SherbyNet - /page/wikibook/entry.mu

<!DOCTYPE html>

<title>Acceleration</title>

</head>

<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Acceleration</span></span>

</h1>

</header>

/* start https://en.wikipedia.org/ */

.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}}

/* end https://en.wikipedia.org/ */

</style><div role="note" class="hatnote navigation-not-searchable">This article is about acceleration in physics. For other uses, see <a href="Acceleration_(disambiguation)" class="mw-disambig" title="Acceleration (disambiguation)">Acceleration (disambiguation)</a>.</div>

<div role="note" class="hatnote navigation-not-searchable">"Accelerate" redirects here. For other uses, see <a href="Accelerate_(disambiguation)" class="mw-disambig" title="Accelerate (disambiguation)">Accelerate (disambiguation)</a>.</div>

</p>

/* start https://en.wikipedia.org/ */

.mw-parser-output .infobox-subbox{padding:0;border:none;margin:-3px;width:auto;min-width:100%;font-size:100%;clear:none;float:none;background-color:transparent}.mw-parser-output .infobox-3cols-child{margin:auto}.mw-parser-output .infobox .navbar{font-size:100%}@media screen{html.skin-theme-clientpref-night .mw-parser-output .infobox-full-data:not(.notheme)>div:not(.notheme)[style]{background:#1f1f23!important;color:#f8f9fa}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .infobox-full-data:not(.notheme)>div:not(.notheme)[style]{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 th,body.skin--responsive .mw-parser-output .infobox-table td{padding-left:inherit;padding-right:inherit}}

/* end https://en.wikipedia.org/ */

</style><table class="infobox"><tbody><tr><th colspan="2" class="infobox-above">Acceleration</th></tr><tr><td colspan="2" class="infobox-image"><div class="infobox-caption"><div style="display: inline-block; line-height: 1.2em; padding: .1em 0;">In vacuum (no <a href="Drag_(physics)" title="Drag (physics)">air resistance</a>), objects attracted by Earth gain speed at a steady rate.</div></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"><b>a</b></td></tr><tr><th scope="row" class="infobox-label"><a href="SI_unit" class="mw-redirect" title="SI unit">SI unit</a></th><td class="infobox-data"><a href="Metre_per_second_squared" title="Metre per second squared">m/s<sup>2</sup>, m·s<sup>−2</sup>, m s<sup>−2</sup></a></td></tr><tr><th scope="row" class="infobox-label"><div style="display: inline-block; line-height: 1.2em; padding: .1em 0;">Derivations from<br>other quantities</div></th><td class="infobox-data"><span class="mwe-math-element mwe-math-element-inline" data-qid="Q11376"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {a} ={\frac {d\mathbf {v} }{dt}}={\frac {d^{2}\mathbf {x} }{dt^{2}}}}">

</mrow>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {a} ={\frac {d\mathbf {v} }{dt}}={\frac {d^{2}\mathbf {x} }{dt^{2}}}}</annotation>

</semantics>

</math></span><img src="./1760b1620e8d07692ecfb879abb37fad5a34f238.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:15.479ex; height:6.009ex;" alt="{\displaystyle \mathbf {a} ={\frac {d\mathbf {v} }{dt}}={\frac {d^{2}\mathbf {x} }{dt^{2}}}}" loading="lazy"></span></td></tr><tr><th scope="row" class="infobox-label"><a href="Dimensional_analysis#Formulation" title="Dimensional analysis">Dimension</a></th><td class="infobox-data"><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {L}}{\mathsf {T}}^{-2}}">

</mrow>

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathsf {L}}{\mathsf {T}}^{-2}}</annotation>

</semantics>

</math></span><img src="./e2639355e0e107e11e7d06c0f772f155c24a0ed1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.176ex; height:2.676ex;" alt="{\displaystyle {\mathsf {L}}{\mathsf {T}}^{-2}}" loading="lazy"></span></td></tr></tbody></table>

/* start https://en.wikipedia.org/ */

.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 dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output .hlist ol ul,.mw-parser-output .hlist ul dl,.mw-parser-output .hlist ul ol,.mw-parser-output .hlist ul ul{display:inline}.mw-parser-output .hlist .mw-empty-li{display:none}.mw-parser-output .hlist dt::after{content:": "}.mw-parser-output .hlist dd::after,.mw-parser-output .hlist li::after{content:" · ";font-weight:bold}.mw-parser-output .hlist dd:last-child::after,.mw-parser-output .hlist dt:last-child::after,.mw-parser-output .hlist li:last-child::after{content:none}.mw-parser-output .hlist dd dd:first-child::before,.mw-parser-output .hlist dd dt:first-child::before,.mw-parser-output .hlist dd li:first-child::before,.mw-parser-output .hlist dt dd:first-child::before,.mw-parser-output .hlist dt dt:first-child::before,.mw-parser-output .hlist dt li:first-child::before,.mw-parser-output .hlist li dd:first-child::before,.mw-parser-output .hlist li dt:first-child::before,.mw-parser-output .hlist li li:first-child::before{content:" (";font-weight:normal}.mw-parser-output .hlist dd dd:last-child::after,.mw-parser-output .hlist dd dt:last-child::after,.mw-parser-output .hlist dd li:last-child::after,.mw-parser-output .hlist dt dd:last-child::after,.mw-parser-output .hlist dt dt:last-child::after,.mw-parser-output .hlist dt li:last-child::after,.mw-parser-output .hlist li dd:last-child::after,.mw-parser-output .hlist li dt:last-child::after,.mw-parser-output .hlist li li:last-child::after{content:")";font-weight:normal}.mw-parser-output .hlist ol{counter-reset:listitem}.mw-parser-output .hlist ol>li{counter-increment:listitem}.mw-parser-output .hlist ol>li::before{content:" "counter(listitem)"\a0 "}.mw-parser-output .hlist dd ol>li:first-child::before,.mw-parser-output .hlist dt ol>li:first-child::before,.mw-parser-output .hlist li ol>li:first-child::before{content:" ("counter(listitem)"\a0 "}

/* end https://en.wikipedia.org/ */

</style><style data-mw-deduplicate="TemplateStyles:r1126788409">

/* start https://en.wikipedia.org/ */

.mw-parser-output .plainlist ol,.mw-parser-output .plainlist ul{line-height:inherit;list-style:none;margin:0;padding:0}.mw-parser-output .plainlist ol li,.mw-parser-output .plainlist ul li{margin-bottom:0}

/* end https://en.wikipedia.org/ */

</style><style data-mw-deduplicate="TemplateStyles:r1246091330">

/* start https://en.wikipedia.org/ */

.mw-parser-output .sidebar{width:22em;float:right;clear:right;margin:0.5em 0 1em 1em;background:var(--background-color-neutral-subtle,#f8f9fa);border:1px solid var(--border-color-base,#a2a9b1);padding:0.2em;text-align:center;line-height:1.4em;font-size:88%;border-collapse:collapse;display:table}body.skin-minerva .mw-parser-output .sidebar{display:table!important;float:right!important;margin:0.5em 0 1em 1em!important}.mw-parser-output .sidebar-subgroup{width:100%;margin:0;border-spacing:0}.mw-parser-output .sidebar-left{float:left;clear:left;margin:0.5em 1em 1em 0}.mw-parser-output .sidebar-none{float:none;clear:both;margin:0.5em 1em 1em 0}.mw-parser-output .sidebar-outer-title{padding:0 0.4em 0.2em;font-size:125%;line-height:1.2em;font-weight:bold}.mw-parser-output .sidebar-top-image{padding:0.4em}.mw-parser-output .sidebar-top-caption,.mw-parser-output .sidebar-pretitle-with-top-image,.mw-parser-output .sidebar-caption{padding:0.2em 0.4em 0;line-height:1.2em}.mw-parser-output .sidebar-pretitle{padding:0.4em 0.4em 0;line-height:1.2em}.mw-parser-output .sidebar-title,.mw-parser-output .sidebar-title-with-pretitle{padding:0.2em 0.8em;font-size:145%;line-height:1.2em}.mw-parser-output .sidebar-title-with-pretitle{padding:0.1em 0.4em}.mw-parser-output .sidebar-image{padding:0.2em 0.4em 0.4em}.mw-parser-output .sidebar-heading{padding:0.1em 0.4em}.mw-parser-output .sidebar-content{padding:0 0.5em 0.4em}.mw-parser-output .sidebar-content-with-subgroup{padding:0.1em 0.4em 0.2em}.mw-parser-output .sidebar-above,.mw-parser-output .sidebar-below{padding:0.3em 0.8em;font-weight:bold}.mw-parser-output .sidebar-collapse .sidebar-above,.mw-parser-output .sidebar-collapse .sidebar-below{border-top:1px solid #aaa;border-bottom:1px solid #aaa}.mw-parser-output .sidebar-navbar{text-align:right;font-size:115%;padding:0 0.4em 0.4em}.mw-parser-output .sidebar-list-title{padding:0 0.4em;text-align:left;font-weight:bold;line-height:1.6em;font-size:105%}.mw-parser-output .sidebar-list-title-c{padding:0 0.4em;text-align:center;margin:0 3.3em}@media(max-width:640px){body.mediawiki .mw-parser-output .sidebar{width:100%!important;clear:both;float:none!important;margin-left:0!important;margin-right:0!important}}body.skin--responsive .mw-parser-output .sidebar a>img{max-width:none!important}@media screen{html.skin-theme-clientpref-night .mw-parser-output .sidebar:not(.notheme) .sidebar-list-title,html.skin-theme-clientpref-night .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle{background:transparent!important}html.skin-theme-clientpref-night .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle a{color:var(--color-progressive)!important}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-list-title,html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle{background:transparent!important}html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle a{color:var(--color-progressive)!important}}@media print{body.ns-0 .mw-parser-output .sidebar{display:none!important}}

/* end https://en.wikipedia.org/ */

</style><style data-mw-deduplicate="TemplateStyles:r1287650570">

/* start https://en.wikipedia.org/ */

.mw-parser-output .cm-sidebar-title{padding-left:0.8em;padding-right:0.8em}.mw-parser-output .cm-sidebar .sidebar-image{padding:0.2em 0 0.6em 0}.mw-parser-output .cm-sidebar .sidebar-caption{padding:0.4em 0;font-style:italic;font-size:90%}.mw-parser-output .cm-sidebar .sidebar-list{margin-bottom:0.4em}.mw-parser-output .cm-sidebar-above{margin:0.4em 0}.mw-parser-output .cm-sidebar .sidebar-list-title{border-bottom:1px solid var(--border-color-divider,black);text-align:center;margin-bottom:0.1em}

