www.phy.duke.edu/~rgb/Class/intro_physics_1/intro_physics_1/node64.html">Physics notes – Kinetic energy in the CM frame</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20070611231147/http://www.phy.duke.edu/~rgb/Class/intro_physics_1/intro_physics_1/node64.html">Archived</a> 2007-06-11 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a>. <a href="/wiki/Duke_University" title="Duke University">Duke</a>.edu. Accessed 2007-11-24.</span> </li> <li id="cite_note-Einstein1922-14"><span class="mw-cite-backlink">^ <a href="#cite_ref-Einstein1922_14-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Einstein1922_14-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFEinstein1922" class="citation book cs1">Einstein, Albert (1922). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=0nIxAQAAMAAJ&q=%22Mass%20and%20energy%20are%20therefore%20essentially%20alike%22"><i>The Meaning of Relativity: Four Lectures Delivered at Princeton University, May, 1921</i></a>. Methuen & Company Limited. pp. <span class="nowrap">51–</span>52.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Meaning+of+Relativity%3A+Four+Lectures+Delivered+at+Princeton+University%2C+May%2C+1921&rft.pages=%3Cspan+class%3D%22nowrap%22%3E51-%3C%2Fspan%3E52&rft.pub=Methuen+%26+Company+Limited&rft.date=1922&rft.aulast=Einstein&rft.aufirst=Albert&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D0nIxAQAAMAAJ%26q%3D%2522Mass%2520and%2520energy%2520are%2520therefore%2520essentially%2520alike%2522&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKinetic+energy" class="Z3988"></span></span> </li> <li id="cite_note-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-15">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFResnick1968" class="citation book cs1">Resnick, Robert (1968). <a rel="nofollow" class="external text" href="https://archive.org/details/robertresnickintroductiontospecialrelativitywiley1968/page/n131/mode/2up?q=%22kinetic+energy%22"><i>Introduction to special relativity</i></a>. New York: Wiley. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-471-71725-6" title="Special:BookSources/978-0-471-71725-6"><bdi>978-0-471-71725-6</bdi></a><span class="reference-accessdate">. Retrieved <span class="nowrap">2024-11-17</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+special+relativity&rft.place=New+York&rft.pub=Wiley&rft.date=1968&rft.isbn=978-0-471-71725-6&rft.aulast=Resnick&rft.aufirst=Robert&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Frobertresnickintroductiontospecialrelativitywiley1968%2Fpage%2Fn131%2Fmode%2F2up%3Fq%3D%2522kinetic%2Benergy%2522&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKinetic+energy" class="Z3988"></span></span> </li> <li id="cite_note-16"><span class="mw-cite-backlink"><b><a href="#cite_ref-16">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFitzpatrick2010" class="citation web cs1">Fitzpatrick, Richard (20 July 2010). <a rel="nofollow" class="external text" href="http://farside.ph.utexas.edu/teaching/qmech/Quantum/node107.html">"Fine Structure of Hydrogen"</a>. <i>Quantum Mechanics</i>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20160825061339/http://farside.ph.utexas.edu/teaching/qmech/Quantum/node107.html">Archived</a> from the original on 25 August 2016<span class="reference-accessdate">. Retrieved <span class="nowrap">20 August</span> 2016</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Quantum+Mechanics&rft.atitle=Fine+Structure+of+Hydrogen&rft.date=2010-07-20&rft.aulast=Fitzpatrick&rft.aufirst=Richard&rft_id=http%3A%2F%2Ffarside.ph.utexas.edu%2Fteaching%2Fqmech%2FQuantum%2Fnode107.