Jerk (physics) - Wikipedia

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<div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Physiological effects and human perception</span> </div> </a> <ul id="toc-Physiological_effects_and_human_perception-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Force,_acceleration,_and_jerk" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Force,_acceleration,_and_jerk"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Force, acceleration, and jerk</span> </div> </a> <ul id="toc-Force,_acceleration,_and_jerk-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-In_an_idealized_setting" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#In_an_idealized_setting"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>In an idealized setting</span> </div> </a> <ul id="toc-In_an_idealized_setting-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-In_rotation" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#In_rotation"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>In rotation</span> </div> </a> <ul id="toc-In_rotation-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-In_elastically_deformable_matter" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#In_elastically_deformable_matter"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>In elastically deformable matter</span> </div> </a> <ul id="toc-In_elastically_deformable_matter-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-In_the_geometric_design_of_roads_and_tracks" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#In_the_geometric_design_of_roads_and_tracks"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>In the geometric design of roads and tracks</span> </div> </a> <ul id="toc-In_the_geometric_design_of_roads_and_tracks-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-In_motion_control" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#In_motion_control"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>In motion control</span> </div> </a> <button aria-controls="toc-In_motion_control-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 In motion control subsection</span> </button> <ul id="toc-In_motion_control-sublist" class="vector-toc-list"> <li id="toc-In_manufacturing" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#In_manufacturing"> <div class="vector-toc-text"> <span class="vector-toc-numb">8.1</span> <span>In manufacturing</span> </div> </a> <ul id="toc-In_manufacturing-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Further_derivatives" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Further_derivatives"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>Further derivatives</span> </div> </a> <ul id="toc-Further_derivatives-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">10</span> <span>See also</span> </div> </a> <ul id="toc-See_also-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">11</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">12</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" > <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">Jerk (physics)</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" 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Available in 42 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-42" 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">42 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%B1%D8%AC%D9%81%D8%A9_(%D9%81%D9%8A%D8%B2%D9%8A%D8%A7%D8%A1)" title="رجفة (فيزياء) – Arabic" lang="ar" hreflang="ar" data-title="رجفة (فيزياء)" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%9C%E0%A6%BE%E0%A6%B0%E0%A7%8D%E0%A6%95" title="জার্ক – Bangla" lang="bn" hreflang="bn" data-title="জার্ক" 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%A0%D1%8B%D0%B2%D0%BE%D0%BA_(%D0%BA%D1%96%D0%BD%D0%B5%D0%BC%D0%B0%D1%82%D1%8B%D0%BA%D0%B0)" title="Рывок (кінематыка) – Belarusian" lang="be" hreflang="be" data-title="Рывок (кінематыка)" data-language-autonym="Беларуская" data-language-local-name="Belarusian" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Sobreacceleraci%C3%B3" title="Sobreacceleració – Catalan" lang="ca" hreflang="ca" data-title="Sobreacceleració" 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%A2%D0%B0%D0%BB%D0%BF%C4%83%D0%BD%D1%83_(%D0%BA%D0%B8%D0%BD%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)" title="Талпăну (кинематика) – Chuvash" lang="cv" hreflang="cv" data-title="Талпăну (кинематика)" 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/Ryv" title="Ryv – Czech" lang="cs" hreflang="cs" data-title="Ryv" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Ruck" title="Ruck – German" lang="de" hreflang="de" data-title="Ruck" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%A1%CF%85%CE%B8%CE%BC%CF%8C%CF%82_%CE%BC%CE%B5%CF%84%CE%B1%CE%B2%CE%BF%CE%BB%CE%AE%CF%82_%CF%84%CE%B7%CF%82_%CE%B5%CF%80%CE%B9%CF%84%CE%AC%CF%87%CF%85%CE%BD%CF%83%CE%B7%CF%82" title="Ρυθμός μεταβολής της επιτάχυνσης – Greek" lang="el" hreflang="el" data-title="Ρυθμός μεταβολής της επιτάχυνσης" 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/Sobreaceleraci%C3%B3n" title="Sobreaceleración – Spanish" lang="es" hreflang="es" data-title="Sobreaceleración" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Gainazelerazio" title="Gainazelerazio – Basque" lang="eu" hreflang="eu" data-title="Gainazelerazio" 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%AC%D9%87%D8%B4_(%D9%81%DB%8C%D8%B2%DB%8C%DA%A9)" title="جهش (فیزیک) – Persian" lang="fa" hreflang="fa" data-title="جهش (فیزیک)" 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%80-coup" title="À-coup – French" lang="fr" hreflang="fr" data-title="À-coup" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EA%B0%80%EA%B0%80%EC%86%8D%EB%8F%84" title="가가속도 – Korean" lang="ko" hreflang="ko" data-title="가가속도" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%9D%E0%A4%9F%E0%A4%95%E0%A4%BE" title="झटका – Hindi" lang="hi" hreflang="hi" data-title="झटका" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Sentakan_(fisika)" title="Sentakan (fisika) – Indonesian" lang="id" hreflang="id" data-title="Sentakan (fisika)" 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-it badge-Q70893996 mw-list-item" title=""><a href="https://it.wikipedia.org/wiki/Strappo_(cinematica)" title="Strappo (cinematica) – Italian" lang="it" hreflang="it" data-title="Strappo (cinematica)" 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%A0%D7%AA%D7%99%D7%A8%D7%94" title="נתירה – Hebrew" lang="he" hreflang="he" data-title="נתירה" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Mutatio_accelerationis" title="Mutatio accelerationis – Latin" lang="la" hreflang="la" data-title="Mutatio accelerationis" data-language-autonym="Latina" data-language-local-name="Latin" class="interlanguage-link-target"><span>Latina</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/R%C3%A1ndul%C3%A1s_(fizika)" title="Rándulás (fizika) – Hungarian" lang="hu" hreflang="hu" data-title="Rándulás (fizika)" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-mr mw-list-item"><a href="https://mr.wikipedia.org/wiki/%E0%A4%B9%E0%A4%BF%E0%A4%B8%E0%A4%95%E0%A4%BE_(%E0%A4%AD%E0%A5%8C%E0%A4%A4%E0%A4%BF%E0%A4%95%E0%A5%80)" title="हिसका (भौतिकी) – Marathi" lang="mr" hreflang="mr" data-title="हिसका (भौतिकी)" 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/Sentakan" title="Sentakan – Malay" lang="ms" hreflang="ms" data-title="Sentakan" 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-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Ruk_(natuurkunde)" title="Ruk (natuurkunde) – Dutch" lang="nl" hreflang="nl" data-title="Ruk (natuurkunde)" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E8%BA%8D%E5%BA%A6" title="躍度 – Japanese" lang="ja" hreflang="ja" data-title="躍度" 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-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Silkinish-jerk_(fizika)" title="Silkinish-jerk (fizika) – Uzbek" lang="uz" hreflang="uz" data-title="Silkinish-jerk (fizika)" 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-nds mw-list-item"><a href="https://nds.wikipedia.org/wiki/Ruck_(Kinetik)" title="Ruck (Kinetik) – Low German" lang="nds" hreflang="nds" data-title="Ruck (Kinetik)" data-language-autonym="Plattdüütsch" data-language-local-name="Low German" class="interlanguage-link-target"><span>Plattdüütsch</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Zryw" title="Zryw – Polish" lang="pl" hreflang="pl" data-title="Zryw" 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/Arrancada" title="Arrancada – Portuguese" lang="pt" hreflang="pt" data-title="Arrancada" 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/Supraaccelera%C8%9Bie" title="Supraaccelerație – Romanian" lang="ro" hreflang="ro" data-title="Supraaccelerație" 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%A0%D1%8B%D0%B2%D0%BE%D0%BA_(%D0%BA%D0%B8%D0%BD%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)" title="Рывок (кинематика) – Russian" lang="ru" hreflang="ru" data-title="Рывок (кинематика)" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Jerk" title="Jerk – Simple English" lang="en-simple" hreflang="en-simple" data-title="Jerk" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Ryv" title="Ryv – Slovak" lang="sk" hreflang="sk" data-title="Ryv" 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/Trzaj_(fizika)" title="Trzaj (fizika) – Slovenian" lang="sl" hreflang="sl" data-title="Trzaj (fizika)" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%A2%D1%80%D0%B7%D0%B0%D1%98_(%D1%84%D0%B8%D0%B7%D0%B8%D0%BA%D0%B0)" title="Трзај (физика) – Serbian" lang="sr" hreflang="sr" data-title="Трзај (физика)" 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/Trzaj" title="Trzaj – Serbo-Croatian" lang="sh" hreflang="sh" data-title="Trzaj" 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/Nyk%C3%A4ys" title="Nykäys – Finnish" lang="fi" hreflang="fi" data-title="Nykäys" 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/Ryck_(fysik)" title="Ryck (fysik) – Swedish" lang="sv" hreflang="sv" data-title="Ryck (fysik)" 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%A4%E0%AE%BF%E0%AE%9F%E0%AF%81%E0%AE%95%E0%AF%8D%E0%AE%95%E0%AE%AE%E0%AF%8D_(%E0%AE%87%E0%AE%AF%E0%AE%B1%E0%AF%8D%E0%AE%AA%E0%AE%BF%E0%AE%AF%E0%AE%B2%E0%AF%8D)" title="திடுக்கம் (இயற்பியல்) – Tamil" lang="ta" hreflang="ta" data-title="திடுக்கம் (இயற்பியல்)" data-language-autonym="தமிழ்" data-language-local-name="Tamil" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Sars%C4%B1m_(fizik)" title="Sarsım (fizik) – Turkish" lang="tr" hreflang="tr" data-title="Sarsım (fizik)" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%A0%D0%B8%D0%B2%D0%BE%D0%BA_(%D0%BA%D1%96%D0%BD%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)" title="Ривок (кінематика) – Ukrainian" lang="uk" hreflang="uk" data-title="Ривок (кінематика)" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/Jerk_(%C4%91%E1%BA%A1i_l%C6%B0%E1%BB%A3ng_v%E1%BA%ADt_l%C3%BD)" title="Jerk (đại lượng vật lý) – Vietnamese" lang="vi" hreflang="vi" data-title="Jerk (đại lượng vật lý)" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamese" 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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">Jerk</th></tr><tr><td colspan="2" class="infobox-image"><span typeof="mw:File"><a href="/wiki/File:Time_derivatives_of_position.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/5b/Time_derivatives_of_position.svg/200px-Time_derivatives_of_position.svg.png" decoding="async" width="200" height="267" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/5b/Time_derivatives_of_position.svg/300px-Time_derivatives_of_position.