Momentum - Wikipedia

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class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Conservation"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.4</span> <span>Conservation</span> </div> </a> <ul id="toc-Conservation-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Dependence_on_reference_frame" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Dependence_on_reference_frame"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.5</span> <span>Dependence on reference frame</span> </div> </a> <ul id="toc-Dependence_on_reference_frame-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Application_to_collisions" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Application_to_collisions"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.6</span> <span>Application to collisions</span> </div> </a> <ul id="toc-Application_to_collisions-sublist" class="vector-toc-list"> <li id="toc-Elastic_collisions" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Elastic_collisions"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.6.1</span> <span>Elastic collisions</span> </div> </a> <ul id="toc-Elastic_collisions-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Inelastic_collisions" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Inelastic_collisions"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.6.2</span> <span>Inelastic collisions</span> </div> </a> <ul id="toc-Inelastic_collisions-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Multiple_dimensions" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Multiple_dimensions"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.7</span> <span>Multiple dimensions</span> </div> </a> <ul id="toc-Multiple_dimensions-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Objects_of_variable_mass" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Objects_of_variable_mass"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.8</span> <span>Objects of variable mass</span> </div> </a> <ul id="toc-Objects_of_variable_mass-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Generalized" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Generalized"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Generalized</span> </div> </a> <button aria-controls="toc-Generalized-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Generalized subsection</span> </button> <ul id="toc-Generalized-sublist" class="vector-toc-list"> <li id="toc-Lagrangian_mechanics" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Lagrangian_mechanics"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Lagrangian mechanics</span> </div> </a> <ul id="toc-Lagrangian_mechanics-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Hamiltonian_mechanics" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Hamiltonian_mechanics"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Hamiltonian mechanics</span> </div> </a> <ul id="toc-Hamiltonian_mechanics-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Symmetry_and_conservation" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Symmetry_and_conservation"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3</span> <span>Symmetry and conservation</span> </div> </a> <ul id="toc-Symmetry_and_conservation-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Momentum_density" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Momentum_density"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Momentum density</span> </div> </a> <button aria-controls="toc-Momentum_density-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Momentum density subsection</span> </button> <ul id="toc-Momentum_density-sublist" class="vector-toc-list"> <li id="toc-In_deformable_bodies_and_fluids" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#In_deformable_bodies_and_fluids"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>In deformable bodies and fluids</span> </div> </a> <ul id="toc-In_deformable_bodies_and_fluids-sublist" class="vector-toc-list"> <li id="toc-Conservation_in_a_continuum" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Conservation_in_a_continuum"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1.1</span> <span>Conservation in a continuum</span> </div> </a> <ul id="toc-Conservation_in_a_continuum-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Acoustic_waves" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Acoustic_waves"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1.2</span> <span>Acoustic waves</span> </div> </a> <ul id="toc-Acoustic_waves-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-In_electromagnetics" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#In_electromagnetics"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>In electromagnetics</span> </div> </a> <ul id="toc-In_electromagnetics-sublist" class="vector-toc-list"> <li id="toc-Particle_in_a_field" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Particle_in_a_field"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2.1</span> <span>Particle in a field</span> </div> </a> <ul id="toc-Particle_in_a_field-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Conservation_2" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Conservation_2"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2.2</span> <span>Conservation</span> </div> </a> <ul id="toc-Conservation_2-sublist" class="vector-toc-list"> <li id="toc-Vacuum" class="vector-toc-list-item vector-toc-level-4"> <a class="vector-toc-link" href="#Vacuum"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2.2.1</span> <span>Vacuum</span> </div> </a> <ul id="toc-Vacuum-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Media" class="vector-toc-list-item vector-toc-level-4"> <a class="vector-toc-link" href="#Media"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2.2.2</span> <span>Media</span> </div> </a> <ul id="toc-Media-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </li> </ul> </li> <li id="toc-Non-classical" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Non-classical"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Non-classical</span> </div> </a> <button aria-controls="toc-Non-classical-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Non-classical subsection</span> </button> <ul id="toc-Non-classical-sublist" class="vector-toc-list"> <li id="toc-Quantum_mechanical" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Quantum_mechanical"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>Quantum mechanical</span> </div> </a> <ul id="toc-Quantum_mechanical-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Relativistic" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Relativistic"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.2</span> <span>Relativistic</span> </div> </a> <ul id="toc-Relativistic-sublist" class="vector-toc-list"> <li id="toc-Lorentz_invariance" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Lorentz_invariance"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.2.1</span> <span>Lorentz invariance</span> </div> </a> <ul id="toc-Lorentz_invariance-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Four-vector_formulation" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Four-vector_formulation"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.2.2</span> <span>Four-vector formulation</span> </div> </a> <ul id="toc-Four-vector_formulation-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </li> <li id="toc-History_of_the_concept" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#History_of_the_concept"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>History of the concept</span> </div> </a> <button aria-controls="toc-History_of_the_concept-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle History of the concept subsection</span> </button> <ul id="toc-History_of_the_concept-sublist" class="vector-toc-list"> <li id="toc-Impetus" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Impetus"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.1</span> <span>Impetus</span> </div> </a> <ul id="toc-Impetus-sublist" class="vector-toc-list"> <li id="toc-John_Philoponus" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#John_Philoponus"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.1.1</span> <span>John Philoponus</span> </div> </a> <ul id="toc-John_Philoponus-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Ibn_Sīnā" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Ibn_Sīnā"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.1.2</span> <span>Ibn Sīnā</span> </div> </a> <ul id="toc-Ibn_Sīnā-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Peter_Olivi,_Jean_Buridan" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Peter_Olivi,_Jean_Buridan"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.1.3</span> <span>Peter Olivi, Jean Buridan</span> </div> </a> <ul id="toc-Peter_Olivi,_Jean_Buridan-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Quantity_of_motion" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Quantity_of_motion"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.2</span> <span>Quantity of motion</span> </div> </a> <ul id="toc-Quantity_of_motion-sublist" class="vector-toc-list"> <li id="toc-René_Descartes" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#René_Descartes"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.2.1</span> <span>René Descartes</span> </div> </a> <ul id="toc-René_Descartes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Christiaan_Huygens" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Christiaan_Huygens"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.2.2</span> <span>Christiaan Huygens</span> </div> </a> <ul id="toc-Christiaan_Huygens-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Momentum" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Momentum"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.3</span> <span>Momentum</span> </div> </a> <ul id="toc-Momentum-sublist" class="vector-toc-list"> <li id="toc-John_Wallis" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#John_Wallis"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.3.1</span> <span>John Wallis</span> </div> </a> <ul id="toc-John_Wallis-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Gottfried_Leibniz" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Gottfried_Leibniz"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.3.2</span> <span>Gottfried Leibniz</span> </div> </a> <ul id="toc-Gottfried_Leibniz-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Isaac_Newton" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Isaac_Newton"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.3.3</span> <span>Isaac Newton</span> </div> </a> <ul id="toc-Isaac_Newton-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-John_Jennings" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#John_Jennings"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.3.4</span> <span>John Jennings</span> </div> </a> <ul id="toc-John_Jennings-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Bibliography" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Bibliography"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Bibliography</span> </div> </a> <ul id="toc-Bibliography-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>External links</span> </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Toggle the table of contents</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Momentum</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. Available in 97 languages" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-97" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">97 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-af mw-list-item"><a href="https://af.wikipedia.org/wiki/Momentum" title="Momentum – Afrikaans" lang="af" hreflang="af" data-title="Momentum" data-language-autonym="Afrikaans" data-language-local-name="Afrikaans" class="interlanguage-link-target"><span>Afrikaans</span></a></li><li class="interlanguage-link interwiki-als mw-list-item"><a href="https://als.wikipedia.org/wiki/Impuls_(Physik)" title="Impuls (Physik) – Alemannic" lang="gsw" hreflang="gsw" data-title="Impuls (Physik)" data-language-autonym="Alemannisch" data-language-local-name="Alemannic" class="interlanguage-link-target"><span>Alemannisch</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%B2%D8%AE%D9%85_%D8%A7%D9%84%D8%AD%D8%B1%D9%83%D8%A9" title="زخم الحركة – Arabic" lang="ar" hreflang="ar" data-title="زخم الحركة" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-as mw-list-item"><a href="https://as.wikipedia.org/wiki/%E0%A6%AD%E0%A7%B0%E0%A6%AC%E0%A7%87%E0%A6%97" title="ভৰবেগ – Assamese" lang="as" hreflang="as" data-title="ভৰবেগ" data-language-autonym="অসমীয়া" data-language-local-name="Assamese" class="interlanguage-link-target"><span>অসমীয়া</span></a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/Cantid%C3%A1_de_movimientu" title="Cantidá de movimientu – Asturian" lang="ast" hreflang="ast" data-title="Cantidá de movimientu" data-language-autonym="Asturianu" data-language-local-name="Asturian" class="interlanguage-link-target"><span>Asturianu</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/%C4%B0mpuls" title="İmpuls – Azerbaijani" lang="az" hreflang="az" data-title="İmpuls" data-language-autonym="Azərbaycanca" data-language-local-name="Azerbaijani" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%AD%E0%A6%B0%E0%A6%AC%E0%A7%87%E0%A6%97" title="ভরবেগ – Bangla" lang="bn" hreflang="bn" data-title="ভরবেগ" data-language-autonym="বাংলা" data-language-local-name="Bangla" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-zh-min-nan mw-list-item"><a href="https://zh-min-nan.wikipedia.org/wiki/%C5%AAn-t%C5%8Dng-li%C5%8Dng" title="Ūn-tōng-liōng – Minnan" lang="nan" hreflang="nan" data-title="Ūn-tōng-liōng" data-language-autonym="閩南語 / Bân-lâm-gú" data-language-local-name="Minnan" class="interlanguage-link-target"><span>閩南語 / Bân-lâm-gú</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%86%D0%BC%D0%BF%D1%83%D0%BB%D1%8C%D1%81" title="Імпульс – Belarusian" lang="be" hreflang="be" data-title="Імпульс" data-language-autonym="Беларуская" data-language-local-name="Belarusian" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-be-x-old mw-list-item"><a href="https://be-tarask.wikipedia.org/wiki/%D0%86%D0%BC%D0%BF%D1%83%D0%BB%D1%8C%D1%81" title="Імпульс – Belarusian (Taraškievica orthography)" lang="be-tarask" hreflang="be-tarask" data-title="Імпульс" data-language-autonym="Беларуская (тарашкевіца)" data-language-local-name="Belarusian (Taraškievica orthography)" class="interlanguage-link-target"><span>Беларуская (тарашкевіца)</span></a></li><li class="interlanguage-link interwiki-bh mw-list-item"><a href="https://bh.wikipedia.org/wiki/%E0%A4%B5%E0%A5%87%E0%A4%97%E0%A4%AE%E0%A4%BE%E0%A4%A8" title="वेगमान – Bhojpuri" lang="bh" hreflang="bh" data-title="वेगमान" data-language-autonym="भोजपुरी" data-language-local-name="Bhojpuri" class="interlanguage-link-target"><span>भोजपुरी</span></a></li><li class="interlanguage-link interwiki-bcl mw-list-item"><a href="https://bcl.wikipedia.org/wiki/Momentum" title="Momentum – Central Bikol" lang="bcl" hreflang="bcl" data-title="Momentum" data-language-autonym="Bikol Central" data-language-local-name="Central Bikol" class="interlanguage-link-target"><span>Bikol Central</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%98%D0%BC%D0%BF%D1%83%D0%BB%D1%81_(%D0%BC%D0%B5%D1%85%D0%B0%D0%BD%D0%B8%D0%BA%D0%B0)" title="Импулс (механика) – Bulgarian" lang="bg" hreflang="bg" data-title="Импулс (механика)" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Koli%C4%8Dina_kretanja" title="Količina kretanja – Bosnian" lang="bs" hreflang="bs" data-title="Količina kretanja" data-language-autonym="Bosanski" data-language-local-name="Bosnian" class="interlanguage-link-target"><span>Bosanski</span></a></li><li class="interlanguage-link interwiki-bxr mw-list-item"><a href="https://bxr.wikipedia.org/wiki/%D0%98%D0%BC%D0%BF%D1%83%D0%BB%D1%8C%D1%81" title="Импульс – Russia Buriat" lang="bxr" hreflang="bxr" data-title="Импульс" data-language-autonym="Буряад" data-language-local-name="Russia Buriat" class="interlanguage-link-target"><span>Буряад</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Quantitat_de_moviment" title="Quantitat de moviment – Catalan" lang="ca" hreflang="ca" data-title="Quantitat de moviment" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%98%D0%BC%D0%BF%D1%83%D0%BB%D1%8C%D1%81" title="Импульс – Chuvash" lang="cv" hreflang="cv" data-title="Импульс" data-language-autonym="Чӑвашла" data-language-local-name="Chuvash" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Hybnost" title="Hybnost – Czech" lang="cs" hreflang="cs" data-title="Hybnost" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-sn mw-list-item"><a href="https://sn.wikipedia.org/wiki/Runhanhira" title="Runhanhira – Shona" lang="sn" hreflang="sn" data-title="Runhanhira" data-language-autonym="ChiShona" data-language-local-name="Shona" class="interlanguage-link-target"><span>ChiShona</span></a></li><li class="interlanguage-link interwiki-cy mw-list-item"><a href="https://cy.wikipedia.org/wiki/Momentwm" title="Momentwm – Welsh" lang="cy" hreflang="cy" data-title="Momentwm" data-language-autonym="Cymraeg" data-language-local-name="Welsh" class="interlanguage-link-target"><span>Cymraeg</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Impuls_(fysik)" title="Impuls (fysik) – Danish" lang="da" hreflang="da" data-title="Impuls (fysik)" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Impuls" title="Impuls – German" lang="de" hreflang="de" data-title="Impuls" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Impulss" title="Impulss – Estonian" lang="et" hreflang="et" data-title="Impulss" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%9F%CF%81%CE%BC%CE%AE" title="Ορμή – Greek" lang="el" hreflang="el" data-title="Ορμή" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Cantidad_de_movimiento" title="Cantidad de movimiento – Spanish" lang="es" hreflang="es" data-title="Cantidad de movimiento" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Movokvanto" title="Movokvanto – Esperanto" lang="eo" hreflang="eo" data-title="Movokvanto" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Momentu_lineal" title="Momentu lineal – Basque" lang="eu" hreflang="eu" data-title="Momentu lineal" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%AA%DA%A9%D8%A7%D9%86%D9%87" title="تکانه – Persian" lang="fa" hreflang="fa" data-title="تکانه" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Quantit%C3%A9_de_mouvement" title="Quantité de mouvement – French" lang="fr" hreflang="fr" data-title="Quantité de mouvement" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-fy mw-list-item"><a href="https://fy.wikipedia.org/wiki/Ympuls_(natuerkunde)" title="Ympuls (natuerkunde) – Western Frisian" lang="fy" hreflang="fy" data-title="Ympuls (natuerkunde)" data-language-autonym="Frysk" data-language-local-name="Western Frisian" class="interlanguage-link-target"><span>Frysk</span></a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/M%C3%B3iminteam" title="Móiminteam – Irish" lang="ga" hreflang="ga" data-title="Móiminteam" data-language-autonym="Gaeilge" data-language-local-name="Irish" class="interlanguage-link-target"><span>Gaeilge</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Cantidade_de_movemento" title="Cantidade de movemento – Galician" lang="gl" hreflang="gl" data-title="Cantidade de movemento" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-gu mw-list-item"><a href="https://gu.wikipedia.org/wiki/%E0%AA%B5%E0%AB%87%E0%AA%97%E0%AA%AE%E0%AA%BE%E0%AA%A8_%E0%AA%B8%E0%AA%82%E0%AA%B0%E0%AA%95%E0%AB%8D%E0%AA%B7%E0%AA%A3%E0%AA%A8%E0%AB%8B_%E0%AA%A8%E0%AA%BF%E0%AA%AF%E0%AA%AE" title="વેગમાન સંરક્ષણનો નિયમ – Gujarati" lang="gu" hreflang="gu" data-title="વેગમાન સંરક્ષણનો નિયમ" data-language-autonym="ગુજરાતી" data-language-local-name="Gujarati" class="interlanguage-link-target"><span>ગુજરાતી</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%9A%B4%EB%8F%99%EB%9F%89" title="운동량 – Korean" lang="ko" hreflang="ko" data-title="운동량" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D4%BB%D5%B4%D5%BA%D5%B8%D6%82%D5%AC%D5%BD_(%D5%B7%D5%A1%D6%80%D5%AA%D5%B4%D5%A1%D5%B6_%D6%84%D5%A1%D5%B6%D5%A1%D5%AF)" title="Իմպուլս (շարժման քանակ) – Armenian" lang="hy" hreflang="hy" data-title="Իմպուլս (շարժման քանակ)" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%B8%E0%A4%82%E0%A4%B5%E0%A5%87%E0%A4%97_(%E0%A4%AD%E0%A5%8C%E0%A4%A4%E0%A4%BF%E0%A4%95%E0%A5%80)" title="संवेग (भौतिकी) – Hindi" lang="hi" hreflang="hi" data-title="संवेग (भौतिकी)" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Koli%C4%8Dina_gibanja" title="Količina gibanja – Croatian" lang="hr" hreflang="hr" data-title="Količina gibanja" data-language-autonym="Hrvatski" data-language-local-name="Croatian" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Momentum" title="Momentum – Indonesian" lang="id" hreflang="id" data-title="Momentum" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/Skri%C3%B0%C3%BEungi" title="Skriðþungi – Icelandic" lang="is" hreflang="is" data-title="Skriðþungi" data-language-autonym="Íslenska" data-language-local-name="Icelandic" class="interlanguage-link-target"><span>Íslenska</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Quantit%C3%A0_di_moto" title="Quantità di moto – Italian" lang="it" hreflang="it" data-title="Quantità di moto" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%AA%D7%A0%D7%A2" title="תנע – Hebrew" lang="he" hreflang="he" data-title="תנע" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-jv mw-list-item"><a href="https://jv.wikipedia.org/wiki/Mom%C3%A8ntum" title="Momèntum – Javanese" lang="jv" hreflang="jv" data-title="Momèntum" data-language-autonym="Jawa" data-language-local-name="Javanese" class="interlanguage-link-target"><span>Jawa</span></a></li><li class="interlanguage-link interwiki-kn mw-list-item"><a href="https://kn.wikipedia.org/wiki/%E0%B2%86%E0%B2%B5%E0%B3%87%E0%B2%97_(%E0%B2%AD%E0%B3%8C%E0%B2%A4%E0%B2%B6%E0%B2%BE%E0%B2%B8%E0%B3%8D%E0%B2%A4%E0%B3%8D%E0%B2%B0)" title="ಆವೇಗ (ಭೌತಶಾಸ್ತ್ರ) – Kannada" lang="kn" hreflang="kn" data-title="ಆವೇಗ (ಭೌತಶಾಸ್ತ್ರ)" data-language-autonym="ಕನ್ನಡ" data-language-local-name="Kannada" class="interlanguage-link-target"><span>ಕನ್ನಡ</span></a></li><li class="interlanguage-link interwiki-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/%E1%83%98%E1%83%9B%E1%83%9E%E1%83%A3%E1%83%9A%E1%83%A1%E1%83%98" title="იმპულსი – Georgian" lang="ka" hreflang="ka" data-title="იმპულსი" data-language-autonym="ქართული" data-language-local-name="Georgian" class="interlanguage-link-target"><span>ქართული</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%94%D0%B5%D0%BD%D0%B5_%D0%B8%D0%BC%D0%BF%D1%83%D0%BB%D1%8C%D1%81%D1%96" title="Дене импульсі – Kazakh" lang="kk" hreflang="kk" data-title="Дене импульсі" data-language-autonym="Қазақша" data-language-local-name="Kazakh" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-ht mw-list-item"><a href="https://ht.wikipedia.org/wiki/Elan" title="Elan – Haitian Creole" lang="ht" hreflang="ht" data-title="Elan" data-language-autonym="Kreyòl ayisyen" data-language-local-name="Haitian Creole" class="interlanguage-link-target"><span>Kreyòl ayisyen</span></a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Momentum" title="Momentum – Latin" lang="la" hreflang="la" data-title="Momentum" data-language-autonym="Latina" data-language-local-name="Latin" class="interlanguage-link-target"><span>Latina</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Impulss" title="Impulss – Latvian" lang="lv" hreflang="lv" data-title="Impulss" data-language-autonym="Latviešu" data-language-local-name="Latvian" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Judesio_kiekis" title="Judesio kiekis – Lithuanian" lang="lt" hreflang="lt" data-title="Judesio kiekis" data-language-autonym="Lietuvių" data-language-local-name="Lithuanian" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Lend%C3%BClet" title="Lendület – Hungarian" lang="hu" hreflang="hu" data-title="Lendület" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%98%D0%BC%D0%BF%D1%83%D0%BB%D1%81_(%D0%BC%D0%B5%D1%85%D0%B0%D0%BD%D0%B8%D0%BA%D0%B0)" title="Импулс (механика) – Macedonian" lang="mk" hreflang="mk" data-title="Импулс (механика)" data-language-autonym="Македонски" data-language-local-name="Macedonian" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%86%E0%B4%95%E0%B5%8D%E0%B4%95%E0%B4%82" title="ആക്കം – Malayalam" lang="ml" hreflang="ml" data-title="ആക്കം" data-language-autonym="മലയാളം" data-language-local-name="Malayalam" class="interlanguage-link-target"><span>മലയാളം</span></a></li><li class="interlanguage-link interwiki-mr mw-list-item"><a href="https://mr.wikipedia.org/wiki/%E0%A4%B8%E0%A4%82%E0%A4%B5%E0%A5%87%E0%A4%97" title="संवेग – Marathi" lang="mr" hreflang="mr" data-title="संवेग" data-language-autonym="मराठी" data-language-local-name="Marathi" class="interlanguage-link-target"><span>मराठी</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Momentum" title="Momentum – Malay" lang="ms" hreflang="ms" data-title="Momentum" data-language-autonym="Bahasa Melayu" data-language-local-name="Malay" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-mn mw-list-item"><a href="https://mn.wikipedia.org/wiki/%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82" title="Момент – Mongolian" lang="mn" hreflang="mn" data-title="Момент" data-language-autonym="Монгол" data-language-local-name="Mongolian" class="interlanguage-link-target"><span>Монгол</span></a></li><li class="interlanguage-link interwiki-my mw-list-item"><a href="https://my.wikipedia.org/wiki/%E1%80%A1%E1%80%9F%E1%80%AF%E1%80%94%E1%80%BA" title="အဟုန် – Burmese" lang="my" hreflang="my" data-title="အဟုန်" data-language-autonym="မြန်မာဘာသာ" data-language-local-name="Burmese" class="interlanguage-link-target"><span>မြန်မာဘာသာ</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Impuls_(natuurkunde)" title="Impuls (natuurkunde) – Dutch" lang="nl" hreflang="nl" data-title="Impuls (natuurkunde)" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ne mw-list-item"><a href="https://ne.wikipedia.org/wiki/%E0%A4%AA%E0%A4%B0%E0%A4%BF%E0%A4%B5%E0%A5%87%E0%A4%97" title="परिवेग – Nepali" lang="ne" hreflang="ne" data-title="परिवेग" data-language-autonym="नेपाली" data-language-local-name="Nepali" class="interlanguage-link-target"><span>नेपाली</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E9%81%8B%E5%8B%95%E9%87%8F" title="運動量 – Japanese" lang="ja" hreflang="ja" data-title="運動量" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Bevegelsesmengde" title="Bevegelsesmengde – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Bevegelsesmengde" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/R%C3%B8rslemengd" title="Rørslemengd – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Rørslemengd" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegian Nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-oc mw-list-item"><a href="https://oc.wikipedia.org/wiki/Quantitat_de_movement" title="Quantitat de movement – Occitan" lang="oc" hreflang="oc" data-title="Quantitat de movement" data-language-autonym="Occitan" data-language-local-name="Occitan" class="interlanguage-link-target"><span>Occitan</span></a></li><li class="interlanguage-link interwiki-om mw-list-item"><a href="https://om.wikipedia.org/wiki/Furguga" title="Furguga – Oromo" lang="om" hreflang="om" data-title="Furguga" data-language-autonym="Oromoo" data-language-local-name="Oromo" class="interlanguage-link-target"><span>Oromoo</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Impuls" title="Impuls – Uzbek" lang="uz" hreflang="uz" data-title="Impuls" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="Uzbek" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></a></li><li class="interlanguage-link interwiki-pa mw-list-item"><a href="https://pa.wikipedia.org/wiki/%E0%A8%B8%E0%A9%B0%E0%A8%B5%E0%A9%87%E0%A8%97" title="ਸੰਵੇਗ – Punjabi" lang="pa" hreflang="pa" data-title="ਸੰਵੇਗ" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="Punjabi" class="interlanguage-link-target"><span>ਪੰਜਾਬੀ</span></a></li><li class="interlanguage-link interwiki-pnb mw-list-item"><a href="https://pnb.wikipedia.org/wiki/%D9%85%D9%88%D9%85%D9%86%D9%B9%D9%85" title="مومنٹم – Western Punjabi" lang="pnb" hreflang="pnb" data-title="مومنٹم" data-language-autonym="پنجابی" data-language-local-name="Western Punjabi" class="interlanguage-link-target"><span>پنجابی</span></a></li><li class="interlanguage-link interwiki-pms mw-list-item"><a href="https://pms.wikipedia.org/wiki/Quantit%C3%A0_%C3%ABd_moviment" title="Quantità ëd moviment – Piedmontese" lang="pms" hreflang="pms" data-title="Quantità ëd moviment" data-language-autonym="Piemontèis" data-language-local-name="Piedmontese" class="interlanguage-link-target"><span>Piemontèis</span></a></li><li class="interlanguage-link interwiki-nds mw-list-item"><a href="https://nds.wikipedia.org/wiki/Impuls_(Physik)" title="Impuls (Physik) – Low German" lang="nds" hreflang="nds" data-title="Impuls (Physik)" data-language-autonym="Plattdüütsch" data-language-local-name="Low German" class="interlanguage-link-target"><span>Plattdüütsch</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/P%C4%99d_(fizyka)" title="Pęd (fizyka) – Polish" lang="pl" hreflang="pl" data-title="Pęd (fizyka)" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Momento_linear" title="Momento linear – Portuguese" lang="pt" hreflang="pt" data-title="Momento linear" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Impuls" title="Impuls – Romanian" lang="ro" hreflang="ro" data-title="Impuls" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%98%D0%BC%D0%BF%D1%83%D0%BB%D1%8C%D1%81" title="Импульс – Russian" lang="ru" hreflang="ru" data-title="Импульс" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Vrulli" title="Vrulli – Albanian" lang="sq" hreflang="sq" data-title="Vrulli" data-language-autonym="Shqip" data-language-local-name="Albanian" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-si mw-list-item"><a href="https://si.wikipedia.org/wiki/%E0%B6%9C%E0%B6%B8%E0%B7%8A%E2%80%8D%E0%B6%BA%E0%B6%AD%E0%B7%8F%E0%B7%80%E0%B6%BA" title="ගම්‍යතාවය – Sinhala" lang="si" hreflang="si" data-title="ගම්‍යතාවය" data-language-autonym="සිංහල" data-language-local-name="Sinhala" class="interlanguage-link-target"><span>සිංහල</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Momentum" title="Momentum – Simple English" lang="en-simple" hreflang="en-simple" data-title="Momentum" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Hybnos%C5%A5" title="Hybnosť – Slovak" lang="sk" hreflang="sk" data-title="Hybnosť" data-language-autonym="Slovenčina" data-language-local-name="Slovak" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Gibalna_koli%C4%8Dina" title="Gibalna količina – Slovenian" lang="sl" hreflang="sl" data-title="Gibalna količina" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%DA%95%D8%A7%D9%88%DB%95%D8%B4" title="ڕاوەش – Central Kurdish" lang="ckb" hreflang="ckb" data-title="ڕاوەش" data-language-autonym="کوردی" data-language-local-name="Central Kurdish" class="interlanguage-link-target"><span>کوردی</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%98%D0%BC%D0%BF%D1%83%D0%BB%D1%81" title="Импулс – Serbian" lang="sr" hreflang="sr" data-title="Импулс" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Impuls" title="Impuls – Serbo-Croatian" lang="sh" hreflang="sh" data-title="Impuls" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="Serbo-Croatian" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-su mw-list-item"><a href="https://su.wikipedia.org/wiki/Mom%C3%A9ntum" title="Moméntum – Sundanese" lang="su" hreflang="su" data-title="Moméntum" data-language-autonym="Sunda" data-language-local-name="Sundanese" class="interlanguage-link-target"><span>Sunda</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Liikem%C3%A4%C3%A4r%C3%A4" title="Liikemäärä – Finnish" lang="fi" hreflang="fi" data-title="Liikemäärä" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/R%C3%B6relsem%C3%A4ngd" title="Rörelsemängd – Swedish" lang="sv" hreflang="sv" data-title="Rörelsemängd" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-tl mw-list-item"><a href="https://tl.wikipedia.org/wiki/Momentum" title="Momentum – Tagalog" lang="tl" hreflang="tl" data-title="Momentum" data-language-autonym="Tagalog" data-language-local-name="Tagalog" class="interlanguage-link-target"><span>Tagalog</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%89%E0%AE%A8%E0%AF%8D%E0%AE%A4%E0%AE%AE%E0%AF%8D" title="உந்தம் – Tamil" lang="ta" hreflang="ta" data-title="உந்தம்" data-language-autonym="தமிழ்" data-language-local-name="Tamil" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-tt mw-list-item"><a href="https://tt.wikipedia.org/wiki/%C4%B0mpuls" title="İmpuls – Tatar" lang="tt" hreflang="tt" data-title="İmpuls" data-language-autonym="Татарча / tatarça" data-language-local-name="Tatar" class="interlanguage-link-target"><span>Татарча / tatarça</span></a></li><li class="interlanguage-link interwiki-te mw-list-item"><a href="https://te.wikipedia.org/wiki/%E0%B0%A6%E0%B1%8D%E0%B0%B0%E0%B0%B5%E0%B1%8D%E0%B0%AF%E0%B0%B5%E0%B1%87%E0%B0%97%E0%B0%82" title="ద్రవ్యవేగం – Telugu" lang="te" hreflang="te" data-title="ద్రవ్యవేగం" data-language-autonym="తెలుగు" data-language-local-name="Telugu" class="interlanguage-link-target"><span>తెలుగు</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B9%82%E0%B8%A1%E0%B9%80%E0%B8%A1%E0%B8%99%E0%B8%95%E0%B8%B1%E0%B8%A1" title="โมเมนตัม – Thai" lang="th" hreflang="th" data-title="โมเมนตัม" data-language-autonym="ไทย" data-language-local-name="Thai" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Momentum" title="Momentum – Turkish" lang="tr" hreflang="tr" data-title="Momentum" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%86%D0%BC%D0%BF%D1%83%D0%BB%D1%8C%D1%81_(%D0%BC%D0%B5%D1%85%D0%B0%D0%BD%D1%96%D0%BA%D0%B0)" title="Імпульс (механіка) – Ukrainian" lang="uk" hreflang="uk" data-title="Імпульс (механіка)" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-ur mw-list-item"><a href="https://ur.wikipedia.org/wiki/%D9%85%D8%B9%DB%8C%D8%A7%D8%B1_%D8%AD%D8%B1%DA%A9%D8%AA" title="معیار حرکت – Urdu" lang="ur" hreflang="ur" data-title="معیار حرکت" data-language-autonym="اردو" data-language-local-name="Urdu" class="interlanguage-link-target"><span>اردو</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/%C4%90%E1%BB%99ng_l%C6%B0%E1%BB%A3ng" title="Động lượng – Vietnamese" lang="vi" hreflang="vi" data-title="Động lượng" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamese" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-zh-classical mw-list-item"><a href="https://zh-classical.wikipedia.org/wiki/%E5%8B%95%E9%87%8F" title="動量 – Literary Chinese" lang="lzh" hreflang="lzh" data-title="動量" data-language-autonym="文言" data-language-local-name="Literary Chinese" class="interlanguage-link-target"><span>文言</span></a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E5%8A%A8%E9%87%8F" title="动量 – Wu" lang="wuu" hreflang="wuu" data-title="动量" data-language-autonym="吴语" data-language-local-name="Wu" class="interlanguage-link-target"><span>吴语</span></a></li><li class="interlanguage-link interwiki-yi mw-list-item"><a href="https://yi.wikipedia.org/wiki/%D7%90%D7%99%D7%9E%D7%A4%D7%A2%D7%98" title="אימפעט – Yiddish" lang="yi" hreflang="yi" data-title="אימפעט" data-language-autonym="ייִדיש" data-language-local-name="Yiddish" class="interlanguage-link-target"><span>ייִדיש</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a 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navigation-not-searchable">This article is about linear momentum. Not to be confused with <a href="/wiki/Angular_momentum" title="Angular momentum">angular momentum</a> or <a href="/wiki/Moment_(physics)" title="Moment (physics)">moment (physics)</a>.</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">This article is about momentum in physics. For other uses, see <a href="/wiki/Momentum_(disambiguation)" class="mw-disambig" title="Momentum (disambiguation)">Momentum (disambiguation)</a>.</div> <p class="mw-empty-elt"> </p> <style data-mw-deduplicate="TemplateStyles:r1257001546">.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){background:#1f1f23!important;color:#f8f9fa}}@media(min-width:640px){body.skin--responsive .mw-parser-output .infobox-table{display:table!important}body.skin--responsive .mw-parser-output .infobox-table>caption{display:table-caption!important}body.skin--responsive .mw-parser-output .infobox-table>tbody{display:table-row-group}body.skin--responsive .mw-parser-output .infobox-table tr{display:table-row!important}body.skin--responsive .mw-parser-output .infobox-table th,body.skin--responsive .mw-parser-output .infobox-table td{padding-left:inherit;padding-right:inherit}}</style><table class="infobox"><tbody><tr><th colspan="2" class="infobox-above">Momentum</th></tr><tr><td colspan="2" class="infobox-image"><span class="mw-default-size" typeof="mw:File/Frameless"><a href="/wiki/File:Billard.JPG" class="mw-file-description" title="A pool break-off shot"><img alt="A pool break-off shot" src="//upload.wikimedia.org/wikipedia/commons/thumb/8/86/Billard.JPG/220px-Billard.JPG" decoding="async" width="220" height="147" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/86/Billard.JPG/330px-Billard.JPG 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/86/Billard.JPG/440px-Billard.JPG 2x" data-file-width="3456" data-file-height="2304" /></a></span><div class="infobox-caption">Momentum of a <a href="/wiki/Pool_(cue_sports)" title="Pool (cue sports)">pool</a> cue ball is transferred to the racked balls after collision.</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"><i>p</i>, <b>p</b></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/SI_unit" class="mw-redirect" title="SI unit">SI unit</a></th><td class="infobox-data">kg⋅m/s</td></tr><tr><th scope="row" class="infobox-label"><div style="display: inline-block; line-height: 1.2em; padding: .1em 0;">Other units</div></th><td class="infobox-data"><a href="/wiki/Slug_(unit)" title="Slug (unit)">slug</a>⋅<a href="/wiki/Foot_per_second" title="Foot per second">ft/s</a></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Conserved_quantity" title="Conserved quantity">Conserved</a>?</th><td class="infobox-data">Yes</td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Dimensional_analysis#Formulation" title="Dimensional analysis">Dimension</a></th><td class="infobox-data"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {M}}{\mathsf {L}}{\mathsf {T}}^{-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">M</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">L</mi> </mrow> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">T</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {M}}{\mathsf {L}}{\mathsf {T}}^{-1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b0634fa9865cdbe994c7bbdcd709cc9fb4ee5ab1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.209ex; height:2.676ex;" alt="{\displaystyle {\mathsf {M}}{\mathsf {L}}{\mathsf {T}}^{-1}}"></span></td></tr></tbody></table> <style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist 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 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style="padding-left:0.9em;padding-right:0.9em;"><a href="/wiki/Classical_mechanics" title="Classical mechanics">Classical mechanics</a></th></tr><tr><td class="sidebar-image"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\textbf {F}}={\frac {d\mathbf {p} }{dt}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext mathvariant="bold">F</mtext> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\textbf {F}}={\frac {d\mathbf {p} }{dt}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c2ad0a6d6780c3abc5247abd82bd8a2249d56ff3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:8.318ex; height:5.509ex;" alt="{\displaystyle {\textbf {F}}={\frac {d\mathbf {p} }{dt}}}"></span><div class="sidebar-caption" style="font-size:90%;padding:0.6em 0;font-style:italic;"><a href="/wiki/Second_law_of_motion" class="mw-redirect" title="Second law of motion">Second law of motion</a></div></td></tr><tr><th class="sidebar-heading" style="font-weight: bold; display:block;margin-bottom:1.0em;"> <div class="hlist"> <ul><li><a href="/wiki/History_of_classical_mechanics" title="History of classical mechanics">History</a></li> <li><a href="/wiki/Timeline_of_classical_mechanics" title="Timeline of classical mechanics">Timeline</a></li> <li><a href="/wiki/List_of_textbooks_on_classical_mechanics_and_quantum_mechanics" title="List of textbooks on classical mechanics and quantum mechanics">Textbooks</a></li></ul> </div></th></tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="border-bottom: 1px solid black;text-align:center;;color: var(--color-base)">Branches</div><div class="sidebar-list-content mw-collapsible-content plainlist" style="padding-top:0.35em;"><div class="hlist"> <ul><li><a href="/wiki/Applied_mechanics" title="Applied mechanics">Applied</a></li> <li><a href="/wiki/Celestial_mechanics" title="Celestial mechanics">Celestial</a></li> <li><a href="/wiki/Continuum_mechanics" title="Continuum mechanics">Continuum</a></li> <li><a href="/wiki/Analytical_dynamics" class="mw-redirect" title="Analytical dynamics">Dynamics</a></li> <li><a href="/wiki/Classical_field_theory" title="Classical field theory">Field theory</a></li> <li><a href="/wiki/Kinematics" title="Kinematics">Kinematics</a></li> <li><a href="/wiki/Kinetics_(physics)" title="Kinetics (physics)">Kinetics</a></li> <li><a href="/wiki/Statics" title="Statics">Statics</a></li> <li><a href="/wiki/Statistical_mechanics" title="Statistical mechanics">Statistical mechanics</a></li></ul> </div></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible"><div class="sidebar-list-title" style="border-bottom: 1px solid black;text-align:center;;color: var(--color-base)">Fundamentals</div><div class="sidebar-list-content mw-collapsible-content plainlist" style="padding-top:0.35em;"><div class="hlist"> <ul><li><a href="/wiki/Acceleration" title="Acceleration">Acceleration</a></li> <li><a href="/wiki/Angular_momentum" title="Angular momentum">Angular momentum</a></li> <li><a href="/wiki/Couple_(mechanics)" title="Couple (mechanics)">Couple</a></li> <li><a href="/wiki/D%27Alembert%27s_principle" title="D'Alembert's principle">D'Alembert's principle</a></li> <li><a href="/wiki/Energy" title="Energy">Energy</a> <ul><li><a href="/wiki/Kinetic_energy#Newtonian_kinetic_energy" title="Kinetic energy">kinetic</a></li> <li><a href="/wiki/Potential_energy" title="Potential energy">potential</a></li></ul></li> <li><a href="/wiki/Force" title="Force">Force</a></li> <li><a href="/wiki/Frame_of_reference" title="Frame of reference">Frame of reference</a></li> <li><a href="/wiki/Inertial_frame_of_reference" title="Inertial frame of reference">Inertial frame of reference</a></li> <li><a href="/wiki/Impulse_(physics)" title="Impulse (physics)">Impulse</a></li> <li><span class="nowrap"><a href="/wiki/Inertia" title="Inertia">Inertia</a> / <a href="/wiki/Moment_of_inertia" title="Moment of inertia">Moment of inertia</a></span></li> <li><a href="/wiki/Mass" title="Mass">Mass</a></li> <li><br /><a href="/wiki/Mechanical_power_(physics)" class="mw-redirect" title="Mechanical power (physics)">Mechanical power</a></li> <li><a href="/wiki/Work_(physics)" title="Work (physics)">Mechanical work</a></li> <li><br /><a href="/wiki/Moment_(physics)" title="Moment (physics)">Moment</a></li> <li><a class="mw-selflink selflink">Momentum</a></li> <li><a href="/wiki/Space" title="Space">Space</a></li> <li><a href="/wiki/Speed" title="Speed">Speed</a></li> <li><a href="/wiki/Time" title="Time">Time</a></li> <li><a href="/wiki/Torque" title="Torque">Torque</a></li> <li><a href="/wiki/Velocity" title="Velocity">Velocity</a></li> <li><a href="/wiki/Virtual_work" title="Virtual work">Virtual work</a></li></ul> </div></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="border-bottom: 1px solid black;text-align:center;;color: var(--color-base)">Formulations</div><div class="sidebar-list-content mw-collapsible-content plainlist" style="padding-top:0.35em;"> <ul><li><div style="display: inline-block; line-height: 1.2em; padding: .1em 0;"><b><a href="/wiki/Newton%27s_laws_of_motion" title="Newton's laws of motion">Newton's laws of motion</a></b></div></li> <li><div style="display: inline-block; line-height: 1.2em; padding: .1em 0;"><b><a href="/wiki/Analytical_mechanics" title="Analytical mechanics">Analytical mechanics</a></b> <div class="plainlist"><ul><li><a href="/wiki/Lagrangian_mechanics" title="Lagrangian mechanics">Lagrangian mechanics</a></li><li><a href="/wiki/Hamiltonian_mechanics" title="Hamiltonian mechanics">Hamiltonian mechanics</a></li><li><a href="/wiki/Routhian_mechanics" title="Routhian mechanics">Routhian mechanics</a></li><li><a href="/wiki/Hamilton%E2%80%93Jacobi_equation" title="Hamilton–Jacobi equation">Hamilton–Jacobi equation</a></li><li><a href="/wiki/Appell%27s_equation_of_motion" title="Appell's equation of motion">Appell's equation of motion</a></li><li><a href="/wiki/Koopman%E2%80%93von_Neumann_classical_mechanics" title="Koopman–von Neumann classical mechanics">Koopman–von Neumann mechanics</a></li></ul></div></div></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="border-bottom: 1px solid black;text-align:center;;color: var(--color-base)">Core topics</div><div class="sidebar-list-content mw-collapsible-content plainlist" style="padding-top:0.35em;"><div class="hlist"> <ul><li><a href="/wiki/Damping" title="Damping">Damping</a></li> <li><a href="/wiki/Displacement_(geometry)" title="Displacement (geometry)">Displacement</a></li> <li><a href="/wiki/Equations_of_motion" title="Equations of motion">Equations of motion</a></li> <li><a href="/wiki/Euler%27s_laws_of_motion" title="Euler's laws of motion"><span class="wrap">Euler's laws of motion</span></a></li> <li><a href="/wiki/Fictitious_force" title="Fictitious force">Fictitious force</a></li> <li><a href="/wiki/Friction" title="Friction">Friction</a></li> <li><a href="/wiki/Harmonic_oscillator" title="Harmonic oscillator">Harmonic oscillator</a></li></ul> </div> <ul><li><span class="nowrap"><a href="/wiki/Inertial_frame_of_reference" title="Inertial frame of reference">Inertial</a> / <a href="/wiki/Non-inertial_reference_frame" title="Non-inertial reference frame">Non-inertial reference frame</a></span></li></ul> <div class="hlist"> <ul><li><a href="/wiki/Motion" title="Motion">Motion</a> (<a href="/wiki/Linear_motion" title="Linear motion">linear</a>)</li> <li><a href="/wiki/Newton%27s_law_of_universal_gravitation" title="Newton's law of universal gravitation"><span class="wrap">Newton's law of universal gravitation</span></a></li> <li><a href="/wiki/Newton%27s_laws_of_motion" title="Newton's laws of motion">Newton's laws of motion</a></li> <li><a href="/wiki/Relative_velocity" title="Relative velocity">Relative velocity</a></li> <li><a href="/wiki/Rigid_body" title="Rigid body">Rigid body</a> <ul><li><a href="/wiki/Rigid_body_dynamics" title="Rigid body dynamics">dynamics</a></li> <li><a href="/wiki/Euler%27s_equations_(rigid_body_dynamics)" title="Euler's equations (rigid body dynamics)">Euler's equations</a></li></ul></li> <li><a href="/wiki/Simple_harmonic_motion" title="Simple harmonic motion">Simple harmonic motion</a></li> <li><a href="/wiki/Vibration" title="Vibration">Vibration</a></li></ul> </div></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="border-bottom: 1px solid black;text-align:center;;color: var(--color-base)"><a href="/wiki/Rotation_around_a_fixed_axis" title="Rotation around a fixed axis">Rotation</a></div><div class="sidebar-list-content mw-collapsible-content plainlist" style="padding-top:0.35em;"><div class="hlist"> <ul><li><a href="/wiki/Circular_motion" title="Circular motion">Circular motion</a></li> <li><a href="/wiki/Rotating_reference_frame" title="Rotating reference frame">Rotating reference frame</a></li> <li><a href="/wiki/Centripetal_force" title="Centripetal force">Centripetal force</a></li> <li><a href="/wiki/Centrifugal_force" title="Centrifugal force">Centrifugal force</a> <ul><li><a href="/wiki/Reactive_centrifugal_force" title="Reactive centrifugal force">reactive</a></li></ul></li> <li><a href="/wiki/Coriolis_force" title="Coriolis force">Coriolis force</a></li> <li><a href="/wiki/Pendulum_(mechanics)" title="Pendulum (mechanics)">Pendulum</a></li> <li><a href="/wiki/Tangential_speed" title="Tangential speed">Tangential speed</a></li> <li><a href="/wiki/Rotational_frequency" title="Rotational frequency">Rotational frequency</a></li></ul> </div> <ul><li><a href="/wiki/Angular_acceleration" title="Angular acceleration">Angular acceleration</a> / <a href="/wiki/Angular_displacement" title="Angular displacement">displacement</a> / <a href="/wiki/Angular_frequency" title="Angular frequency">frequency</a> / <a href="/wiki/Angular_velocity" title="Angular velocity">velocity</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list 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//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/31px-Symbol_category_class.svg.png 2x" data-file-width="180" data-file-height="185" /></span></span> <a href="/wiki/Category:Classical_mechanics" title="Category:Classical mechanics">Category</a></span></li></ul></td></tr><tr><td class="sidebar-navbar"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1239400231">.