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKinetic+energy" class="Z3988"></span></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPhysics_Classroom2000" class="citation web cs1">Physics Classroom (2000). <a rel="nofollow" class="external text" href="http://www.physicsclassroom.com/class/energy/Lesson-1/Kinetic-Energy">"Kinetic Energy"</a><span class="reference-accessdate">. Retrieved <span class="nowrap">2015-07-19</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Kinetic+Energy&rft.date=2000&rft.au=Physics+Classroom&rft_id=http%3A%2F%2Fwww.physicsclassroom.com%2Fclass%2Fenergy%2FLesson-1%2FKinetic-Energy&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKinetic+energy" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSchool_of_Mathematics_and_Statistics,_University_of_St_Andrews2000" class="citation web cs1">School of Mathematics and Statistics, University of St Andrews (2000). <a rel="nofollow" class="external text" href="http://www-history.mcs.st-andrews.ac.uk/Mathematicians/Coriolis.html">"Biography of Gaspard-Gustave de Coriolis (1792–1843)"</a><span class="reference-accessdate">. Retrieved <span class="nowrap">2006-03-03</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Biography+of+Gaspard-Gustave+de+Coriolis+%281792%E2%80%931843%29&rft.date=2000&rft.au=School+of+Mathematics+and+Statistics%2C+University+of+St+Andrews&rft_id=http%3A%2F%2Fwww-history.mcs.st-andrews.ac.uk%2FMathematicians%2FCoriolis.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKinetic+energy" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSerwayJewett,_John_W.2004" class="citation book cs1">Serway, Raymond A.; Jewett, John W. (2004). <a rel="nofollow" class="external text" href="https://archive.org/details/physicssciengv2p00serw"><i>Physics for Scientists and Engineers</i></a> (6th ed.). Brooks/Cole. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-534-40842-7" title="Special:BookSources/0-534-40842-7"><bdi>0-534-40842-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Physics+for+Scientists+and+Engineers&rft.edition=6th&rft.pub=Brooks%2FCole&rft.date=2004&rft.isbn=0-534-40842-7&rft.aulast=Serway&rft.aufirst=Raymond+A.&rft.au=Jewett%2C+John+W.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fphysicssciengv2p00serw&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKinetic+energy" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFTipler2004" class="citation book cs1">Tipler, Paul (2004). <i>Physics for Scientists and Engineers: Mechanics, Oscillations and Waves, Thermodynamics</i> (5th ed.). W. H. Freeman. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-7167-0809-4" title="Special:BookSources/0-7167-0809-4"><bdi>0-7167-0809-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Physics+for+Scientists+and+Engineers%3A+Mechanics%2C+Oscillations+and+Waves%2C+Thermodynamics&rft.edition=5th&rft.pub=W.+H.+Freeman&rft.date=2004&rft.isbn=0-7167-0809-4&rft.aulast=Tipler&rft.aufirst=Paul&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKinetic+energy" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFTiplerLlewellyn,_Ralph2002" class="citation book cs1">Tipler, Paul; Llewellyn, Ralph (2002). <i>Modern Physics</i> (4th ed.). W. H. Freeman. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-7167-4345-0" title="Special:BookSources/0-7167-4345-0"><bdi>0-7167-4345-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Modern+Physics&rft.edition=4th&rft.pub=W.+H.+Freeman&rft.date=2002&rft.isbn=0-7167-4345-0&rft.aulast=Tipler&rft.aufirst=Paul&rft.au=Llewellyn%2C+Ralph&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKinetic+energy" class="Z3988"></span></li></ul> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div> <ul><li><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Commons-logo.