/* end https://en.wikipedia.org/ */

</style><table class="sidebar sidebar-collapse nomobile nowraplinks cm-sidebar"><tbody><tr><td class="sidebar-pretitle">Part of a series on</td></tr><tr><th class="sidebar-title-with-pretitle cm-sidebar-title"><a href="Classical_mechanics" title="Classical mechanics">Classical mechanics</a></th></tr><tr><td class="sidebar-image"><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\textbf {F}}={\frac {d\mathbf {p} }{dt}}}">

</mrow>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\textbf {F}}={\frac {d\mathbf {p} }{dt}}}</annotation>

</semantics>

</math></span><img src="./c2ad0a6d6780c3abc5247abd82bd8a2249d56ff3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:8.318ex; height:5.509ex;" alt="{\displaystyle {\textbf {F}}={\frac {d\mathbf {p} }{dt}}}" loading="lazy"></span><div class="sidebar-caption"><a href="Second_law_of_motion" class="mw-redirect" title="Second law of motion">Second law of motion</a></div></td></tr><tr><th class="sidebar-heading cm-sidebar-above">

<ul><li><a href="History_of_classical_mechanics" title="History of classical mechanics">History</a></li>

<li><a href="Timeline_of_classical_mechanics" title="Timeline of classical mechanics">Timeline</a></li>

<li><a href="List_of_textbooks_on_classical_mechanics_and_quantum_mechanics" title="List of textbooks on classical mechanics and quantum mechanics">Textbooks</a></li></ul>

</div></th></tr><tr><td class="sidebar-content">

<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)"><div class="sidebar-list-title-c">Branches</div></div><div class="sidebar-list-content mw-collapsible-content plainlist"><div class="hlist">

<ul><li><a href="Applied_mechanics" title="Applied mechanics">Applied</a></li>

<li><a href="Celestial_mechanics" title="Celestial mechanics">Celestial</a></li>

<li><a href="Continuum_mechanics" title="Continuum mechanics">Continuum</a></li>

<li><a href="Dynamics_(mechanics)" title="Dynamics (mechanics)">Dynamics</a></li>

<li><a href="Classical_field_theory" title="Classical field theory">Field theory</a></li>

<li><a href="Kinematics" title="Kinematics">Kinematics</a></li>

<li><a href="Kinetics_(physics)" title="Kinetics (physics)">Kinetics</a></li>

<li><a href="Statics" title="Statics">Statics</a></li>

<li><a href="Statistical_mechanics" title="Statistical mechanics">Statistical mechanics</a></li></ul>

</div></div></div></td>

</tr><tr><td class="sidebar-content">

<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)"><div class="sidebar-list-title-c">Fundamentals</div></div><div class="sidebar-list-content mw-collapsible-content plainlist"><div class="hlist">

<ul>

<li><a href="Angular_momentum" title="Angular momentum">Angular momentum</a></li>

<li><a href="Couple_(mechanics)" title="Couple (mechanics)">Couple</a></li>

<li><a href="D'Alembert's_principle" title="D'Alembert's principle">D'Alembert's principle</a></li>

<li><a href="Energy" title="Energy">Energy</a>

<ul><li><a href="Kinetic_energy#Newtonian_kinetic_energy" title="Kinetic energy">kinetic</a></li>

<li><a href="Potential_energy" title="Potential energy">potential</a></li></ul></li>

<li><a href="Force" title="Force">Force</a></li>

<li><a href="Frame_of_reference" title="Frame of reference">Frame of reference</a></li>

<li><a href="Inertial_frame_of_reference" title="Inertial frame of reference">Inertial frame of reference</a></li>

<li><a href="Impulse_(physics)" title="Impulse (physics)">Impulse</a></li>

<li><span class="nowrap"><a href="Inertia" title="Inertia">Inertia</a> / <a href="Moment_of_inertia" title="Moment of inertia">Moment of inertia</a></span></li>

<li><br><a href="Mechanical_power_(physics)" class="mw-redirect" title="Mechanical power (physics)">Mechanical power</a></li>

<li><a href="Work_(physics)" title="Work (physics)">Mechanical work</a></li>

<li><br><a href="Moment_(physics)" title="Moment (physics)">Moment</a></li>

<li><a href="Momentum" title="Momentum">Momentum</a></li>

<li><a href="Space" title="Space">Space</a></li>

<li><a href="Speed" title="Speed">Speed</a></li>

<li><a href="Torque" title="Torque">Torque</a></li>

<li><a href="Velocity" title="Velocity">Velocity</a></li>

<li><a href="Virtual_work" title="Virtual work">Virtual work</a></li></ul>

</div></div></div></td>

</tr><tr><td class="sidebar-content">

<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)"><div class="sidebar-list-title-c">Formulations</div></div><div class="sidebar-list-content mw-collapsible-content plainlist">

<ul><li><div style="display: inline-block; line-height: 1.2em; padding: .1em 0;"><b><a href="Newton's_laws_of_motion" title="Newton's laws of motion">Newton's laws of motion</a></b></div></li>

<li><div style="display: inline-block; line-height: 1.2em; padding: .1em 0;"><b><a href="Analytical_mechanics" title="Analytical mechanics">Analytical mechanics</a></b> <div class="plainlist"><ul><li><a href="Lagrangian_mechanics" title="Lagrangian mechanics">Lagrangian mechanics</a></li><li><a href="Hamiltonian_mechanics" title="Hamiltonian mechanics">Hamiltonian mechanics</a></li><li><a href="Routhian_mechanics" title="Routhian mechanics">Routhian mechanics</a></li><li><a href="Hamilton%E2%80%93Jacobi_equation" title="Hamilton–Jacobi equation">Hamilton–Jacobi equation</a></li><li><a href="Appell's_equation_of_motion" title="Appell's equation of motion">Appell's equation of motion</a></li><li><a href="Koopman%E2%80%93von_Neumann_classical_mechanics" title="Koopman–von Neumann classical mechanics">Koopman–von Neumann mechanics</a></li></ul></div></div></li></ul></div></div></td>

</tr><tr><td class="sidebar-content">

<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)"><div class="sidebar-list-title-c">Core topics</div></div><div class="sidebar-list-content mw-collapsible-content plainlist"><div class="hlist">

<ul><li><a href="Damping" title="Damping">Damping</a></li>

<li><a href="Displacement_(geometry)" title="Displacement (geometry)">Displacement</a></li>

<li><a href="Equations_of_motion" title="Equations of motion">Equations of motion</a></li>

<li><a href="Euler's_laws_of_motion" title="Euler's laws of motion"><span class="wrap">Euler's laws of motion</span></a></li>

<li><a href="Fictitious_force" title="Fictitious force">Fictitious force</a></li>

<li><a href="Friction" title="Friction">Friction</a></li>

<li><a href="Harmonic_oscillator" title="Harmonic oscillator">Harmonic oscillator</a></li></ul>

</div>

<ul><li><span class="nowrap"><a href="Inertial_frame_of_reference" title="Inertial frame of reference">Inertial</a> / <a href="Non-inertial_reference_frame" title="Non-inertial reference frame">Non-inertial reference frame</a></span></li></ul>

<ul><li><a href="Motion" title="Motion">Motion</a> (<a href="Linear_motion" title="Linear motion">linear</a>)</li>

<li><a href="Newton's_law_of_universal_gravitation" title="Newton's law of universal gravitation"><span class="wrap">Newton's law of universal gravitation</span></a></li>

<li><a href="Newton's_laws_of_motion" title="Newton's laws of motion">Newton's laws of motion</a></li>

<li><a href="Relative_velocity" title="Relative velocity">Relative velocity</a></li>

<li><a href="Rigid_body" title="Rigid body">Rigid body</a>

<ul><li><a href="Rigid_body_dynamics" title="Rigid body dynamics">dynamics</a></li>

<li><a href="Euler's_equations_(rigid_body_dynamics)" title="Euler's equations (rigid body dynamics)">Euler's equations</a></li></ul></li>

<li><a href="Simple_harmonic_motion" title="Simple harmonic motion">Simple harmonic motion</a></li>

<li><a href="Vibration" title="Vibration">Vibration</a></li></ul>

</div></div></div></td>

</tr><tr><td class="sidebar-content">

<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)"><div class="sidebar-list-title-c"><a href="Rotation_around_a_fixed_axis" title="Rotation around a fixed axis">Rotation</a></div></div><div class="sidebar-list-content mw-collapsible-content plainlist"><div class="hlist">

<ul><li><a href="Circular_motion" title="Circular motion">Circular motion</a></li>

<li><a href="Rotating_reference_frame" title="Rotating reference frame">Rotating reference frame</a></li>

<li><a href="Centripetal_force" title="Centripetal force">Centripetal force</a></li>

<li><a href="Centrifugal_force" title="Centrifugal force">Centrifugal force</a>

<ul><li><a href="Reactive_centrifugal_force" title="Reactive centrifugal force">reactive</a></li></ul></li>

<li><a href="Coriolis_force" title="Coriolis force">Coriolis force</a></li>

<li><a href="Pendulum_(mechanics)" title="Pendulum (mechanics)">Pendulum</a></li>

<li><a href="Tangential_speed" title="Tangential speed">Tangential speed</a></li>

<li><a href="Rotational_frequency" title="Rotational frequency">Rotational frequency</a></li></ul>

</div>

<ul><li><a href="Angular_acceleration" title="Angular acceleration">Angular acceleration</a> / <a href="Angular_displacement" title="Angular displacement">displacement</a> / <a href="Angular_frequency" title="Angular frequency">frequency</a> / <a href="Angular_velocity" title="Angular velocity">velocity</a></li></ul></div></div></td>

</tr><tr><td class="sidebar-content">

<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)"><div class="sidebar-list-title-c">Scientists</div></div><div class="sidebar-list-content mw-collapsible-content plainlist"><div class="hlist">

<ul><li><a href="Johannes_Kepler" title="Johannes Kepler">Kepler</a></li>

<li><a href="Galileo_Galilei" title="Galileo Galilei">Galileo</a></li>

<li><a href="Christiaan_Huygens" title="Christiaan Huygens">Huygens</a></li>

<li><a href="Isaac_Newton" title="Isaac Newton">Newton</a></li>

<li><a href="Jeremiah_Horrocks" title="Jeremiah Horrocks">Horrocks</a></li>

<li><a href="Edmond_Halley" title="Edmond Halley">Halley</a></li>

<li><a href="Pierre_Louis_Maupertuis" title="Pierre Louis Maupertuis">Maupertuis</a></li>