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}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Classical_mechanics" title="Template:Classical mechanics"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Classical_mechanics" title="Template talk:Classical mechanics"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Classical_mechanics" title="Special:EditPage/Template:Classical mechanics"><abbr title="Edit this template">e</abbr></a></li></ul></div></td></tr></tbody></table> <p>In <a href="/wiki/Newtonian_mechanics" class="mw-redirect" title="Newtonian mechanics">Newtonian mechanics</a>, <b>momentum</b> (<abbr title="plural form">pl.</abbr>: <b>momenta</b> or <b>momentums</b>; more specifically <b>linear momentum</b> or <b>translational momentum</b>) is the <a href="/wiki/Multiplication" title="Multiplication">product</a> of the <a href="/wiki/Mass" title="Mass">mass</a> and <a href="/wiki/Velocity" title="Velocity">velocity</a> of an object. It is a <a href="/wiki/Euclidean_vector" title="Euclidean vector">vector</a> quantity, possessing a magnitude and a direction. If <span class="texhtml"><i>m</i></span> is an object's mass and <span class="texhtml"><b>v</b></span> is its velocity (also a vector quantity), then the object's momentum <span class="texhtml"><b>p</b></span> (from Latin <i><a href="https://en.wiktionary.org/wiki/pello#Latin" class="extiw" title="wikt:pello">pellere</a></i> "push, drive") is: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {p} =m\mathbf {v} .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> <mo>=</mo> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {p} =m\mathbf {v} .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/21233e5c3bb1a4db8ed4a3ceb873f166a495c7f9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.682ex; height:2.009ex;" alt="{\displaystyle \mathbf {p} =m\mathbf {v} .}"></span> In the <a href="/wiki/International_System_of_Units" title="International System of Units">International System of Units</a> (SI), the <a href="/wiki/Unit_of_measurement" title="Unit of measurement">unit of measurement</a> of momentum is the <a href="/wiki/Kilogram" title="Kilogram">kilogram</a> <a href="/wiki/Metre_per_second" title="Metre per second">metre per second</a> (kg⋅m/s), which is <a href="/wiki/Dimensional_analysis" title="Dimensional analysis">dimensionally equivalent</a> to the <a href="/wiki/Newton-second" title="Newton-second">newton-second</a>. </p><p><a href="/wiki/Newton%27s_laws_of_motion#Newton's_second_law" title="Newton's laws of motion">Newton's second law of motion</a> states that the rate of change of a body's momentum is equal to the net force acting on it. Momentum depends on the <a href="/wiki/Frame_of_reference" title="Frame of reference">frame of reference</a>, but in any inertial frame it is a <i>conserved</i> quantity, meaning that if a <a href="/wiki/Closed_system" title="Closed system">closed system</a> is not affected by external forces, its total momentum does not change. Momentum is also conserved in <a href="/wiki/Special_relativity" title="Special relativity">special relativity</a> (with a modified formula) and, in a modified form, in <a href="/wiki/Electrodynamics" class="mw-redirect" title="Electrodynamics">electrodynamics</a>, <a href="/wiki/Quantum_mechanics" title="Quantum mechanics">quantum mechanics</a>, <a href="/wiki/Quantum_field_theory" title="Quantum field theory">quantum field theory</a>, and <a href="/wiki/General_relativity" title="General relativity">general relativity</a>. It is an expression of one of the fundamental symmetries of space and time: <a href="/wiki/Translational_symmetry" title="Translational symmetry">translational symmetry</a>. </p><p>Advanced formulations of classical mechanics, <a href="/wiki/Lagrangian_mechanics" title="Lagrangian mechanics">Lagrangian</a> and <a href="/wiki/Hamiltonian_mechanics" title="Hamiltonian mechanics">Hamiltonian mechanics</a>, allow one to choose coordinate systems that incorporate symmetries and constraints. In these systems the conserved quantity is <b>generalized momentum</b>, and in general this is different from the <b>kinetic</b> momentum defined above. The concept of generalized momentum is carried over into quantum mechanics, where it becomes an operator on a <a href="/wiki/Wave_function" title="Wave function">wave function</a>. The momentum and position operators are related by the <a href="/wiki/Heisenberg_uncertainty_principle" class="mw-redirect" title="Heisenberg uncertainty principle">Heisenberg uncertainty principle</a>. </p><p>In continuous systems such as <a href="/wiki/Electromagnetic_field" title="Electromagnetic field">electromagnetic fields</a>, <a href="/wiki/Fluid_dynamics" title="Fluid dynamics">fluid dynamics</a> and <a href="/wiki/Deformable_bodies" class="mw-redirect" title="Deformable bodies">deformable bodies</a>, a <b>momentum density</b> can be defined as momentum per volume (a <a href="/wiki/Volume-specific_quantity" class="mw-redirect" title="Volume-specific quantity">volume-specific quantity</a>). A <a href="/wiki/Continuum_(physics)" class="mw-redirect" title="Continuum (physics)">continuum</a> version of the conservation of momentum leads to equations such as the <a href="/wiki/Navier%E2%80%93Stokes_equations" title="Navier–Stokes equations">Navier–Stokes equations</a> for fluids or the <a href="/wiki/Cauchy_momentum_equation" title="Cauchy momentum equation">Cauchy momentum equation</a> for deformable solids or fluids. </p> <style data-mw-deduplicate="TemplateStyles:r886046785">.mw-parser-output .toclimit-2 .toclevel-1 ul,.mw-parser-output .toclimit-3 .toclevel-2 ul,.mw-parser-output .toclimit-4 .toclevel-3 ul,.mw-parser-output .toclimit-5 .toclevel-4 ul,.mw-parser-output .toclimit-6 .toclevel-5 ul,.mw-parser-output .toclimit-7 .toclevel-6 ul{display:none}</style><div class="toclimit-3"><meta property="mw:PageProp/toc" /></div> <div class="mw-heading mw-heading2"><h2 id="Classical">Classical</h2></div> <p>Momentum is a <a href="/wiki/Vector_quantity" title="Vector quantity">vector quantity</a>: it has both magnitude and direction. Since momentum has a direction, it can be used to predict the resulting direction and speed of motion of objects after they collide. Below, the basic properties of momentum are described in one dimension. The vector equations are almost identical to the scalar equations (see <a class="mw-selflink-fragment" href="#Multiple_dimensions">multiple dimensions</a>). </p> <div class="mw-heading mw-heading3"><h3 id="Single_particle">Single particle</h3></div> <p>The momentum of a particle is conventionally represented by the letter <span class="texhtml"><i>p</i></span>. It is the product of two quantities, the particle's <a href="/wiki/Mass" title="Mass">mass</a> (represented by the letter <span class="texhtml"><i>m</i></span>) and its <a href="/wiki/Velocity" title="Velocity">velocity</a> (<span class="texhtml"><i>v</i></span>):<sup id="cite_ref-FeynmanCh9_1-0" class="reference"><a href="#cite_note-FeynmanCh9-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> <span class="mwe-math-element" data-qid="Q41273"><a href="/w/index.php?title=Special:MathWikibase&qid=Q41273" style="color:inherit;"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p=mv.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>=</mo> <mi>m</mi> <mi>v</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p=mv.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd84eb5cc8d8ca0cefd6f93d8a00fbacb4da17f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:8.172ex; height:2.009ex;" alt="{\displaystyle p=mv.}"></a></span> </p><p>The unit of momentum is the product of the units of mass and velocity. In <a href="/wiki/SI_units" class="mw-redirect" title="SI units">SI units</a>, if the mass is in kilograms and the velocity is in meters per second then the momentum is in kilogram meters per second (kg⋅m/s). In <a href="/wiki/Centimetre%E2%80%93gram%E2%80%93second_system_of_units" title="Centimetre–gram–second system of units">cgs units</a>, if the mass is in grams and the velocity in centimeters per second, then the momentum is in gram centimeters per second (g⋅cm/s). </p><p>Being a vector, momentum has magnitude and direction. For example, a 1 kg model airplane, traveling due north at 1 m/s in straight and level flight, has a momentum of 1 kg⋅m/s due north measured with reference to the ground. </p> <div class="mw-heading mw-heading3"><h3 id="Many_particles">Many particles</h3></div> <p>The momentum of a system of particles is the vector sum of their momenta. If two particles have respective masses <span class="texhtml"><var style="padding-right: 1px;">m</var><sub>1</sub></span> and <span class="texhtml"><var style="padding-right: 1px;">m</var><sub>2</sub></span>, and velocities <span class="texhtml"><var style="padding-right: 1px;">v</var><sub>1</sub></span> and <span class="texhtml"><var style="padding-right: 1px;">v</var><sub>2</sub></span>, the total momentum is <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}p&=p_{1}+p_{2}\\&=m_{1}v_{1}+m_{2}v_{2}\,.\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>p</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mspace width="thinmathspace" /> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}p&=p_{1}+p_{2}\\&=m_{1}v_{1}+m_{2}v_{2}\,.\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fe17d5389e60187d1d65004d9512046244c37be" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:19.447ex; height:5.843ex;" alt="{\displaystyle {\begin{aligned}p&=p_{1}+p_{2}\\&=m_{1}v_{1}+m_{2}v_{2}\,.\end{aligned}}}"></span> The momenta of more than two particles can be added more generally with the following: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p=\sum _{i}m_{i}v_{i}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p=\sum _{i}m_{i}v_{i}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7bb4d4327b75bf0c95cec88e11852c64191d24da" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; margin-left: -0.089ex; width:13.514ex; height:5.509ex;" alt="{\displaystyle p=\sum _{i}m_{i}v_{i}.}"></span> </p><p>A system of particles has a <a href="/wiki/Center_of_mass" title="Center of mass">center of mass</a>, a point determined by the weighted sum of their positions: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{\text{cm}}={\frac {m_{1}r_{1}+m_{2}r_{2}+\cdots }{m_{1}+m_{2}+\cdots }}={\frac {\sum _{i}m_{i}r_{i}}{\sum _{i}m_{i}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>cm</mtext> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <mo>⋯</mo> </mrow> <mrow> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <mo>⋯</mo> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{\text{cm}}={\frac {m_{1}r_{1}+m_{2}r_{2}+\cdots }{m_{1}+m_{2}+\cdots }}={\frac {\sum _{i}m_{i}r_{i}}{\sum _{i}m_{i}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9e385b6c3a7d2d5318264932b6f2b9c7b2b5c215" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:39.024ex; height:6.509ex;" alt="{\displaystyle r_{\text{cm}}={\frac {m_{1}r_{1}+m_{2}r_{2}+\cdots }{m_{1}+m_{2}+\cdots }}={\frac {\sum _{i}m_{i}r_{i}}{\sum _{i}m_{i}}}.}"></span> </p><p>If one or more of the particles is moving, the center of mass of the system will generally be moving as well (unless the system is in pure rotation around it). If the total mass of the particles is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}"></span>, and the center of mass is moving at velocity <span class="texhtml"><var style="padding-right: 1px;">v</var><sub>cm</sub></span>, the momentum of the system is: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p=mv_{\text{cm}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>=</mo> <mi>m</mi> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>cm</mtext> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p=mv_{\text{cm}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/820e8ac543167e213ad98d35a031075ca69004c7" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:10.503ex; height:2.009ex;" alt="{\displaystyle p=mv_{\text{cm}}.}"></span> </p><p>This is known as <a href="/wiki/Euler%27s_laws_of_motion" title="Euler's laws of motion">Euler's first law</a>.<sup id="cite_ref-BookRags_2-0" class="reference"><a href="#cite_note-BookRags-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-McGillKing_3-0" class="reference"><a href="#cite_note-McGillKing-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Relation_to_force">Relation to force</h3></div> <p>If the net force <span class="texhtml mvar" style="font-style:italic;">F</span> applied to a particle is constant, and is applied for a time interval <span class="texhtml">Δ<var style="padding-right: 1px;">t</var></span>, the momentum of the particle changes by an amount <span class="mwe-math-element" data-qid="Q2397319"><a href="/w/index.php?title=Special:MathWikibase&qid=Q2397319" style="color:inherit;"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta p=F\Delta t\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mi>p</mi> <mo>=</mo> <mi>F</mi> <mi mathvariant="normal">Δ</mi> <mi>t</mi> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta p=F\Delta t\,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/96ed84409d35164bb09b2e3d994d04d3eccd4e98" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.754ex; height:2.509ex;" alt="{\displaystyle \Delta p=F\Delta t\,.}"></a></span> </p><p>In differential form, this is <a href="/wiki/Newton%27s_second_law" class="mw-redirect" title="Newton's second law">Newton's second law</a>; the rate of change of the momentum of a particle is equal to the instantaneous force <span class="texhtml mvar" style="font-style:italic;">F</span> acting on it,<sup id="cite_ref-FeynmanCh9_1-1" class="reference"><a href="#cite_note-FeynmanCh9-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F={\frac {{\text{d}}p}{{\text{d}}t}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>d</mtext> </mrow> <mi>p</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>d</mtext> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F={\frac {{\text{d}}p}{{\text{d}}t}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/29216db2c3868257343422b890cd3e53d29f526a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:8.784ex; height:5.509ex;" alt="{\displaystyle F={\frac {{\text{d}}p}{{\text{d}}t}}.}"></span> </p><p>If the net force experienced by a particle changes as a function of time, <span class="texhtml"><var style="padding-right: 1px;">F</var>(<var style="padding-right: 1px;">t</var>)</span>, the change in momentum (or <a href="/wiki/Impulse_(physics)" title="Impulse (physics)">impulse</a> <span class="texhtml mvar" style="font-style:italic;">J</span>) between times <span class="texhtml"><var style="padding-right: 1px;">t</var><sub>1</sub></span> and <span class="texhtml"><var style="padding-right: 1px;">t</var><sub>2</sub></span> is <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta p=J=\int _{t_{1}}^{t_{2}}F(t)\,{\text{d}}t\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mi>p</mi> <mo>=</mo> <mi>J</mi> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </msubsup> <mi>F</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>d</mtext> </mrow> <mi>t</mi> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta p=J=\int _{t_{1}}^{t_{2}}F(t)\,{\text{d}}t\,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/07a6e49c02b9c9c03218ff822a7378f8ce79a54e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:23.225ex; height:6.509ex;" alt="{\displaystyle \Delta p=J=\int _{t_{1}}^{t_{2}}F(t)\,{\text{d}}t\,.}"></span> </p><p>Impulse is measured in the <a href="/wiki/SI_derived_unit" title="SI derived unit">derived units</a> of the <a href="/wiki/Newton_second" class="mw-redirect" title="Newton second">newton second</a> (1 N⋅s = 1 kg⋅m/s) or <a href="/wiki/Dyne" title="Dyne">dyne</a> second (1 dyne⋅s = 1 g⋅cm/s) </p><p>Under the assumption of constant mass <span class="texhtml mvar" style="font-style:italic;">m</span>, it is equivalent to write </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F={\frac {{\text{d}}(mv)}{{\text{d}}t}}=m{\frac {{\text{d}}v}{{\text{d}}t}}=ma,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>d</mtext> </mrow> <mo stretchy="false">(</mo> <mi>m</mi> <mi>v</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>d</mtext> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>d</mtext> </mrow> <mi>v</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>d</mtext> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mi>m</mi> <mi>a</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F={\frac {{\text{d}}(mv)}{{\text{d}}t}}=m{\frac {{\text{d}}v}{{\text{d}}t}}=ma,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7265791c5837fec0dbe14d5184af30389ca79b01" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:27.356ex; height:5.843ex;" alt="{\displaystyle F={\frac {{\text{d}}(mv)}{{\text{d}}t}}=m{\frac {{\text{d}}v}{{\text{d}}t}}=ma,}"></span> </p><p>hence the net force is equal to the mass of the particle times its <a href="/wiki/Acceleration" title="Acceleration">acceleration</a>.<sup id="cite_ref-FeynmanCh9_1-2" class="reference"><a href="#cite_note-FeynmanCh9-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p><p><i>Example</i>: A model airplane of mass 1 kg accelerates from rest to a velocity of 6 m/s due north in 2 s. The net force required to produce this acceleration is 3 <a href="/wiki/Newton_(unit)" title="Newton (unit)">newtons</a> due north. The change in momentum is 6 kg⋅m/s due north. The rate of change of momentum is 3 (kg⋅m/s)/s due north which is numerically equivalent to 3 newtons. </p> <div class="mw-heading mw-heading3"><h3 id="Conservation">Conservation</h3></div> <p>In a <a href="/wiki/Closed_system" title="Closed system">closed system</a> (one that does not exchange any matter with its surroundings and is not acted on by external forces) the total momentum remains constant. This fact, known as the <b>law of conservation of momentum</b>, is implied by <a href="/wiki/Newton%27s_laws_of_motion" title="Newton's laws of motion">Newton's laws of motion</a>.<sup id="cite_ref-FeynmanCh10_4-0" class="reference"><a href="#cite_note-FeynmanCh10-4"><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> Suppose, for example, that two particles interact. As explained by the third law, the forces between them are equal in magnitude but opposite in direction. If the particles are numbered 1 and 2, the second law states that <span class="texhtml"><var style="padding-right: 1px;">F</var><sub>1</sub> = <style data-mw-deduplicate="TemplateStyles:r1214402035">.mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}</style><span class="sfrac">⁠<span class="tion"><span class="num">d<var style="padding-right: 1px;">p</var><sub>1</sub></span><span class="sr-only">/</span><span class="den">d<var style="padding-right: 1px;">t</var></span></span>⁠</span></span> and <span class="texhtml"><var style="padding-right: 1px;">F</var><sub>2</sub> = <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">d<var style="padding-right: 1px;">p</var><sub>2</sub></span><span class="sr-only">/</span><span class="den">d<var style="padding-right: 1px;">t</var></span></span>⁠</span></span>. Therefore, </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {{\text{d}}p_{1}}{{\text{d}}t}}=-{\frac {{\text{d}}p_{2}}{{\text{d}}t}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>d</mtext> </mrow> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>d</mtext> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>d</mtext> </mrow> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>d</mtext> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {{\text{d}}p_{1}}{{\text{d}}t}}=-{\frac {{\text{d}}p_{2}}{{\text{d}}t}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c6b498a548d127e73e01cb37428f2a38c2c2b636" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:14.258ex; height:5.509ex;" alt="{\displaystyle {\frac {{\text{d}}p_{1}}{{\text{d}}t}}=-{\frac {{\text{d}}p_{2}}{{\text{d}}t}},}"></span> with the negative sign indicating that the forces oppose. Equivalently, </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\text{d}}{{\text{d}}t}}\left(p_{1}+p_{2}\right)=0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mtext>d</mtext> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>d</mtext> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\text{d}}{{\text{d}}t}}\left(p_{1}+p_{2}\right)=0.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a867b9dc0465dca892fc5b85171326187f0851a2" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:17.36ex; height:5.509ex;" alt="{\displaystyle {\frac {\text{d}}{{\text{d}}t}}\left(p_{1}+p_{2}\right)=0.}"></span> </p><p>If the velocities of the particles are <span class="texhtml"><var style="padding-right: 1px;">v</var><sub>A1</sub></span> and <span class="texhtml"><var style="padding-right: 1px;">v</var><sub>B1</sub></span> before the interaction, and afterwards they are <span class="texhtml"><var style="padding-right: 1px;">v</var><sub>A2</sub></span> and <span class="texhtml"><var style="padding-right: 1px;">v</var><sub>B2</sub></span>, then </p><p><span class="mwe-math-element" data-qid="Q2305665"><a href="/w/index.php?title=Special:MathWikibase&qid=Q2305665" style="color:inherit;"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m_{A}v_{A1}+m_{B}v_{B1}=m_{A}v_{A2}+m_{B}v_{B2}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> <mn>1</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> <mn>2</mn> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m_{A}v_{A1}+m_{B}v_{B1}=m_{A}v_{A2}+m_{B}v_{B2}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d6e18b5524cbf54c82a2f40cdfd9ab8a9af12bdc" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:37.164ex; height:2.343ex;" alt="{\displaystyle m_{A}v_{A1}+m_{B}v_{B1}=m_{A}v_{A2}+m_{B}v_{B2}.