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/12px-Commons-logo.svg.png" decoding="async" width="12" height="16" class="mw-file-element" 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<li><a href="/wiki/Thermodynamic_potential" title="Thermodynamic potential">Thermodynamic potential</a></li> <li><a href="/wiki/Thermodynamic_state" title="Thermodynamic state">Thermodynamic state</a></li> <li><a href="/wiki/Thermodynamic_system" title="Thermodynamic system">Thermodynamic system</a></li> <li><a href="/wiki/Thermodynamic_temperature" title="Thermodynamic temperature">Thermodynamic temperature</a></li> <li><a href="/wiki/Volume_(thermodynamics)" title="Volume (thermodynamics)">Volume (thermodynamics)</a></li> <li><a href="/wiki/Work_(physics)" title="Work (physics)">Work</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Types</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Binding_energy" title="Binding energy">Binding</a> <ul><li><a href="/wiki/Nuclear_binding_energy" title="Nuclear binding energy">Nuclear</a></li></ul></li> <li><a href="/wiki/Chemical_energy" title="Chemical energy">Chemical</a></li> <li><a href="/wiki/Dark_energy" title="Dark energy">Dark</a></li> <li><a href="/wiki/Elastic_energy" title="Elastic energy">Elastic</a></li> <li><a href="/wiki/Electric_potential_energy" title="Electric potential energy">Electric potential energy</a></li> <li><a href="/wiki/Electrical_energy" title="Electrical energy">Electrical</a></li> <li><a href="/wiki/Gravitational_energy" title="Gravitational energy">Gravitational</a> <ul><li><a href="/wiki/Gravitational_binding_energy" title="Gravitational binding energy">Binding</a></li></ul></li> <li><a href="/wiki/Interatomic_potential" title="Interatomic potential">Interatomic potential</a></li> <li><a href="/wiki/Internal_energy" title="Internal energy">Internal</a></li> <li><a href="/wiki/Ionization_energy" title="Ionization energy">Ionization</a></li> <li><a class="mw-selflink selflink">Kinetic</a></li> <li><a href="/wiki/Magnetic_energy" title="Magnetic energy">Magnetic</a></li> <li><a href="/wiki/Mechanical_energy" title="Mechanical energy">Mechanical</a></li> <li><a href="/wiki/Negative_energy" title="Negative energy">Negative</a></li> <li><a href="/wiki/Phantom_energy" title="Phantom energy">Phantom</a></li> <li><a href="/wiki/Potential_energy" title="Potential energy">Potential</a></li> <li><a href="/wiki/Quantum_chromodynamics_binding_energy" title="Quantum chromodynamics binding energy">Quantum chromodynamics binding energy</a></li> <li><a href="/wiki/Quantum_fluctuation" title="Quantum fluctuation">Quantum fluctuation</a></li> <li><a href="/wiki/Quantum_potential" title="Quantum potential">Quantum potential</a></li> <li><a href="/wiki/Quintessence_(physics)" title="Quintessence (physics)">Quintessence</a></li> <li><a href="/wiki/Radiant_energy" title="Radiant energy">Radiant</a></li> <li><a href="/wiki/Rest_energy" class="mw-redirect" title="Rest energy">Rest</a></li> <li><a href="/wiki/Sound_energy" title="Sound energy">Sound</a></li> <li><a href="/wiki/Surface_energy" title="Surface energy">Surface</a></li> <li><a href="/wiki/Thermal_energy" title="Thermal energy">Thermal</a></li> <li><a href="/wiki/Vacuum_energy" title="Vacuum energy">Vacuum</a></li> <li><a href="/wiki/Zero-point_energy" title="Zero-point energy">Zero-point</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Energy_carrier" title="Energy carrier">Energy carriers</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Electric_battery" title="Electric battery">Battery</a></li> <li><a href="/wiki/Capacitor" title="Capacitor">Capacitor</a></li> <li><a href="/wiki/Electricity" title="Electricity">Electricity</a></li> <li><a href="/wiki/Enthalpy" title="Enthalpy">Enthalpy</a></li> <li><a href="/wiki/Fuel" title="Fuel">Fuel</a> <ul><li><a