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/5b/Time_derivatives_of_position.svg/400px-Time_derivatives_of_position.svg.png 2x" data-file-width="512" data-file-height="683" /></a></span><div class="infobox-caption">Time-derivatives of position, including jerk</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"><span class="texhtml mvar" style="font-style:italic;">j</span>, <span class="texhtml"><b>j</b></span>, <span class="texhtml"><i>ȷ</i><span style="position:relative; margin-right:-0.75em; right:0.75em; bottom:0.75em;;"><small>→</small></span></span></td></tr><tr><th scope="row" class="infobox-label">In <a href="/wiki/SI_base_unit" title="SI base unit"><span class="wrap">SI base units</span></a></th><td class="infobox-data"><a href="/wiki/Meter" class="mw-redirect" title="Meter">m</a>·<a href="/wiki/Second" title="Second">s</a><sup>-3</sup>, <a href="/wiki/Meter" class="mw-redirect" title="Meter">m</a>/<a href="/wiki/Second" title="Second">s</a><sup>3</sup></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Dimensional_analysis#Formulation" title="Dimensional analysis">Dimension</a></th><td class="infobox-data"><b>L</b> <b>T</b><sup>−3</sup></td></tr></tbody></table> <p><b>Jerk</b> (also known as <b>jolt</b>) is the rate of change of an object's <a href="/wiki/Acceleration" title="Acceleration">acceleration</a> over time. It is a <a href="/wiki/Vector_quantity" title="Vector quantity">vector quantity</a> (having both magnitude and direction). Jerk is most commonly denoted by the symbol <span class="texhtml mvar" style="font-style:italic;">j</span> and expressed in m/s<sup>3</sup> (<a href="/wiki/SI_unit" class="mw-redirect" title="SI unit">SI units</a>) or <a href="/wiki/Standard_gravity" title="Standard gravity">standard gravities</a> per second (<i>g</i><sub>0</sub>/s). </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Expressions">Expressions</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Jerk_(physics)&action=edit&section=1" title="Edit section: Expressions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>As a vector, jerk <span class="texhtml"><b>j</b></span> can be expressed as the first <a href="/wiki/Time_derivative" title="Time derivative">time derivative</a> of <a href="/wiki/Acceleration" title="Acceleration">acceleration</a>, <a href="/wiki/Second_derivative" title="Second derivative">second time derivative</a> of <a href="/wiki/Velocity" title="Velocity">velocity</a>, and <a href="/wiki/Third_derivative" title="Third derivative">third time derivative</a> of <a href="/wiki/Position_(vector)" class="mw-redirect" title="Position (vector)">position</a>: </p><p><span class="mwe-math-element" data-qid="Q497332"><a href="/w/index.php?title=Special:MathWikibase&qid=Q497332" style="color:inherit;"><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 {j} (t)={\frac {\mathrm {d} \mathbf {a} (t)}{\mathrm {d} t}}={\frac {\mathrm {d} ^{2}\mathbf {v} (t)}{\mathrm {d} t^{2}}}={\frac {\mathrm {d} ^{3}\mathbf {r} (t)}{\mathrm {d} t^{3}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">j</mi> </mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {j} (t)={\frac {\mathrm {d} \mathbf {a} (t)}{\mathrm {d} t}}={\frac {\mathrm {d} ^{2}\mathbf {v} (t)}{\mathrm {d} t^{2}}}={\frac {\mathrm {d} ^{3}\mathbf {r} (t)}{\mathrm {d} t^{3}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47cbc022c17db48975670f9a51bef95678fa8620" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; margin-left: -0.164ex; width:33.178ex; height:6.343ex;" alt="{\displaystyle \mathbf {j} (t)={\frac {\mathrm {d} \mathbf {a} (t)}{\mathrm {d} t}}={\frac {\mathrm {d} ^{2}\mathbf {v} (t)}{\mathrm {d} t^{2}}}={\frac {\mathrm {d} ^{3}\mathbf {r} (t)}{\mathrm {d} t^{3}}}}"></a></span> </p><p>Where: </p> <ul><li><span class="texhtml"><b>a</b></span> is acceleration</li> <li><span class="texhtml"><b>v</b></span> is velocity</li> <li><span class="texhtml"><b>r</b></span> is position</li> <li><span class="texhtml mvar" style="font-style:italic;">t</span> is time</li></ul> <p>Third-order <a href="/wiki/Differential_equations" class="mw-redirect" title="Differential equations">differential equations</a> of the form <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 J\left({\overset {\mathbf {...} }{x}},{\ddot {x}},{\dot {x}},x\right)=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>J</mi> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo mathvariant="bold">.</mo> <mo mathvariant="bold">.</mo> <mo mathvariant="bold">.</mo> </mrow> </mover> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo>¨</mo> </mover> </mrow> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mo>,</mo> <mi>x</mi> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle J\left({\overset {\mathbf {...} }{x}},{\ddot {x}},{\dot {x}},x\right)=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ce419f5dcb355c5710ee591b231c684be85827d9" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:16.914ex; height:3.343ex;" alt="{\displaystyle J\left({\overset {\mathbf {...} }{x}},{\ddot {x}},{\dot {x}},x\right)=0}"></span> are sometimes called <i>jerk equations</i>. When converted to an equivalent system of three ordinary <a href="/wiki/Differential_equation#Ordinary_differential_equations" title="Differential equation">first-order</a> <a href="/wiki/Differential_equation#Non-linear_differential_equations" title="Differential equation">non-linear</a> differential equations, jerk equations are the minimal setting for solutions showing <a href="/wiki/Chaos_theory" title="Chaos theory">chaotic behaviour</a>. This condition generates mathematical interest in <i>jerk systems</i>. Systems involving fourth-order derivatives or higher are accordingly called <i>hyperjerk systems</i>.<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> <div class="mw-heading mw-heading2"><h2 id="Physiological_effects_and_human_perception">Physiological effects and human perception</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Jerk_(physics)&action=edit&section=2" title="Edit section: Physiological effects and human perception"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></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/G-force#Human_tolerance" title="G-force">Human tolerance of g-force</a> and <a href="/wiki/Motion_simulator#HumanPhysiologyResponseToMotion" title="Motion simulator">How human physiology processes and responds to motion</a></div> <p>Human body position is controlled by balancing the forces of <a href="/wiki/Antagonist_(muscle)" class="mw-redirect" title="Antagonist (muscle)">antagonistic muscles</a>. In balancing a given force, such as holding up a weight, the <a href="/wiki/Postcentral_gyrus" title="Postcentral gyrus">postcentral gyrus</a> establishes a <a href="/wiki/Control_loop" title="Control loop">control loop</a> to achieve the desired <a href="/wiki/Mechanical_equilibrium" title="Mechanical equilibrium">equilibrium</a>. If the force changes too quickly, the muscles cannot relax or tense fast enough and overshoot in either direction, causing a temporary loss of control. The reaction time for responding to changes in force depends on physiological limitations and the <a href="/wiki/Attention" title="Attention">attention</a> level of the brain: an <i>expected</i> change will be stabilized faster than a <i>sudden</i> decrease or increase of load. </p><p>To avoid vehicle passengers losing control over body motion and getting injured, it is necessary to limit the exposure to both the maximum force (acceleration) <i>and</i> maximum jerk, since time is needed to adjust muscle tension and adapt to even limited stress changes. Sudden changes in acceleration can cause injuries such as <a href="/wiki/Whiplash_(medicine)" title="Whiplash (medicine)">whiplash</a>.<sup id="cite_ref-thetartan2007_2-0" class="reference"><a href="#cite_note-thetartan2007-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> Excessive jerk may also result in an uncomfortable ride, even at levels that do not cause injury. Engineers expend considerable design effort minimizing "jerky motion" on <a href="/wiki/Elevator" title="Elevator">elevators</a>, <a href="/wiki/Tram" title="Tram">trams</a>, and other conveyances. </p><p>For example, consider the effects of acceleration and jerk when riding in a car: </p> <ul><li>Skilled and experienced drivers can accelerate smoothly, but beginners often provide a <i>jerky</i> ride. When changing gears in a car with a foot-operated clutch, the accelerating force is limited by engine power, but an inexperienced driver can cause severe jerk because of intermittent force closure over the clutch.</li> <li>The feeling of being pressed into the seats in a high-powered sports car is due to the acceleration. As the car launches from rest, there is a large positive jerk as its acceleration rapidly increases. After the launch, there is a small, sustained negative jerk as the force of air resistance increases with the car's velocity, gradually decreasing acceleration and reducing the force pressing the passenger into the seat. When the car reaches its top speed, the acceleration has reached 0 and remains constant, after which there is no jerk until the driver decelerates or changes direction.</li> <li>When braking suddenly or during collisions, passengers whip forward with an initial acceleration that is larger than during the rest of the braking process because muscle tension regains control of the body quickly after the onset of braking or impact. These effects are not modeled in vehicle testing because <a href="/wiki/Cadaver" title="Cadaver">cadavers</a> and <a href="/wiki/Crash_test_dummy" title="Crash test dummy">crash test dummies</a> do not have active muscle control.</li> <li>To minimize the jerk, curves along roads are designed to be <a href="/wiki/Euler_spiral" title="Euler spiral">clothoids</a> as are railroad curves and <a href="/wiki/Vertical_loop" title="Vertical loop">roller coaster loops</a>.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Force,_acceleration,_and_jerk"><span id="Force.2C_acceleration.2C_and_jerk"></span>Force, acceleration, and jerk</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Jerk_(physics)&action=edit&section=3" title="Edit section: Force, acceleration, and jerk"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>For a constant mass <span class="texhtml mvar" style="font-style:italic;">m</span>, acceleration <span class="texhtml mvar" style="font-style:italic;">a</span> is directly proportional to force <span class="texhtml mvar" style="font-style:italic;">F</span> according to <a href="/wiki/Newton%27s_second_law" class="mw-redirect" title="Newton's second law">Newton's second law of motion</a>: <span class="mwe-math-element" data-qid="Q2397319"><a href="/w/index.php?title=Special:MathWikibase&qid=Q2397319" style="color:inherit;"><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} =m\mathbf {a} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mo>=</mo> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {F} =m\mathbf {a} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d0c60fab89e8c3193952047dc565bcf8d233d115" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.121ex; height:2.176ex;" alt="{\displaystyle \mathbf {F} =m\mathbf {a} }"></a></span> </p><p>In <a href="/wiki/Classical_mechanics" title="Classical mechanics">classical mechanics</a> of rigid bodies, there are no <i>forces</i> associated with the derivatives of acceleration; however, physical systems experience oscillations and deformations as a result of jerk. In designing the <a href="/wiki/Hubble_Space_Telescope" title="Hubble Space Telescope">Hubble Space Telescope</a>, <a href="/wiki/NASA" title="NASA">NASA</a> set limits on both jerk and <a href="/wiki/Jounce" class="mw-redirect" title="Jounce">jounce</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 <a href="/wiki/Abraham%E2%80%93Lorentz_force" title="Abraham–Lorentz force">Abraham–Lorentz force</a> is the <a href="/wiki/Recoil" title="Recoil">recoil</a> force on an accelerating charged particle emitting radiation. This force is proportional to the particle's jerk and to the square of its <a href="/wiki/Charge_(physics)" title="Charge (physics)">charge</a>. The <a href="/wiki/Wheeler%E2%80%93Feynman_absorber_theory" title="Wheeler–Feynman absorber theory">Wheeler–Feynman absorber theory</a> is a more advanced theory, applicable in a relativistic and quantum environment, and accounting for <a href="/wiki/Self-energy" title="Self-energy">self-energy</a>. </p> <div class="mw-heading mw-heading2"><h2 id="In_an_idealized_setting">In an idealized setting</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Jerk_(physics)&action=edit&section=4" title="Edit