<li><a href="Daniel_Bernoulli" title="Daniel Bernoulli">Daniel Bernoulli</a></li>

<li><a href="Johann_Bernoulli" title="Johann Bernoulli">Johann Bernoulli</a></li>

<li><a href="Leonhard_Euler" title="Leonhard Euler">Euler</a></li>

<li><a href="Jean_le_Rond_d'Alembert" title="Jean le Rond d'Alembert">d'Alembert</a></li>

<li><a href="Alexis_Clairaut" title="Alexis Clairaut">Clairaut</a></li>

<li><a href="Joseph-Louis_Lagrange" title="Joseph-Louis Lagrange">Lagrange</a></li>

<li><a href="Pierre-Simon_Laplace" title="Pierre-Simon Laplace">Laplace</a></li>

<li><a href="Sim%C3%A9on_Denis_Poisson" title="Siméon Denis Poisson">Poisson</a></li>

<li><a href="William_Rowan_Hamilton" title="William Rowan Hamilton">Hamilton</a></li>

<li><a href="Carl_Gustav_Jacob_Jacobi" title="Carl Gustav Jacob Jacobi">Jacobi</a></li>

<li><a href="Augustin-Louis_Cauchy" title="Augustin-Louis Cauchy">Cauchy</a></li>

<li><a href="Edward_Routh" title="Edward Routh">Routh</a></li>

<li><a href="Joseph_Liouville" title="Joseph Liouville">Liouville</a></li>

<li><a href="Paul_%C3%89mile_Appell" title="Paul Émile Appell">Appell</a></li>

<li><a href="Josiah_Willard_Gibbs" title="Josiah Willard Gibbs">Gibbs</a></li>

<li><a href="Bernard_Koopman" title="Bernard Koopman">Koopman</a></li>

<li><a href="John_von_Neumann" title="John von Neumann">von Neumann</a></li></ul>

</div></div></div></td>

</tr><tr><td class="sidebar-below hlist">

<ul><li><span class="nowrap"><span class="nowrap"><span class="skin-invert-image noviewer" typeof="mw:File"></span> </span><a href="Portal%3APhysics" title="Portal:Physics">Physics portal</a></span></li>

<li><span class="nowrap"><span class="noviewer" typeof="mw:File"><span title="Category"></span></span> Category</span></li></ul></td></tr><tr><td class="sidebar-navbar"><style data-mw-deduplicate="TemplateStyles:r1239400231">

/* start https://en.wikipedia.org/ */

.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}

/* end https://en.wikipedia.org/ */

</style></td></tr></tbody></table>

<p>In <a href="Mechanics" title="Mechanics">mechanics</a>, <b>acceleration</b> is the <a href="Rate_(mathematics)" title="Rate (mathematics)">rate</a> of change of the <a href="Velocity" title="Velocity">velocity</a> of an object with respect to time. Acceleration is one of several components of <a href="Kinematics" title="Kinematics">kinematics</a>, the study of <a href="Motion" title="Motion">motion</a>. Accelerations are <a href="Euclidean_vector" title="Euclidean vector">vector</a> quantities (in that they have <a href="Magnitude_(mathematics)" title="Magnitude (mathematics)">magnitude</a> and <a href="Direction_(geometry)" title="Direction (geometry)">direction</a>).<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> The orientation of an object's acceleration is given by the orientation of the <i>net</i> <a href="Force" title="Force">force</a> acting on that object. The magnitude of an object's acceleration, as described by <a href="Newton's_second_law" class="mw-redirect" title="Newton's second law">Newton's second law</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> is the combined effect of two causes:

</p>

<ul><li>the net balance of all external <a href="Force" title="Force">forces</a> acting onto that object — magnitude is <a href="Direct_proportionality" class="mw-redirect" title="Direct proportionality">directly proportional</a> to this net resulting force;</li>

<li>that object's <a href="Mass" title="Mass">mass</a>, depending on the materials out of which it is made — magnitude is <a href="Inverse_proportionality" class="mw-redirect" title="Inverse proportionality">inversely proportional</a> to the object's mass.</li></ul>

<p>The <a href="International_System_of_Units" title="International System of Units">SI</a> unit for acceleration is <a href="Metre_per_second_squared" title="Metre per second squared">metre per second squared</a> (<span class="nowrap">m⋅s<sup>−2</sup></span>, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {\tfrac {m}{s^{2}}} }">

<mfrac>

<msup>

</mrow>

</msup>

</mfrac>

</mstyle>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {\tfrac {m}{s^{2}}} }</annotation>

</semantics>

</math></span><img src="./98dfac31f9b284074d7828a73a97246a48a9f41d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.505ex; width:2.316ex; height:3.509ex;" alt="{\displaystyle \mathrm {\tfrac {m}{s^{2}}} }" loading="lazy"></span>).

</p><p>For example, when a <a href="Vehicle" title="Vehicle">vehicle</a> starts from a <a href="https://en.wiktionary.org/wiki/standstill" class="extiw external" title="wikt:standstill">standstill</a> (zero velocity, in an <a href="Inertial_frame_of_reference" title="Inertial frame of reference">inertial frame of reference</a>) and travels in a straight line at increasing speeds, it is accelerating in the direction of travel. If the vehicle turns, an acceleration occurs toward the new direction and changes its motion vector. The acceleration of the vehicle in its current direction of motion is called a linear (or tangential during <a href="Circular_motion" title="Circular motion">circular motions</a>) acceleration, the <a href="Reaction_(physics)" title="Reaction (physics)">reaction</a> to which the passengers on board experience as a force pushing them back into their seats. When changing direction, the effecting acceleration is called radial (or centripetal during circular motions) acceleration, the reaction to which the passengers experience as a <a href="Centrifugal_force" title="Centrifugal force">centrifugal force</a>. If the speed of the vehicle decreases, this is an acceleration in the opposite direction of the velocity vector (mathematically a <a href="Negative_number" title="Negative number">negative</a>, if the movement is unidimensional and the velocity is positive), sometimes called <b>deceleration</b><sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> or <b>retardation</b>, and passengers experience the reaction to deceleration as an <a href="Inertia" title="Inertia">inertial</a> force pushing them forward. Such negative accelerations are often achieved by <a href="Retrorocket" title="Retrorocket">retrorocket</a> burning in <a href="Spacecraft" title="Spacecraft">spacecraft</a>.<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> Both acceleration and deceleration are treated the same, as they are both changes in velocity. Each of these accelerations (tangential, radial, deceleration) is felt by passengers until their relative (differential) velocity are neutralised in <a href="Frame_of_reference" title="Frame of reference">reference</a> to the acceleration due to change in speed.

</p>

<div class="mw-heading mw-heading2"><h2 id="Definition_and_properties">Definition and properties</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Average_acceleration">Average acceleration</h3></div>

<p>An object's average acceleration over a period of <a href="Time_in_physics" title="Time in physics">time</a> is its change in <a href="Velocity" title="Velocity">velocity</a>, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta \mathbf {v} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Delta \mathbf {v} }</annotation>

</semantics>

</math></span><img src="./0ae2621f980eefffc0a3f3e806bfce57a5be867a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.347ex; height:2.176ex;" alt="{\displaystyle \Delta \mathbf {v} }" loading="lazy"></span>, divided by the duration of the period, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta t}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Delta t}</annotation>

</semantics>

</math></span><img src="./8c28867ecd34e2caed12cf38feadf6a81a7ee542.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.775ex; height:2.176ex;" alt="{\displaystyle \Delta t}" loading="lazy"></span>. Mathematically,

<mover>

</mrow>

</mover>

</mrow>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\bar {\mathbf {a} }}={\frac {\Delta \mathbf {v} }{\Delta t}}.}</annotation>

</semantics>

</math></span></span>

</p>

<div class="mw-heading mw-heading3"><h3 id="Instantaneous_acceleration">Instantaneous acceleration</h3></div>

<p>Instantaneous acceleration, meanwhile, is the <a href="Limit_of_a_function" title="Limit of a function">limit</a> of the average acceleration over an <a href="Infinitesimal" title="Infinitesimal">infinitesimal</a> interval of time. In the terms of <a href="Calculus" title="Calculus">calculus</a>, instantaneous acceleration is the <a href="Derivative" title="Derivative">derivative</a> of the velocity vector with respect to time:

</mrow>

</mrow>

</mrow>

</munder>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {a} =\lim _{{\Delta t}\to 0}{\frac {\Delta \mathbf {v} }{\Delta t}}={\frac {d\mathbf {v} }{dt}}.}</annotation>

</semantics>

</math></span></span>

As acceleration is defined as the derivative of velocity, <span class="texhtml"><b>v</b></span>, with respect to time <span class="texhtml mvar" style="font-style:italic;">t</span> and velocity is defined as the derivative of position, <span class="texhtml"><b>x</b></span>, with respect to time, acceleration can be thought of as the <a href="Second_derivative" title="Second derivative">second derivative</a> of <span class="texhtml"><b>x</b></span> with respect to <span class="texhtml mvar" style="font-style:italic;">t</span>:

</mrow>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {a} ={\frac {d\mathbf {v} }{dt}}={\frac {d^{2}\mathbf {x} }{dt^{2}}}.}</annotation>

</semantics>

</math></span></span>

</p><p>(Here and elsewhere, if <a href="Rectilinear_motion" class="mw-redirect" title="Rectilinear motion">motion is in a straight line</a>, <a href="Euclidean_vector" title="Euclidean vector">vector</a> quantities can be substituted by <a href="Scalar_(physics)" title="Scalar (physics)">scalars</a> in the equations.)

</p><p>By the <a href="Fundamental_theorem_of_calculus" title="Fundamental theorem of calculus">fundamental theorem of calculus</a>, it can be seen that the <a href="Integral" title="Integral">integral</a> of the acceleration function <span class="texhtml"><i>a</i>(<i>t</i>)</span> is the velocity function <span class="texhtml"><i>v</i>(<i>t</i>)</span>; that is, the area under the curve of an acceleration vs. time (<span class="texhtml mvar" style="font-style:italic;">a</span> vs. <span class="texhtml mvar" style="font-style:italic;">t</span>) graph corresponds to the change of velocity.