href="/wiki/Fossil_fuel" title="Fossil fuel">Fossil</a></li> <li><a href="/wiki/Fuel_oil" title="Fuel oil">Oil</a></li></ul></li> <li><a href="/wiki/Heat" title="Heat">Heat</a> <ul><li><a href="/wiki/Latent_heat" title="Latent heat">Latent heat</a></li></ul></li> <li><a href="/wiki/Hydrogen" title="Hydrogen">Hydrogen</a> <ul><li><a href="/wiki/Hydrogen_fuel" class="mw-redirect" title="Hydrogen fuel">Hydrogen fuel</a></li></ul></li> <li><a href="/wiki/Mechanical_wave" title="Mechanical wave">Mechanical wave</a></li> <li><a href="/wiki/Radiation" title="Radiation">Radiation</a></li> <li><a href="/wiki/Sound_wave" class="mw-redirect" title="Sound wave">Sound wave</a></li> <li><a href="/wiki/Work_(physics)" title="Work (physics)">Work</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Primary_energy" title="Primary energy">Primary energy</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Bioenergy" title="Bioenergy">Bioenergy</a></li> <li><a href="/wiki/Fossil_fuel" title="Fossil fuel">Fossil fuel</a> <ul><li><a href="/wiki/Coal" title="Coal">Coal</a></li> <li><a href="/wiki/Natural_gas" title="Natural gas">Natural gas</a></li> <li><a href="/wiki/Petroleum" title="Petroleum">Petroleum</a></li></ul></li> <li><a href="/wiki/Geothermal_energy" title="Geothermal energy">Geothermal</a></li> <li><a href="/wiki/Gravitational_energy" title="Gravitational energy">Gravitational</a></li> <li><a href="/wiki/Hydropower" title="Hydropower">Hydropower</a></li> <li><a href="/wiki/Marine_energy" title="Marine energy">Marine</a></li> <li><a href="/wiki/Nuclear_fuel" title="Nuclear fuel">Nuclear fuel</a> <ul><li><a href="/wiki/Natural_uranium" title="Natural uranium">Natural uranium</a></li></ul></li> <li><a href="/wiki/Radiant_energy" title="Radiant energy">Radiant</a></li> <li><a href="/wiki/Solar_energy" title="Solar energy">Solar</a></li> <li><a href="/wiki/Wind_power" title="Wind power">Wind</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Energy_system" title="Energy system">Energy system</a><br />components</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Biomass" title="Biomass">Biomass</a></li> <li><a href="/wiki/Electric_power" title="Electric power">Electric power</a></li> <li><a href="/wiki/Electricity_delivery" title="Electricity delivery">Electricity delivery</a></li> <li><a href="/wiki/Energy_engineering" title="Energy engineering">Energy engineering</a></li> <li><a href="/wiki/Fossil_fuel_power_station" title="Fossil fuel power station">Fossil fuel power station</a> <ul><li><a href="/wiki/Cogeneration" title="Cogeneration">Cogeneration</a></li> <li><a href="/wiki/Integrated_gasification_combined_cycle" title="Integrated gasification combined cycle">Integrated gasification combined cycle</a></li></ul></li> <li><a href="/wiki/Geothermal_power" title="Geothermal power">Geothermal power</a></li> <li><a href="/wiki/Hydropower" title="Hydropower">Hydropower</a> <ul><li><a href="/wiki/Hydroelectricity" title="Hydroelectricity">Hydroelectricity</a></li> <li><a href="/wiki/Tidal_power" title="Tidal power">Tidal power</a></li> <li><a href="/wiki/Wave_farm" class="mw-redirect" title="Wave farm">Wave farm</a></li></ul></li> <li><a href="/wiki/Nuclear_power" title="Nuclear power">Nuclear power</a> <ul><li><a href="/wiki/Nuclear_power_plant" title="Nuclear power plant">Nuclear power plant</a></li> <li><a href="/wiki/Radioisotope_thermoelectric_generator" title="Radioisotope thermoelectric generator">Radioisotope thermoelectric generator</a></li></ul></li> <li><a href="/wiki/Oil_refinery" title="Oil refinery">Oil refinery</a></li> <li><a href="/wiki/Solar_power" title="Solar power">Solar power</a> <ul><li><a href="/wiki/Concentrated_solar_power" title="Concentrated solar power">Concentrated