section: In an idealized setting"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Discontinuities in acceleration do not occur in real-world environments because of <a href="/wiki/Deformation_(engineering)" title="Deformation (engineering)">deformation</a>, <a href="/wiki/Quantum_mechanics" title="Quantum mechanics">quantum mechanics</a> effects, and other causes. However, a jump-discontinuity in acceleration and, accordingly, unbounded jerk are feasible in an idealized setting, such as an idealized <a href="/wiki/Point_mass" class="mw-redirect" title="Point mass">point mass</a> moving along a <a href="/wiki/Piecewise" class="mw-redirect" title="Piecewise">piecewise</a> <a href="/wiki/Smooth_function" class="mw-redirect" title="Smooth function">smooth</a>, whole continuous path. The jump-discontinuity occurs at points where the path is not smooth. Extrapolating from these idealized settings, one can qualitatively describe, explain and predict the effects of jerk in real situations. </p><p>Jump-discontinuity in acceleration can be modeled using a <a href="/wiki/Dirac_delta" class="mw-redirect" title="Dirac delta">Dirac delta function</a> in jerk, scaled to the height of the jump. Integrating jerk over time across the Dirac delta yields the jump-discontinuity. </p><p>For example, consider a path along an arc of radius <span class="texhtml mvar" style="font-style:italic;">r</span>, which <a href="/wiki/Tangent#Tangent_line_to_a_curve" title="Tangent">tangentially</a> connects to a straight line. The whole path is continuous, and its pieces are smooth. Now assume a point particle moves with constant speed along this path, so its <a href="/wiki/Acceleration#Tangential_and_centripetal_acceleration" title="Acceleration">tangential acceleration</a> is zero. The <a href="/wiki/Acceleration#Tangential_and_centripetal_acceleration" title="Acceleration">centripetal acceleration</a> given by <span class="texhtml"><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"><i>v</i><sup>2</sup></span><span class="sr-only">/</span><span class="den"><i>r</i></span></span>⁠</span></span> is normal to the arc and inward. When the particle passes the connection of pieces, it experiences a jump-discontinuity in acceleration given by <span class="texhtml"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num"><i>v</i><sup>2</sup></span><span class="sr-only">/</span><span class="den"><i>r</i></span></span>⁠</span></span>, and it undergoes a jerk that can be modeled by a Dirac delta, scaled to the jump-discontinuity. </p><p>For a more tangible example of discontinuous acceleration, consider an ideal <a href="/wiki/Spring%E2%80%93mass_system" class="mw-redirect" title="Spring–mass system">spring–mass system</a> with the mass oscillating on an idealized surface with friction. The force on the mass is equal to the <a href="/wiki/Vector_sum" class="mw-redirect" title="Vector sum">vector sum</a> of the spring force and the <a href="/wiki/Friction" title="Friction">kinetic frictional force</a>. When the velocity changes sign (at the maximum and minimum <a href="/wiki/Displacement_(vector)" class="mw-redirect" title="Displacement (vector)">displacements</a>), the magnitude of the force on the mass changes by twice the magnitude of the frictional force, because the spring force is continuous and the frictional force reverses direction with velocity. The jump in acceleration equals the force on the mass divided by the mass. That is, each time the mass passes through a minimum or maximum displacement, the mass experiences a discontinuous acceleration, and the jerk contains a Dirac delta until the mass stops. The static friction force adapts to the residual spring force, establishing equilibrium with zero net force and zero velocity. </p><p>Consider the example of a braking and decelerating car. The <a href="/wiki/Brake_pad" title="Brake pad">brake pads</a> generate kinetic <a href="/wiki/Frictional_force" class="mw-redirect" title="Frictional force">frictional forces</a> and constant braking <a href="/wiki/Torque" title="Torque">torques</a> on the <a href="/wiki/Disk_brake" class="mw-redirect" title="Disk brake">disks</a> (or <a href="/wiki/Drum_brake" title="Drum brake">drums</a>) of the wheels. Rotational velocity decreases linearly to zero with constant angular deceleration. The frictional force, torque, and car deceleration suddenly reach zero, which indicates a Dirac delta in physical jerk. The Dirac delta is smoothed down by the real environment, the cumulative effects of which are analogous to damping of the physiologically perceived jerk. This example neglects the effects of tire sliding, suspension dipping, real deflection of all ideally rigid mechanisms, etc. </p><p>Another example of significant jerk, analogous to the first example, is the cutting of a rope with a particle on its end. Assume the particle is oscillating in a circular path with non-zero centripetal acceleration. When the rope is cut, the particle's path changes abruptly to a straight path, and the force in the inward direction changes suddenly to zero. Imagine a monomolecular fiber cut by a laser; the particle would experience very high rates of jerk because of the extremely short cutting time. </p> <div class="mw-heading mw-heading2"><h2 id="In_rotation">In rotation</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Jerk_(physics)&action=edit&section=5" title="Edit section: In rotation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Chronogrammes_croix_malte_4_branches_complet_EN.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/5a/Chronogrammes_croix_malte_4_branches_complet_EN.svg/290px-Chronogrammes_croix_malte_4_branches_complet_EN.svg.png" decoding="async" width="290" height="483" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/5a/Chronogrammes_croix_malte_4_branches_complet_EN.svg/435px-Chronogrammes_croix_malte_4_branches_complet_EN.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/5a/Chronogrammes_croix_malte_4_branches_complet_EN.svg/580px-Chronogrammes_croix_malte_4_branches_complet_EN.svg.png 2x" data-file-width="503" data-file-height="838" /></a><figcaption>Timing diagram over one revolution for angle, angular velocity, angular acceleration, and angular jerk</figcaption></figure> <p>Consider a rigid body rotating about a fixed axis in an <a href="/wiki/Inertial_frame_of_reference#Newton's_inertial_frame_of_reference" title="Inertial frame of reference">inertial reference frame</a>. If its angular position as a function of time is <span class="texhtml"><i>θ</i>(<i>t</i>)</span>, the angular velocity, acceleration, and jerk can be expressed as follows: </p> <ul><li><a href="/wiki/Angular_velocity" title="Angular velocity">Angular velocity</a>, <span class="mwe-math-element" data-qid="Q161635"><a href="/w/index.php?title=Special:MathWikibase&qid=Q161635" style="color:inherit;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega (t)={\dot {\theta }}(t)={\frac {\mathrm {d} \theta (t)}{\mathrm {d} t}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ω</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>θ</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>θ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega (t)={\dot {\theta }}(t)={\frac {\mathrm {d} \theta (t)}{\mathrm {d} t}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fc8661fccf91229833de95dbf00272d7f5ed9de9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:20.165ex; height:5.843ex;" alt="{\displaystyle \omega (t)={\dot {\theta }}(t)={\frac {\mathrm {d} \theta (t)}{\mathrm {d} t}}}"></a></span>, is the time derivative of <span class="texhtml"><i>θ</i>(<i>t</i>)</span>.</li> <li><a href="/wiki/Angular_acceleration" title="Angular acceleration">Angular acceleration</a>, <span class="mwe-math-element" data-qid="Q186300"><a href="/w/index.php?title=Special:MathWikibase&qid=Q186300" style="color:inherit;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha (t)={\dot {\omega }}(t)={\frac {\mathrm {d} \omega (t)}{\mathrm {d} t}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ω</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>ω</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha (t)={\dot {\omega }}(t)={\frac {\mathrm {d} \omega (t)}{\mathrm {d} t}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3a065b0710faf461b0bb0b126a4bf525ba0e5015" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:20.652ex; height:5.843ex;" alt="{\displaystyle \alpha (t)={\dot {\omega }}(t)={\frac {\mathrm {d} \omega (t)}{\mathrm {d} t}}}"></a></span>, is the time derivative of <span class="texhtml"><i>ω</i>(<i>t</i>)</span>.</li> <li>Angular jerk, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \zeta (t)={\dot {\alpha }}(t)={\ddot {\omega }}(t)={\overset {...}{\theta }}(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ζ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>α</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ω</mi> <mo>¨</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>θ</mi> <mrow> <mo>.</mo> <mo>.</mo> <mo>.</mo> </mrow> </mover> </mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \zeta (t)={\dot {\alpha }}(t)={\ddot {\omega }}(t)={\overset {...}{\theta }}(t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fa364c3bdef39090cb7b6afecd5782caf1b49d07" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.292ex; height:3.676ex;" alt="{\displaystyle \zeta (t)={\dot {\alpha }}(t)={\ddot {\omega }}(t)={\overset {...}{\theta }}(t)}"></span>, is the time derivative of <span class="texhtml"><i>α</i>(<i>t</i>)</span>.</li></ul> <p>Angular acceleration equals the <a href="/wiki/Torque" title="Torque">torque</a> acting on the body, divided by the body's <a href="/wiki/Moment_of_inertia" title="Moment of inertia">moment of inertia</a> with respect to the momentary axis of rotation. A change in torque results in angular jerk. </p><p>The general case of a rotating rigid body can be modeled using kinematic <a href="/wiki/Screw_theory" title="Screw theory">screw theory</a>, which includes one axial <a href="/wiki/Pseudovector" title="Pseudovector">vector</a>, angular velocity <span class="texhtml"><b>Ω</b>(<i>t</i>)</span>, and one polar <a href="/wiki/Pseudovector" title="Pseudovector">vector</a>, linear velocity <span class="texhtml"><b>v</b>(<i>t</i>)</span>. From this, the angular acceleration is defined 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 {\boldsymbol {\alpha }}(t)={\frac {\mathrm {d} }{\mathrm {d} t}}{\boldsymbol {\omega }}(t)={\dot {\boldsymbol {\omega }}}(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">α</mi> </mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ω</mi> </mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold-italic">ω</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {\alpha }}(t)={\frac {\mathrm {d} }{\mathrm {d} t}}{\boldsymbol {\omega }}(t)={\dot {\boldsymbol {\omega }}}(t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/986960e684f3b2c844a536c87caf59778e0cf1e9" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:22.218ex; height:5.509ex;" alt="{\displaystyle {\boldsymbol {\alpha }}(t)={\frac {\mathrm {d} }{\mathrm {d} t}}{\boldsymbol {\omega }}(t)={\dot {\boldsymbol {\omega }}}(t)}"></span> </p><p>and the angular jerk is given by <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 {\boldsymbol {\zeta }}(t)={\frac {\mathrm {d} }{\mathrm {d} t}}{\boldsymbol {\alpha }}(t)={\dot {\boldsymbol {\alpha }}}(t)={\ddot {\boldsymbol {\omega }}}(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ζ</mi> </mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">α</mi> </mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold-italic">α</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold-italic">ω</mi> <mo>¨</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {\zeta }}(t)={\frac {\mathrm {d} }{\mathrm {d} t}}{\boldsymbol {\alpha }}(t)={\dot {\boldsymbol {\alpha }}}(t)={\ddot {\boldsymbol {\omega }}}(t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/016b6003032a44af442050442a863f560993913e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:29.277ex; height:5.509ex;" alt="{\displaystyle {\boldsymbol {\zeta }}(t)={\frac {\mathrm {d} }{\mathrm {d} t}}{\boldsymbol {\alpha }}(t)={\dot {\boldsymbol {\alpha }}}(t)={\ddot {\boldsymbol {\omega }}}(t)}"></span> </p><p>taking the angular acceleration from <a href="/wiki/Angular_acceleration#Particle_in_three_dimensions" title="Angular acceleration">Angular acceleration#Particle in three dimensions</a> as <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 {\boldsymbol {\alpha }}={\frac {d{\boldsymbol {\omega }}}{dt}}={\frac {\mathbf {r} \times \mathbf {a} }{r^{2}}}-{\frac {2}{r}}{\frac {dr}{dt}}{\boldsymbol {\omega }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">α</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ω</mi> </mrow> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> </mrow> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mi>r</mi> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>r</mi> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ω</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {\alpha }}={\frac {d{\boldsymbol {\omega }}}{dt}}={\frac {\mathbf {r} \times \mathbf {a} }{r^{2}}}-{\frac {2}{r}}{\frac {dr}{dt}}{\boldsymbol {\omega }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9e7819e55e2ea3ad52ea1afc9d775f1efd9d6764" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:27.373ex; height:5.676ex;" alt="{\displaystyle {\boldsymbol {\alpha }}={\frac {d{\boldsymbol {\omega }}}{dt}}={\frac {\mathbf {r} \times \mathbf {a} }{r^{2}}}-{\frac {2}{r}}{\frac {dr}{dt}}{\boldsymbol {\omega }}}"></span>, 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}{\boldsymbol {\zeta }}={\frac {d{\boldsymbol {\alpha }}}{dt}}={\frac {1}{r^{2}}}\left(\mathbf {r} \times {\frac {d\mathbf {a} }{dt}}+{\frac {d\mathbf {r} }{dt}}\times \mathbf {a} \right)-{\frac {2}{r^{3}}}{\frac {dr}{dt}}\left(\mathbf {r} \times \mathbf {a} \right)\\\\+{\frac {2}{r^{2}}}\left({\frac {dr}{dt}}\right)^{2}{\boldsymbol {\omega }}-{\frac {2}{r}}{\frac {d^{2}r}{dt^{2}}}{\boldsymbol {\omega }}-{\frac {2}{r}}{\frac {dr}{dt}}{\frac {d{\boldsymbol {\omega }}}{dt}}\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ζ</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">α</mi> </mrow> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> </mrow> <mo>)</mo> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>r</mi> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd /> </mtr> <mtr> <mtd> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>r</mi> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ω</mi> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mi>r</mi> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>r</mi> </mrow> <mrow> <mi>d</mi> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ω</mi> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mi>r</mi> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>r</mi> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ω</mi> </mrow> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\boldsymbol {\zeta }}={\frac {d{\boldsymbol {\alpha }}}{dt}}={\frac {1}{r^{2}}}\left(\mathbf {r} \times {\frac {d\mathbf {a} }{dt}}+{\frac {d\mathbf {r} }{dt}}\times \mathbf {a} \right)-{\frac {2}{r^{3}}}{\frac {dr}{dt}}\left(\mathbf {r} \times \mathbf {a} \right)\\\\+{\frac {2}{r^{2}}}\left({\frac {dr}{dt}}\right)^{2}{\boldsymbol {\omega }}-{\frac {2}{r}}{\frac {d^{2}r}{dt^{2}}}{\boldsymbol {\omega }}-{\frac {2}{r}}{\frac {dr}{dt}}{\frac {d{\boldsymbol {\omega }}}{dt}}\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/383d94d0c296144f09a3ca64f57d7e67ed600653" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -7.338ex; width:52.474ex; height:15.843ex;" alt="{\displaystyle {\begin{aligned}{\boldsymbol {\zeta }}={\frac {d{\boldsymbol {\alpha }}}{dt}}={\frac {1}{r^{2}}}\left(\mathbf {r} \times {\frac {d\mathbf {a} }{dt}}+{\frac {d\mathbf {r} }{dt}}\times \mathbf {a} \right)-{\frac {2}{r^{3}}}{\frac {dr}{dt}}\left(\mathbf {r} \times \mathbf {a} \right)\\\\+{\frac {2}{r^{2}}}\left({\frac {dr}{dt}}\right)^{2}{\boldsymbol {\omega }}-{\frac {2}{r}}{\frac {d^{2}r}{dt^{2}}}{\boldsymbol {\omega }}-{\frac {2}{r}}{\frac {dr}{dt}}{\frac {d{\boldsymbol {\omega }}}{dt}}\end{aligned}}}"></span> </p><p>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 {\frac {d{\boldsymbol {\omega }}}{dt}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ω</mi> </mrow> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {d{\boldsymbol {\omega }}}{dt}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5646506363fff327882ef6988e30a4e699bfbeb1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:3.721ex; height:5.509ex;" alt="{\displaystyle {\frac {d{\boldsymbol {\omega }}}{dt}}}"></span> we can have the last item as <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 {2}{r}}{\frac {dr}{dt}}{\frac {d{\boldsymbol {\omega }}}{dt}}&=-{\frac {2}{r}}{\frac {dr}{dt}}\left({\frac {\mathbf {r} \times \mathbf {a} }{r^{2}}}-{\frac {2}{r}}{\frac {dr}{dt}}{\boldsymbol {\omega }}\right)\\\\&=-{\frac {2}{r^{3}}}{\frac {dr}{dt}}\left(\mathbf {r} \times \mathbf {a} \right)+{\frac {4}{r^{2}}}\left({\frac {dr}{dt}}\right)^{2}{\boldsymbol {\omega }}\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> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mi>r</mi> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>r</mi> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ω</mi> </mrow> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mi>r</mi> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>r</mi> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> </mrow> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mi>r</mi> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>r</mi> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ω</mi> </mrow> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd /> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>r</mi> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>4</mn> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>r</mi> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ω</mi> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}-{\frac {2}{r}}{\frac {dr}{dt}}{\frac {d{\boldsymbol {\omega }}}{dt}}&=-{\frac {2}{r}}{\frac {dr}{dt}}\left({\frac {\mathbf {r} \times \mathbf {a} }{r^{2}}}-{\frac {2}{r}}{\frac {dr}{dt}}{\boldsymbol {\omega }}\right)\\\\&=-{\frac {2}{r^{3}}}{\frac {dr}{dt}}\left(\mathbf {r} \times \mathbf {a} \right)+{\frac {4}{r^{2}}}\left({\frac {dr}{dt}}\right)^{2}{\boldsymbol {\omega }}\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e90add0d6b4bc744c96605b1267777310bdf26fa" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -7.338ex; width:44.789ex; height:15.843ex;" alt="{\displaystyle {\begin{aligned}-{\frac {2}{r}}{\frac {dr}{dt}}{\frac {d{\boldsymbol {\omega }}}{dt}}&=-{\frac {2}{r}}{\frac {dr}{dt}}\left({\frac {\mathbf {r} \times \mathbf {a} }{r^{2}}}-{\frac {2}{r}}{\frac {dr}{dt}}{\boldsymbol {\omega }}\right)\\\\&=-{\frac {2}{r^{3}}}{\frac {dr}{dt}}\left(\mathbf {r} \times \mathbf {a} \right)+{\frac {4}{r^{2}}}\left({\frac {dr}{dt}}\right)^{2}{\boldsymbol {\omega }}\end{aligned}}}"></span>, and we finally 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 {\begin{aligned}{\boldsymbol {\zeta }}={\frac {\mathbf {r} \times \mathbf {j} }{r^{2}}}+{\frac {\mathbf {v} \times \mathbf {a} }{r^{2}}}-{\frac {4}{r^{3}}}{\frac {dr}{dt}}\left(\mathbf {r} \times \mathbf {a} \right)+{\frac {6}{r^{2}}}\left({\frac {dr}{dt}}\right)^{2}{\boldsymbol {\omega }}-{\frac {2}{r}}{\frac {d^{2}r}{dt^{2}}}{\boldsymbol {\omega }}\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ζ</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">j</mi> </mrow> </mrow> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> </mrow> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>4</mn> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>r</mi> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>6</mn> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>r</mi> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ω</mi> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mi>r</mi> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>r</mi> </mrow> <mrow> <mi>d</mi> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ω</mi> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\boldsymbol {\zeta }}={\frac {\mathbf {r} \times \mathbf {j} }{r^{2}}}+{\frac {\mathbf {v} \times \mathbf {a} }{r^{2}}}-{\frac {4}{r^{3}}}{\frac {dr}{dt}}\left(\mathbf {r} \times \mathbf {a} \right)+{\frac {6}{r^{2}}}\left({\frac {dr}{dt}}\right)^{2}{\boldsymbol {\omega }}-{\frac {2}{r}}{\frac {d^{2}r}{dt^{2}}}{\boldsymbol {\omega }}\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8f3c17d5bedf7f362d8829458368b0bbfb68f234" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:61.891ex; height:6.509ex;" alt="{\displaystyle {\begin{aligned}{\boldsymbol {\zeta }}={\frac {\mathbf {r} \times \mathbf {j} }{r^{2}}}+{\frac {\mathbf {v} \times \mathbf {a} }{r^{2}}}-{\frac {4}{r^{3}}}{\frac {dr}{dt}}\left(\mathbf {r} \times \mathbf {a} \right)+{\frac {6}{r^{2}}}\left({\frac {dr}{dt}}\right)^{2}{\boldsymbol {\omega }}-{\frac {2}{r}}{\frac {d^{2}r}{dt^{2}}}{\boldsymbol {\omega }}\end{aligned}}}"></span> </p><p>or vice versa, 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 \left(\mathbf {r} \times \mathbf {a} \right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(\mathbf {r} \times \mathbf {a} \right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/76aaccd28ae281df67245380c0467b073c7bd508" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.051ex; height:2.843ex;" alt="{\displaystyle \left(\mathbf {r} \times \mathbf {a} \right)}"></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 {\boldsymbol {\alpha }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">α</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {\alpha }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a585d2bb19071162720ea56a7b087dab3ec17156" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.769ex; height:1.676ex;" alt="{\displaystyle {\boldsymbol {\alpha }}}"></span>: <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}{\boldsymbol {\zeta }}={\frac {\mathbf {r} \times \mathbf {j} }{r^{2}}}+{\frac {\mathbf {v} \times \mathbf {a} }{r^{2}}}-{\frac {4}{r}}{\frac {dr}{dt}}{\boldsymbol {\alpha }}-{\frac {2}{r^{2}}}\left({\frac {dr}{dt}}\right)^{2}{\boldsymbol {\omega }}-{\frac {2}{r}}{\frac {d^{2}r}{dt^{2}}}{\boldsymbol {\omega }}\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ζ</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">j</mi> </mrow> </mrow> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> </mrow> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>4</mn> <mi>r</mi> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>r</mi> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">α</mi> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>r</mi> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ω</mi> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mi>r</mi> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>r</mi> </mrow> <mrow> <mi>d</mi> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ω</mi> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\boldsymbol {\zeta }}={\frac {\mathbf {r} \times \mathbf {j} }{r^{2}}}+{\frac {\mathbf {v} \times \mathbf {a} }{r^{2}}}-{\frac {4}{r}}{\frac {dr}{dt}}{\boldsymbol {\alpha }}-{\frac {2}{r^{2}}}\left({\frac {dr}{dt}}\right)^{2}{\boldsymbol {\omega }}-{\frac {2}{r}}{\frac {d^{2}r}{dt^{2}}}{\boldsymbol {\omega }}\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54e97ae64314d705ffcf404d42a76276014515d9" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:55.281ex; height:6.509ex;" alt="{\displaystyle {\begin{aligned}{\boldsymbol {\zeta }}={\frac {\mathbf {r} \times \mathbf {j} }{r^{2}}}+{\frac {\mathbf {v} \times \mathbf {a} }{r^{2}}}-{\frac {4}{r}}{\frac {dr}{dt}}{\boldsymbol {\alpha }}-{\frac {2}{r^{2}}}\left({\frac {dr}{dt}}\right)^{2}{\boldsymbol {\omega }}-{\frac {2}{r}}{\frac {d^{2}r}{dt^{2}}}{\boldsymbol {\omega }}\end{aligned}}}"></span> </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Animiertes_Prinzip_Malteserkreuzgetriebe_3D.