}"></a></span> </p><p>This law holds no matter how complicated the force is between particles. Similarly, if there are several particles, the momentum exchanged between each pair of particles adds to zero, so the total change in momentum is zero. The conservation of the total momentum of a number of interacting particles can be expressed as <sup id="cite_ref-FeynmanCh10_4-1" class="reference"><a href="#cite_note-FeynmanCh10-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m_{A}v_{A}+m_{B}v_{B}+m_{C}v_{C}+...=constant.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> </msub> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> </msub> <mo>+</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>=</mo> <mi>c</mi> <mi>o</mi> <mi>n</mi> <mi>s</mi> <mi>t</mi> <mi>a</mi> <mi>n</mi> <mi>t</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m_{A}v_{A}+m_{B}v_{B}+m_{C}v_{C}+...=constant.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/088011a3698be3ca05479ffb4d85412fbd9df153" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:40.97ex; height:2.343ex;" alt="{\displaystyle m_{A}v_{A}+m_{B}v_{B}+m_{C}v_{C}+...=constant.}"></span> </p><p>This conservation law applies to all interactions, including <a href="/wiki/Collision" title="Collision">collisions</a> (both <a href="/wiki/Elastic_collision" title="Elastic collision">elastic</a> and <a href="/wiki/Inelastic_collision" title="Inelastic collision">inelastic</a>) and separations caused by explosive forces.<sup id="cite_ref-FeynmanCh10_4-2" class="reference"><a href="#cite_note-FeynmanCh10-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> It can also be generalized to situations where Newton's laws do not hold, for example in the <a href="/wiki/Theory_of_relativity" title="Theory of relativity">theory of relativity</a> and in <a href="/wiki/Classical_electromagnetism" title="Classical electromagnetism">electrodynamics</a>.<sup id="cite_ref-Goldstein54_6-0" class="reference"><a href="#cite_note-Goldstein54-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Dependence_on_reference_frame">Dependence on reference frame</h3></div> <p>Momentum is a measurable quantity, and the measurement depends on the <a href="/wiki/Frame_of_reference" title="Frame of reference">frame of reference</a>. For example: if an aircraft of mass 1000 kg is flying through the air at a speed of 50 m/s its momentum can be calculated to be 50,000 kg.m/s. If the aircraft is flying into a headwind of 5 m/s its speed relative to the surface of the Earth is only 45 m/s and its momentum can be calculated to be 45,000 kg.m/s. Both calculations are equally correct. In both frames of reference, any change in momentum will be found to be consistent with the relevant laws of physics. </p><p>Suppose <span class="texhtml mvar" style="font-style:italic;">x</span> is a position in an inertial frame of reference. From the point of view of another frame of reference, moving at a constant speed <span class="texhtml mvar" style="font-style:italic;">u</span> relative to the other, the position (represented by a primed coordinate) changes with time as </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x'=x-ut\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mo>′</mo> </msup> <mo>=</mo> <mi>x</mi> <mo>−</mo> <mi>u</mi> <mi>t</mi> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x'=x-ut\,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/05b7bb71b62bcd46e6e5e85be6c2624e0ad3bf27" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:12.486ex; height:2.676ex;" alt="{\displaystyle x'=x-ut\,.}"></span> </p><p>This is called a <a href="/wiki/Galilean_transformation" title="Galilean transformation">Galilean transformation</a>. </p><p>If a particle is moving at speed <span class="texhtml"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">d<var style="padding-right: 1px;">x</var></span><span class="sr-only">/</span><span class="den">d<var style="padding-right: 1px;">t</var></span></span>⁠</span> = <var style="padding-right: 1px;">v</var></span> in the first frame of reference, in the second, it is moving at speed </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v'={\frac {{\text{d}}x'}{{\text{d}}t}}=v-u\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>v</mi> <mo>′</mo> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>d</mtext> </mrow> <msup> <mi>x</mi> <mo>′</mo> </msup> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>d</mtext> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mi>v</mi> <mo>−</mo> <mi>u</mi> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v'={\frac {{\text{d}}x'}{{\text{d}}t}}=v-u\,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ecbb23d542254786df69a4a380385a23ab7db0d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:18.484ex; height:5.676ex;" alt="{\displaystyle v'={\frac {{\text{d}}x'}{{\text{d}}t}}=v-u\,.}"></span> </p><p>Since <span class="texhtml mvar" style="font-style:italic;">u</span> does not change, the second reference frame is also an inertial frame and the accelerations are the same: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a'={\frac {{\text{d}}v'}{{\text{d}}t}}=a\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>a</mi> <mo>′</mo> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>d</mtext> </mrow> <msup> <mi>v</mi> <mo>′</mo> </msup> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>d</mtext> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mi>a</mi> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a'={\frac {{\text{d}}v'}{{\text{d}}t}}=a\,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32794263e1cd078fca081a5e366c1a4ce9026128" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:14.316ex; height:5.676ex;" alt="{\displaystyle a'={\frac {{\text{d}}v'}{{\text{d}}t}}=a\,.}"></span> </p><p>Thus, momentum is conserved in both reference frames. Moreover, as long as the force has the same form, in both frames, Newton's second law is unchanged. Forces such as Newtonian gravity, which depend only on the scalar distance between objects, satisfy this criterion. This independence of reference frame is called Newtonian relativity or <a href="/wiki/Galilean_invariance" title="Galilean invariance">Galilean invariance</a>.<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> </p><p>A change of reference frame, can, often, simplify calculations of motion. For example, in a collision of two particles, a reference frame can be chosen, where, one particle begins at rest. Another, commonly used reference frame, is the <a href="/wiki/Center_of_mass_frame" class="mw-redirect" title="Center of mass frame">center of mass frame</a> – one that is moving with the center of mass. In this frame, the total momentum is zero. </p> <div class="mw-heading mw-heading3"><h3 id="Application_to_collisions">Application to collisions</h3></div> <p>If two particles, each of known momentum, collide and coalesce, the law of conservation of momentum can be used to determine the momentum of the coalesced body. If the outcome of the collision is that the two particles separate, the law is not sufficient to determine the momentum of each particle. If the momentum of one particle after the collision is known, the law can be used to determine the momentum of the other particle. Alternatively if the combined <a href="/wiki/Kinetic_energy" title="Kinetic energy">kinetic energy</a> after the collision is known, the law can be used to determine the momentum of each particle after the collision.<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> Kinetic energy is usually not conserved. If it is conserved, the collision is called an <i><a href="/wiki/Elastic_collision" title="Elastic collision">elastic collision</a></i>; if not, it is an <i><a href="/wiki/Inelastic_collision" title="Inelastic collision">inelastic collision</a></i>. </p> <div class="mw-heading mw-heading4"><h4 id="Elastic_collisions">Elastic collisions</h4></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Elastic_collision" title="Elastic collision">Elastic collision</a></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Elastischer_sto%C3%9F.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c6/Elastischer_sto%C3%9F.gif/220px-Elastischer_sto%C3%9F.gif" decoding="async" width="220" height="26" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c6/Elastischer_sto%C3%9F.gif/330px-Elastischer_sto%C3%9F.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c6/Elastischer_sto%C3%9F.gif/440px-Elastischer_sto%C3%9F.gif 2x" data-file-width="500" data-file-height="60" /></a><figcaption>Elastic collision of equal masses</figcaption></figure> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Elastischer_sto%C3%9F3.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e5/Elastischer_sto%C3%9F3.gif/220px-Elastischer_sto%C3%9F3.gif" decoding="async" width="220" height="28" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e5/Elastischer_sto%C3%9F3.gif/330px-Elastischer_sto%C3%9F3.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e5/Elastischer_sto%C3%9F3.gif/440px-Elastischer_sto%C3%9F3.gif 2x" data-file-width="500" data-file-height="63" /></a><figcaption>Elastic collision of unequal masses</figcaption></figure> <p>An elastic collision is one in which no <a href="/wiki/Kinetic_energy" title="Kinetic energy">kinetic energy</a> is transformed into heat or some other form of energy. Perfectly elastic collisions can occur when the objects do not touch each other, as for example in atomic or nuclear scattering where electric repulsion keeps the objects apart. A <a href="/wiki/Gravity_assist" title="Gravity assist">slingshot maneuver</a> of a satellite around a planet can also be viewed as a perfectly elastic collision. A collision between two <a href="/wiki/Pool_billiards" class="mw-redirect" title="Pool billiards">pool</a> balls is a good example of an <i>almost</i> totally elastic collision, due to their high <a href="/wiki/Stiffness" title="Stiffness">rigidity</a>, but when bodies come in contact there is always some <a href="/wiki/Dissipation" title="Dissipation">dissipation</a>.<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> </p><p>A head-on elastic collision between two bodies can be represented by velocities in one dimension, along a line passing through the bodies. If the velocities are <span class="texhtml"><var style="padding-right: 1px;">v</var><sub>A1</sub></span> and <span class="texhtml"><var style="padding-right: 1px;">v</var><sub>B1</sub></span> before the collision and <span class="texhtml"><var style="padding-right: 1px;">v</var><sub>A2</sub></span> and <span class="texhtml"><var style="padding-right: 1px;">v</var><sub>B2</sub></span> after, the equations expressing conservation of momentum and kinetic energy are: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}m_{A}v_{A1}+m_{B}v_{B1}&=m_{A}v_{A2}+m_{B}v_{B2}\\{\tfrac {1}{2}}m_{A}v_{A1}^{2}+{\tfrac {1}{2}}m_{B}v_{B1}^{2}&={\tfrac {1}{2}}m_{A}v_{A2}^{2}+{\tfrac {1}{2}}m_{B}v_{B2}^{2}\,.\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> <mn>2</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <msubsup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <msubsup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <msubsup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <msubsup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mspace width="thinmathspace" /> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}m_{A}v_{A1}+m_{B}v_{B1}&=m_{A}v_{A2}+m_{B}v_{B2}\\{\tfrac {1}{2}}m_{A}v_{A1}^{2}+{\tfrac {1}{2}}m_{B}v_{B1}^{2}&={\tfrac {1}{2}}m_{A}v_{A2}^{2}+{\tfrac {1}{2}}m_{B}v_{B2}^{2}\,.\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d39be080c25e5354eeca0a90d15fb34897ef03e2" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:44.935ex; height:6.509ex;" alt="{\displaystyle {\begin{aligned}m_{A}v_{A1}+m_{B}v_{B1}&=m_{A}v_{A2}+m_{B}v_{B2}\\{\tfrac {1}{2}}m_{A}v_{A1}^{2}+{\tfrac {1}{2}}m_{B}v_{B1}^{2}&={\tfrac {1}{2}}m_{A}v_{A2}^{2}+{\tfrac {1}{2}}m_{B}v_{B2}^{2}\,.\end{aligned}}}"></span> </p><p>A change of reference frame can simplify analysis of a collision. For example, suppose there are two bodies of equal mass <span class="texhtml mvar" style="font-style:italic;">m</span>, one stationary and one approaching the other at a speed <span class="texhtml mvar" style="font-style:italic;">v</span> (as in the figure). The center of mass is moving at speed <span class="texhtml"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num"><var style="padding-right: 1px;">v</var></span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span></span> and both bodies are moving towards it at speed <span class="texhtml"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num"><var style="padding-right: 1px;">v</var></span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span></span>. Because of the symmetry, after the collision both must be moving away from the center of mass at the same speed. Adding the speed of the center of mass to both, we find that the body that was moving is now stopped and the other is moving away at speed <span class="texhtml mvar" style="font-style:italic;">v</span>. The bodies have exchanged their velocities. Regardless of the velocities of the bodies, a switch to the center of mass frame leads us to the same conclusion. Therefore, the final velocities are given by<sup id="cite_ref-FeynmanCh10_4-3" class="reference"><a href="#cite_note-FeynmanCh10-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}v_{A2}&=v_{B1}\\v_{B2}&=v_{A1}\,.\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> <mn>1</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> <mn>1</mn> </mrow> </msub> <mspace width="thinmathspace" /> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}v_{A2}&=v_{B1}\\v_{B2}&=v_{A1}\,.\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/242d3b584b063c7b9438da956be0e06a893df288" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:11.727ex; height:5.843ex;" alt="{\displaystyle {\begin{aligned}v_{A2}&=v_{B1}\\v_{B2}&=v_{A1}\,.\end{aligned}}}"></span> </p><p>In general, when the initial velocities are known, the final velocities are given by<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}v_{A2}&=\left({\frac {m_{A}-m_{B}}{m_{A}+m_{B}}}\right)v_{A1}+\left({\frac {2m_{B}}{m_{A}+m_{B}}}\right)v_{B1}\\v_{B2}&=\left({\frac {m_{B}-m_{A}}{m_{A}+m_{B}}}\right)v_{B1}+\left({\frac {2m_{A}}{m_{A}+m_{B}}}\right)v_{A1}\,.\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> </mrow> <mrow> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> </mrow> <mrow> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> <mn>1</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> </mrow> <mrow> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> </mrow> <mrow> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> <mn>1</mn> </mrow> </msub> <mspace width="thinmathspace" /> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}v_{A2}&=\left({\frac {m_{A}-m_{B}}{m_{A}+m_{B}}}\right)v_{A1}+\left({\frac {2m_{B}}{m_{A}+m_{B}}}\right)v_{B1}\\v_{B2}&=\left({\frac {m_{B}-m_{A}}{m_{A}+m_{B}}}\right)v_{B1}+\left({\frac {2m_{A}}{m_{A}+m_{B}}}\right)v_{A1}\,.\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2da536fb0316fb5a6608a4d8fb6466a94593356b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.671ex; width:47.017ex; height:12.509ex;" alt="{\displaystyle {\begin{aligned}v_{A2}&=\left({\frac {m_{A}-m_{B}}{m_{A}+m_{B}}}\right)v_{A1}+\left({\frac {2m_{B}}{m_{A}+m_{B}}}\right)v_{B1}\\v_{B2}&=\left({\frac {m_{B}-m_{A}}{m_{A}+m_{B}}}\right)v_{B1}+\left({\frac {2m_{A}}{m_{A}+m_{B}}}\right)v_{A1}\,.\end{aligned}}}"></span> </p><p>If one body has much greater mass than the other, its velocity will be little affected by a collision while the other body will experience a large change. </p> <div class="mw-heading mw-heading4"><h4 id="Inelastic_collisions">Inelastic collisions</h4></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Inelastic_collision" title="Inelastic collision">Inelastic collision</a></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Inelastischer_sto%C3%9F.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/de/Inelastischer_sto%C3%9F.gif/220px-Inelastischer_sto%C3%9F.gif" decoding="async" width="220" height="26" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/de/Inelastischer_sto%C3%9F.gif/330px-Inelastischer_sto%C3%9F.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/de/Inelastischer_sto%C3%9F.gif/440px-Inelastischer_sto%C3%9F.gif 2x" data-file-width="500" data-file-height="60" /></a><figcaption>a perfectly inelastic collision between equal masses</figcaption></figure> <p>In an inelastic collision, some of the kinetic energy of the colliding bodies is converted into other forms of energy (such as <a href="/wiki/Heat" title="Heat">heat</a> or <a href="/wiki/Sound" title="Sound">sound</a>). Examples include <a href="/wiki/Traffic_collisions" class="mw-redirect" title="Traffic collisions">traffic collisions</a>,<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> in which the effect of loss of kinetic energy can be seen in the damage to the vehicles; electrons losing some of their energy to atoms (as in the <a href="/wiki/Franck%E2%80%93Hertz_experiment" title="Franck–Hertz experiment">Franck–Hertz experiment</a>);<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> and <a href="/wiki/Particle_accelerator" title="Particle accelerator">particle accelerators</a> in which the kinetic energy is converted into mass in the form of new particles. </p><p>In a perfectly inelastic collision (such as a bug hitting a windshield), both bodies have the same motion afterwards. A head-on inelastic collision between two bodies can be represented by velocities in one dimension, along a line passing through the bodies. If the velocities are <span class="texhtml"><var style="padding-right: 1px;">v</var><sub>A1</sub></span> and <span class="texhtml"><var style="padding-right: 1px;">v</var><sub>B1</sub></span> before the collision then in a perfectly inelastic collision both bodies will be travelling with velocity <span class="texhtml mvar" style="font-style:italic;">v</span><sub>2</sub> after the collision. The equation expressing conservation of momentum is: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}m_{A}v_{A1}+m_{B}v_{B1}&=\left(m_{A}+m_{B}\right)v_{2}\,.\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mspace width="thinmathspace" /> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}m_{A}v_{A1}+m_{B}v_{B1}&=\left(m_{A}+m_{B}\right)v_{2}\,.\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c6b061d520fd1927a26fdc0cfc236ae3775a202d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:35.837ex; height:2.843ex;" alt="{\displaystyle {\begin{aligned}m_{A}v_{A1}+m_{B}v_{B1}&=\left(m_{A}+m_{B}\right)v_{2}\,.\end{aligned}}}"></span> </p><p>If one body is motionless to begin with (e.g. <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u_{2}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u_{2}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7776a43002c1908e8e8a771112e9535166d5f27d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.645ex; height:2.509ex;" alt="{\displaystyle u_{2}=0}"></span>), the equation for conservation of momentum is </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m_{A}v_{A1}=\left(m_{A}+m_{B}\right)v_{2}\,,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mspace width="thinmathspace" /> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m_{A}v_{A1}=\left(m_{A}+m_{B}\right)v_{2}\,,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b227d560bbc813ebacb4ef0677540440b349fff" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.296ex; height:2.843ex;" alt="{\displaystyle m_{A}v_{A1}=\left(m_{A}+m_{B}\right)v_{2}\,,}"></span> </p><p>so </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v_{2}={\frac {m_{A}}{m_{A}+m_{B}}}v_{A1}\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mrow> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> <mn>1</mn> </mrow> </msub> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v_{2}={\frac {m_{A}}{m_{A}+m_{B}}}v_{A1}\,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f51ffab923d9a04db8d436ae9b0d8a7ac1fd2b21" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:20.43ex; height:5.009ex;" alt="{\displaystyle v_{2}={\frac {m_{A}}{m_{A}+m_{B}}}v_{A1}\,.}"></span> </p><p>In a different situation, if the frame of reference is moving at the final velocity such that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v_{2}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v_{2}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ff598dc268fa4d52edf5eea2c524356cfb6472b6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.443ex; height:2.509ex;" alt="{\displaystyle v_{2}=0}"></span>, the objects would be brought to rest by a perfectly inelastic collision and 100% of the kinetic energy is converted to other forms of energy. In this instance the initial velocities of the bodies would be non-zero, or the bodies would have to be massless. </p><p>One measure of the inelasticity of the collision is the <a href="/wiki/Coefficient_of_restitution" title="Coefficient of restitution">coefficient of restitution</a> <span class="texhtml"><var style="padding-right: 1px;">C</var><sub>R</sub></span>, defined as the ratio of relative velocity of separation to relative velocity of approach. In applying this measure to a ball bouncing from a solid surface, this can be easily measured using the following formula:<sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{\text{R}}={\sqrt {\frac {\text{bounce height}}{\text{drop height}}}}\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>R</mtext> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mtext>bounce height</mtext> <mtext>drop height</mtext> </mfrac> </msqrt> </mrow> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{\text{R}}={\sqrt {\frac {\text{bounce height}}{\text{drop height}}}}\,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/89c352c513f4323754caefd5d0c9f301ee18107a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:24.413ex; height:7.509ex;" alt="{\displaystyle C_{\text{R}}={\sqrt {\frac {\text{bounce height}}{\text{drop height}}}}\,.}"></span> </p><p>The momentum and energy equations also apply to the motions of objects that begin together and then move apart. For example, an <a href="/wiki/Explosion" title="Explosion">explosion</a> is the result of a chain reaction that transforms potential energy stored in chemical, mechanical, or nuclear form into kinetic energy, acoustic energy, and electromagnetic radiation. <a href="/wiki/Rocket" title="Rocket">Rockets</a> also make use of conservation of momentum: propellant is thrust outward, gaining momentum, and an equal and opposite momentum is imparted to the rocket.<sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Multiple_dimensions">Multiple dimensions</h3></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Elastischer_sto%C3%9F_2D.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/2c/Elastischer_sto%C3%9F_2D.gif/220px-Elastischer_sto%C3%9F_2D.gif" decoding="async" width="220" height="126" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/2c/Elastischer_sto%C3%9F_2D.gif/330px-Elastischer_sto%C3%9F_2D.