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {\Delta v} =\int \mathbf {a} \,dt.}</annotation>

</semantics>

</math></span></span>

</p><p>Likewise, the integral of the <a href="Jerk_(physics)" title="Jerk (physics)">jerk</a> function <span class="texhtml"><i>j</i>(<i>t</i>)</span>, the derivative of the acceleration function, can be used to find the change of acceleration at a certain time:

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {\Delta a} =\int \mathbf {j} \,dt.}</annotation>

</semantics>

</math></span></span>

</p>

<div class="mw-heading mw-heading3"><h3 id="Units">Units</h3></div>

<p>Acceleration has the <a href="Dimensional_analysis" title="Dimensional analysis">dimensions</a> of velocity (L/T) divided by time, i.e. <a href="Length" title="Length">L</a> <a href="Time" title="Time">T</a><sup>−2</sup>. The <a href="International_System_of_Units" title="International System of Units">SI</a> unit of acceleration is the <a href="Metre_per_second_squared" title="Metre per second squared">metre per second squared</a> (m s<sup>−2</sup>); or "metre per second per second", as the velocity in metres per second changes by the acceleration value, every second.

</p>

<div class="mw-heading mw-heading3"><h3 id="Other_forms">Other forms</h3></div>

<p>An object moving in a circular motion—such as a satellite orbiting the Earth—is accelerating due to the change of direction of motion, although its speed may be constant. In this case it is said to be undergoing <i>centripetal</i> (directed towards the center) acceleration.

</p><p><a href="Proper_acceleration" title="Proper acceleration">Proper acceleration</a>, the acceleration of a body relative to a free-fall condition, is measured by an instrument called an <a href="Accelerometer" title="Accelerometer">accelerometer</a>.

</p><p>In <a href="Classical_mechanics" title="Classical mechanics">classical mechanics</a>, for a body with constant mass, the (vector) acceleration of the body's center of mass is proportional to the net <a href="Force" title="Force">force</a> vector (i.e. sum of all forces) acting on it (<a href="Newton's_laws_of_motion#Newton's_second_law" title="Newton's laws of motion">Newton's second law</a>):

</mrow>

</mrow>

</mrow>

<mfrac>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {F} =m\mathbf {a} \quad \implies \quad \mathbf {a} ={\frac {\mathbf {F} }{m}},}</annotation>

</semantics>

</math></span></span>

where <span class="texhtml"><b>F</b></span> is the net force acting on the body, <span class="texhtml mvar" style="font-style:italic;">m</span> is the <a href="Mass" title="Mass">mass</a> of the body, and <span class="texhtml"><b>a</b></span> is the center-of-mass acceleration. As speeds approach the <a href="Speed_of_light" title="Speed of light">speed of light</a>, <a href="Special_relativity" title="Special relativity">relativistic effects</a> become increasingly large.

</p>

<div class="mw-heading mw-heading2"><h2 id="Tangential_and_centripetal_acceleration">Tangential and centripetal acceleration</h2></div>

<div role="note" class="hatnote navigation-not-searchable">See also: <a href="Centripetal_force#Local_coordinates" title="Centripetal force">Centripetal force § Local coordinates</a>, and <a href="Tangential_velocity" class="mw-redirect" title="Tangential velocity">Tangential velocity</a></div>

<p>The velocity of a particle moving on a curved path as a <a href="Function_(mathematics)" title="Function (mathematics)">function</a> of time can be written as:

</mrow>

<mfrac>

<mrow>

</mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {v} (t)=v(t){\frac {\mathbf {v} (t)}{v(t)}}=v(t)\mathbf {u} _{\mathrm {t} }(t),}</annotation>

</semantics>

</math></span></span>

with <span class="texhtml"><i>v</i>(<i>t</i>)</span> equal to the speed of travel along the path, and

<msub>

</mrow>

</mrow>

</msub>

<mfrac>

<mrow>

</mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {u} _{\mathrm {t} }={\frac {\mathbf {v} (t)}{v(t)}}\,,}</annotation>

</semantics>

</math></span></span>

a <a href="Differential_geometry_of_curves" class="mw-redirect" title="Differential geometry of curves">unit vector tangent</a> to the path pointing in the direction of motion at the chosen moment in time. Taking into account both the changing speed <span class="texhtml"><i>v</i>(<i>t</i>)</span> and the changing direction of <span class="texhtml"><b>u</b><sub><i>t</i></sub></span>, the acceleration of a particle moving on a curved path can be written using the <a href="Chain_rule" title="Chain rule">chain rule</a> of differentiation<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> for the product of two functions of time as:

</p><p><span class="mwe-math-element mwe-math-element-block"><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{alignedat}{3}\mathbf {a} &={\frac {d\mathbf {v} }{dt}}\\&={\frac {dv}{dt}}\mathbf {u} _{\mathrm {t} }+v(t){\frac {d\mathbf {u} _{\mathrm {t} }}{dt}}\\&={\frac {dv}{dt}}\mathbf {u} _{\mathrm {t} }+{\frac {v^{2}}{r}}\mathbf {u} _{\mathrm {n} }\ ,\end{alignedat}}}">

<mtr>

<mtd>

</mrow>

</mtd>

<mtd>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

<msub>

</mrow>

</mrow>

</msub>

<mfrac>

<mrow>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

<msub>

</mrow>

</mrow>

</msub>

<mfrac>

<msup>

</mrow>

</msup>

</mfrac>

</mrow>

<msub>

</mrow>

</mrow>

</msub>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{alignedat}{3}\mathbf {a} &={\frac {d\mathbf {v} }{dt}}\\&={\frac {dv}{dt}}\mathbf {u} _{\mathrm {t} }+v(t){\frac {d\mathbf {u} _{\mathrm {t} }}{dt}}\\&={\frac {dv}{dt}}\mathbf {u} _{\mathrm {t} }+{\frac {v^{2}}{r}}\mathbf {u} _{\mathrm {n} }\ ,\end{alignedat}}}</annotation>

</semantics>

</math></span></span>

</p><p>where <span class="texhtml"><b>u</b><sub>n</sub></span> is the unit (inward) <a href="Differential_geometry_of_curves" class="mw-redirect" title="Differential geometry of curves">normal vector</a> to the particle's trajectory (also called <i>the principal normal</i>), and <span class="texhtml"><b>r</b></span> is its instantaneous <a href="Curvature#Curvature_of_plane_curves" title="Curvature">radius of curvature</a> based upon the <a href="Osculating_circle#Mathematical_description" title="Osculating circle">osculating circle</a> at time <span class="texhtml mvar" style="font-style:italic;">t</span>. The components

</p>

<msub>

</mrow>

</mrow>

</msub>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</mrow>

</msub>

<mfrac>

<msup>

</mrow>

</msup>

</mfrac>

</mrow>

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {a} _{\mathrm {t} }={\frac {dv}{dt}}\mathbf {u} _{\mathrm {t} }\quad {\text{and}}\quad \mathbf {a} _{\mathrm {c} }={\frac {v^{2}}{r}}\mathbf {u} _{\mathrm {n} }}</annotation>

</semantics>

</math></span><img src="./8cc7ccce3437ffaa274f0018f50b895faa209dd1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:30.209ex; height:5.843ex;" alt="{\displaystyle \mathbf {a} _{\mathrm {t} }={\frac {dv}{dt}}\mathbf {u} _{\mathrm {t} }\quad {\text{and}}\quad \mathbf {a} _{\mathrm {c} }={\frac {v^{2}}{r}}\mathbf {u} _{\mathrm {n} }}" loading="lazy"></span></dd></dl>

<p>are called the <a href="Tangential_acceleration" class="mw-redirect" title="Tangential acceleration">tangential acceleration</a> and the normal or radial acceleration (or centripetal acceleration in circular motion, see also <a href="Circular_motion" title="Circular motion">circular motion</a> and <a href="Centripetal_force" title="Centripetal force">centripetal force</a>), respectively.

</p><p>Geometrical analysis of three-dimensional space curves, which explains tangent, (principal) normal and binormal, is described by the <a href="Frenet%E2%80%93Serret_formulas" title="Frenet–Serret formulas">Frenet–Serret formulas</a>.<sup id="cite_ref-Andrews_8-0" class="reference"><a href="#cite_note-Andrews-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Chand_9-0" class="reference"><a href="#cite_note-Chand-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup>

</p>

<div class="mw-heading mw-heading2"><h2 id="Special_cases">Special cases</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Uniform_acceleration">Uniform acceleration</h3></div>

<div role="note" class="hatnote navigation-not-searchable">See also: <a href="Torricelli's_equation" title="Torricelli's equation">Torricelli's equation</a></div>

<p><i>Uniform</i> or <i>constant</i> acceleration is a type of motion in which the <a href="Velocity" title="Velocity">velocity</a> of an object changes by an equal amount in every equal time period.