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/8c/Animiertes_Prinzip_Malteserkreuzgetriebe_3D.gif/170px-Animiertes_Prinzip_Malteserkreuzgetriebe_3D.gif" decoding="async" width="170" height="249" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/8c/Animiertes_Prinzip_Malteserkreuzgetriebe_3D.gif/255px-Animiertes_Prinzip_Malteserkreuzgetriebe_3D.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/8/8c/Animiertes_Prinzip_Malteserkreuzgetriebe_3D.gif 2x" data-file-width="300" data-file-height="440" /></a><figcaption>Animation showing a four-position external <a href="/wiki/Geneva_drive" title="Geneva drive">Geneva drive</a> in operation</figcaption></figure> <p>For example, consider a <a href="/wiki/Geneva_drive" title="Geneva drive">Geneva drive</a>, a device used for creating intermittent rotation of a driven wheel (the blue wheel in the animation) by continuous rotation of a driving wheel (the red wheel in the animation). During one cycle of the driving wheel, the driven wheel's angular position <span class="texhtml mvar" style="font-style:italic;">θ</span> changes by 90 degrees and then remains constant. Because of the finite thickness of the driving wheel's fork (the slot for the driving pin), this device generates a discontinuity in the angular acceleration <span class="texhtml mvar" style="font-style:italic;">α</span>, and an unbounded angular jerk <span class="texhtml mvar" style="font-style:italic;">ζ</span> in the driven wheel. </p><p>Jerk does not preclude the Geneva drive from being used in applications such as movie projectors and <a href="/wiki/Cam_(mechanism)" title="Cam (mechanism)">cams</a>. In movie projectors, the film advances frame-by-frame, but the projector operation has low noise and is highly reliable because of the low film load (only a small section of film weighing a few grams is driven), the moderate speed (2.4 m/s), and the low friction. </p> <style data-mw-deduplicate="TemplateStyles:r1237032888/mw-parser-output/.tmulti">.mw-parser-output .tmulti .multiimageinner{display:flex;flex-direction:column}.mw-parser-output .tmulti .trow{display:flex;flex-direction:row;clear:left;flex-wrap:wrap;width:100%;box-sizing:border-box}.mw-parser-output .tmulti .tsingle{margin:1px;float:left}.mw-parser-output .tmulti .theader{clear:both;font-weight:bold;text-align:center;align-self:center;background-color:transparent;width:100%}.mw-parser-output .tmulti .thumbcaption{background-color:transparent}.mw-parser-output .tmulti .text-align-left{text-align:left}.mw-parser-output .tmulti .text-align-right{text-align:right}.mw-parser-output .tmulti .text-align-center{text-align:center}@media all and (max-width:720px){.mw-parser-output .tmulti .thumbinner{width:100%!important;box-sizing:border-box;max-width:none!important;align-items:center}.mw-parser-output .tmulti .trow{justify-content:center}.mw-parser-output .tmulti .tsingle{float:none!important;max-width:100%!important;box-sizing:border-box;text-align:center}.mw-parser-output .tmulti .tsingle .thumbcaption{text-align:left}.mw-parser-output .tmulti .trow>.thumbcaption{text-align:center}}@media screen{html.skin-theme-clientpref-night .mw-parser-output .tmulti .multiimageinner img{background-color:white}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .tmulti .multiimageinner img{background-color:white}}</style><div class="thumb tmulti tleft"><div class="thumbinner multiimageinner" style="width:304px;max-width:304px"><div class="trow"><div class="theader" style="text-align:center">Dual cam drives</div></div><div class="trow"><div class="tsingle" style="width:302px;max-width:302px"><div class="thumbimage"><span typeof="mw:File"><a href="/wiki/File:Cames_conjuguees_rotation_intermittente_un_sixieme_de_tour.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c9/Cames_conjuguees_rotation_intermittente_un_sixieme_de_tour.svg/300px-Cames_conjuguees_rotation_intermittente_un_sixieme_de_tour.svg.png" decoding="async" width="300" height="91" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c9/Cames_conjuguees_rotation_intermittente_un_sixieme_de_tour.svg/450px-Cames_conjuguees_rotation_intermittente_un_sixieme_de_tour.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c9/Cames_conjuguees_rotation_intermittente_un_sixieme_de_tour.svg/600px-Cames_conjuguees_rotation_intermittente_un_sixieme_de_tour.svg.png 2x" data-file-width="496" data-file-height="150" /></a></span></div><div class="thumbcaption"><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">6</span></span>⁠</span> per revolution</div></div></div><div class="trow"><div class="tsingle" style="width:302px;max-width:302px"><div class="thumbimage"><span typeof="mw:File"><a href="/wiki/File:Cames_conjuguees_rotation_intermittente_un_tiers_de_tour.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/68/Cames_conjuguees_rotation_intermittente_un_tiers_de_tour.svg/300px-Cames_conjuguees_rotation_intermittente_un_tiers_de_tour.svg.png" decoding="async" width="300" height="72" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/68/Cames_conjuguees_rotation_intermittente_un_tiers_de_tour.svg/450px-Cames_conjuguees_rotation_intermittente_un_tiers_de_tour.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/68/Cames_conjuguees_rotation_intermittente_un_tiers_de_tour.svg/600px-Cames_conjuguees_rotation_intermittente_un_tiers_de_tour.svg.png 2x" data-file-width="721" data-file-height="172" /></a></span></div><div class="thumbcaption"><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">3</span></span>⁠</span> per revolution</div></div></div></div></div><p> With <a href="/wiki/Cam_(mechanism)" title="Cam (mechanism)">cam drive</a> systems, use of a dual cam can avoid the jerk of a single cam; however, the dual cam is bulkier and more expensive. The dual-cam system has two cams on one axle that shifts a second axle by a fraction of a revolution. The graphic shows step drives of one-sixth and one-third rotation per one revolution of the driving axle. There is no radial clearance because two arms of the stepped wheel are always in contact with the double cam. Generally, combined contacts may be used to avoid the jerk (and wear and noise) associated with a single follower (such as a single follower gliding along a slot and changing its contact point from one side of the slot to the other can be avoided by using two followers sliding along the same slot, one side each).</p><div style="clear:both;" class=""></div> <div class="mw-heading mw-heading2"><h2 id="In_elastically_deformable_matter">In elastically deformable matter</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Jerk_(physics)&action=edit&section=6" title="Edit section: In elastically deformable matter"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1237032888/mw-parser-output/.tmulti"><div class="thumb tmulti tright"><div class="thumbinner multiimageinner" style="width:408px;max-width:408px"><div class="trow"><div class="theader" style="text-align:">Compression wave patterns</div></div><div class="trow"><div class="tsingle" style="width:202px;max-width:202px"><div class="thumbimage"><span typeof="mw:File"><a href="/wiki/File:Onde_compression_impulsion_1d_30_petit.gif" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/62/Onde_compression_impulsion_1d_30_petit.gif/200px-Onde_compression_impulsion_1d_30_petit.gif" decoding="async" width="200" height="151" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/62/Onde_compression_impulsion_1d_30_petit.gif/300px-Onde_compression_impulsion_1d_30_petit.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/6/62/Onde_compression_impulsion_1d_30_petit.gif 2x" data-file-width="305" data-file-height="231" /></a></span></div><div class="thumbcaption">Plane wave</div></div><div class="tsingle" style="width:202px;max-width:202px"><div class="thumbimage"><span typeof="mw:File"><a href="/wiki/File:Ondes_cisaillement_2d_20_petit.gif" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/3/31/Ondes_cisaillement_2d_20_petit.gif/200px-Ondes_cisaillement_2d_20_petit.gif" decoding="async" width="200" height="151" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/31/Ondes_cisaillement_2d_20_petit.gif/300px-Ondes_cisaillement_2d_20_petit.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/3/31/Ondes_cisaillement_2d_20_petit.gif 2x" data-file-width="305" data-file-height="231" /></a></span></div><div class="thumbcaption">Cylindrical symmetry</div></div></div></div></div> <p>An <a href="/wiki/Elastic_Deformation" class="mw-redirect" title="Elastic Deformation">elastically deformable</a> mass deforms under an applied force (or acceleration); the <a href="/wiki/Deformation_(engineering)" title="Deformation (engineering)">deformation</a> is a function of its <a href="/wiki/Stiffness" title="Stiffness">stiffness</a> and the magnitude of the force. If the change in force is slow, the jerk is small, and the <a href="/wiki/Wave_propagation" class="mw-redirect" title="Wave propagation">propagation</a> of deformation is considered instantaneous as compared to the change in acceleration. The distorted body acts as if it were in a <a href="/wiki/Quasistatic_loading" title="Quasistatic loading">quasistatic regime</a>, and only a changing force (nonzero jerk) can cause propagation of mechanical waves (or <a href="/wiki/Electromagnetic_wave" class="mw-redirect" title="Electromagnetic wave">electromagnetic waves</a> for a charged particle); therefore, for nonzero to high jerk, a <a href="/wiki/Shock_wave" title="Shock wave">shock wave</a> and its propagation through the body should be considered. </p><p>The propagation of deformation is shown in the graphic "Compression wave patterns" as a compressional <a href="/wiki/Plane_wave" title="Plane wave">plane wave</a> through an elastically deformable material. Also shown, for angular jerk, are the deformation waves propagating in a circular pattern, which causes <a href="/wiki/Shear_stress" title="Shear stress">shear stress</a> and possibly other <a href="/wiki/Normal_mode" title="Normal mode">modes</a> of <a href="/wiki/Vibration" title="Vibration">vibration</a>. The reflection of waves along the boundaries cause constructive <a href="/wiki/Wave_interference" title="Wave interference">interference patterns</a> (not pictured), producing stresses that may exceed the material's limits. The deformation waves may cause vibrations, which can lead to noise, wear, and failure, especially in cases of resonance. </p> <figure class="mw-default-size mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/File:Acceleration_et_deformation_elastique.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4f/Acceleration_et_deformation_elastique.svg/220px-Acceleration_et_deformation_elastique.svg.png" decoding="async" width="220" height="153" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4f/Acceleration_et_deformation_elastique.svg/330px-Acceleration_et_deformation_elastique.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4f/Acceleration_et_deformation_elastique.svg/440px-Acceleration_et_deformation_elastique.svg.png 2x" data-file-width="305" data-file-height="212" /></a><figcaption>Pole with massive top</figcaption></figure> <p>The graphic captioned "Pole with massive top" shows a block connected to an elastic pole and a massive top. The pole bends when the block accelerates, and when the acceleration stops, the top will oscillate (<a href="/wiki/Damping_ratio" class="mw-redirect" title="Damping ratio">damped</a>) under the regime of pole stiffness. One could argue that a greater (periodic) jerk might excite a larger amplitude of oscillation because small oscillations are damped before reinforcement by a shock wave. One can also argue that a larger jerk might increase the probability of exciting a <a href="/wiki/Resonance" title="Resonance">resonant mode</a> because the larger wave components of the shock wave have higher frequencies and <a href="/wiki/Fourier_series" title="Fourier series">Fourier coefficients</a>. </p> <div style="clear:both;" class=""></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Chronogrammes_loi_sinusoidale_par_partie_en_vitesse.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d0/Chronogrammes_loi_sinusoidale_par_partie_en_vitesse.svg/220px-Chronogrammes_loi_sinusoidale_par_partie_en_vitesse.svg.png" decoding="async" width="220" height="294" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d0/Chronogrammes_loi_sinusoidale_par_partie_en_vitesse.svg/330px-Chronogrammes_loi_sinusoidale_par_partie_en_vitesse.