solar power</a></li> <li><a href="/wiki/Photovoltaic_system" title="Photovoltaic system">Photovoltaic system</a></li></ul></li> <li><a href="/wiki/Solar_thermal_energy" title="Solar thermal energy">Solar thermal energy</a> <ul><li><a href="/wiki/Solar_furnace" title="Solar furnace">Solar furnace</a></li> <li><a href="/wiki/Solar_power_tower" title="Solar power tower">Solar power tower</a></li></ul></li> <li><a href="/wiki/Wind_power" title="Wind power">Wind power</a> <ul><li><a href="/wiki/Airborne_wind_energy" title="Airborne wind energy">Airborne wind energy</a></li> <li><a href="/wiki/Wind_farm" title="Wind farm">Wind farm</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Use and<br /><a href="/wiki/Energy_supply" title="Energy supply">supply</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Efficient_energy_use" title="Efficient energy use">Efficient energy use</a> <ul><li><a href="/wiki/Energy_efficiency_in_agriculture" title="Energy efficiency in agriculture">Agriculture</a></li> <li><a href="/wiki/Power_usage_effectiveness" title="Power usage effectiveness">Computing</a></li> <li><a href="/wiki/Energy_efficiency_in_transport" title="Energy efficiency in transport">Transport</a></li></ul></li> <li><a href="/wiki/Energy_conservation" title="Energy conservation">Energy conservation</a></li> <li><a href="/wiki/Energy_consumption" title="Energy consumption">Energy consumption</a></li> <li><a href="/wiki/Energy_policy" title="Energy policy">Energy policy</a> <ul><li><a href="/wiki/Energy_development" title="Energy development">Energy development</a></li></ul></li> <li><a href="/wiki/Energy_security" title="Energy security">Energy security</a></li> <li><a href="/wiki/Energy_storage" title="Energy storage">Energy storage</a></li> <li><a href="/wiki/Renewable_energy" title="Renewable energy">Renewable energy</a></li> <li><a href="/wiki/Sustainable_energy" title="Sustainable energy">Sustainable energy</a></li> <li><a href="/wiki/World_energy_supply_and_consumption" title="World energy supply and consumption">World energy supply and consumption</a></li> <li><a href="/wiki/Energy_in_Africa" title="Energy in Africa">Africa</a></li> <li><a href="/wiki/Energy_in_Asia" class="mw-redirect" title="Energy in Asia">Asia</a></li> <li><a href="/wiki/Energy_in_Australia" title="Energy in Australia">Australia</a></li> <li><a href="/wiki/Energy_policy_of_Canada" title="Energy policy of Canada">Canada</a></li> <li><a href="/wiki/Energy_in_Europe" title="Energy in Europe">Europe</a></li> <li><a href="/wiki/Energy_in_Mexico" title="Energy in Mexico">Mexico</a></li> <li><a href="/wiki/Energy_in_South_America" title="Energy in South America">South America</a></li> <li><a href="/wiki/Energy_in_the_United_States" title="Energy in the United States">United States</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Misc.</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Carbon_footprint" title="Carbon footprint">Carbon footprint</a></li> <li><a href="/wiki/Energy_democracy" title="Energy democracy">Energy democracy</a></li> <li><a href="/wiki/Energy_recovery" title="Energy recovery">Energy recovery</a></li> <li><a href="/wiki/Energy_recycling" title="Energy recycling">Energy recycling</a></li> <li><a href="/wiki/Jevons_paradox" title="Jevons paradox">Jevons paradox</a></li> <li><a href="/wiki/Waste-to-energy" title="Waste-to-energy">Waste-to-energy</a> <ul><li><a href="/wiki/Waste-to-energy_plant" title="Waste-to-energy plant">Waste-to-energy plant</a></li></ul></li></ul> </div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div> <ul><li><span class="noviewer" typeof="mw:File"><span title="Category"><img alt="" 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Kinetic energy - Wikipedia