</p><p>A frequently cited example of uniform acceleration is that of an object in <a href="Free_fall" title="Free fall">free fall</a> in a uniform gravitational field. The acceleration of a falling body in the absence of resistances to motion is dependent only on the <a href="Gravitational_field" title="Gravitational field">gravitational field</a> strength <a href="Standard_gravity" title="Standard gravity"><span class="texhtml">g</span></a> (also called <i>acceleration due to gravity</i>). By <a href="Newton's_second_law" class="mw-redirect" title="Newton's second law">Newton's second law</a> the <a href="Force" title="Force">force</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {F_{g}} }">

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {F_{g}} }</annotation>

</semantics>

</math></span><img src="./63d24e9fb0bb3ba45643f3e889d122827f6ac039.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.86ex; height:2.843ex;" alt="{\displaystyle \mathbf {F_{g}} }" loading="lazy"></span> acting on a body is given by:

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {F_{g}} =m\mathbf {g} .}</annotation>

</semantics>

</math></span></span>

</p><p>Because of the simple analytic properties of the case of constant acceleration, there are simple formulas relating the <a href="Displacement_(vector)" class="mw-redirect" title="Displacement (vector)">displacement</a>, initial and time-dependent <a href="Velocity" title="Velocity">velocities</a>, and acceleration to the <a href="Time_in_physics" title="Time in physics">time elapsed</a>:<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>

<mtr>

<mtd>

</mrow>

</mtd>

<mtd>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

<mfrac>

</mfrac>

</mstyle>

</mrow>

</mrow>

<msup>

</mrow>

</msup>

<msub>

</mrow>

</mrow>

</msub>

<mfrac>

</mfrac>

</mstyle>

</mrow>

<mrow>

<mrow>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

</mrow>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

</mrow>

</mtd>

<mtd>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

<msup>

</mrow>

</msup>

</mrow>

</mtd>

<mtd>

<msup>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msup>

</mrow>

</mrow>

<msub>

</mrow>

</mrow>

</msub>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\mathbf {s} (t)&=\mathbf {s} _{0}+\mathbf {v} _{0}t+{\tfrac {1}{2}}\mathbf {a} t^{2}=\mathbf {s} _{0}+{\tfrac {1}{2}}\left(\mathbf {v} _{0}+\mathbf {v} (t)\right)t\\\mathbf {v} (t)&=\mathbf {v} _{0}+\mathbf {a} t\\{v^{2}}(t)&={v_{0}}^{2}+2\mathbf {a\cdot } [\mathbf {s} (t)-\mathbf {s} _{0}],\end{aligned}}}</annotation>

</semantics>

</math></span></span>

</p><p>where

</p>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle t}</annotation>

</semantics>

</math></span><img src="./65658b7b223af9e1acc877d848888ecdb4466560.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}" loading="lazy"></span> is the elapsed time,</li>

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {s} _{0}}</annotation>

</semantics>

</math></span><img src="./1f0bd2e93a5e3dd1ebd7fc7d1efbad1cf1a916aa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.11ex; height:2.009ex;" alt="{\displaystyle \mathbf {s} _{0}}" loading="lazy"></span> is the initial displacement from the origin,</li>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {s} (t)}</annotation>

</semantics>

</math></span><img src="./fde0f017af7e73213f4726ead5baf36a0512260b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.705ex; height:2.843ex;" alt="{\displaystyle \mathbf {s} (t)}" loading="lazy"></span> is the displacement from the origin at time <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle t}</annotation>

</semantics>

</math></span><img src="./65658b7b223af9e1acc877d848888ecdb4466560.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}" loading="lazy"></span>,</li>

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {v} _{0}}</annotation>

</semantics>

</math></span><img src="./ace1ee3c9fc209269fe95de85fb09c25823dd1e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.465ex; height:2.009ex;" alt="{\displaystyle \mathbf {v} _{0}}" loading="lazy"></span> is the initial velocity,</li>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {v} (t)}</annotation>

</semantics>

</math></span><img src="./ee594765ca30167f80394d3349307a445782d012.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.06ex; height:2.843ex;" alt="{\displaystyle \mathbf {v} (t)}" loading="lazy"></span> is the velocity at time <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle t}</annotation>

</semantics>

</math></span><img src="./65658b7b223af9e1acc877d848888ecdb4466560.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}" loading="lazy"></span>, and</li>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {a} }</annotation>

</semantics>

</math></span><img src="./1a957216653a9ee0d0133dcefd13fb75e36b8b9d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.299ex; height:1.676ex;" alt="{\displaystyle \mathbf {a} }" loading="lazy"></span> is the uniform rate of acceleration.</li></ul>

<p>In particular, the motion can be resolved into two orthogonal parts, one of constant velocity and the other according to the above equations. As <a href="Galileo" class="mw-redirect" title="Galileo">Galileo</a> showed, the net result is parabolic motion, which describes, e.g., the trajectory of a projectile in vacuum near the surface of Earth.<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup>

</p>

<div class="mw-heading mw-heading3"><h3 id="Circular_motion">Circular motion</h3></div>

/* start https://en.wikipedia.org/ */

.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 span:not(.skin-invert-image):not(.skin-invert):not(.bg-transparent) img{background-color:white}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .tmulti .multiimageinner span:not(.skin-invert-image):not(.skin-invert):not(.bg-transparent) img{background-color:white}}

/* end https://en.wikipedia.org/ */

</style><div class="thumb tmulti tright"><div class="thumbinner multiimageinner" style="width:462px;max-width:462px"><div class="trow"><div class="tsingle" style="width:102px;max-width:102px"><div class="thumbimage"><span typeof="mw:File"></span></div><div class="thumbcaption">Position vector <b>r</b>, always points radially from the origin.</div></div><div class="tsingle" style="width:152px;max-width:152px"><div class="thumbimage"><span typeof="mw:File"></span></div><div class="thumbcaption">Velocity vector <b>v</b>, always tangent to the path of motion.</div></div><div class="tsingle" style="width:202px;max-width:202px"><div class="thumbimage"><span typeof="mw:File"></span></div><div class="thumbcaption">Acceleration vector <b>a</b>, not parallel to the radial motion but offset by the angular and Coriolis accelerations, nor tangent to the path but offset by the centripetal and radial accelerations.</div></div></div><div class="trow" style="display:flex"><div class="thumbcaption">Kinematic vectors in plane <a href="Polar_coordinates" class="mw-redirect" title="Polar coordinates">polar coordinates</a>. Notice the setup is not restricted to 2d space, but may represent the <a href="Osculating_plane" title="Osculating plane">osculating plane</a> plane in a point of an arbitrary curve in any higher dimension.</div></div></div></div>

<p>In uniform <a href="Circular_motion" title="Circular motion">circular motion</a>, that is moving with constant <i>speed</i> along a circular path, a particle experiences an acceleration resulting from the change of the direction of the velocity vector, while its magnitude remains constant. The derivative of the location of a point on a curve with respect to time, i.e. its velocity, turns out to be always exactly tangential to the curve, respectively orthogonal to the radius in this point. Since in uniform motion the velocity in the tangential direction does not change, the acceleration must be in radial direction, pointing to the center of the circle. This acceleration constantly changes the direction of the velocity to be tangent in the neighbouring point, thereby rotating the velocity vector along the circle.

</p>

<ul><li>For a given speed <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle v}</annotation>

</semantics>

</math></span><img src="./e07b00e7fc0847fbd16391c778d65bc25c452597.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle v}" loading="lazy"></span>, the magnitude of this geometrically caused acceleration (centripetal acceleration) is inversely proportional to the radius <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle r}</annotation>

</semantics>

</math></span><img src="./0d1ecb613aa2984f0576f70f86650b7c2a132538.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}" loading="lazy"></span> of the circle, and increases as the square of this speed: <span class="mwe-math-element mwe-math-element-block" data-qid="Q2248131"><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 a_{c}={\frac {v^{2}}{r}}\,.}">

<msub>

</mrow>

</msub>

<mfrac>

<msup>

</mrow>

</msup>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a_{c}={\frac {v^{2}}{r}}\,.}</annotation>

</semantics>

</math></span></span></li>

<li>For a given <a href="Angular_velocity" title="Angular velocity">angular velocity</a> <span class="mwe-math-element mwe-math-element-inline"><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 }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \omega }</annotation>

</semantics>

</math></span><img src="./48eff443f9de7a985bb94ca3bde20813ea737be8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.446ex; height:1.676ex;" alt="{\displaystyle \omega }" loading="lazy"></span>, the centripetal acceleration is directly proportional to radius <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle r}</annotation>

</semantics>

</math></span><img src="./0d1ecb613aa2984f0576f70f86650b7c2a132538.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}" loading="lazy"></span>. This is due to the dependence of velocity <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle v}</annotation>

</semantics>

</math></span><img src="./e07b00e7fc0847fbd16391c778d65bc25c452597.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle v}" loading="lazy"></span> on the radius <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle r}</annotation>

</semantics>

</math></span><img src="./0d1ecb613aa2984f0576f70f86650b7c2a132538.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}" loading="lazy"></span>. <span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v=\omega r.}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle v=\omega r.}</annotation>

</semantics>

</math></span></span></li></ul>

<p>Expressing centripetal acceleration vector in polar components, where <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {r} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {r} }</annotation>

</semantics>

</math></span><img src="./eca0f46511c4c986c48b254073732c0bd98ae0c1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.102ex; height:1.676ex;" alt="{\displaystyle \mathbf {r} }" loading="lazy"></span> is a vector from the centre of the circle to the particle with magnitude equal to this distance, and considering the orientation of the acceleration towards the center, yields

<msub>

</mrow>

</msub>

</mrow>

<mfrac>

<msup>

</mrow>

</msup>

<mrow>

</mrow>

</mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

</mrow>

<mrow>

</mrow>

</mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {a_{c}} =-{\frac {v^{2}}{|\mathbf {r} |}}\cdot {\frac {\mathbf {r} }{|\mathbf {r} |}}\,.}</annotation>

</semantics>

</math></span></span>

</p><p>As usual in rotations, the speed <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle v}</annotation>

</semantics>

</math></span><img src="./e07b00e7fc0847fbd16391c778d65bc25c452597.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle v}" loading="lazy"></span> of a particle may be expressed as an <a href="Angular_velocity" title="Angular velocity"><i>angular speed</i></a> with respect to a point at the distance <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle r}</annotation>

</semantics>

</math></span><img src="./0d1ecb613aa2984f0576f70f86650b7c2a132538.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}" loading="lazy"></span> as

<mfrac>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \omega ={\frac {v}{r}}.}</annotation>

</semantics>

</math></span></span>

</p><p>Thus <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {a_{c}} =-\omega ^{2}\mathbf {r} \,.}">

<msub>

</mrow>

</msub>

</mrow>

<msup>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {a_{c}} =-\omega ^{2}\mathbf {r} \,.}</annotation>

</semantics>

</math></span><img src="./e04e2c9f9a5b24de7419347a42b60da7f99af078.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.914ex; height:3.009ex;" alt="{\displaystyle \mathbf {a_{c}} =-\omega ^{2}\mathbf {r} \,.}" loading="lazy"></span>

</p><p>This acceleration and the mass of the particle determine the necessary <a href="Centripetal_force" title="Centripetal force">centripetal force</a>, directed <i>toward</i> the centre of the circle, as the net force acting on this particle to keep it in this uniform circular motion. The so-called '<a href="Centrifugal_force" title="Centrifugal force">centrifugal force</a>', appearing to act outward on the body, is a so-called <a href="Pseudo_force" class="mw-redirect" title="Pseudo force">pseudo force</a> experienced in the <a href="Frame_of_reference" title="Frame of reference">frame of reference</a> of the body in circular motion, due to the body's <a href="Linear_momentum" class="mw-redirect" title="Linear momentum">linear momentum</a>, a vector tangent to the circle of motion.