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/2/2c/Elastischer_sto%C3%9F_2D.gif 2x" data-file-width="350" data-file-height="200" /></a><figcaption>Two-dimensional elastic collision. There is no motion perpendicular to the image, so only two components are needed to represent the velocities and momenta. The two blue vectors represent velocities after the collision and add vectorially to get the initial (red) velocity.</figcaption></figure> <p>Real motion has both direction and velocity and must be represented by a <a href="/wiki/Vector_(geometry)" class="mw-redirect" title="Vector (geometry)">vector</a>. In a coordinate system with <span class="texhtml"><var style="padding-right: 1px;">x</var>, <var style="padding-right: 1px;">y</var>, <var style="padding-right: 1px;">z</var></span> axes, velocity has components <span class="texhtml"><var style="padding-right: 1px;">v</var><sub><var style="padding-right: 1px;">x</var></sub></span> in the <span class="texhtml mvar" style="font-style:italic;">x</span>-direction, <span class="texhtml"><var style="padding-right: 1px;">v</var><sub><var style="padding-right: 1px;">y</var></sub></span> in the <span class="texhtml mvar" style="font-style:italic;">y</span>-direction, <span class="texhtml"><var style="padding-right: 1px;">v</var><sub><var style="padding-right: 1px;">z</var></sub></span> in the <span class="texhtml mvar" style="font-style:italic;">z</span>-direction. The vector is represented by a boldface symbol:<sup id="cite_ref-FeynmanCh11_15-0" class="reference"><a href="#cite_note-FeynmanCh11-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {v} =\left(v_{x},v_{y},v_{z}\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {v} =\left(v_{x},v_{y},v_{z}\right).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87828c8eca817d5a29bbc10ed280a19589fab149" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:16.027ex; height:3.009ex;" alt="{\displaystyle \mathbf {v} =\left(v_{x},v_{y},v_{z}\right).}"></span> </p><p>Similarly, the momentum is a vector quantity and is represented by a boldface symbol: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {p} =\left(p_{x},p_{y},p_{z}\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {p} =\left(p_{x},p_{y},p_{z}\right).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f811d2e927c0b4fcc89d88be76237af3ac53748d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:16.227ex; height:3.009ex;" alt="{\displaystyle \mathbf {p} =\left(p_{x},p_{y},p_{z}\right).}"></span> </p><p>The equations in the previous sections, work in vector form if the scalars <span class="texhtml">p</span> and <span class="texhtml">v</span> are replaced by vectors <span class="texhtml"><b>p</b></span> and <span class="texhtml"><b>v</b></span>. Each vector equation represents three scalar equations. For example, </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {p} =m\mathbf {v} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> <mo>=</mo> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {p} =m\mathbf {v} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a271a96e7b925fd39686375167c76d406e87c813" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.035ex; height:2.009ex;" alt="{\displaystyle \mathbf {p} =m\mathbf {v} }"></span> </p><p>represents three equations:<sup id="cite_ref-FeynmanCh11_15-1" class="reference"><a href="#cite_note-FeynmanCh11-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}p_{x}&=mv_{x}\\p_{y}&=mv_{y}\\p_{z}&=mv_{z}.\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>m</mi> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>m</mi> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>m</mi> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}p_{x}&=mv_{x}\\p_{y}&=mv_{y}\\p_{z}&=mv_{z}.\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/048790c67b2f03abfdefa3c80144a78fa965a15c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:11.008ex; height:9.176ex;" alt="{\displaystyle {\begin{aligned}p_{x}&=mv_{x}\\p_{y}&=mv_{y}\\p_{z}&=mv_{z}.\end{aligned}}}"></span> </p><p>The kinetic energy equations are exceptions to the above replacement rule. The equations are still one-dimensional, but each scalar represents the <a href="/wiki/Magnitude_(mathematics)#Euclidean_vector_space" title="Magnitude (mathematics)">magnitude of the vector</a>, for example, </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v^{2}=v_{x}^{2}+v_{y}^{2}+v_{z}^{2}\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msubsup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v^{2}=v_{x}^{2}+v_{y}^{2}+v_{z}^{2}\,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bdc305729a263c2bbcdab32f6f765118536af299" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:18.659ex; height:3.343ex;" alt="{\displaystyle v^{2}=v_{x}^{2}+v_{y}^{2}+v_{z}^{2}\,.}"></span> </p><p>Each vector equation represents three scalar equations. Often coordinates can be chosen so that only two components are needed, as in the figure. Each component can be obtained separately and the results combined to produce a vector result.<sup id="cite_ref-FeynmanCh11_15-2" class="reference"><a href="#cite_note-FeynmanCh11-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> </p><p>A simple construction involving the center of mass frame can be used to show that if a stationary elastic sphere is struck by a moving sphere, the two will head off at right angles after the collision (as in the figure).<sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Objects_of_variable_mass">Objects of variable mass</h3></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Variable-mass_system" title="Variable-mass system">Variable-mass system</a></div> <p>The concept of momentum plays a fundamental role in explaining the behavior of variable-mass objects such as a <a href="/wiki/Rocket" title="Rocket">rocket</a> ejecting fuel or a <a href="/wiki/Star" title="Star">star</a> <a href="/wiki/Accretion_(astrophysics)" title="Accretion (astrophysics)">accreting</a> gas. In analyzing such an object, one treats the object's mass as a function that varies with time: <span class="texhtml"><var style="padding-right: 1px;">m</var>(<var style="padding-right: 1px;">t</var>)</span>. The momentum of the object at time <span class="texhtml mvar" style="font-style:italic;">t</span> is therefore <span class="texhtml"><var style="padding-right: 1px;">p</var>(<var style="padding-right: 1px;">t</var>) = <var style="padding-right: 1px;">m</var>(<var style="padding-right: 1px;">t</var>)<var style="padding-right: 1px;">v</var>(<var style="padding-right: 1px;">t</var>)</span>. One might then try to invoke Newton's second law of motion by saying that the external force <span class="texhtml mvar" style="font-style:italic;">F</span> on the object is related to its momentum <span class="texhtml"><var style="padding-right: 1px;">p</var>(<var style="padding-right: 1px;">t</var>)</span> by <span class="texhtml"><var style="padding-right: 1px;">F</var> = <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">d<var style="padding-right: 1px;">p</var></span><span class="sr-only">/</span><span class="den">d<var style="padding-right: 1px;">t</var></span></span>⁠</span></span>, but this is incorrect, as is the related expression found by applying the product rule to <span class="texhtml"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num"><var style="padding-right: 1px;">d</var>(<var style="padding-right: 1px;">m</var><var style="padding-right: 1px;">v</var>)</span><span class="sr-only">/</span><span class="den">d<var style="padding-right: 1px;">t</var></span></span>⁠</span></span>:<sup id="cite_ref-kleppner135_17-0" class="reference"><a href="#cite_note-kleppner135-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F=m(t){\frac {{\text{d}}v}{{\text{d}}t}}+v(t){\frac {{\text{d}}m}{{\text{d}}t}}.{\text{(incorrect)}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo>=</mo> <mi>m</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>d</mtext> </mrow> <mi>v</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>d</mtext> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mi>v</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>d</mtext> </mrow> <mi>m</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>d</mtext> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>.</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>(incorrect)</mtext> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F=m(t){\frac {{\text{d}}v}{{\text{d}}t}}+v(t){\frac {{\text{d}}m}{{\text{d}}t}}.{\text{(incorrect)}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a345e1181b8dcd52d40ec0b3b1f4c2855c44330f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:35.341ex; height:5.509ex;" alt="{\displaystyle F=m(t){\frac {{\text{d}}v}{{\text{d}}t}}+v(t){\frac {{\text{d}}m}{{\text{d}}t}}.{\text{(incorrect)}}}"></span> </p><p>This equation does not correctly describe the motion of variable-mass objects. The correct equation is </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F=m(t){\frac {{\text{d}}v}{{\text{d}}t}}-u{\frac {{\text{d}}m}{{\text{d}}t}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo>=</mo> <mi>m</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>d</mtext> </mrow> <mi>v</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>d</mtext> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>−</mo> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>d</mtext> </mrow> <mi>m</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>d</mtext> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F=m(t){\frac {{\text{d}}v}{{\text{d}}t}}-u{\frac {{\text{d}}m}{{\text{d}}t}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/95af30150307a77c42d87687fd27a91c9928ecec" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:21.771ex; height:5.509ex;" alt="{\displaystyle F=m(t){\frac {{\text{d}}v}{{\text{d}}t}}-u{\frac {{\text{d}}m}{{\text{d}}t}},}"></span> </p><p>where <span class="texhtml mvar" style="font-style:italic;">u</span> is the velocity of the ejected/accreted mass <i>as seen in the object's rest frame</i>.<sup id="cite_ref-kleppner135_17-1" class="reference"><a href="#cite_note-kleppner135-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> This is distinct from <span class="texhtml mvar" style="font-style:italic;">v</span>, which is the velocity of the object itself as seen in an inertial frame. </p><p>This equation is derived by keeping track of both the momentum of the object as well as the momentum of the ejected/accreted mass (<span class="texhtml">d<var style="padding-right: 1px;">m</var></span>). When considered together, the object and the mass (<span class="texhtml">d<var style="padding-right: 1px;">m</var></span>) constitute a closed system in which total momentum is conserved. </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(t+{\text{d}}t)=(m-{\text{d}}m)(v+{\text{d}}v)+{\text{d}}m(v-u)=mv+m{\text{d}}v-u{\text{d}}m=P(t)+m{\text{d}}v-u{\text{d}}m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>d</mtext> </mrow> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>m</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>d</mtext> </mrow> <mi>m</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>v</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>d</mtext> </mrow> <mi>v</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>d</mtext> </mrow> <mi>m</mi> <mo stretchy="false">(</mo> <mi>v</mi> <mo>−</mo> <mi>u</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>m</mi> <mi>v</mi> <mo>+</mo> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>d</mtext> </mrow> <mi>v</mi> <mo>−</mo> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>d</mtext> </mrow> <mi>m</mi> <mo>=</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>d</mtext> </mrow> <mi>v</mi> <mo>−</mo> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>d</mtext> </mrow> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(t+{\text{d}}t)=(m-{\text{d}}m)(v+{\text{d}}v)+{\text{d}}m(v-u)=mv+m{\text{d}}v-u{\text{d}}m=P(t)+m{\text{d}}v-u{\text{d}}m}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/84a76792a96d7c1b01b2f5c86b87859e984bdcb7" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:87.333ex; height:2.843ex;" alt="{\displaystyle P(t+{\text{d}}t)=(m-{\text{d}}m)(v+{\text{d}}v)+{\text{d}}m(v-u)=mv+m{\text{d}}v-u{\text{d}}m=P(t)+m{\text{d}}v-u{\text{d}}m}"></span> </p> <div class="mw-heading mw-heading2"><h2 id="Generalized">Generalized</h2></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Analytical_mechanics" title="Analytical mechanics">Analytical mechanics</a></div> <p>Newton's laws can be difficult to apply to many kinds of motion because the motion is limited by <i>constraints</i>. For example, a bead on an abacus is constrained to move along its wire and a pendulum bob is constrained to swing at a fixed distance from the pivot. Many such constraints can be incorporated by changing the normal <a href="/wiki/Cartesian_coordinates" class="mw-redirect" title="Cartesian coordinates">Cartesian coordinates</a> to a set of <i><a href="/wiki/Generalized_coordinates" title="Generalized coordinates">generalized coordinates</a></i> that may be fewer in number.<sup id="cite_ref-18" class="reference"><a href="#cite_note-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> Refined mathematical methods have been developed for solving mechanics problems in generalized coordinates. They introduce a <i>generalized momentum</i>, also known as the <i>canonical momentum</i> or <i>conjugate momentum</i>, that extends the concepts of both linear momentum and <a href="/wiki/Angular_momentum" title="Angular momentum">angular momentum</a>. To distinguish it from generalized momentum, the product of mass and velocity is also referred to as <i>mechanical momentum</i>, <i>kinetic momentum</i> or <i>kinematic momentum</i>.<sup id="cite_ref-Goldstein54_6-1" class="reference"><a href="#cite_note-Goldstein54-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-19" class="reference"><a href="#cite_note-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-FeynmanQM_20-0" class="reference"><a href="#cite_note-FeynmanQM-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> The two main methods are described below. </p> <div class="mw-heading mw-heading3"><h3 id="Lagrangian_mechanics">Lagrangian mechanics</h3></div> <p>In <a href="/wiki/Lagrangian_mechanics" title="Lagrangian mechanics">Lagrangian mechanics</a>, a Lagrangian is defined as the difference between the kinetic energy <span class="texhtml mvar" style="font-style:italic;">T</span> and the <a href="/wiki/Potential_energy" title="Potential energy">potential energy</a> <span class="texhtml mvar" style="font-style:italic;">V</span>: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {L}}=T-V\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mo>=</mo> <mi>T</mi> <mo>−</mo> <mi>V</mi> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}}=T-V\,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cb009b20edbf4244a71a1f8564ba40a89e57fd64" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:12ex; height:2.343ex;" alt="{\displaystyle {\mathcal {L}}=T-V\,.}"></span> </p><p>If the generalized coordinates are represented as a vector <span class="texhtml"><b>q</b> = (<var style="padding-right: 1px;">q</var><sub>1</sub>, <var style="padding-right: 1px;">q</var><sub>2</sub>, ... , <var style="padding-right: 1px;">q</var><sub><var style="padding-right: 1px;">N</var></sub>) </span> and time differentiation is represented by a dot over the variable, then the equations of motion (known as the Lagrange or <a href="/wiki/Euler%E2%80%93Lagrange_equation" title="Euler–Lagrange equation">Euler–Lagrange equations</a>) are a set of <span class="texhtml mvar" style="font-style:italic;">N</span> equations:<sup id="cite_ref-21" class="reference"><a href="#cite_note-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\text{d}}{{\text{d}}t}}\left({\frac {\partial {\mathcal {L}}}{\partial {\dot {q}}_{j}}}\right)-{\frac {\partial {\mathcal {L}}}{\partial q_{j}}}=0\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mtext>d</mtext> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>d</mtext> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>q</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>=</mo> <mn>0</mn> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\text{d}}{{\text{d}}t}}\left({\frac {\partial {\mathcal {L}}}{\partial {\dot {q}}_{j}}}\right)-{\frac {\partial {\mathcal {L}}}{\partial q_{j}}}=0\,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e704f970e1677b0dfcbc5b1e3766c632a3e306c4" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:23.714ex; height:7.509ex;" alt="{\displaystyle {\frac {\text{d}}{{\text{d}}t}}\left({\frac {\partial {\mathcal {L}}}{\partial {\dot {q}}_{j}}}\right)-{\frac {\partial {\mathcal {L}}}{\partial q_{j}}}=0\,.}"></span> </p><p>If a coordinate <span class="texhtml"><var style="padding-right: 1px;">q</var><sub><var style="padding-right: 1px;">i</var></sub></span> is not a Cartesian coordinate, the associated generalized momentum component <span class="texhtml"><var style="padding-right: 1px;">p</var><sub><var style="padding-right: 1px;">i</var></sub></span> does not necessarily have the dimensions of linear momentum. Even if <span class="texhtml"><var style="padding-right: 1px;">q</var><sub><var style="padding-right: 1px;">i</var></sub></span> is a Cartesian coordinate, <span class="texhtml"><var style="padding-right: 1px;">p</var><sub><var style="padding-right: 1px;">i</var></sub></span> will not be the same as the mechanical momentum if the potential depends on velocity.<sup id="cite_ref-Goldstein54_6-2" class="reference"><a href="#cite_note-Goldstein54-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> Some sources represent the kinematic momentum by the symbol <span class="texhtml"><b>Π</b></span>.<sup id="cite_ref-Lerner_22-0" class="reference"><a href="#cite_note-Lerner-22"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup> </p><p>In this mathematical framework, a generalized momentum is associated with the generalized coordinates. Its components are defined as </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{j}={\frac {\partial {\mathcal {L}}}{\partial {\dot {q}}_{j}}}\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>q</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{j}={\frac {\partial {\mathcal {L}}}{\partial {\dot {q}}_{j}}}\,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7a4539399149807e96a002329571bf09ab728257" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; margin-left: -0.089ex; width:10.742ex; height:6.343ex;" alt="{\displaystyle p_{j}={\frac {\partial {\mathcal {L}}}{\partial {\dot {q}}_{j}}}\,.}"></span> </p><p>Each component <span class="texhtml"><var style="padding-right: 1px;">p</var><sub><var style="padding-right: 1px;">j</var></sub></span> is said to be the <i>conjugate momentum</i> for the coordinate <span class="texhtml"><var style="padding-right: 1px;">q</var><sub><var style="padding-right: 1px;">j</var></sub></span>. </p><p>Now if a given coordinate <span class="texhtml"><var style="padding-right: 1px;">q</var><sub><var style="padding-right: 1px;">i</var></sub></span> does not appear in the Lagrangian (although its time derivative might appear), then <span class="texhtml"><var style="padding-right: 1px;">p</var><sub><var style="padding-right: 1px;">j</var></sub></span> is constant. This is the generalization of the conservation of momentum.<sup id="cite_ref-Goldstein54_6-3" class="reference"><a href="#cite_note-Goldstein54-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p><p>Even if the generalized coordinates are just the ordinary spatial coordinates, the conjugate momenta are not necessarily the ordinary momentum coordinates. An example is found in the section on electromagnetism. </p> <div class="mw-heading mw-heading3"><h3 id="Hamiltonian_mechanics">Hamiltonian mechanics</h3></div> <p>In <a href="/wiki/Hamiltonian_mechanics" title="Hamiltonian mechanics">Hamiltonian mechanics</a>, the Lagrangian (a function of generalized coordinates and their derivatives) is replaced by a Hamiltonian that is a function of generalized coordinates and momentum. The Hamiltonian is defined as </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {H}}\left(\mathbf {q} ,\mathbf {p} ,t\right)=\mathbf {p} \cdot {\dot {\mathbf {q} }}-{\mathcal {L}}\left(\mathbf {q} ,{\dot {\mathbf {q} }},t\right)\,,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">H</mi> </mrow> </mrow> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">q</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> <mo>,</mo> <mi>t</mi> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">q</mi> </mrow> <mo>˙</mo> </mover> </mrow> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">q</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">q</mi> </mrow> <mo>˙</mo> </mover> </mrow> </mrow> <mo>,</mo> <mi>t</mi> </mrow> <mo>)</mo> </mrow> <mspace width="thinmathspace" /> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {H}}\left(\mathbf {q} ,\mathbf {p} ,t\right)=\mathbf {p} \cdot {\dot {\mathbf {q} }}-{\mathcal {L}}\left(\mathbf {q} ,{\dot {\mathbf {q} }},t\right)\,,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f54145f5d85289262062b635cd5583d713c364c0" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.447ex; height:2.843ex;" alt="{\displaystyle {\mathcal {H}}\left(\mathbf {q} ,\mathbf {p} ,t\right)=\mathbf {p} \cdot {\dot {\mathbf {q} }}-{\mathcal {L}}\left(\mathbf {q} ,{\dot {\mathbf {q} }},t\right)\,,}"></span> </p><p>where the momentum is obtained by differentiating the Lagrangian as above. The Hamiltonian equations of motion are<sup id="cite_ref-23" class="reference"><a href="#cite_note-23"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}{\dot {q}}_{i}&={\frac {\partial {\mathcal {H}}}{\partial p_{i}}}\\-{\dot {p}}_{i}&={\frac {\partial {\mathcal {H}}}{\partial q_{i}}}\\-{\frac {\partial {\mathcal {L}}}{\partial t}}&={\frac {{\text{d}}{\mathcal {H}}}{{\text{d}}t}}\,.\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>q</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">H</mi> </mrow> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>p</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">H</mi> </mrow> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>d</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">H</mi> </mrow> </mrow> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>d</mtext> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <mspace width="thinmathspace" /> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\dot {q}}_{i}&={\frac {\partial {\mathcal {H}}}{\partial p_{i}}}\\-{\dot {p}}_{i}&={\frac {\partial {\mathcal {H}}}{\partial q_{i}}}\\-{\frac {\partial {\mathcal {L}}}{\partial t}}&={\frac {{\text{d}}{\mathcal {H}}}{{\text{d}}t}}\,.\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/83e21a7536d8c0de930c9a9333673946a18c52da" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -8.009ex; margin-bottom: -0.329ex; width:14.542ex; height:17.843ex;" alt="{\displaystyle {\begin{aligned}{\dot {q}}_{i}&={\frac {\partial {\mathcal {H}}}{\partial p_{i}}}\\-{\dot {p}}_{i}&={\frac {\partial {\mathcal {H}}}{\partial q_{i}}}\\-{\frac {\partial {\mathcal {L}}}{\partial t}}&={\frac {{\text{d}}{\mathcal {H}}}{{\text{d}}t}}\,.\end{aligned}}}"></span> </p><p>As in Lagrangian mechanics, if a generalized coordinate does not appear in the Hamiltonian, its conjugate momentum component is conserved.