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d0/Chronogrammes_loi_sinusoidale_par_partie_en_vitesse.svg/440px-Chronogrammes_loi_sinusoidale_par_partie_en_vitesse.svg.png 2x" data-file-width="610" data-file-height="814" /></a><figcaption>Sinusoidal acceleration profile</figcaption></figure><p> To reduce the amplitude of excited stress waves and vibrations, one can limit jerk by shaping motion and making the acceleration continuous with slopes as flat as possible. Due to limitations of abstract models, algorithms for reducing vibrations include higher derivatives, such as <a href="/wiki/Jounce" class="mw-redirect" title="Jounce">jounce</a>, or suggest continuous regimes for both acceleration and jerk. One concept for limiting jerk is to shape acceleration and deceleration sinusoidally with zero acceleration in between (see graphic captioned "Sinusoidal acceleration profile"), making the speed appear sinusoidal with constant maximum speed. The jerk, however, will remain discontinuous at the points where acceleration enters and leaves the zero phases.</p><div style="clear:both;" class=""></div> <div class="mw-heading mw-heading2"><h2 id="In_the_geometric_design_of_roads_and_tracks">In the geometric design of roads and tracks</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Jerk_(physics)&action=edit&section=7" title="Edit section: In the geometric design of roads and tracks"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Easement_curve.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/54/Easement_curve.svg/220px-Easement_curve.svg.png" decoding="async" width="220" height="123" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/54/Easement_curve.svg/330px-Easement_curve.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/54/Easement_curve.svg/440px-Easement_curve.svg.png 2x" data-file-width="575" data-file-height="321" /></a><figcaption>A <a href="/wiki/Track_transition_curve" title="Track transition curve">track transition curve</a> limits jerk. The transition is shown in red between the blue straight line and green arc.</figcaption></figure> <p>Roads and tracks are designed to limit the jerk caused by changes in their curvature. Design standards for <a href="/wiki/High-speed_rail" title="High-speed rail">high-speed rail</a> vary from 0.2 m/s<sup>3</sup> to 0.6 m/s<sup>3</sup>.<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> <a href="/wiki/Track_transition_curves" class="mw-redirect" title="Track transition curves">Track transition curves</a> limit the jerk when transitioning from a straight line to a curve, or vice versa. Recall that in constant-speed motion along an arc, acceleration is zero in the tangential direction and nonzero in the inward normal direction. Transition curves gradually increase the curvature and, consequently, the centripetal acceleration. </p><p>An <a href="/wiki/Euler_spiral" title="Euler spiral">Euler spiral</a>, the theoretically optimum transition curve, linearly increases centripetal acceleration and results in constant jerk (see graphic). In real-world applications, the plane of the track is inclined (<a href="/wiki/Cant_(road/rail)" class="mw-redirect" title="Cant (road/rail)">cant</a>) along the curved sections. The incline causes vertical acceleration, which is a design consideration for wear on the track and embankment. The Wiener Kurve (Viennese Curve) is a patented curve designed to minimize this wear.<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><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> </p><p><a href="/wiki/Rollercoaster" class="mw-redirect" title="Rollercoaster">Rollercoasters</a><sup id="cite_ref-thetartan2007_2-1" class="reference"><a href="#cite_note-thetartan2007-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> are also designed with track transitions to limit jerk. When entering a loop, acceleration values can reach around 4<i>g</i> (40 m/s<sup>2</sup>), and riding in this high acceleration environment is only possible with track transitions. S-shaped curves, such as figure eights, also use track transitions for smooth rides. </p> <div class="mw-heading mw-heading2"><h2 id="In_motion_control">In motion control</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Jerk_(physics)&action=edit&section=8" title="Edit section: In motion control"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In <a href="/wiki/Motion_control" title="Motion control">motion control</a>, the design focus is on straight, linear motion, with the need to move a system from one steady position to another (point-to-point motion). The design concern from a jerk perspective is vertical jerk; the jerk from tangential acceleration is effectively zero since linear motion is non-rotational. </p><p>Motion control applications include passenger <a href="/wiki/Elevator" title="Elevator">elevators</a> and machining tools. Limiting vertical jerk is considered essential for elevator riding convenience.<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/List_of_ISO_standards" title="List of ISO standards">ISO 8100-34</a><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> specifies measurement methods for elevator ride quality with respect to jerk, acceleration, vibration, and noise; however, the standard does not specify levels for acceptable or unacceptable ride quality. It is reported<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> that most passengers rate a vertical jerk of 2 m/s<sup>3</sup> as acceptable and 6 m/s<sup>3</sup> as intolerable. For hospitals, 0.7 m/s<sup>3</sup> is the recommended limit. </p><p>A primary design goal for motion control is to minimize the transition time without exceeding speed, acceleration, or jerk limits. Consider a third-order motion-control profile with quadratic ramping and deramping phases in velocity (see figure). </p> <figure class="mw-default-size mw-halign-center" typeof="mw:File"><a href="/wiki/File:Schematic_diagram_of_Jerk,_Acceleration,_and_Speed.svg" class="mw-file-description" title="This picture shows a schematic diagram of jerk, acceleration, and speed, assuming all three are limited in their magnitude, when linearly going from one point to another, which are sufficiently far apart to reach the respective maxima."><img alt="This picture shows a schematic diagram of jerk, acceleration, and speed, assuming all three are limited in their magnitude, when linearly going from one point to another, which are sufficiently far apart to reach the respective maxima." src="//upload.wikimedia.org/wikipedia/commons/thumb/4/43/Schematic_diagram_of_Jerk%2C_Acceleration%2C_and_Speed.svg/720px-Schematic_diagram_of_Jerk%2C_Acceleration%2C_and_Speed.svg.png" decoding="async" width="720" height="480" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/43/Schematic_diagram_of_Jerk%2C_Acceleration%2C_and_Speed.svg/1080px-Schematic_diagram_of_Jerk%2C_Acceleration%2C_and_Speed.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/43/Schematic_diagram_of_Jerk%2C_Acceleration%2C_and_Speed.svg/1440px-Schematic_diagram_of_Jerk%2C_Acceleration%2C_and_Speed.svg.png 2x" data-file-width="720" data-file-height="480" /></a><figcaption>This picture shows a schematic diagram of jerk, acceleration, and speed, assuming all three are limited in their magnitude, when linearly going from one point to another, which are sufficiently far apart to reach the respective maxima.</figcaption></figure> <p>This motion profile consists of the following seven segments: </p> <ol><li>Acceleration build up — positive jerk limit; linear increase in acceleration to the positive acceleration limit; quadratic increase in velocity</li> <li>Upper acceleration limit — zero jerk; linear increase in velocity</li> <li>Acceleration ramp down — negative jerk limit; linear decrease in acceleration; (negative) quadratic increase in velocity, approaching the desired velocity limit</li> <li>Velocity limit — zero jerk; zero acceleration</li> <li>Deceleration build up — negative jerk limit; linear decrease in acceleration to the negative acceleration limit; (negative) quadratic decrease in velocity</li> <li>Lower deceleration limit — zero jerk; linear decrease in velocity</li> <li>Deceleration ramp down — positive jerk limit; linear increase in acceleration to zero; quadratic decrease in velocity; approaching the desired position at zero speed and zero acceleration</li></ol> <p>Segment four's time period (constant velocity) varies with distance between the two positions. If this distance is so small that omitting segment four would not suffice, then segments two and six (constant acceleration) could be equally reduced, and the constant velocity limit would not be reached. If this modification does not sufficiently reduce the crossed distance, then segments one, three, five, and seven could be shortened by an equal amount, and the constant acceleration limits would not be reached. </p><p>Other motion profile strategies are used, such as minimizing the square of jerk for a given transition time<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> and, as discussed above, sinusoidal-shaped acceleration profiles. Motion profiles are tailored for specific applications including machines, people movers, chain hoists, automobiles, and robotics. </p> <div class="mw-heading mw-heading3"><h3 id="In_manufacturing">In manufacturing</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Jerk_(physics)&action=edit&section=9" title="Edit section: In manufacturing"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Jerk is an important consideration in <a href="/wiki/Manufacturing" title="Manufacturing">manufacturing</a> processes. Rapid changes in acceleration of a cutting tool can lead to premature tool wear and result in uneven cuts; consequently, modern <a href="/wiki/Motion_control" title="Motion control">motion controllers</a> include jerk limitation features. In mechanical engineering, jerk, in addition to velocity and acceleration, is considered in the development of cam profiles because of <a href="/wiki/Tribology" title="Tribology">tribological</a> implications and the ability of the actuated body to follow the cam profile without <a href="/wiki/Machining_vibrations" title="Machining vibrations">chatter</a>.<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> Jerk is often considered when vibration is a concern. A device that measures jerk is called a "jerkmeter". </p> <div class="mw-heading mw-heading2"><h2 id="Further_derivatives">Further derivatives</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Jerk_(physics)&action=edit&section=10" title="Edit section: Further derivatives"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Fourth,_fifth,_and_sixth_derivatives_of_position" title="Fourth, fifth, and sixth derivatives of position">Fourth, fifth, and sixth derivatives of position</a></div> <p>Further time derivatives have also been named, as snap or jounce (fourth derivative), crackle (fifth derivative), and pop (sixth derivative).The seventh derivative is known as "Bang," as it is a logical continuation to the cycle. The eighth derivative has been referred to as "Boom," and the 9th is known as "Crash."<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><sup id="cite_ref-Visser2004_13-0" class="reference"><a href="#cite_note-Visser2004-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> However, time derivatives of position of higher order than four appear rarely.<sup id="cite_ref-PhysicsFAQ_14-0" class="reference"><a href="#cite_note-PhysicsFAQ-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> </p><p>The terms <i>snap</i>, <i>crackle</i>, and <i>pop</i>‍—‌for the fourth, fifth, and sixth derivatives of position‍—‌were inspired by the advertising mascots <a href="/wiki/Snap,_Crackle,_and_Pop" class="mw-redirect" title="Snap, Crackle, and Pop">Snap, Crackle, and Pop</a>.