</p><p>In a nonuniform circular motion, i.e., the speed along the curved path is changing, the acceleration has a non-zero component tangential to the curve, and is not confined to the <a href="Principal_normal_vector" class="mw-redirect" title="Principal normal vector">principal normal</a>, which directs to the center of the osculating circle, that determines the radius <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle r}</annotation>

</semantics>

</math></span><img src="./0d1ecb613aa2984f0576f70f86650b7c2a132538.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}" loading="lazy"></span> for the centripetal acceleration. The tangential component is given by the angular acceleration <span class="mwe-math-element mwe-math-element-inline"><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 }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation>

</semantics>

</math></span><img src="./b79333175c8b3f0840bfb4ec41b8072c83ea88d3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }" loading="lazy"></span>, i.e., the rate of change <span class="mwe-math-element mwe-math-element-inline"><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 ={\dot {\omega }}}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \alpha ={\dot {\omega }}}</annotation>

</semantics>

</math></span><img src="./5decfde4e2bbd6e948a608dd100f1939a3bfac30.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.032ex; height:2.176ex;" alt="{\displaystyle \alpha ={\dot {\omega }}}" loading="lazy"></span> of the angular speed <span class="mwe-math-element mwe-math-element-inline"><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 }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \omega }</annotation>

</semantics>

</math></span><img src="./48eff443f9de7a985bb94ca3bde20813ea737be8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.446ex; height:1.676ex;" alt="{\displaystyle \omega }" loading="lazy"></span> times the radius <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle r}</annotation>

</semantics>

</math></span><img src="./0d1ecb613aa2984f0576f70f86650b7c2a132538.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}" loading="lazy"></span>. That is,

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a_{t}=r\alpha .}</annotation>

</semantics>

</math></span></span>

</p><p>The sign of the tangential component of the acceleration is determined by the sign of the <a href="Angular_acceleration" title="Angular acceleration">angular acceleration</a> (<span class="mwe-math-element mwe-math-element-inline"><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 }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation>

</semantics>

</math></span><img src="./b79333175c8b3f0840bfb4ec41b8072c83ea88d3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }" loading="lazy"></span>), and the tangent is always directed at right angles to the radius vector.

</p>

<div class="mw-heading mw-heading2"><h2 id="Coordinate_systems">Coordinate systems</h2></div>

<p>In multi-dimensional <a href="Cartesian_coordinate_system" title="Cartesian coordinate system">Cartesian coordinate systems</a>, acceleration is broken up into components that correspond with each dimensional axis of the coordinate system. In a two-dimensional system, where there is an x-axis and a y-axis, corresponding acceleration components are defined as<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><span class="mwe-math-element mwe-math-element-block"><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 a_{x}=dv_{x}/dt=d^{2}x/dt^{2},}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

<msup>

</mrow>

</msup>

</mrow>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a_{x}=dv_{x}/dt=d^{2}x/dt^{2},}</annotation>

</semantics>

</math></span></span> <span class="mwe-math-element mwe-math-element-block"><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 a_{y}=dv_{y}/dt=d^{2}y/dt^{2}.}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

<msup>

</mrow>

</msup>

</mrow>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a_{y}=dv_{y}/dt=d^{2}y/dt^{2}.}</annotation>

</semantics>

</math></span></span>The two-dimensional acceleration vector is then defined as <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\textbf {a}}=<a_{x},a_{y}>}">

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\textbf {a}}=<a_{x},a_{y}>}</annotation>

</semantics>

</math></span><img src="./9fd2e70de96488f95ef8f65e8c4a6a2d3b0bf739.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:14.375ex; height:2.509ex;" alt="{\displaystyle {\textbf {a}}=<a_{x},a_{y}>}" loading="lazy"></span>. The magnitude of this vector is found by the <a href="Euclidean_distance" title="Euclidean distance">distance formula</a> as<span class="mwe-math-element mwe-math-element-block"><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 |a|={\sqrt {a_{x}^{2}+a_{y}^{2}}}.}">

</mrow>

</mrow>

<msqrt>

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</msubsup>

</msqrt>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle |a|={\sqrt {a_{x}^{2}+a_{y}^{2}}}.}</annotation>

</semantics>

</math></span></span>In three-dimensional systems where there is an additional z-axis, the corresponding acceleration component is defined as<span class="mwe-math-element mwe-math-element-block"><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 a_{z}=dv_{z}/dt=d^{2}z/dt^{2}.}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

<msup>

</mrow>

</msup>

</mrow>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a_{z}=dv_{z}/dt=d^{2}z/dt^{2}.}</annotation>

</semantics>

</math></span></span>The three-dimensional acceleration vector is defined as <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\textbf {a}}=<a_{x},a_{y},a_{z}>}">

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\textbf {a}}=<a_{x},a_{y},a_{z}>}</annotation>

</semantics>

</math></span><img src="./a6f5c26e810aeeace73497545b45c5b6f7a7e4e7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:17.64ex; height:2.509ex;" alt="{\displaystyle {\textbf {a}}=<a_{x},a_{y},a_{z}>}" loading="lazy"></span> with its magnitude being determined by<span class="mwe-math-element mwe-math-element-block"><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 |a|={\sqrt {a_{x}^{2}+a_{y}^{2}+a_{z}^{2}}}.}">

</mrow>

</mrow>

<msqrt>

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</msubsup>

</msqrt>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle |a|={\sqrt {a_{x}^{2}+a_{y}^{2}+a_{z}^{2}}}.}</annotation>

</semantics>

</math></span></span>

</p>

<div class="mw-heading mw-heading2"><h2 id="Relation_to_relativity">Relation to relativity</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Special_relativity">Special relativity</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Main articles: <a href="Special_relativity" title="Special relativity">Special relativity</a> and <a href="Acceleration_(special_relativity)" title="Acceleration (special relativity)">Acceleration (special relativity)</a></div>

<p>The special theory of relativity describes the behaviour of objects travelling relative to other objects at speeds approaching that of light in vacuum. <a href="Newtonian_mechanics" class="mw-redirect" title="Newtonian mechanics">Newtonian mechanics</a> is exactly revealed to be an approximation to reality, valid to great accuracy at lower speeds. As the relevant speeds increase toward the speed of light, acceleration no longer follows classical equations.

</p><p>As speeds approach that of light, the acceleration produced by a given force decreases, becoming <a href="Infinitesimally" class="mw-redirect" title="Infinitesimally">infinitesimally</a> small as light speed is approached; an object with mass can approach this speed <a href="Asymptotically" class="mw-redirect" title="Asymptotically">asymptotically</a>, but never reach it.

</p>

<div class="mw-heading mw-heading3"><h3 id="General_relativity">General relativity</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="General_relativity" title="General relativity">General relativity</a></div>

<p>Unless the state of motion of an object is known, it is impossible to distinguish whether an observed force is due to <a href="Gravity" title="Gravity">gravity</a> or to acceleration—gravity and inertial acceleration have identical effects. <a href="Albert_Einstein" title="Albert Einstein">Albert Einstein</a> called this the <a href="Equivalence_principle" title="Equivalence principle">equivalence principle</a>, and said that only observers who feel no force at all—including the force of gravity—are justified in concluding that they are not accelerating.<sup id="cite_ref-Greene_13-0" class="reference"><a href="#cite_note-Greene-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup>

</p>

<div class="mw-heading mw-heading2"><h2 id="Conversions">Conversions</h2></div>

<caption><span class="nowrap">Conversions between common units of acceleration</span>

</caption>

<th scope="col">Base value

</th>

<th scope="col">(<a href="Gal_(unit)" title="Gal (unit)">Gal</a>, or cm/s<sup>2</sup>)

</th>

<th scope="col">(<a href="Foot_per_second_squared" title="Foot per second squared">ft/s<sup>2</sup></a>)

</th>

<th scope="col">(<a href="Metre_per_second_squared" title="Metre per second squared">m/s<sup>2</sup></a>)

</th>

<th scope="col">(<a href="Standard_gravity" title="Standard gravity">Standard gravity</a>, <i>g</i><sub>0</sub>)

</th></tr>

<tr>

</td>

</td>

</td>

</td>

</td></tr>

<tr>

</td>

</td>

</td>

</td>

</td></tr>

<tr>

</td>

</td>

</td>

</td>

</td></tr>

<tr>

</td>

</td>

</td>

</td>

</td></tr></tbody></table>

/* start https://en.wikipedia.org/ */

.mw-parser-output .div-col{margin-top:0.3em;column-width:30em}.mw-parser-output .div-col-small{font-size:90%}.mw-parser-output .div-col-rules{column-rule:1px solid #aaa}.mw-parser-output .div-col dl,.mw-parser-output .div-col ol,.mw-parser-output .div-col ul{margin-top:0}.mw-parser-output .div-col li,.mw-parser-output .div-col dd{page-break-inside:avoid;break-inside:avoid-column}

/* end https://en.wikipedia.org/ */

</style><div class="div-col" style="column-width: 22em;">

<ul><li><a href="Acceleration_(differential_geometry)" title="Acceleration (differential geometry)">Acceleration (differential geometry)</a></li>

<li><a href="Four-vector" title="Four-vector">Four-vector</a>: making the connection between space and time explicit</li>

<li><a href="Gravitational_acceleration" title="Gravitational acceleration">Gravitational acceleration</a></li>

<li><a href="Inertia" title="Inertia">Inertia</a></li>

<li><a href="Orders_of_magnitude_(acceleration)" title="Orders of magnitude (acceleration)">Orders of magnitude (acceleration)</a></li>

<li><a href="Shock_(mechanics)" title="Shock (mechanics)">Shock (mechanics)</a></li>

<li><a href="Shock_and_vibration_data_logger" title="Shock and vibration data logger">Shock and vibration data logger</a> measuring 3-axis acceleration</li>

<li><a href="Space_travel_using_constant_acceleration" class="mw-redirect" title="Space travel using constant acceleration">Space travel using constant acceleration</a></li>

<li><a href="Specific_force" title="Specific force">Specific force</a></li></ul>

</div>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

/* start https://en.wikipedia.org/ */

.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}

/* end https://en.wikipedia.org/ */

</style><div class="reflist">

/* start https://en.wikipedia.org/ */

.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("./mw/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("./mw/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("./mw/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("./mw/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}