<sup id="cite_ref-24" class="reference"><a href="#cite_note-24"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Symmetry_and_conservation">Symmetry and conservation</h3></div> <p>Conservation of momentum is a mathematical consequence of the <a href="/wiki/Homogeneity_(physics)" title="Homogeneity (physics)">homogeneity</a> (shift <a href="/wiki/Symmetry" title="Symmetry">symmetry</a>) of space (position in space is the <a href="/wiki/Canonical_conjugate" class="mw-redirect" title="Canonical conjugate">canonical conjugate</a> quantity to momentum). That is, conservation of momentum is a consequence of the fact that the laws of physics do not depend on position; this is a special case of <a href="/wiki/Noether%27s_theorem" title="Noether's theorem">Noether's theorem</a>.<sup id="cite_ref-25" class="reference"><a href="#cite_note-25"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup> For systems that do not have this symmetry, it may not be possible to define conservation of momentum. Examples where conservation of momentum does not apply include <a href="/wiki/Curved_space" title="Curved space">curved spacetimes</a> in <a href="/wiki/General_relativity" title="General relativity">general relativity</a><sup id="cite_ref-26" class="reference"><a href="#cite_note-26"><span class="cite-bracket">[</span>26<span class="cite-bracket">]</span></a></sup> or <a href="/wiki/Time_crystals" class="mw-redirect" title="Time crystals">time crystals</a> in <a href="/wiki/Condensed_matter_physics" title="Condensed matter physics">condensed matter physics</a>.<sup id="cite_ref-Grossman_2012_27-0" class="reference"><a href="#cite_note-Grossman_2012-27"><span class="cite-bracket">[</span>27<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Cowen_2012_28-0" class="reference"><a href="#cite_note-Cowen_2012-28"><span class="cite-bracket">[</span>28<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Powell_2013_29-0" class="reference"><a href="#cite_note-Powell_2013-29"><span class="cite-bracket">[</span>29<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Gibney_2017_30-0" class="reference"><a href="#cite_note-Gibney_2017-30"><span class="cite-bracket">[</span>30<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Momentum_density">Momentum density</h2></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Mass_flux" title="Mass flux">Mass flux</a></div> <div class="mw-heading mw-heading3"><h3 id="In_deformable_bodies_and_fluids">In deformable bodies and fluids</h3></div> <div class="mw-heading mw-heading4"><h4 id="Conservation_in_a_continuum">Conservation in a continuum</h4></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Cauchy_momentum_equation" title="Cauchy momentum equation">Cauchy momentum equation</a></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Equation_motion_body.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f0/Equation_motion_body.svg/220px-Equation_motion_body.svg.png" decoding="async" width="220" height="203" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f0/Equation_motion_body.svg/330px-Equation_motion_body.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f0/Equation_motion_body.svg/440px-Equation_motion_body.svg.png 2x" data-file-width="316" data-file-height="292" /></a><figcaption>Motion of a material body</figcaption></figure> <p>In fields such as <a href="/wiki/Fluid_dynamics" title="Fluid dynamics">fluid dynamics</a> and <a href="/wiki/Solid_mechanics" title="Solid mechanics">solid mechanics</a>, it is not feasible to follow the motion of individual atoms or molecules. Instead, the materials must be approximated by a <a href="/wiki/Continuum_mechanics" title="Continuum mechanics">continuum</a> in which, at each point, there is a particle or <a href="/wiki/Fluid_parcel" title="Fluid parcel">fluid parcel</a> that is assigned the average of the properties of atoms in a small region nearby. In particular, it has a density <span class="texhtml mvar" style="font-style:italic;">ρ</span> and velocity <span class="texhtml"><b>v</b></span> that depend on time <span class="texhtml mvar" style="font-style:italic;">t</span> and position <span class="texhtml"><b>r</b></span>. The momentum per unit volume is <span class="texhtml"><var style="padding-right: 1px;">ρ</var><b>v</b></span>.<sup id="cite_ref-31" class="reference"><a href="#cite_note-31"><span class="cite-bracket">[</span>31<span class="cite-bracket">]</span></a></sup> </p><p>Consider a column of water in <a href="/wiki/Hydrostatic_equilibrium" title="Hydrostatic equilibrium">hydrostatic equilibrium</a>. All the forces on the water are in balance and the water is motionless. On any given drop of water, two forces are balanced. The first is gravity, which acts directly on each atom and molecule inside. The gravitational force per unit volume is <span class="texhtml"><var style="padding-right: 1px;">ρ</var><b>g</b></span>, where <span class="texhtml"><b>g</b></span> is the <a href="/wiki/Gravitational_acceleration" title="Gravitational acceleration">gravitational acceleration</a>. The second force is the sum of all the forces exerted on its surface by the surrounding water. The force from below is greater than the force from above by just the amount needed to balance gravity. The normal force per unit area is the <a href="/wiki/Pressure" title="Pressure">pressure</a> <span class="texhtml mvar" style="font-style:italic;">p</span>. The average force per unit volume inside the droplet is the gradient of the pressure, so the force balance equation is<sup id="cite_ref-FeynmanCh40_32-0" class="reference"><a href="#cite_note-FeynmanCh40-32"><span class="cite-bracket">[</span>32<span class="cite-bracket">]</span></a></sup> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -\nabla p+\rho \mathbf {g} =0\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mi mathvariant="normal">∇</mi> <mi>p</mi> <mo>+</mo> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">g</mi> </mrow> <mo>=</mo> <mn>0</mn> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -\nabla p+\rho \mathbf {g} =0\,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5af9af8a07b2a2fbc21114360572dcd4d4583dc9" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.587ex; height:2.676ex;" alt="{\displaystyle -\nabla p+\rho \mathbf {g} =0\,.}"></span> </p><p>If the forces are not balanced, the droplet accelerates. This acceleration is not simply the partial derivative <span class="texhtml"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num"><i>∂</i><b>v</b></span><span class="sr-only">/</span><span class="den"><i>∂</i><var style="padding-right: 1px;">t</var></span></span>⁠</span></span> because the fluid in a given volume changes with time. Instead, the <a href="/wiki/Material_derivative" title="Material derivative">material derivative</a> is needed:<sup id="cite_ref-33" class="reference"><a href="#cite_note-33"><span class="cite-bracket">[</span>33<span class="cite-bracket">]</span></a></sup> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {D}{Dt}}\equiv {\frac {\partial }{\partial t}}+\mathbf {v} \cdot {\boldsymbol {\nabla }}\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>D</mi> <mrow> <mi>D</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>≡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">∇</mi> </mrow> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {D}{Dt}}\equiv {\frac {\partial }{\partial t}}+\mathbf {v} \cdot {\boldsymbol {\nabla }}\,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06e6300772911a8adf746640cd17a3873349ee6e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:18.883ex; height:5.509ex;" alt="{\displaystyle {\frac {D}{Dt}}\equiv {\frac {\partial }{\partial t}}+\mathbf {v} \cdot {\boldsymbol {\nabla }}\,.}"></span> </p><p>Applied to any physical quantity, the material derivative includes the rate of change at a point and the changes due to <a href="/wiki/Advection" title="Advection">advection</a> as fluid is carried past the point. Per unit volume, the rate of change in momentum is equal to <span class="texhtml"><var style="padding-right: 1px;">ρ</var><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num"><i>D</i><b>v</b></span><span class="sr-only">/</span><span class="den"><i>D</i><var style="padding-right: 1px;">t</var></span></span>⁠</span></span>. This is equal to the net force on the droplet. </p><p>Forces that can change the momentum of a droplet include the gradient of the pressure and gravity, as above. In addition, surface forces can deform the droplet. In the simplest case, a <a href="/wiki/Shear_stress" title="Shear stress">shear stress</a> <span class="texhtml mvar" style="font-style:italic;">τ</span>, exerted by a force parallel to the surface of the droplet, is proportional to the rate of deformation or <a href="/wiki/Strain_rate" title="Strain rate">strain rate</a>. Such a shear stress occurs if the fluid has a velocity gradient because the fluid is moving faster on one side than another. If the speed in the <span class="texhtml mvar" style="font-style:italic;">x</span> direction varies with <span class="texhtml mvar" style="font-style:italic;">z</span>, the tangential force in direction <span class="texhtml mvar" style="font-style:italic;">x</span> per unit area normal to the <span class="texhtml mvar" style="font-style:italic;">z</span> direction is </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{zx}=-\mu {\frac {\partial v_{x}}{\partial z}}\,,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> <mi>x</mi> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>z</mi> </mrow> </mfrac> </mrow> <mspace width="thinmathspace" /> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{zx}=-\mu {\frac {\partial v_{x}}{\partial z}}\,,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/63aa1a782152c0cdc4ed33f749c22eb19c3adad0" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:15.066ex; height:5.509ex;" alt="{\displaystyle \sigma _{zx}=-\mu {\frac {\partial v_{x}}{\partial z}}\,,}"></span> </p><p>where <span class="texhtml mvar" style="font-style:italic;">μ</span> is the <a href="/wiki/Viscosity" title="Viscosity">viscosity</a>. This is also a <a href="/wiki/Flux" title="Flux">flux</a>, or flow per unit area, of <span class="texhtml mvar" style="font-style:italic;">x</span>-momentum through the surface.<sup id="cite_ref-34" class="reference"><a href="#cite_note-34"><span class="cite-bracket">[</span>34<span class="cite-bracket">]</span></a></sup> </p><p>Including the effect of viscosity, the momentum balance equations for the <a href="/wiki/Incompressible_flow" title="Incompressible flow">incompressible flow</a> of a <a href="/wiki/Newtonian_fluid" title="Newtonian fluid">Newtonian fluid</a> are </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho {\frac {D\mathbf {v} }{Dt}}=-{\boldsymbol {\nabla }}p+\mu \nabla ^{2}\mathbf {v} +\rho \mathbf {g} .\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> </mrow> <mrow> <mi>D</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">∇</mi> </mrow> <mi>p</mi> <mo>+</mo> <mi>μ</mi> <msup> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>+</mo> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">g</mi> </mrow> <mo>.</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho {\frac {D\mathbf {v} }{Dt}}=-{\boldsymbol {\nabla }}p+\mu \nabla ^{2}\mathbf {v} +\rho \mathbf {g} .\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8f63d8f029a0627ff37ef9c1f252acb31f9eca5b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:28.732ex; height:5.176ex;" alt="{\displaystyle \rho {\frac {D\mathbf {v} }{Dt}}=-{\boldsymbol {\nabla }}p+\mu \nabla ^{2}\mathbf {v} +\rho \mathbf {g} .\,}"></span> </p><p>These are known as the <a href="/wiki/Navier%E2%80%93Stokes_equations" title="Navier–Stokes equations">Navier–Stokes equations</a>.<sup id="cite_ref-35" class="reference"><a href="#cite_note-35"><span class="cite-bracket">[</span>35<span class="cite-bracket">]</span></a></sup> </p><p>The momentum balance equations can be extended to more general materials, including solids. For each surface with normal in direction <span class="texhtml mvar" style="font-style:italic;">i</span> and force in direction <span class="texhtml mvar" style="font-style:italic;">j</span>, there is a stress component <span class="texhtml"><var style="padding-right: 1px;">σ</var><sub><var style="padding-right: 1px;">i</var><var style="padding-right: 1px;">j</var></sub></span>. The nine components make up the <a href="/wiki/Cauchy_stress_tensor" title="Cauchy stress tensor">Cauchy stress tensor</a> <span class="texhtml"><b>σ</b></span>, which includes both pressure and shear. The local conservation of momentum is expressed by the <a href="/wiki/Cauchy_momentum_equation" title="Cauchy momentum equation">Cauchy momentum equation</a>: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho {\frac {D\mathbf {v} }{Dt}}={\boldsymbol {\nabla }}\cdot {\boldsymbol {\sigma }}+\mathbf {f} \,,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> </mrow> <mrow> <mi>D</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">∇</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">σ</mi> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">f</mi> </mrow> <mspace width="thinmathspace" /> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho {\frac {D\mathbf {v} }{Dt}}={\boldsymbol {\nabla }}\cdot {\boldsymbol {\sigma }}+\mathbf {f} \,,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c2138aab218677da33d90206fdb945fda9e630f0" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:18.899ex; height:5.176ex;" alt="{\displaystyle \rho {\frac {D\mathbf {v} }{Dt}}={\boldsymbol {\nabla }}\cdot {\boldsymbol {\sigma }}+\mathbf {f} \,,}"></span> </p><p>where <span class="texhtml"><b>f</b></span> is the <a href="/wiki/Body_force" title="Body force">body force</a>.<sup id="cite_ref-36" class="reference"><a href="#cite_note-36"><span class="cite-bracket">[</span>36<span class="cite-bracket">]</span></a></sup> </p><p>The Cauchy momentum equation is broadly applicable to <a href="/wiki/Deformation_(mechanics)" class="mw-redirect" title="Deformation (mechanics)">deformations</a> of solids and liquids. The relationship between the stresses and the strain rate depends on the properties of the material (see <a href="/wiki/Viscosity#Types_of_viscosity" title="Viscosity">Types of viscosity</a>). </p> <div class="mw-heading mw-heading4"><h4 id="Acoustic_waves">Acoustic waves</h4></div> <p>A disturbance in a medium gives rise to oscillations, or <a href="/wiki/Wave" title="Wave">waves</a>, that propagate away from their source. In a fluid, small changes in pressure <span class="texhtml mvar" style="font-style:italic;">p</span> can often be described by the <a href="/wiki/Acoustic_wave_equation" title="Acoustic wave equation">acoustic wave equation</a>: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial ^{2}p}{\partial t^{2}}}=c^{2}\nabla ^{2}p\,,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>p</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>=</mo> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>p</mi> <mspace width="thinmathspace" /> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial ^{2}p}{\partial t^{2}}}=c^{2}\nabla ^{2}p\,,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c8b5b90da3fdb0bbb9a7f465a9ccfb1c10a26932" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:14.756ex; height:6.009ex;" alt="{\displaystyle {\frac {\partial ^{2}p}{\partial t^{2}}}=c^{2}\nabla ^{2}p\,,}"></span> </p><p>where <span class="texhtml mvar" style="font-style:italic;">c</span> is the <a href="/wiki/Speed_of_sound" title="Speed of sound">speed of sound</a>. In a solid, similar equations can be obtained for propagation of pressure (<a href="/wiki/P-wave" class="mw-redirect" title="P-wave">P-waves</a>) and shear (<a href="/wiki/S-waves" class="mw-redirect" title="S-waves">S-waves</a>).<sup id="cite_ref-37" class="reference"><a href="#cite_note-37"><span class="cite-bracket">[</span>37<span class="cite-bracket">]</span></a></sup> </p><p>The flux, or transport per unit area, of a momentum component <span class="texhtml"><var style="padding-right: 1px;">ρ</var><var style="padding-right: 1px;">v</var><sub><var style="padding-right: 1px;">j</var></sub></span> by a velocity <span class="texhtml"><var style="padding-right: 1px;">v</var><sub><var style="padding-right: 1px;">i</var></sub></span> is equal to <span class="texhtml"><var style="padding-right: 1px;">ρ</var><var style="padding-right: 1px;">v</var><sub><var style="padding-right: 1px;">j</var></sub><var style="padding-right: 1px;">v</var><sub><var style="padding-right: 1px;">j</var></sub></span>.<sup class="noprint Inline-Template" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Accuracy_dispute#Disputed_statement" title="Wikipedia:Accuracy dispute"><span title="so proportional to v_j^2 and not dependent on v_i? (April 2023)">dubious</span></a> – <a href="/wiki/Talk:Momentum#Dubious" title="Talk:Momentum">discuss</a></i>]</sup> In the linear approximation that leads to the above acoustic equation, the time average of this flux is zero. However, nonlinear effects can give rise to a nonzero average.<sup id="cite_ref-38" class="reference"><a href="#cite_note-38"><span class="cite-bracket">[</span>38<span class="cite-bracket">]</span></a></sup> It is possible for momentum flux to occur even though the wave itself does not have a mean momentum.<sup id="cite_ref-39" class="reference"><a href="#cite_note-39"><span class="cite-bracket">[</span>39<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="In_electromagnetics">In electromagnetics</h3></div> <div class="mw-heading mw-heading4"><h4 id="Particle_in_a_field">Particle in a field</h4></div> <p>In <a href="/wiki/Maxwell%27s_equations" title="Maxwell's equations">Maxwell's equations</a>, the forces between particles are mediated by electric and magnetic fields. The electromagnetic force (<i><a href="/wiki/Lorentz_force" title="Lorentz force">Lorentz force</a></i>) on a particle with charge <span class="texhtml mvar" style="font-style:italic;">q</span> due to a combination of <a href="/wiki/Electric_field" title="Electric field">electric field</a> <span class="texhtml"><b>E</b></span> and <a href="/wiki/Magnetic_field" title="Magnetic field">magnetic field</a> <span class="texhtml"><b>B</b></span> is </p><p><span class="mwe-math-element" data-qid="Q849919"><a href="/w/index.php?title=Special:MathWikibase&qid=Q849919" style="color:inherit;"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {F} =q(\mathbf {E} +\mathbf {v} \times \mathbf {B} ).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mo>=</mo> <mi>q</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {F} =q(\mathbf {E} +\mathbf {v} \times \mathbf {B} ).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/98b985e6332ac21f2eb2b6aeefae9135fe6fe3c6" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.057ex; height:2.843ex;" alt="{\displaystyle \mathbf {F} =q(\mathbf {E} +\mathbf {v} \times \mathbf {B} ).}"></a></span> </p><p>(in <a href="/wiki/International_System_of_Units" title="International System of Units">SI units</a>).<sup id="cite_ref-40" class="reference"><a href="#cite_note-40"><span class="cite-bracket">[</span>40<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 2">: 2 </span></sup> It has an <a href="/wiki/Electric_potential" title="Electric potential">electric potential</a> <span class="texhtml"><var style="padding-right: 1px;">φ</var>(<b>r</b>, <var style="padding-right: 1px;">t</var>)</span> and <a href="/wiki/Magnetic_vector_potential" title="Magnetic vector potential">magnetic vector potential</a> <span class="texhtml"><b>A</b>(<b>r</b>, <var style="padding-right: 1px;">t</var>)</span>.<sup id="cite_ref-Lerner_22-1" class="reference"><a href="#cite_note-Lerner-22"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup> In the non-relativistic regime, its generalized momentum is </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {P} =m\mathbf {\mathbf {v} } +q\mathbf {A} ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">P</mi> </mrow> <mo>=</mo> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> </mrow> <mo>+</mo> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {P} =m\mathbf {\mathbf {v} } +q\mathbf {A} ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6d598dfaaef9d1d457882afd31e9f7aa402a24b4" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.953ex; height:2.509ex;" alt="{\displaystyle \mathbf {P} =m\mathbf {\mathbf {v} } +q\mathbf {A} ,}"></span> </p><p>while in relativistic mechanics this becomes </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {P} =\gamma m\mathbf {\mathbf {v} } +q\mathbf {A} .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">P</mi> </mrow> <mo>=</mo> <mi>γ</mi> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> </mrow> <mo>+</mo> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {P} =\gamma m\mathbf {\mathbf {v} } +q\mathbf {A} .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9adf040ba5faf7e2c8b4b2d9126842872c80bcd2" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.215ex; height:2.676ex;" alt="{\displaystyle \mathbf {P} =\gamma m\mathbf {\mathbf {v} } +q\mathbf {A} .}"></span> </p><p>The quantity <span class="texhtml"><var style="padding-right: 1px;">V</var> = <var style="padding-right: 1px;">q</var><b>A</b></span> is sometimes called the <i>potential momentum</i>.<sup id="cite_ref-41" class="reference"><a href="#cite_note-41"><span class="cite-bracket">[</span>41<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-42" class="reference"><a href="#cite_note-42"><span class="cite-bracket">[</span>42<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-43" class="reference"><a href="#cite_note-43"><span class="cite-bracket">[</span>43<span class="cite-bracket">]</span></a></sup> It is the momentum due to the interaction of the particle with the electromagnetic fields. The name is an analogy with the potential energy <span class="texhtml"><var style="padding-right: 1px;">U</var> = <var style="padding-right: 1px;">q</var><var style="padding-right: 1px;">φ</var></span>, which is the energy due to the interaction of the particle with the electromagnetic fields. These quantities form a four-vector, so the analogy is consistent; besides, the concept of potential momentum is important in explaining the so-called <a href="/wiki/Hidden_momentum" title="Hidden momentum">hidden momentum</a> of the electromagnetic fields.<sup id="cite_ref-44" class="reference"><a href="#cite_note-44"><span class="cite-bracket">[</span>44<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading4"><h4 id="Conservation_2">Conservation</h4></div> <p>In Newtonian mechanics, the law of conservation of momentum can be derived from the <a href="/wiki/Law_of_action_and_reaction" class="mw-redirect" title="Law of action and reaction">law of action and reaction</a>, which states that every force has a reciprocating equal and opposite force. Under some circumstances, moving charged particles can exert forces on each other in non-opposite directions.<sup id="cite_ref-Griffiths_45-0" class="reference"><a href="#cite_note-Griffiths-45"><span class="cite-bracket">[</span>45<span class="cite-bracket">]</span></a></sup> Nevertheless, the combined momentum of the particles and the electromagnetic field is conserved. </p> <div class="mw-heading mw-heading5"><h5 id="Vacuum">Vacuum</h5></div> <p>The Lorentz force imparts a momentum to the particle, so by Newton's second law the particle must impart a momentum to the electromagnetic fields.