<sup id="cite_ref-Visser2004_13-1" class="reference"><a href="#cite_note-Visser2004-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Jerk_(physics)&action=edit&section=11" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Geomagnetic_jerk" title="Geomagnetic jerk">Geomagnetic jerk</a></li> <li><a href="/wiki/Shock_(mechanics)" title="Shock (mechanics)">Shock (mechanics)</a></li> <li><a href="/wiki/Yank_(physics)" class="mw-redirect" title="Yank (physics)">Yank</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Jerk_(physics)&action=edit&section=12" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></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 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Neurosci</i>. <b>4</b> (11): 2745–2754. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1523%2FJNEUROSCI.04-11-02745.1984">10.1523/JNEUROSCI.04-11-02745.1984</a></span>. <a href="/wiki/PMC_(identifier)" class="mw-redirect" title="PMC (identifier)">PMC</a> <span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC6564718">6564718</a></span>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/6502203">6502203</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=J.+Neurosci.&rft.atitle=An+organizing+principle+for+a+class+of+voluntary+movements&rft.volume=4&rft.issue=11&rft.pages=2745-2754&rft.date=1984&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC6564718%23id-name%3DPMC&rft_id=info%3Apmid%2F6502203&rft_id=info%3Adoi%2F10.1523%2FJNEUROSCI.04-11-02745.1984&rft.aulast=Hogan&rft.aufirst=Neville&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC6564718&rfr_id=info%3Asid%2Fen.wikipedia.org%3AJerk+%28physics%29" class="Z3988"></span></span> </li> <li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text">Blair, G., "Making the Cam", <i>Race Engine Technology </i> 10, September–October 2005</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="CITEREFThompson2011" class="citation conference cs1">Thompson, Peter M. (March 2011). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20170304041659/https://info.aiaa.org/Regions/Western/Orange_County/Newsletters/Presentations%20Posted%20by%20Enrique%20P.%20Castro/AIAAOC_SnapCracklePop_docx.pdf">"Snap, Crackle, and Pop"</a> <span class="cs1-format">(PDF)</span>. <i>Proc of AIAA Southern California Aerospace Systems and Technology Conference</i>. p. 1. Archived from <a rel="nofollow" class="external text" href="https://info.aiaa.org/Regions/Western/Orange_County/Newsletters/Presentations%20Posted%20by%20Enrique%20P.%20Castro/AIAAOC_SnapCracklePop_docx.pdf">the original</a> <span class="cs1-format">(PDF)</span> on 2017-03-04<span class="reference-accessdate">. Retrieved <span class="nowrap">29 February</span> 2020</span>. <q>The common names for the first three derivatives are velocity, acceleration, and jerk. The not so common names for the next three derivatives are snap, crackle, and pop.</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=conference&rft.atitle=Snap%2C+Crackle%2C+and+Pop&rft.btitle=Proc+of+AIAA+Southern+California+Aerospace+Systems+and+Technology+Conference&rft.pages=1&rft.date=2011-03&rft.aulast=Thompson&rft.aufirst=Peter+M.&rft_id=https%3A%2F%2Finfo.aiaa.org%2FRegions%2FWestern%2FOrange_County%2FNewsletters%2FPresentations%2520Posted%2520by%2520Enrique%2520P.%2520Castro%2FAIAAOC_SnapCracklePop_docx.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AJerk+%28physics%29" class="Z3988"></span></span> </li> <li id="cite_note-Visser2004-13"><span class="mw-cite-backlink">^ <a href="#cite_ref-Visser2004_13-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Visser2004_13-1"><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="CITEREFVisser2004" class="citation journal cs1">Visser, Matt (31 March 2004). "Jerk, snap and the cosmological equation of state". <i><a href="/wiki/Classical_and_Quantum_Gravity" title="Classical and Quantum Gravity">Classical and Quantum Gravity</a></i>. <b>21</b> (11): 2603–2616. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/gr-qc/0309109">gr-qc/0309109</a></span>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2004CQGra..21.2603V">2004CQGra..21.2603V</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1088%2F0264-9381%2F21%2F11%2F006">10.1088/0264-9381/21/11/006</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0264-9381">0264-9381</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:10468158">10468158</a>. <q>Snap [the fourth time derivative] is also sometimes called jounce. The fifth and sixth time derivatives are sometimes somewhat facetiously referred to as crackle and pop.</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Classical+and+Quantum+Gravity&rft.atitle=Jerk%2C+snap+and+the+cosmological+equation+of+state&rft.volume=21&rft.issue=11&rft.pages=2603-2616&rft.date=2004-03-31&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A10468158%23id-name%3DS2CID&rft_id=info%3Abibcode%2F2004CQGra..21.2603V&rft_id=info%3Aarxiv%2Fgr-qc%2F0309109&rft.issn=0264-9381&rft_id=info%3Adoi%2F10.1088%2F0264-9381%2F21%2F11%2F006&rft.aulast=Visser&rft.aufirst=Matt&rfr_id=info%3Asid%2Fen.wikipedia.org%3AJerk+%28physics%29" class="Z3988"></span></span> </li> <li id="cite_note-PhysicsFAQ-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-PhysicsFAQ_14-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGragertGibbs1998" class="citation web cs1">Gragert, Stephanie; Gibbs, Philip (November 1998). <a rel="nofollow" class="external text" href="http://math.ucr.edu/home/baez/physics/General/jerk.html">"What is the term used for the third derivative of position?"</a>. <i>Usenet Physics and Relativity FAQ</i>. Math Dept., <a href="/wiki/University_of_California,_Riverside" title="University of California, Riverside">University of California, Riverside</a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20161130132227/http://math.ucr.edu/home/baez/physics/General/jerk.html">Archived</a> from the original on 2016-11-30<span class="reference-accessdate">. Retrieved <span class="nowrap">2015-10-24</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Usenet+Physics+and+Relativity+FAQ&rft.atitle=What+is+the+term+used+for+the+third+derivative+of+position%3F&rft.date=1998-11&rft.aulast=Gragert&rft.aufirst=Stephanie&rft.au=Gibbs%2C+Philip&rft_id=http%3A%2F%2Fmath.ucr.edu%2Fhome%2Fbaez%2Fphysics%2FGeneral%2Fjerk.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AJerk+%28physics%29" class="Z3988"></span></span> </li> </ol></div></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSprott2003" class="citation book cs1">Sprott JC (2003). <i>Chaos and Time-Series Analysis</i>. Oxford University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-19-850839-5" title="Special:BookSources/0-19-850839-5"><bdi>0-19-850839-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Chaos+and+Time-Series+Analysis&rft.pub=Oxford+University+Press&rft.date=2003&rft.isbn=0-19-850839-5&rft.au=Sprott+JC&rfr_id=info%3Asid%2Fen.wikipedia.org%3AJerk+%28physics%29" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSprott_JC1997" class="citation journal cs1">Sprott JC (1997). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20100613054556/http://sprott.physics.wisc.edu/pubs/paper229.pdf">"Some simple chaotic jerk functions"</a> <span class="cs1-format">(PDF)</span>. <i>Am J Phys</i>. <b>65</b> (6): 537–43. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1997AmJPh..65..537S">1997AmJPh..65..537S</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1119%2F1.18585">10.1119/1.18585</a>. Archived from <a rel="nofollow" class="external text" href="http://sprott.physics.wisc.edu/pubs/paper229.pdf">the original</a> <span class="cs1-format">(PDF)</span> on 2010-06-13<span class="reference-accessdate">. Retrieved <span class="nowrap">2009-09-28</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Am+J+Phys&rft.atitle=Some+simple+chaotic+jerk+functions&rft.volume=65&rft.issue=6&rft.pages=537-43&rft.date=1997&rft_id=info%3Adoi%2F10.1119%2F1.18585&rft_id=info%3Abibcode%2F1997AmJPh..65..537S&rft.au=Sprott+JC&rft_id=http%3A%2F%2Fsprott.physics.wisc.edu%2Fpubs%2Fpaper229.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AJerk+%28physics%29" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBlair_G2005" class="citation journal cs1">Blair G (2005). <a rel="nofollow" class="external text" href="http://www.profblairandassociates.com/pdfs/Camshaft%20RET%20010.pdf">"Making the Cam"</a> <span class="cs1-format">(PDF)</span>. <i>Race Engine Technology</i> (10). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20080515204208/http://www.profblairandassociates.com/pdfs/Camshaft%20RET%20010.pdf">Archived</a> <span class="cs1-format">(PDF)</span> from the original on 2008-05-15<span class="reference-accessdate">. Retrieved <span class="nowrap">2009-09-29</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Race+Engine+Technology&rft.atitle=Making+the+Cam&rft.issue=10&rft.date=2005&rft.au=Blair+G&rft_id=http%3A%2F%2Fwww.profblairandassociates.com%2Fpdfs%2FCamshaft%2520RET%2520010.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AJerk+%28physics%29" class="Z3988"></span></li></ul> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Jerk_(physics)&action=edit&section=13" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a rel="nofollow" class="external text" href="http://math.ucr.edu/home/baez/physics/General/jerk.html">What is the term used for the third derivative of position?</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20161130132227/http://math.ucr.edu/home/baez/physics/General/jerk.html">Archived</a> 2016-11-30 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a>, description of jerk in the <a rel="nofollow" class="external text" href="http://math.ucr.edu/home/baez/physics/index.html">Usenet Physics FAQ</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20110623030258/http://math.ucr.edu/home/baez/physics/index.html">Archived</a> 2011-06-23 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a></li> <li><a rel="nofollow" class="external text" href="https://performance-motion-devices.s3.amazonaws.com/pdfs/White%20Papers/Mathematics_of_Motion_Control_Profiles.pdf">Mathematics of Motion Control Profiles</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20201002055616/https://performance-motion-devices.s3.amazonaws.com/pdfs/White%20Papers/Mathematics_of_Motion_Control_Profiles.pdf">Archived</a> 2020-10-02 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a></li> <li><a rel="nofollow" class="external text" href="https://www.scribd.com/document/273374901/Elevator-Ride-Quality">Elevator-Ride-Quality</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20220328085506/https://www.scribd.com/document/273374901/Elevator-Ride-Quality">Archived</a> 2022-03-28 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a></li> <li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20140826120525/http://www.schindler.com/content/ie/internet/en/mobility-solutions/products/elevators/schindler-5300/_jcr_content/rightPar/downloadlist/downloadList/3_1340031711862.download.asset.3_1340031711862/05SML9039_Inform_Sheet_EN.pdf">Elevator manufacturer brochure</a></li> <li><a rel="nofollow" class="external text" href="https://depatisnet.dpma.de/DepatisNet/depatisnet?window=1&space=menu&content=treffer&action=pdf&docid=AT000000412975B">Patent of <i>Wiener Kurve</i></a></li> <li><span class="languageicon">(in German)</span> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20160313052948/http://mplusm.at/ifg/download/presle-05.pdf">Description of <i>Wiener Kurve</i></a></li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist 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href="/wiki/Classical_mechanics" title="Classical mechanics">Classical mechanics</a> <a href="/wiki/International_System_of_Units" title="International System of Units">SI units</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0;"><table class="wikitable" style="text-align:center;line-height:0.9;border-collapse:collapse;margin:auto;border:none;background:none;"> <tbody><tr> <td colspan="4" style="border:none;backgound:none; font-weight:bold;">Linear/translational quantities</td> <td rowspan="12" style="border:none;backgound:none;"></td> <td colspan="4" style="border:none;backgound:none; font-weight:bold;">Angular/rotational quantities</td> </tr> <tr> <th style="font-weight:normal;font-size:80%;">Dimensions</th> <th