/* end https://en.wikipedia.org/ */

</style><cite id="CITEREFBondi1980" class="citation book cs1">Bondi, Hermann (1980). <a rel="nofollow" class="external text" href="https://archive.org/details/relativitycommon0000bond/page/3"><i>Relativity and Common Sense</i></a>. Courier Dover Publications. pp. <a rel="nofollow" class="external text" href="https://archive.org/details/relativitycommon0000bond/page/3">3</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-486-24021-3</bdi>.</cite></span>

</li>

<li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><cite id="CITEREFLehrman1998" class="citation book cs1">Lehrman, Robert L. (1998). <a rel="nofollow" class="external text" href="https://archive.org/details/physicseasyway00lehr_0/page/27"><i>Physics the Easy Way</i></a>. Barron's Educational Series. pp. <a rel="nofollow" class="external text" href="https://archive.org/details/physicseasyway00lehr_0/page/27">27</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-7641-0236-3</bdi>.</cite></span>

</li>

<li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><cite id="CITEREFCrew2008" class="citation book cs1">Crew, Henry (2008). <i>The Principles of Mechanics</i>. BiblioBazaar, LLC. p. 43. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-559-36871-4</bdi>.</cite></span>

</li>

<li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><cite id="CITEREFP._SmithR._C._Smith1991" class="citation book cs1">P. Smith; R. C. Smith (1991). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=Zzh_unG7OAsC"><i>Mechanics</i></a> (2nd, illustrated, reprinted ed.). John Wiley & Sons. p. 39. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-471-92737-2</bdi>.</cite> <a rel="nofollow" class="external text" href="https://books.google.com/books?id=Zzh_unG7OAsC&pg=PA39">Extract of page 39</a></span>

</li>

<li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><cite id="CITEREFJohn_D._CutnellKenneth_W._Johnson2014" class="citation book cs1">John D. Cutnell; Kenneth W. Johnson (2014). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=PJWDBgAAQBAJ"><i>Physics, Volume One: Chapters 1-17, Volume 1</i></a> (1st0, illustrated ed.). John Wiley & Sons. p. 36. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1-118-83688-0</bdi>.</cite> <a rel="nofollow" class="external text" href="https://books.google.com/books?id=PJWDBgAAQBAJ&pg=PA36">Extract of page 36</a></span>

</li>

<li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><cite id="CITEREFRaymond_A._SerwayChris_VuilleJerry_S._Faughn2008" class="citation book cs1">Raymond A. Serway; Chris Vuille; Jerry S. Faughn (2008). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=CX0u0mIOZ44C&pg=PA32"><i>College Physics, Volume 10</i></a>. Cengage. p. 32. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9780495386933</bdi>.</cite></span>

</li>

<li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"><cite id="CITEREFWeisstein" class="citation web cs1">Weisstein, Eric W. <a rel="nofollow" class="external text" href="http://mathworld.wolfram.com/ChainRule.html">"Chain Rule"</a>. <i>Wolfram MathWorld</i>. Wolfram Research<span class="reference-accessdate">. Retrieved <span class="nowrap">2 August</span> 2016</span>.</cite></span>

</li>

<li id="cite_note-Andrews-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-Andrews_8-0">^</a></b></span> <span class="reference-text"><cite id="CITEREFLarry_C._AndrewsRonald_L._Phillips2003" class="citation book cs1">Larry C. Andrews; Ronald L. Phillips (2003). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=MwrDfvrQyWYC&q=particle+%22planar+motion%22&pg=PA164"><i>Mathematical Techniques for Engineers and Scientists</i></a>. SPIE Press. p. 164. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-8194-4506-3</bdi>.</cite></span>

</li>

<li id="cite_note-Chand-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-Chand_9-0">^</a></b></span> <span class="reference-text"><cite id="CITEREFCh_V_Ramana_MurthyNC_Srinivas2001" class="citation book cs1">Ch V Ramana Murthy; NC Srinivas (2001). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=Q0Pvv4vWOlQC&pg=PA337"><i>Applied Mathematics</i></a>. New Delhi: S. Chand & Co. p. 337. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-81-219-2082-7</bdi>.</cite></span>

</li>

<li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text"><cite id="CITEREFKeith_Johnson2001" class="citation book cs1">Keith Johnson (2001). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=D4nrQDzq1jkC&q=suvat&pg=PA135"><i>Physics for you: revised national curriculum edition for GCSE</i></a> (4th ed.). Nelson Thornes. p. 135. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-7487-6236-1</bdi>.</cite></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"><cite id="CITEREFDavid_C._CassidyGerald_James_HoltonF._James_Rutherford2002" class="citation book cs1">David C. Cassidy; Gerald James Holton; F. James Rutherford (2002). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=iPsKvL_ATygC&q=parabolic+arc+uniform-acceleration+galileo&pg=PA146"><i>Understanding physics</i></a>. Birkhäuser. p. 146. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-387-98756-9</bdi>.</cite></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"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.feynmanlectures.caltech.edu/I_09.html">"The Feynman Lectures on Physics Vol. I Ch. 9: Newton's Laws of Dynamics"</a>. <i>www.feynmanlectures.caltech.edu</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2024-01-04</span></span>.</cite></span>

</li>

<li id="cite_note-Greene-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-Greene_13-0">^</a></b></span> <span class="reference-text"><cite id="CITEREFGreene2005" class="citation book cs1"><a href="Brian_Greene" title="Brian Greene">Greene, Brian</a> (8 February 2005). <a href="The_Fabric_of_the_Cosmos" title="The Fabric of the Cosmos"><i>The Fabric of the Cosmos: Space, Time, and the Texture of Reality</i></a>. Vintage. p. 67. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-375-72720-5</bdi>.</cite></span>

</li>

</ol></div></div>

<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>

/* start https://en.wikipedia.org/ */

.mw-parser-output .side-box{margin:4px 0;box-sizing:border-box;border:1px solid #aaa;font-size:88%;line-height:1.25em;background-color:var(--background-color-interactive-subtle,#f8f9fa);display:flow-root}.mw-parser-output .infobox .side-box{font-size:100%}.mw-parser-output .side-box-abovebelow,.mw-parser-output .side-box-text{padding:0.25em 0.9em}.mw-parser-output .side-box-image{padding:2px 0 2px 0.9em;text-align:center}.mw-parser-output .side-box-imageright{padding:2px 0.9em 2px 0;text-align:center}@media(min-width:500px){.mw-parser-output .side-box-flex{display:flex;align-items:center}.mw-parser-output .side-box-text{flex:1;min-width:0}}@media(min-width:720px){.mw-parser-output .side-box{width:238px}.mw-parser-output .side-box-right{clear:right;float:right;margin-left:1em}.mw-parser-output .side-box-left{margin-right:1em}}

/* end https://en.wikipedia.org/ */

</style><style data-mw-deduplicate="TemplateStyles:r1237033735">

/* start https://en.wikipedia.org/ */

@media print{body.ns-0 .mw-parser-output .sistersitebox{display:none!important}}@media screen{html.skin-theme-clientpref-night .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}

/* end https://en.wikipedia.org/ */

</style><div class="side-box side-box-right sistersitebox">

<div class="side-box-text plainlist">Wikimedia Commons has media related to <span style="font-weight: bold; font-style: italic;"><a href="https://commons.wikimedia.org/wiki/Category:Acceleration" class="extiw external" title="commons:Category:Acceleration">Acceleration</a></span>.</div></div>

</div>

<ul><li><a rel="nofollow" class="external text" href="http://www.unitjuggler.com/convert-acceleration-from-ms2-to-fts2.html">Acceleration Calculator</a> Simple acceleration unit converter</li></ul>

/* start https://en.wikipedia.org/ */

.mw-parser-output .navbox{box-sizing:border-box;border:1px solid #a2a9b1;width:100%;clear:both;font-size:88%;text-align:center;padding:1px;margin:1em auto 0}.mw-parser-output .navbox .navbox{margin-top:0}.mw-parser-output .navbox+.navbox,.mw-parser-output .navbox+.navbox-styles+.navbox{margin-top:-1px}.mw-parser-output .navbox-inner,.mw-parser-output .navbox-subgroup{width:100%}.mw-parser-output .navbox-group,.mw-parser-output .navbox-title,.mw-parser-output .navbox-abovebelow{padding:0.25em 1em;line-height:1.5em;text-align:center}.mw-parser-output .navbox-group{white-space:nowrap;text-align:right}.mw-parser-output .navbox,.mw-parser-output .navbox-subgroup{background-color:#fdfdfd}.mw-parser-output .navbox-list{line-height:1.5em;border-color:#fdfdfd}.mw-parser-output .navbox-list-with-group{text-align:left;border-left-width:2px;border-left-style:solid}.mw-parser-output tr+tr>.navbox-abovebelow,.mw-parser-output tr+tr>.navbox-group,.mw-parser-output tr+tr>.navbox-image,.mw-parser-output tr+tr>.navbox-list{border-top:2px solid #fdfdfd}.mw-parser-output .navbox-title{background-color:#ccf}.mw-parser-output .navbox-abovebelow,.mw-parser-output .navbox-group,.mw-parser-output .navbox-subgroup .navbox-title{background-color:#ddf}.mw-parser-output .navbox-subgroup .navbox-group,.mw-parser-output .navbox-subgroup .navbox-abovebelow{background-color:#e6e6ff}.mw-parser-output .navbox-even{background-color:#f7f7f7}.mw-parser-output .navbox-odd{background-color:transparent}.mw-parser-output .navbox .hlist td dl,.mw-parser-output .navbox .hlist td ol,.mw-parser-output .navbox .hlist td ul,.mw-parser-output .navbox td.hlist dl,.mw-parser-output .navbox td.hlist ol,.mw-parser-output .navbox td.hlist ul{padding:0.125em 0}.mw-parser-output .navbox .navbar{display:block;font-size:100%}.mw-parser-output .navbox-title .navbar{float:left;text-align:left;margin-right:0.5em}body.skin--responsive .mw-parser-output .navbox-image img{max-width:none!important}@media print{body.ns-0 .mw-parser-output .navbox{display:none!important}}

/* end https://en.wikipedia.org/ */

</style></div><div role="navigation" class="navbox" aria-labelledby="Kinematics14" style="padding:3px"><table class="nowraplinks mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Kinematics14" style="font-size:114%;margin:0 4em"><a href="Kinematics" title="Kinematics">Kinematics</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li>← <a href="Integral" title="Integral">Integrate</a> … <a href="Derivative" title="Derivative">Differentiate</a> →</li></ul>