<sup id="cite_ref-Jackson238_46-0" class="reference"><a href="#cite_note-Jackson238-46"><span class="cite-bracket">[</span>46<span class="cite-bracket">]</span></a></sup> </p><p>In a vacuum, the momentum per unit volume is </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {g} ={\frac {1}{\mu _{0}c^{2}}}\mathbf {E} \times \mathbf {B} \,,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">g</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> <mspace width="thinmathspace" /> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {g} ={\frac {1}{\mu _{0}c^{2}}}\mathbf {E} \times \mathbf {B} \,,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8c371084d5c2924bf8320bad230b7a1fac7b1460" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:17.321ex; height:5.843ex;" alt="{\displaystyle \mathbf {g} ={\frac {1}{\mu _{0}c^{2}}}\mathbf {E} \times \mathbf {B} \,,}"></span> </p><p>where <span class="texhtml"><var style="padding-right: 1px;">μ</var><sub>0</sub></span> is the <a href="/wiki/Vacuum_permeability" title="Vacuum permeability">vacuum permeability</a> and <span class="texhtml mvar" style="font-style:italic;">c</span> is the <a href="/wiki/Speed_of_light" title="Speed of light">speed of light</a>. The momentum density is proportional to the <a href="/wiki/Poynting_vector" title="Poynting vector">Poynting vector</a> <span class="texhtml"><b>S</b></span> which gives the directional rate of energy transfer per unit area:<sup id="cite_ref-Jackson238_46-1" class="reference"><a href="#cite_note-Jackson238-46"><span class="cite-bracket">[</span>46<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-FeynmanCh27_47-0" class="reference"><a href="#cite_note-FeynmanCh27-47"><span class="cite-bracket">[</span>47<span class="cite-bracket">]</span></a></sup> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {g} ={\frac {\mathbf {S} }{c^{2}}}\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">g</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">S</mi> </mrow> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {g} ={\frac {\mathbf {S} }{c^{2}}}\,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c0ddc861f562f9087b493317dc68094b75683f84" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:8.366ex; height:5.676ex;" alt="{\displaystyle \mathbf {g} ={\frac {\mathbf {S} }{c^{2}}}\,.}"></span> </p><p>If momentum is to be conserved over the volume <span class="texhtml mvar" style="font-style:italic;">V</span> over a region <span class="texhtml mvar" style="font-style:italic;">Q</span>, changes in the momentum of matter through the Lorentz force must be balanced by changes in the momentum of the electromagnetic field and outflow of momentum. If <span class="texhtml"><b>P</b><sub>mech</sub></span> is the momentum of all the particles in <span class="texhtml mvar" style="font-style:italic;">Q</span>, and the particles are treated as a continuum, then Newton's second law gives </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {{\text{d}}\mathbf {P} _{\text{mech}}}{{\text{d}}t}}=\iiint \limits _{Q}\left(\rho \mathbf {E} +\mathbf {J} \times \mathbf {B} \right){\text{d}}V\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>d</mtext> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">P</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>mech</mtext> </mrow> </msub> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>d</mtext> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <munder> <mo>∭</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>Q</mi> </mrow> </munder> <mrow> <mo>(</mo> <mrow> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">J</mi> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>d</mtext> </mrow> <mi>V</mi> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {{\text{d}}\mathbf {P} _{\text{mech}}}{{\text{d}}t}}=\iiint \limits _{Q}\left(\rho \mathbf {E} +\mathbf {J} \times \mathbf {B} \right){\text{d}}V\,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fde8523263b682f31bc9d09789f5cd90c27dfedb" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.338ex; width:34.248ex; height:7.843ex;" alt="{\displaystyle {\frac {{\text{d}}\mathbf {P} _{\text{mech}}}{{\text{d}}t}}=\iiint \limits _{Q}\left(\rho \mathbf {E} +\mathbf {J} \times \mathbf {B} \right){\text{d}}V\,.}"></span> </p><p>The electromagnetic momentum is </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {P} _{\text{field}}={\frac {1}{\mu _{0}c^{2}}}\iiint \limits _{Q}\mathbf {E} \times \mathbf {B} \,dV\,,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">P</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>field</mtext> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <munder> <mo>∭</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>Q</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <mi>V</mi> <mspace width="thinmathspace" /> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {P} _{\text{field}}={\frac {1}{\mu _{0}c^{2}}}\iiint \limits _{Q}\mathbf {E} \times \mathbf {B} \,dV\,,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7419ebbb4464057c334f377e37d3780f6b5ef82f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.338ex; width:29.869ex; height:7.676ex;" alt="{\displaystyle \mathbf {P} _{\text{field}}={\frac {1}{\mu _{0}c^{2}}}\iiint \limits _{Q}\mathbf {E} \times \mathbf {B} \,dV\,,}"></span> </p><p>and the equation for conservation of each component <span class="texhtml mvar" style="font-style:italic;">i</span> of the momentum is </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\text{d}}{{\text{d}}t}}\left(\mathbf {P} _{\text{mech}}+\mathbf {P} _{\text{field}}\right)_{i}=\iint \limits _{\sigma }\left(\sum \limits _{j}T_{ij}n_{j}\right){\text{d}}\Sigma \,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mtext>d</mtext> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>d</mtext> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <msub> <mrow> <mo>(</mo> <mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">P</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>mech</mtext> </mrow> </msub> <mo>+</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">P</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>field</mtext> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <munder> <mo>∬</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>σ</mi> </mrow> </munder> <mrow> <mo>(</mo> <mrow> <munder> <mo movablelimits="false">∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </munder> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>d</mtext> </mrow> <mi mathvariant="normal">Σ</mi> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\text{d}}{{\text{d}}t}}\left(\mathbf {P} _{\text{mech}}+\mathbf {P} _{\text{field}}\right)_{i}=\iint \limits _{\sigma }\left(\sum \limits _{j}T_{ij}n_{j}\right){\text{d}}\Sigma \,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/109082aa9621f1778eda695cd09e4ceb4184cf86" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.838ex; width:43.2ex; height:8.176ex;" alt="{\displaystyle {\frac {\text{d}}{{\text{d}}t}}\left(\mathbf {P} _{\text{mech}}+\mathbf {P} _{\text{field}}\right)_{i}=\iint \limits _{\sigma }\left(\sum \limits _{j}T_{ij}n_{j}\right){\text{d}}\Sigma \,.}"></span> </p><p>The term on the right is an integral over the surface area <span class="texhtml mvar" style="font-style:italic;">Σ</span> of the surface <span class="texhtml mvar" style="font-style:italic;">σ</span> representing momentum flow into and out of the volume, and <span class="texhtml"><var style="padding-right: 1px;">n</var><sub>j</sub></span> is a component of the surface normal of <span class="texhtml mvar" style="font-style:italic;">S</span>. The quantity <span class="texhtml"><var style="padding-right: 1px;">T</var><sub><var style="padding-right: 1px;">i</var><var style="padding-right: 1px;">j</var></sub></span> is called the <a href="/wiki/Maxwell_stress_tensor" title="Maxwell stress tensor">Maxwell stress tensor</a>, defined as<sup id="cite_ref-Jackson238_46-2" class="reference"><a href="#cite_note-Jackson238-46"><span class="cite-bracket">[</span>46<span class="cite-bracket">]</span></a></sup> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T_{ij}\equiv \epsilon _{0}\left(E_{i}E_{j}-{\frac {1}{2}}\delta _{ij}E^{2}\right)+{\frac {1}{\mu _{0}}}\left(B_{i}B_{j}-{\frac {1}{2}}\delta _{ij}B^{2}\right)\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>≡</mo> <msub> <mi>ϵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <msup> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <msup> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{ij}\equiv \epsilon _{0}\left(E_{i}E_{j}-{\frac {1}{2}}\delta _{ij}E^{2}\right)+{\frac {1}{\mu _{0}}}\left(B_{i}B_{j}-{\frac {1}{2}}\delta _{ij}B^{2}\right)\,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8efd050fa2761fbc80dbb43034fe8a3b038aabf7" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:53.842ex; height:6.176ex;" alt="{\displaystyle T_{ij}\equiv \epsilon _{0}\left(E_{i}E_{j}-{\frac {1}{2}}\delta _{ij}E^{2}\right)+{\frac {1}{\mu _{0}}}\left(B_{i}B_{j}-{\frac {1}{2}}\delta _{ij}B^{2}\right)\,.}"></span> </p> <div class="mw-heading mw-heading5"><h5 id="Media">Media</h5></div> <p>The above results are for the <i>microscopic</i> Maxwell equations, applicable to electromagnetic forces in a vacuum (or on a very small scale in media). It is more difficult to define momentum density in media because the division into electromagnetic and mechanical is arbitrary. The definition of electromagnetic momentum density is modified to </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {g} ={\frac {1}{c^{2}}}\mathbf {E} \times \mathbf {H} ={\frac {\mathbf {S} }{c^{2}}}\,,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">g</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">H</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">S</mi> </mrow> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mspace width="thinmathspace" /> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {g} ={\frac {1}{c^{2}}}\mathbf {E} \times \mathbf {H} ={\frac {\mathbf {S} }{c^{2}}}\,,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cca3cb60e3dc5a35e1770d30eeb906a0bf3ffaf2" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:21.051ex; height:5.676ex;" alt="{\displaystyle \mathbf {g} ={\frac {1}{c^{2}}}\mathbf {E} \times \mathbf {H} ={\frac {\mathbf {S} }{c^{2}}}\,,}"></span> </p><p>where the H-field <span class="texhtml"><b>H</b></span> is related to the B-field and the <a href="/wiki/Magnetization" title="Magnetization">magnetization</a> <span class="texhtml"><b>M</b></span> by </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {B} =\mu _{0}\left(\mathbf {H} +\mathbf {M} \right)\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> <mo>=</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">H</mi> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">M</mi> </mrow> </mrow> <mo>)</mo> </mrow> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {B} =\mu _{0}\left(\mathbf {H} +\mathbf {M} \right)\,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b7fa74e0861419d936cab17363805189acd3179e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.542ex; height:2.843ex;" alt="{\displaystyle \mathbf {B} =\mu _{0}\left(\mathbf {H} +\mathbf {M} \right)\,.}"></span> </p><p>The electromagnetic stress tensor depends on the properties of the media.<sup id="cite_ref-Jackson238_46-3" class="reference"><a href="#cite_note-Jackson238-46"><span class="cite-bracket">[</span>46<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Non-classical">Non-classical</h2></div> <div class="mw-heading mw-heading3"><h3 id="Quantum_mechanical">Quantum mechanical</h3></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Further information: <a href="/wiki/Momentum_operator" title="Momentum operator">Momentum operator</a></div> <p>In <a href="/wiki/Quantum_mechanics" title="Quantum mechanics">quantum mechanics</a>, momentum is defined as a <a href="/wiki/Self-adjoint_operator" title="Self-adjoint operator">self-adjoint operator</a> on the <a href="/wiki/Wave_function" title="Wave function">wave function</a>. The <a href="/wiki/Werner_Heisenberg" title="Werner Heisenberg">Heisenberg</a> <a href="/wiki/Uncertainty_principle" title="Uncertainty principle">uncertainty principle</a> defines limits on how accurately the momentum and position of a single observable system can be known at once. In quantum mechanics, position and momentum are <a href="/wiki/Conjugate_variables" title="Conjugate variables">conjugate variables</a>. </p><p>For a single particle described in the position basis the momentum operator can be written as </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {p} ={\hbar \over i}\nabla =-i\hbar \nabla \,,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi class="MJX-variant">ℏ</mi> <mi>i</mi> </mfrac> </mrow> <mi mathvariant="normal">∇</mi> <mo>=</mo> <mo>−</mo> <mi>i</mi> <mi class="MJX-variant">ℏ</mi> <mi mathvariant="normal">∇</mi> <mspace width="thinmathspace" /> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {p} ={\hbar \over i}\nabla =-i\hbar \nabla \,,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2ad37fde62d56189418758cf5dc239a5d1e5cb59" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:18.648ex; height:5.343ex;" alt="{\displaystyle \mathbf {p} ={\hbar \over i}\nabla =-i\hbar \nabla \,,}"></span> </p><p>where <span class="texhtml">∇</span> is the <a href="/wiki/Gradient" title="Gradient">gradient</a> operator, <span class="texhtml mvar" style="font-style:italic;">ħ</span> is the <a href="/wiki/Reduced_Planck_constant" class="mw-redirect" title="Reduced Planck constant">reduced Planck constant</a>, and <span class="texhtml mvar" style="font-style:italic;">i</span> is the <a href="/wiki/Imaginary_unit" title="Imaginary unit">imaginary unit</a>. This is a commonly encountered form of the momentum operator, though the momentum operator in other bases can take other forms. For example, in <a href="/wiki/Momentum_space" class="mw-redirect" title="Momentum space">momentum space</a> the momentum operator is represented by the <a href="/wiki/Eigenvalue" class="mw-redirect" title="Eigenvalue">eigenvalue</a> equation </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {p} \psi (p)=p\psi (p)\,,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>p</mi> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {p} \psi (p)=p\psi (p)\,,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b928c9445f9939bd3c0ba212c0b619750e259699" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.771ex; height:2.843ex;" alt="{\displaystyle \mathbf {p} \psi (p)=p\psi (p)\,,}"></span> </p><p>where the operator <span class="texhtml"><b>p</b></span> acting on a wave eigenfunction <span class="texhtml"><var style="padding-right: 1px;">ψ</var>(<var style="padding-right: 1px;">p</var>)</span> yields that wave function multiplied by the eigenvalue <span class="texhtml mvar" style="font-style:italic;">p</span>, in an analogous fashion to the way that the position operator acting on a wave function <span class="texhtml"><var style="padding-right: 1px;">ψ</var>(<var style="padding-right: 1px;">x</var>)</span> yields that wave function multiplied by the eigenvalue <span class="texhtml mvar" style="font-style:italic;">x</span>. </p><p>For both massive and massless objects, relativistic momentum is related to the <a href="/wiki/Phase_constant" class="mw-redirect" title="Phase constant">phase constant</a> <span class="texhtml mvar" style="font-style:italic;">β</span> by<sup id="cite_ref-48" class="reference"><a href="#cite_note-48"><span class="cite-bracket">[</span>48<span class="cite-bracket">]</span></a></sup> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p=\hbar \beta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>=</mo> <mi class="MJX-variant">ℏ</mi> <mi>β</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p=\hbar \beta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/808e88a516cd633249ab865b1c22a2d377d5ca00" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:6.996ex; height:2.509ex;" alt="{\displaystyle p=\hbar \beta }"></span> </p><p><a href="/wiki/Electromagnetic_radiation" title="Electromagnetic radiation">Electromagnetic radiation</a> (including <a href="/wiki/Light" title="Light">visible light</a>, <a href="/wiki/Ultraviolet" title="Ultraviolet">ultraviolet</a> light, and <a href="/wiki/Radio_waves" class="mw-redirect" title="Radio waves">radio waves</a>) is carried by <a href="/wiki/Photons" class="mw-redirect" title="Photons">photons</a>. Even though photons (the particle aspect of light) have no mass, they still carry momentum. This leads to applications such as the <a href="/wiki/Solar_sail" title="Solar sail">solar sail</a>. The calculation of the momentum of light within <a href="/wiki/Dielectric" title="Dielectric">dielectric</a> media is somewhat controversial (see <a href="/wiki/Abraham%E2%80%93Minkowski_controversy" title="Abraham–Minkowski controversy">Abraham–Minkowski controversy</a>).<sup id="cite_ref-49" class="reference"><a href="#cite_note-49"><span class="cite-bracket">[</span>49<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-50" class="reference"><a href="#cite_note-50"><span class="cite-bracket">[</span>50<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Relativistic">Relativistic</h3></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Mass_in_special_relativity" title="Mass in special relativity">Mass in special relativity</a> and <a href="/wiki/Tests_of_relativistic_energy_and_momentum" title="Tests of relativistic energy and momentum">Tests of relativistic energy and momentum</a></div> <div class="mw-heading mw-heading4"><h4 id="Lorentz_invariance">Lorentz invariance</h4></div> <p>Newtonian physics assumes that <a href="/wiki/Absolute_time_and_space" class="mw-redirect" title="Absolute time and space">absolute time and space</a> exist outside of any observer; this gives rise to <a href="/wiki/Galilean_invariance" title="Galilean invariance">Galilean invariance</a>. It also results in a prediction that the <a href="/wiki/Speed_of_light" title="Speed of light">speed of light</a> can vary from one reference frame to another. This is contrary to what has been observed. In the <a href="/wiki/Special_theory_of_relativity" class="mw-redirect" title="Special theory of relativity">special theory of relativity</a>, Einstein keeps the postulate that the equations of motion do not depend on the reference frame, but assumes that the speed of light <span class="texhtml mvar" style="font-style:italic;">c</span> is invariant. As a result, position and time in two reference frames are related by the <a href="/wiki/Lorentz_transformation" title="Lorentz transformation">Lorentz transformation</a> instead of the <a href="/wiki/Galilean_transformation" title="Galilean transformation">Galilean transformation</a>.<sup id="cite_ref-RindlerCh2_51-0" class="reference"><a href="#cite_note-RindlerCh2-51"><span class="cite-bracket">[</span>51<span class="cite-bracket">]</span></a></sup> </p><p>Consider, for example, one reference frame moving relative to another at velocity <span class="texhtml mvar" style="font-style:italic;">v</span> in the <span class="texhtml mvar" style="font-style:italic;">x</span> direction. The Galilean transformation gives the coordinates of the moving frame as </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}t'&=t\\x'&=x-vt\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msup> <mi>t</mi> <mo>′</mo> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>t</mi> </mtd> </mtr> <mtr> <mtd> <msup> <mi>x</mi> <mo>′</mo> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>x</mi> <mo>−</mo> <mi>v</mi> <mi>t</mi> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}t'&=t\\x'&=x-vt\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c037154888facbaaea129b6cb93719cc067fa800" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:12.002ex; height:5.843ex;" alt="{\displaystyle {\begin{aligned}t'&=t\\x'&=x-vt\end{aligned}}}"></span> </p><p>while the Lorentz transformation gives<sup id="cite_ref-FeynmanCh15_52-0" class="reference"><a href="#cite_note-FeynmanCh15-52"><span class="cite-bracket">[</span>52<span class="cite-bracket">]</span></a></sup> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}t'&=\gamma \left(t-{\frac {vx}{c^{2}}}\right)\\x'&=\gamma \left(x-vt\right)\,\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msup> <mi>t</mi> <mo>′</mo> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>γ</mi> <mrow> <mo>(</mo> <mrow> <mi>t</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>v</mi> <mi>x</mi> </mrow> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <msup> <mi>x</mi> <mo>′</mo> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>γ</mi> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>−</mo> <mi>v</mi> <mi>t</mi> </mrow> <mo>)</mo> </mrow> <mspace width="thinmathspace" /> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}t'&=\gamma \left(t-{\frac {vx}{c^{2}}}\right)\\x'&=\gamma \left(x-vt\right)\,\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bd5d38963e4c0c8c9b0aec9aca30e6c49ab68c34" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.788ex; margin-bottom: -0.217ex; width:17.908ex; height:9.176ex;" alt="{\displaystyle {\begin{aligned}t'&=\gamma \left(t-{\frac {vx}{c^{2}}}\right)\\x'&=\gamma \left(x-vt\right)\,\end{aligned}}}"></span> </p><p>where <span class="texhtml mvar" style="font-style:italic;">γ</span> is the <a href="/wiki/Lorentz_factor" title="Lorentz factor">Lorentz factor</a>: </p><p><span class="mwe-math-element" data-qid="Q599404"><a href="/w/index.php?title=Special:MathWikibase&qid=Q599404" style="color:inherit;"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma ={\frac {1}{\sqrt {1-v^{2}/c^{2}}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>γ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma ={\frac {1}{\sqrt {1-v^{2}/c^{2}}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cfe4e10cca53d8ca26a83eaf41c74c23b3ce4219" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:17.576ex; height:6.509ex;" alt="{\displaystyle \gamma ={\frac {1}{\sqrt {1-v^{2}/c^{2}}}}.}"></a></span> </p><p>Newton's second law, with mass fixed, is not invariant under a Lorentz transformation. However, it can be made invariant by making the <i>inertial mass</i> <span class="texhtml mvar" style="font-style:italic;">m</span> of an object a function of velocity: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m=\gamma m_{0}\,;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo>=</mo> <mi>γ</mi> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mspace width="thinmathspace" /> <mo>;</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m=\gamma m_{0}\,;}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e966a59aaab70d4dd8855d02c0da45afc9c97e2c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.53ex; height:2.176ex;" alt="{\displaystyle m=\gamma m_{0}\,;}"></span> </p><p><span class="texhtml"><var style="padding-right: 1px;">m</var><sub>0</sub></span> is the object's <a href="/wiki/Invariant_mass" title="Invariant mass">invariant mass</a>.<sup id="cite_ref-Rindler_53-0" class="reference"><a href="#cite_note-Rindler-53"><span class="cite-bracket">[</span>53<span class="cite-bracket">]</span></a></sup> </p><p>The modified momentum, </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {p} =\gamma m_{0}\mathbf {v} \,,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> <mo>=</mo> <mi>γ</mi> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mspace width="thinmathspace" /> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {p} =\gamma m_{0}\mathbf {v} \,,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d2c196a3d5f67016106e1b305fc89cf06e914b8a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.386ex; height:2.176ex;" alt="{\displaystyle \mathbf {p} =\gamma m_{0}\mathbf {v} \,,}"></span> </p><p>obeys Newton's second law: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {F} ={\frac {d\mathbf {p} }{dt}}\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {F} ={\frac {d\mathbf {p} }{dt}}\,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/99ecbac6df650ec669de4173a064ab3824920644" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:9.352ex; height:5.509ex;" alt="{\displaystyle \mathbf {F} ={\frac {d\mathbf {p} }{dt}}\,.