style="font-weight:normal;">1</th> <th style="font-weight:normal;">L</th> <th style="font-weight:normal;">L<sup>2</sup></th> <th style="font-weight:normal;font-size:80%;">Dimensions</th> <th style="font-weight:normal;">1</th> <th style="font-weight:normal;"><span class="texhtml"><i>θ</i></span></th> <th style="font-weight:normal;"><span class="texhtml"><i>θ</i></span><sup>2</sup></th> </tr> <tr> <th style="font-weight:normal;">T</th> <td><a href="/wiki/Time" title="Time">time</a>: <span class="texhtml"><i>t</i></span><br /><a href="/wiki/Second" title="Second">s</a></td> <td><a href="/wiki/Absement" title="Absement">absement</a>: <span class="texhtml"><b>A</b></span><br /><a href="/wiki/Meter_second" class="mw-redirect" title="Meter second">m s</a></td> <td></td> <th style="font-weight:normal;">T</th> <td><a href="/wiki/Time" title="Time">time</a>: <span class="texhtml"><i>t</i></span><br /><a href="/wiki/Second" title="Second">s</a></td> <td></td> <td></td> </tr> <tr> <th style="font-weight:normal;">1</th> <td></td> <td><a href="/wiki/Distance" title="Distance">distance</a>: <span class="texhtml"><i>d</i></span>, <span class="nowrap"><a href="/wiki/Position_(vector)" class="mw-redirect" title="Position (vector)">position</a>: <span class="texhtml"><b>r</b></span>, <span class="texhtml"><b>s</b></span>, <span class="texhtml"><b>x</b></span></span>, <a href="/wiki/Displacement_(vector)" class="mw-redirect" title="Displacement (vector)">displacement</a><br /><a href="/wiki/Metre" title="Metre">m</a></td> <td><a href="/wiki/Area" title="Area">area</a>: <span class="texhtml"><i>A</i></span><br /><a href="/wiki/Square_metre" title="Square metre">m<sup>2</sup></a></td> <th style="font-weight:normal;">1</th> <td></td> <td><a href="/wiki/Angle" title="Angle">angle</a>: <span class="texhtml"><i>θ</i></span>, <a href="/wiki/Angular_displacement" title="Angular displacement">angular displacement</a>: <span class="texhtml"><i><b>θ</b></i></span><br /><a href="/wiki/Radian" title="Radian">rad</a></td> <td><span class="nowrap"><a href="/wiki/Solid_angle" title="Solid angle">solid angle</a>: <span class="texhtml">Ω</span><br /><a href="/wiki/Steradian" title="Steradian">rad<sup>2</sup>, sr</a></span></td> </tr> <tr> <th style="font-weight:normal;">T<sup>−1</sup></th> <td><span class="nowrap"><a href="/wiki/Frequency" title="Frequency">frequency</a>: <span class="texhtml"><i>f</i></span></span><br /><a href="/wiki/Inverse_second" title="Inverse second">s<sup>−1</sup></a>, <a href="/wiki/Hertz" title="Hertz">Hz</a></td> <td><a href="/wiki/Speed" title="Speed">speed</a>: <span class="texhtml"><i>v</i></span>, <a href="/wiki/Velocity" title="Velocity">velocity</a>: <span class="texhtml"><b>v</b></span><br /><a href="/wiki/Metre_per_second" title="Metre per second">m s<sup>−1</sup></a></td> <td><a href="/wiki/Kinematic_viscosity" class="mw-redirect" title="Kinematic viscosity">kinematic viscosity</a>: <span class="texhtml"><i>ν</i></span>,<br /><a href="/wiki/Specific_angular_momentum" title="Specific angular momentum">specific angular momentum</a>: <span class="texhtml"><b>h</b></span><br />m<sup>2</sup> s<sup>−1</sup></td> <th style="font-weight:normal;">T<sup>−1</sup></th> <td><span class="nowrap"><a href="/wiki/Frequency" title="Frequency">frequency</a>: <span class="texhtml"><i>f</i></span></span>, <span class="nowrap"><a href="/wiki/Rotational_speed" class="mw-redirect" title="Rotational speed">rotational speed</a>: <span class="texhtml"><i>n</i></span></span>, <span class="nowrap"><a href="/wiki/Rotational_velocity" class="mw-redirect" title="Rotational velocity">rotational velocity</a>: <span class="texhtml"><i><b>n</b></i></span></span><br /><a href="/wiki/Inverse_second" title="Inverse second">s<sup>−1</sup></a>, <a href="/wiki/Hertz" title="Hertz">Hz</a></td> <td><a href="/wiki/Angular_speed" class="mw-redirect" title="Angular speed">angular speed</a>: <span class="texhtml"><i>ω</i></span>, <a href="/wiki/Angular_velocity" title="Angular velocity">angular velocity</a>: <span class="texhtml"><i><b>ω</b></i></span><br /><a href="/wiki/Radian_per_second" title="Radian per second">rad<span style="letter-spacing:0.1em"> </span>s<sup>−1</sup></a></td> <td></td> </tr> <tr> <th style="font-weight:normal;">T<sup>−2</sup></th> <td></td> <td><a href="/wiki/Acceleration" title="Acceleration">acceleration</a>: <span class="texhtml"><b>a</b></span><br /><a href="/wiki/Metre_per_second_squared" title="Metre per second squared">m s<sup>−2</sup></a></td> <td></td> <th style="font-weight:normal;">T<sup>−2</sup></th> <td><span class="nowrap"><a href="/wiki/Rotational_acceleration" class="mw-redirect" title="Rotational acceleration">rotational acceleration</a></span><br /><a href="/wiki/Inverse_square_second" class="mw-redirect" title="Inverse square second">s<sup>−2</sup></a></td> <td><a href="/wiki/Angular_acceleration" title="Angular acceleration">angular acceleration</a>: <span class="texhtml"><i><b>α</b></i></span><br /><a href="/wiki/Radian_per_second_squared" class="mw-redirect" title="Radian per second squared">rad<span style="letter-spacing:0.1em"> </span>s<sup>−2</sup></a></td> <td></td> </tr> <tr> <th style="font-weight:normal;">T<sup>−3</sup></th> <td></td> <td><a class="mw-selflink selflink">jerk</a>: <span class="texhtml"><b>j</b></span><br />m s<sup>−3</sup></td> <td></td> <th style="font-weight:normal;">T<sup>−3</sup></th> <td></td> <td><a class="mw-selflink-fragment" href="#Jerk_in_rotation">angular jerk</a>: <span class="texhtml"><i><b>ζ</b></i></span><br />rad<span style="letter-spacing:0.1em"> </span>s<sup>−3</sup></td> <td></td> </tr> <tr style="border-top: 3px double #a2a9b1;"> <th style="font-weight:normal;">M</th> <td><a href="/wiki/Mass" title="Mass">mass</a>: <span class="texhtml"><i>m</i></span><br /><a href="/wiki/Kilogram" title="Kilogram">kg</a></td> <td>weighted position: <span class="texhtml"><i>M</i> ⟨<i>x</i>⟩ = ∑ <i>m</i> <i>x</i></span> </td> <td></td> <th style="font-weight:normal;">ML<sup>2</sup></th> <td><a href="/wiki/Moment_of_inertia" title="Moment of inertia">moment of inertia</a>: <span class="texhtml"><i>I</i></span><br /><a href="/wiki/Kilogram_square_metre" class="mw-redirect" title="Kilogram square metre">kg<span style="letter-spacing:0.1em"> </span>m<sup>2</sup></a></td> <td></td> <td></td> </tr> <tr> <th style="font-weight:normal;">MT<sup>−1</sup></th> <td><a href="/wiki/Mass_flow_rate" title="Mass flow rate">Mass flow rate</a>: <span class="texhtml"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\dot {m}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>m</mi> <mo>˙</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\dot {m}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ad59b9876301e8fb75b9ddbf08de594b87251d3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:2.176ex;" alt="{\displaystyle {\dot {m}}}"></span></span><br /><a href="/wiki/Kilogram_per_second" class="mw-redirect" title="Kilogram per second">kg<span style="letter-spacing:0.1em"> </span>s<sup>−1</sup></a></td> <td><a href="/wiki/Momentum" title="Momentum">momentum</a>: <span class="texhtml"><b>p</b></span>, <a href="/wiki/Impulse_(physics)" title="Impulse (physics)">impulse</a>: <span class="texhtml"><b>J</b></span><br /><a href="/wiki/Kilogram_metre_per_second" class="mw-redirect" title="Kilogram metre per second">kg<span style="letter-spacing:0.1em"> </span>m s<sup>−1</sup></a>, <a href="/wiki/Newton_second" class="mw-redirect" title="Newton second">N s</a></td> <td><a href="/wiki/Action_(physics)" title="Action (physics)">action</a>: <span class="texhtml">𝒮</span>, <a href="/wiki/Absement#Applications" title="Absement">actergy</a>: <span class="texhtml">ℵ</span><br /><a href="/wiki/Kilogram_square_metre_per_second" class="mw-redirect" title="Kilogram square metre per second">kg<span style="letter-spacing:0.1em"> </span>m<sup>2</sup> s<sup>−1</sup></a>, <a href="/wiki/Joule-second" title="Joule-second">J s</a></td> <th style="font-weight:normal;">ML<sup>2</sup>T<sup>−1</sup></th> <td></td> <td><a href="/wiki/Angular_momentum" title="Angular momentum">angular momentum</a>: <span class="texhtml"><b>L</b></span>, <a href="/wiki/List_of_equations_in_classical_mechanics#Derived_dynamic_quantities" title="List of equations in classical mechanics">angular impulse</a>: <span class="texhtml">Δ<b>L</b></span><br /><a href="/wiki/Kilogram_square_metre_per_second" class="mw-redirect" title="Kilogram square metre per second">kg<span style="letter-spacing:0.1em"> </span>m<sup>2</sup> s<sup>−1</sup></a></td> <td><a href="/wiki/Action_(physics)" title="Action (physics)">action</a>: <span class="texhtml">𝒮</span>, <a href="/wiki/Absement#Applications" title="Absement">actergy</a>: <span class="texhtml">ℵ</span><br /><a href="/wiki/Kilogram_square_metre_per_second" class="mw-redirect" title="Kilogram square metre per second">kg<span style="letter-spacing:0.1em"> </span>m<sup>2</sup> s<sup>−1</sup></a>, <a href="/wiki/Joule-second" title="Joule-second">J s</a></td> </tr> <tr> <th style="font-weight:normal;">MT<sup>−2</sup></th> <td></td> <td><a href="/wiki/Force" title="Force">force</a>: <span class="texhtml"><b>F</b></span>, <a href="/wiki/Weight" title="Weight">weight</a>: <span class="texhtml"><b>F</b><sub>g</sub></span><br /><span style="margin-right:0.1em;">kg </span> m s<sup>−2</sup>, <a href="/wiki/Newton_(unit)" title="Newton (unit)">N</a></td> <td><a href="/wiki/Energy" title="Energy">energy</a>: <span class="texhtml"><i>E</i></span>, <a href="/wiki/Work_(physics)" title="Work (physics)">work</a>: <span class="texhtml"><i>W</i></span>, <a href="/wiki/Lagrangian_mechanics" title="Lagrangian mechanics">Lagrangian</a>: <span class="texhtml"><i>L</i></span><br /><span style="margin-right:0.1em;">kg</span> m<sup>2</sup> s<sup>−2</sup>, <a href="/wiki/Joule" title="Joule">J</a></td> <th style="font-weight:normal;">ML<sup>2</sup>T<sup>−2</sup></th> <td></td> <td><a href="/wiki/Torque" title="Torque">torque</a>: <span class="texhtml"><i><b>τ</b></i></span>, <a href="/wiki/Torque#Terminology" title="Torque">moment</a>: <span class="texhtml"><b>M</b></span><br /><span style="margin-right:0.1em;">kg</span> m<sup>2</sup> s<sup>−2</sup>, <a href="/wiki/Newton-metre" title="Newton-metre">N m</a></td> <td><a href="/wiki/Energy" title="Energy">energy</a>: <span class="texhtml"><i>E</i></span>, <a href="/wiki/Work_(physics)" title="Work (physics)">work</a>: <span class="texhtml"><i>W</i></span>, <a href="/wiki/Lagrangian_mechanics" title="Lagrangian mechanics">Lagrangian</a>: <span class="texhtml"><i>L</i></span><br /><span style="margin-right:0.1em;">kg</span> m<sup>2</sup> s<sup>−2</sup>, <a href="/wiki/Joule" title="Joule">J</a></td> </tr> <tr> <th style="font-weight:normal;">MT<sup>−3</sup></th> <td></td> <td><a href="/wiki/Yank_(physics)" class="mw-redirect" title="Yank (physics)">yank</a>: <span class="texhtml"><b>Y</b></span><br /><span style="margin-right:0.1em;">kg</span> m s<sup>−3</sup>, N s<sup>−1</sup></td> <td><a href="/wiki/Power_(physics)" title="Power (physics)">power</a>: <span class="texhtml"><i>P</i></span><br /><span style="margin-right:0.1em;">kg</span> m<sup>2</sup> s<sup>−3</sup>, <a href="/wiki/Watt" title="Watt">W</a></td> <th style="font-weight:normal;">ML<sup>2</sup>T<sup>−3</sup></th> <td></td> <td><a href="/wiki/Rotatum" class="mw-redirect" title="Rotatum">rotatum</a>: <span class="texhtml"><b>P</b></span><br /><span style="margin-right:0.1em;">kg</span> m<sup>2</sup> s<sup>−3</sup>, N m s<sup>−1</sup></td> <td><a href="/wiki/Power_(physics)" title="Power (physics)">power</a>: <span class="texhtml"><i>P</i></span><br /><span style="margin-right:0.1em;">kg</span> m<sup>2 </sup>s<sup>−3</sup>, <a href="/wiki/Watt" title="Watt">W</a></td> </tr> </tbody></table></div></td></tr></tbody></table></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Jerk_(physics)&oldid=1257968173">https://en.wikipedia.org/w/index.php?title=Jerk_(physics)&oldid=1257968173</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Help:Category" title="Help:Category">Categories</a>: <ul><li><a href="/wiki/Category:Acceleration" 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