</div></td></tr><tr><td colspan="2" class="navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Absement" title="Absement">Absement</a></li>

<li><a href="Displacement_(geometry)" title="Displacement (geometry)">Displacement</a> (<a href="Distance" title="Distance">Distance</a>)</li>

<li><a href="Velocity" title="Velocity">Velocity</a> (<a href="Speed" title="Speed">Speed</a>)</li>

<li><a href="Fourth%2C_fifth%2C_and_sixth_derivatives_of_position" title="Fourth, fifth, and sixth derivatives of position">Higher derivatives</a></li></ul>

</div></td></tr></tbody></table></div>

<div class="navbox-styles"></div><div role="navigation" class="navbox" aria-labelledby="Classical_mechanics_SI_units66" style="padding:3px"><table class="nowraplinks mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Classical_mechanics_SI_units66" style="font-size:114%;margin:0 4em"><a href="Classical_mechanics" title="Classical mechanics">Classical mechanics</a> <a href="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;">

<td colspan="4" style="border:none;backgound:none; font-weight:bold;">Linear/translational quantities</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;font-size:80%;">Dimensions</th>

</tr>

<tr>

<td><a href="Absement" title="Absement">absement</a>: <span class="texhtml"><b>A</b></span><br><a href="Meter_second" class="mw-redirect" title="Meter second">m s</a></td>

</tr>

<tr>

<td><a href="Distance" title="Distance">distance</a>: <span class="texhtml"><i>d</i></span>, <span class="nowrap"><a href="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="Displacement_(vector)" class="mw-redirect" title="Displacement (vector)">displacement</a><br><a href="Metre" title="Metre">m</a></td>

<td><a href="Angle" title="Angle">angle</a>: <span class="texhtml"><i>θ</i></span>, <a href="Angular_displacement" title="Angular displacement">angular displacement</a>: <span class="texhtml"><i><b>θ</b></i></span><br><a href="Radian" title="Radian">rad</a></td>

<td><span class="nowrap"><a href="Solid_angle" title="Solid angle">solid angle</a>: <span class="texhtml">Ω</span><br><a href="Steradian" title="Steradian">rad<sup>2</sup>, sr</a></span></td>

</tr>

<tr>

<td><span class="nowrap"><a href="Frequency" title="Frequency">frequency</a>: <span class="texhtml"><i>f</i></span></span><br><a href="Inverse_second" title="Inverse second">s<sup>−1</sup></a>, <a href="Hertz" title="Hertz">Hz</a></td>

<td><a href="Speed" title="Speed">speed</a>: <span class="texhtml"><i>v</i></span>, <a href="Velocity" title="Velocity">velocity</a>: <span class="texhtml"><b>v</b></span><br><a href="Metre_per_second" title="Metre per second">m s<sup>−1</sup></a></td>

<td><a href="Kinematic_viscosity" class="mw-redirect" title="Kinematic viscosity">kinematic viscosity</a>: <span class="texhtml"><i>ν</i></span>,<br><a href="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>

<td><span class="nowrap"><a href="Frequency" title="Frequency">frequency</a>: <span class="texhtml"><i>f</i></span></span>, <span class="nowrap"><a href="Rotational_speed" class="mw-redirect" title="Rotational speed">rotational speed</a>: <span class="texhtml"><i>n</i></span></span>, <span class="nowrap"><a href="Rotational_velocity" class="mw-redirect" title="Rotational velocity">rotational velocity</a>: <span class="texhtml"><i><b>n</b></i></span></span><br><a href="Inverse_second" title="Inverse second">s<sup>−1</sup></a>, <a href="Hertz" title="Hertz">Hz</a></td>

<td><a href="Angular_speed" class="mw-redirect" title="Angular speed">angular speed</a>: <span class="texhtml"><i>ω</i></span>, <a href="Angular_velocity" title="Angular velocity">angular velocity</a>: <span class="texhtml"><i><b>ω</b></i></span><br><a href="Radian_per_second" title="Radian per second">rad<span style="letter-spacing:0.1em"> </span>s<sup>−1</sup></a></td>

</tr>

<tr>

<td><span class="nowrap"><a href="Rotational_acceleration" class="mw-redirect" title="Rotational acceleration">rotational acceleration</a></span><br><a href="Inverse_square_second" class="mw-redirect" title="Inverse square second">s<sup>−2</sup></a></td>

<td><a href="Angular_acceleration" title="Angular acceleration">angular acceleration</a>: <span class="texhtml"><i><b>α</b></i></span><br><a href="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>

</tr>

<tr>

<td><a href="Jerk_(physics)#Jerk_in_rotation" title="Jerk (physics)">angular jerk</a>: <span class="texhtml"><b>ζ</b></span><br>rad<span style="letter-spacing:0.1em"> </span>s<sup>−3</sup></td>

</tr>

<td><a href="Moment_(physics)" title="Moment (physics)">weighted position</a>: <span class="texhtml"><i>M</i> ⟨<i>x</i>⟩ = ∑ <i>m</i> <i>x</i></span> </td>

<td><a href="Moment_of_inertia" title="Moment of inertia">moment of inertia</a>: <span class="texhtml"><i>I</i></span><br><a href="Kilogram_square_metre" class="mw-redirect" title="Kilogram square metre">kg<span style="letter-spacing:0.1em"> </span>m<sup>2</sup></a></td>

</tr>

<tr>

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\dot {m}}}</annotation>

</semantics>

</math></span><img src="./ad59b9876301e8fb75b9ddbf08de594b87251d3b.svg" 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}}}" loading="lazy"></span></span><br><a href="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="Momentum" title="Momentum">momentum</a>: <span class="texhtml"><b>p</b></span>, <a href="Impulse_(physics)" title="Impulse (physics)">impulse</a>: <span class="texhtml"><b>J</b></span><br><a href="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="Newton_second" class="mw-redirect" title="Newton second">N s</a></td>

<td><a href="Action_(physics)" title="Action (physics)">action</a>: <span class="texhtml">𝒮</span>, <a href="Absement#Applications" title="Absement">actergy</a>: <span class="texhtml">ℵ</span><br><a href="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="Joule-second" title="Joule-second">J s</a></td>

<td><a href="Angular_momentum" title="Angular momentum">angular momentum</a>: <span class="texhtml"><b>L</b></span>, <a href="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="Kilogram_square_metre_per_second" class="mw-redirect" title="Kilogram square metre per second">kg<span style="letter-spacing:0.1em"> </span>m rad s<sup>−1</sup></a></td>

</tr>

<tr>

<td><a href="Force" title="Force">force</a>: <span class="texhtml"><b>F</b></span>, <a href="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="Newton_(unit)" title="Newton (unit)">N</a></td>

<td><a href="Energy" title="Energy">energy</a>: <span class="texhtml"><i>E</i></span>, <a href="Work_(physics)" title="Work (physics)">work</a>: <span class="texhtml"><i>W</i></span>, <a href="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="Joule" title="Joule">J</a></td>

<td><a href="Torque" title="Torque">torque</a>: <span class="texhtml"><i><b>τ</b></i></span>, <a href="Torque#Terminology" title="Torque">moment</a>: <span class="texhtml"><b>M</b></span><br><span style="margin-right:0.1em;">kg</span> m rad s<sup>−2</sup>, <a href="Newton-metre" title="Newton-metre">N m</a></td>

</tr>

<tr>

<td><a href="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="Watt" title="Watt">W</a></td>

<td><a href="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 rad s<sup>−3</sup>, N m s<sup>−1</sup></td>

</tr>

</tbody></table></div></td></tr></tbody></table></div>

/* start https://en.wikipedia.org/ */

.mw-parser-output .tooltip-dotted{border-bottom:1px dotted;cursor:help}

/* end https://en.wikipedia.org/ */

</style></div><div role="navigation" class="navbox authority-control" aria-labelledby="Authority_control_databases_frameless&#124;text-top&#124;10px&#124;alt=Edit_this_at_Wikidata&#124;link=https&#58;//www.wikidata.org/wiki/Q11376#identifiers&#124;class=noprint&#124;Edit_this_at_Wikidata1434" style="padding:3px"><table class="nowraplinks hlist mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Authority_control_databases_frameless&#124;text-top&#124;10px&#124;alt=Edit_this_at_Wikidata&#124;link=https&#58;//www.wikidata.org/wiki/Q11376#identifiers&#124;class=noprint&#124;Edit_this_at_Wikidata1434" style="font-size:114%;margin:0 4em">Authority control databases </div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">National</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"><ul><li><span class="uid"><a rel="nofollow" class="external text" href="https://d-nb.info/gnd/4144870-4">Germany</a></span></li><li><span class="uid"><a rel="nofollow" class="external text" href="https://id.loc.gov/authorities/sh85000344">United States</a></span></li><li><span class="uid"><span class="rt-commentedText tooltip tooltip-dotted" title="Accélération (mécanique)"><a rel="nofollow" class="external text" href="https://catalogue.bnf.fr/ark:/12148/cb11978022j">France</a></span></span></li><li><span class="uid"><span class="rt-commentedText tooltip tooltip-dotted" title="Accélération (mécanique)"><a rel="nofollow" class="external text" href="https://data.bnf.fr/ark:/12148/cb11978022j">BnF data</a></span></span></li><li><span class="uid"><a rel="nofollow" class="external text" href="http://www.bncatalogo.cl/F?func=direct&local_base=red10&doc_number=000000074">Chile</a></span><ul><li><span class="uid"><a rel="nofollow" class="external text" href="http://www.bncatalogo.cl/F?func=direct&local_base=red10&doc_number=000000075">2</a></span></li></ul></li><li><span class="uid"><a rel="nofollow" class="external text" href="https://www.nli.org.il/en/authorities/987007292957505171">Israel</a></span></li></ul></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Other</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"><ul><li><span class="uid"><a rel="nofollow" class="external text" href="https://lux.collections.yale.edu/view/concept/0dff72f8-96c2-416a-ab09-a0cc400055e2">Yale LUX</a></span></li></ul></div></td></tr></tbody></table></div></div><div><div class="zim-footer">

This article is issued from <a class="external text" title="Last edited on 2025-04-24" href="https://en.wikipedia.org/wiki/?title=Acceleration&oldid=1287145346">Wikipedia</a>. The text is available under <a class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en">Creative Commons Attribution-Share Alike 4.0</a> unless otherwise noted. Additional terms may apply for the media files.

</div>

</div></div>

</div>

</main>

</div>

</body></html>