}"></span> </p><p>Within the domain of classical mechanics, relativistic momentum closely approximates Newtonian momentum: at low velocity, <span class="texhtml"><var style="padding-right: 1px;">γ</var><var style="padding-right: 1px;">m</var><sub>0</sub><b>v</b></span> is approximately equal to <span class="texhtml"><var style="padding-right: 1px;">m</var><sub>0</sub><b>v</b></span>, the Newtonian expression for momentum. </p> <div class="mw-heading mw-heading4"><h4 id="Four-vector_formulation">Four-vector formulation</h4></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Four-momentum" title="Four-momentum">Four-momentum</a></div> <p>In the theory of special relativity, physical quantities are expressed in terms of <a href="/wiki/Four-vector" title="Four-vector">four-vectors</a> that include time as a fourth coordinate along with the three space coordinates. These vectors are generally represented by capital letters, for example <span class="texhtml"><b>R</b></span> for position. The expression for the <i>four-momentum</i> depends on how the coordinates are expressed. Time may be given in its normal units or multiplied by the speed of light so that all the components of the four-vector have dimensions of length. If the latter scaling is used, an interval of <a href="/wiki/Proper_time" title="Proper time">proper time</a>, <span class="texhtml mvar" style="font-style:italic;">τ</span>, defined by<sup id="cite_ref-54" class="reference"><a href="#cite_note-54"><span class="cite-bracket">[</span>54<span class="cite-bracket">]</span></a></sup> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c^{2}{\text{d}}\tau ^{2}=c^{2}{\text{d}}t^{2}-{\text{d}}x^{2}-{\text{d}}y^{2}-{\text{d}}z^{2}\,,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mtext>d</mtext> </mrow> <msup> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mtext>d</mtext> </mrow> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>d</mtext> </mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>d</mtext> </mrow> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>d</mtext> </mrow> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="thinmathspace" /> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c^{2}{\text{d}}\tau ^{2}=c^{2}{\text{d}}t^{2}-{\text{d}}x^{2}-{\text{d}}y^{2}-{\text{d}}z^{2}\,,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0095543d611a232a4ba8f049fddec7b8264e00ba" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:34.187ex; height:3.009ex;" alt="{\displaystyle c^{2}{\text{d}}\tau ^{2}=c^{2}{\text{d}}t^{2}-{\text{d}}x^{2}-{\text{d}}y^{2}-{\text{d}}z^{2}\,,}"></span> </p><p>is <a href="/wiki/Invariant_(physics)" title="Invariant (physics)">invariant</a> under Lorentz transformations (in this expression and in what follows the <span class="nowrap">(+ − − −)</span> <a href="/wiki/Metric_signature" title="Metric signature">metric signature</a> has been used, different authors use different conventions). Mathematically this invariance can be ensured in one of two ways: by treating the four-vectors as <a href="/wiki/Euclidean_vector" title="Euclidean vector">Euclidean vectors</a> and multiplying time by <span class="texhtml"><a href="/wiki/Imaginary_unit" title="Imaginary unit"><span class="nowrap">√<span style="border-top:1px solid; padding:0 0.1em;">−1</span></span></a></span>; or by keeping time a real quantity and embedding the vectors in a <a href="/wiki/Minkowski_space" title="Minkowski space">Minkowski space</a>.<sup id="cite_ref-55" class="reference"><a href="#cite_note-55"><span class="cite-bracket">[</span>55<span class="cite-bracket">]</span></a></sup> In a Minkowski space, the <a href="/wiki/Scalar_product" class="mw-redirect" title="Scalar product">scalar product</a> of two four-vectors <span class="texhtml"><b>U</b> = (<var style="padding-right: 1px;">U</var><sub>0</sub>, <var style="padding-right: 1px;">U</var><sub>1</sub>, <var style="padding-right: 1px;">U</var><sub>2</sub>, <var style="padding-right: 1px;">U</var><sub>3</sub>)</span> and <span class="texhtml"><b>V</b> = (<var style="padding-right: 1px;">V</var><sub>0</sub>, <var style="padding-right: 1px;">V</var><sub>1</sub>, <var style="padding-right: 1px;">V</var><sub>2</sub>, <var style="padding-right: 1px;">V</var><sub>3</sub>)</span> is defined as </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {U} \cdot \mathbf {V} =U_{0}V_{0}-U_{1}V_{1}-U_{2}V_{2}-U_{3}V_{3}\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">U</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">V</mi> </mrow> <mo>=</mo> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {U} \cdot \mathbf {V} =U_{0}V_{0}-U_{1}V_{1}-U_{2}V_{2}-U_{3}V_{3}\,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7c3dfb58b4e97dd14acfa16692a0a639a3588979" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:38.614ex; height:2.509ex;" alt="{\displaystyle \mathbf {U} \cdot \mathbf {V} =U_{0}V_{0}-U_{1}V_{1}-U_{2}V_{2}-U_{3}V_{3}\,.}"></span> </p><p>In all the coordinate systems, the (<a href="/wiki/Covariance_and_contravariance_of_vectors" title="Covariance and contravariance of vectors">contravariant</a>) relativistic four-velocity is defined by </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {U} \equiv {\frac {{\text{d}}\mathbf {R} }{{\text{d}}\tau }}=\gamma {\frac {{\text{d}}\mathbf {R} }{{\text{d}}t}}\,,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">U</mi> </mrow> <mo>≡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>d</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>d</mtext> </mrow> <mi>τ</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>d</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>d</mtext> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <mspace width="thinmathspace" /> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {U} \equiv {\frac {{\text{d}}\mathbf {R} }{{\text{d}}\tau }}=\gamma {\frac {{\text{d}}\mathbf {R} }{{\text{d}}t}}\,,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0c9b6a92fa2321a83a2113c34322182ff6c2dbb5" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:18.814ex; height:5.509ex;" alt="{\displaystyle \mathbf {U} \equiv {\frac {{\text{d}}\mathbf {R} }{{\text{d}}\tau }}=\gamma {\frac {{\text{d}}\mathbf {R} }{{\text{d}}t}}\,,}"></span> </p><p>and the (contravariant) <a href="/wiki/Four-momentum" title="Four-momentum">four-momentum</a> is </p><p><span class="mwe-math-element" data-qid="Q1068463"><a href="/w/index.php?title=Special:MathWikibase&qid=Q1068463" style="color:inherit;"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {P} =m_{0}\mathbf {U} \,,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">P</mi> </mrow> <mo>=</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">U</mi> </mrow> <mspace width="thinmathspace" /> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {P} =m_{0}\mathbf {U} \,,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/40cec5840d1b8724ce04e95c8b428575e3060fb8" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.11ex; height:2.509ex;" alt="{\displaystyle \mathbf {P} =m_{0}\mathbf {U} \,,}"></a></span> </p><p>where <span class="texhtml"><var style="padding-right: 1px;">m</var><sub>0</sub></span> is the invariant mass. If <span class="texhtml"><b>R</b> = (<var style="padding-right: 1px;">c</var><var style="padding-right: 1px;">t</var>, <var style="padding-right: 1px;">x</var>, <var style="padding-right: 1px;">y</var>, <var style="padding-right: 1px;">z</var>)</span> (in Minkowski space), then </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {P} =\gamma m_{0}\left(c,\mathbf {v} \right)=(mc,\mathbf {p} )\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">P</mi> </mrow> <mo>=</mo> <mi>γ</mi> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <mi>c</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <mi>m</mi> <mi>c</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {P} =\gamma m_{0}\left(c,\mathbf {v} \right)=(mc,\mathbf {p} )\,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2bae79ae206d7d8da13d9d9110efc7403608a6ef" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.438ex; height:2.843ex;" alt="{\displaystyle \mathbf {P} =\gamma m_{0}\left(c,\mathbf {v} \right)=(mc,\mathbf {p} )\,.}"></span> </p><p>Using Einstein's <a href="/wiki/Mass%E2%80%93energy_equivalence" title="Mass–energy equivalence">mass–energy equivalence</a>, <span class="texhtml"><var style="padding-right: 1px;">E</var> = <var style="padding-right: 1px;">m</var><var style="padding-right: 1px;">c</var><sup>2</sup></span>, this can be rewritten as </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {P} =\left({\frac {E}{c}},\mathbf {p} \right)\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">P</mi> </mrow> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>E</mi> <mi>c</mi> </mfrac> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> </mrow> <mo>)</mo> </mrow> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {P} =\left({\frac {E}{c}},\mathbf {p} \right)\,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8675fb30bbff8e7d4152dd5c16e348ba79f9c976" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:14.898ex; height:6.176ex;" alt="{\displaystyle \mathbf {P} =\left({\frac {E}{c}},\mathbf {p} \right)\,.}"></span> </p><p>Thus, conservation of four-momentum is Lorentz-invariant and implies conservation of both mass and energy. </p><p>The magnitude of the momentum four-vector is equal to <span class="texhtml"><var style="padding-right: 1px;">m</var><sub>0</sub><var style="padding-right: 1px;">c</var></span>: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|\mathbf {P} \|^{2}=\mathbf {P} \cdot \mathbf {P} =\gamma ^{2}m_{0}^{2}\left(c^{2}-v^{2}\right)=(m_{0}c)^{2}\,,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">P</mi> </mrow> <msup> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">P</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">P</mi> </mrow> <mo>=</mo> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msubsup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mrow> <mo>(</mo> <mrow> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mi>c</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="thinmathspace" /> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|\mathbf {P} \|^{2}=\mathbf {P} \cdot \mathbf {P} =\gamma ^{2}m_{0}^{2}\left(c^{2}-v^{2}\right)=(m_{0}c)^{2}\,,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a77a8fea2237b1c59ee6bf950a1f0e89775fc402" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:42.862ex; height:3.343ex;" alt="{\displaystyle \|\mathbf {P} \|^{2}=\mathbf {P} \cdot \mathbf {P} =\gamma ^{2}m_{0}^{2}\left(c^{2}-v^{2}\right)=(m_{0}c)^{2}\,,}"></span> </p><p>and is invariant across all reference frames. </p><p>The relativistic energy–momentum relationship holds even for massless particles such as photons; by setting <span class="texhtml"><var style="padding-right: 1px;">m</var><sub>0</sub> = 0</span> it follows that </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E=pc\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo>=</mo> <mi>p</mi> <mi>c</mi> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E=pc\,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c2e55dc867ed9d89166a2c7318f609e0cd5ff911" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.084ex; height:2.509ex;" alt="{\displaystyle E=pc\,.}"></span> </p><p>In a game of relativistic "billiards", if a stationary particle is hit by a moving particle in an elastic collision, the paths formed by the two afterwards will form an acute angle. This is unlike the non-relativistic case where they travel at right angles.<sup id="cite_ref-56" class="reference"><a href="#cite_note-56"><span class="cite-bracket">[</span>56<span class="cite-bracket">]</span></a></sup> </p><p>The four-momentum of a planar wave can be related to a wave four-vector<sup id="cite_ref-57" class="reference"><a href="#cite_note-57"><span class="cite-bracket">[</span>57<span class="cite-bracket">]</span></a></sup> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {P} =\left({\frac {E}{c}},{\vec {\mathbf {p} }}\right)=\hbar \mathbf {K} =\hbar \left({\frac {\omega }{c}},{\vec {\mathbf {k} }}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">P</mi> </mrow> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>E</mi> <mi>c</mi> </mfrac> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mi class="MJX-variant">ℏ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">K</mi> </mrow> <mo>=</mo> <mi class="MJX-variant">ℏ</mi> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ω</mi> <mi>c</mi> </mfrac> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">k</mi> </mrow> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {P} =\left({\frac {E}{c}},{\vec {\mathbf {p} }}\right)=\hbar \mathbf {K} =\hbar \left({\frac {\omega }{c}},{\vec {\mathbf {k} }}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aa7e3358ef32788e1fe4d339c6b47a90b379bddc" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:32.27ex; height:6.176ex;" alt="{\displaystyle \mathbf {P} =\left({\frac {E}{c}},{\vec {\mathbf {p} }}\right)=\hbar \mathbf {K} =\hbar \left({\frac {\omega }{c}},{\vec {\mathbf {k} }}\right)}"></span> </p><p>For a particle, the relationship between temporal components, <span class="texhtml"><var style="padding-right: 1px;">E</var> = <var style="padding-right: 1px;">ħ</var><var style="padding-right: 1px;">ω</var></span>, is the <a href="/wiki/Planck%E2%80%93Einstein_relation" class="mw-redirect" title="Planck–Einstein relation">Planck–Einstein relation</a>, and the relation between spatial components, <span class="texhtml"><b>p</b> = <var style="padding-right: 1px;">ħ</var><b>k</b></span>, describes a <a href="/wiki/De_Broglie" class="mw-redirect" title="De Broglie">de Broglie</a> <a href="/wiki/Matter_wave" title="Matter wave">matter wave</a>. </p> <div class="mw-heading mw-heading2"><h2 id="History_of_the_concept">History of the concept</h2></div> <div class="mw-heading mw-heading3"><h3 id="Impetus">Impetus</h3></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Theory_of_impetus" title="Theory of impetus">Theory of impetus</a></div> <div class="mw-heading mw-heading4"><h4 id="John_Philoponus">John Philoponus</h4></div> <p>In about 530 AD, <a href="/wiki/John_Philoponus" title="John Philoponus">John Philoponus</a> developed a concept of momentum in <i>On Physics</i>, a commentary to <a href="/wiki/Aristotle" title="Aristotle">Aristotle</a>'s <i><a href="/wiki/Physics_(Aristotle)" title="Physics (Aristotle)">Physics</a></i>. Aristotle claimed that everything that is moving must be kept moving by something. For example, a thrown ball must be kept moving by motions of the air. Philoponus pointed out the absurdity in Aristotle's claim that motion of an object is promoted by the same air that is resisting its passage. He proposed instead that an impetus was imparted to the object in the act of throwing it.<sup id="cite_ref-58" class="reference"><a href="#cite_note-58"><span class="cite-bracket">[</span>58<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading4"><h4 id="Ibn_Sīnā"><span id="Ibn_S.C4.ABn.C4.81"></span>Ibn Sīnā</h4></div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:1950_%22Avicenna%22_stamp_of_Iran_(cropped).jpg" class="mw-file-description"><img alt="Engraving of Ibn Sīnā" src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ea/1950_%22Avicenna%22_stamp_of_Iran_%28cropped%29.jpg/150px-1950_%22Avicenna%22_stamp_of_Iran_%28cropped%29.jpg" decoding="async" width="150" height="195" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ea/1950_%22Avicenna%22_stamp_of_Iran_%28cropped%29.jpg/225px-1950_%22Avicenna%22_stamp_of_Iran_%28cropped%29.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/e/ea/1950_%22Avicenna%22_stamp_of_Iran_%28cropped%29.jpg 2x" data-file-width="251" data-file-height="327" /></a><figcaption>Ibn Sīnā<br />(980–1037)</figcaption></figure> <p>In 1020, <a href="/wiki/Avicenna" title="Avicenna">Ibn Sīnā</a> (also known by his <a href="/wiki/Latinisation_of_names" title="Latinisation of names">Latinized</a> name Avicenna) read Philoponus and published his own theory of motion in <i><a href="/wiki/The_Book_of_Healing" title="The Book of Healing">The Book of Healing</a></i>. He agreed that an impetus is imparted to a projectile by the thrower; but unlike Philoponus, who believed that it was a temporary virtue that would decline even in a vacuum, he viewed it as a persistent, requiring external forces such as <a href="/wiki/Air_resistance" class="mw-redirect" title="Air resistance">air resistance</a> to dissipate it.<sup id="cite_ref-Espinoza_59-0" class="reference"><a href="#cite_note-Espinoza-59"><span class="cite-bracket">[</span>59<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Nasr_60-0" class="reference"><a href="#cite_note-Nasr-60"><span class="cite-bracket">[</span>60<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Sayili_61-0" class="reference"><a href="#cite_note-Sayili-61"><span class="cite-bracket">[</span>61<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading4"><h4 id="Peter_Olivi,_Jean_Buridan"><span id="Peter_Olivi.2C_Jean_Buridan"></span>Peter Olivi, Jean Buridan</h4></div> <p>In the 13th and 14th century, <a href="/wiki/Peter_Olivi" class="mw-redirect" title="Peter Olivi">Peter Olivi</a> and <a href="/wiki/Jean_Buridan" title="Jean Buridan">Jean Buridan</a> read and refined the work of Philoponus, and possibly that of Ibn Sīnā.<sup id="cite_ref-Sayili_61-1" class="reference"><a href="#cite_note-Sayili-61"><span class="cite-bracket">[</span>61<span class="cite-bracket">]</span></a></sup> Buridan, who in about 1350 was made rector of the University of Paris, referred to <a href="/wiki/Theory_of_impetus" title="Theory of impetus">impetus</a> being proportional to the weight times the speed. Moreover, Buridan's theory was different from his predecessor's in that he did not consider impetus to be self-dissipating, asserting that a body would be arrested by the forces of air resistance and gravity which might be opposing its impetus.<sup id="cite_ref-62" class="reference"><a href="#cite_note-62"><span class="cite-bracket">[</span>62<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Park_63-0" class="reference"><a href="#cite_note-Park-63"><span class="cite-bracket">[</span>63<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Quantity_of_motion">Quantity of motion<span class="anchor" id="Quantity_of_motion"></span></h3></div> <div class="mw-heading mw-heading4"><h4 id="René_Descartes"><span id="Ren.C3.A9_Descartes"></span>René Descartes</h4></div><p> In <i><a href="/wiki/Principles_of_Philosophy" title="Principles of Philosophy">Principles of Philosophy</a></i> (<i>Principia Philosophiae</i>) from 1644, the French philosopher <a href="/wiki/Ren%C3%A9_Descartes" title="René Descartes">René Descartes</a> defined "quantity of motion" (<i><a href="/wiki/Latin_language" class="mw-redirect" title="Latin language">Latin</a>: quantitas motus</i>) as the product of size and speed,<sup id="cite_ref-:0_64-0" class="reference"><a href="#cite_note-:0-64"><span class="cite-bracket">[</span>64<span class="cite-bracket">]</span></a></sup> and claimed that the total quantity of motion in the universe is conserved.<sup id="cite_ref-:0_64-1" class="reference"><a href="#cite_note-:0-64"><span class="cite-bracket">[</span>64<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-65" class="reference"><a href="#cite_note-65"><span class="cite-bracket">[</span>65<span class="cite-bracket">]</span></a></sup></p><figure typeof="mw:File/Thumb"><a href="/wiki/File:Frans_Hals_-_Portret_van_Ren%C3%A9_Descartes_(cropped)2.jpg" class="mw-file-description"><img alt="Portrait of René Descartes" src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b1/Frans_Hals_-_Portret_van_Ren%C3%A9_Descartes_%28cropped%292.jpg/150px-Frans_Hals_-_Portret_van_Ren%C3%A9_Descartes_%28cropped%292.jpg" decoding="async" width="150" height="153" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b1/Frans_Hals_-_Portret_van_Ren%C3%A9_Descartes_%28cropped%292.jpg/225px-Frans_Hals_-_Portret_van_Ren%C3%A9_Descartes_%28cropped%292.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b1/Frans_Hals_-_Portret_van_Ren%C3%A9_Descartes_%28cropped%292.jpg/299px-Frans_Hals_-_Portret_van_Ren%C3%A9_Descartes_%28cropped%292.jpg 2x" data-file-width="478" data-file-height="489" /></a><figcaption>René Descartes<br />(1596–1650)</figcaption></figure><style data-mw-deduplicate="TemplateStyles:r1244412712">.mw-parser-output .templatequote{overflow:hidden;margin:1em 0;padding:0 32px}.mw-parser-output .templatequotecite{line-height:1.5em;text-align:left;margin-top:0}@media(min-width:500px){.mw-parser-output .templatequotecite{padding-left:1.6em}}</style><blockquote class="templatequote"><p>If x is twice the size of y, and is moving half as fast, then there's the same amount of motion in each.</p></blockquote><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1244412712"><blockquote class="templatequote"><p>[God] created matter, along with its motion ... merely by letting things run their course, he preserves the same amount of motion ... as he put there in the beginning.</p></blockquote> <p>This should not be read as a statement of the modern law of <a href="/wiki/Conservation_of_momentum" class="mw-redirect" title="Conservation of momentum">conservation of momentum</a>, since Descartes had no concept of mass as distinct from weight and size. (The concept of mass, as distinct from weight, was introduced by Newton in 1686.)<sup id="cite_ref-66" class="reference"><a href="#cite_note-66"><span class="cite-bracket">[</span>66<span class="cite-bracket">]</span></a></sup> More important, he believed that it is speed rather than velocity that is conserved. So for Descartes, if a moving object were to bounce off a surface, changing its direction but not its speed, there would be no change in its quantity of motion.<sup id="cite_ref-67" class="reference"><a href="#cite_note-67"><span class="cite-bracket">[</span>67<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-68" class="reference"><a href="#cite_note-68"><span class="cite-bracket">[</span>68<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-69" class="reference"><a href="#cite_note-69"><span class="cite-bracket">[</span>69<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Galileo" class="mw-redirect" title="Galileo">Galileo</a>, in his <i><a href="/wiki/Two_New_Sciences" title="Two New Sciences">Two New Sciences</a></i> (published in 1638), used the <a href="/wiki/Italian_language" title="Italian language">Italian</a> word <span title="Italian-language text"><i lang="it">impeto</i></span> to similarly describe Descartes's quantity of motion. </p> <div class="mw-heading mw-heading4"><h4 id="Christiaan_Huygens">Christiaan Huygens</h4></div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Christiaan_Huygens-painting_(cropped).jpeg" class="mw-file-description"><img alt="Portrait of Christiaan Huygens" src="//upload.wikimedia.org/wikipedia/commons/thumb/b/bc/Christiaan_Huygens-painting_%28cropped%29.jpeg/150px-Christiaan_Huygens-painting_%28cropped%29.jpeg" decoding="async" width="150" height="155" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/bc/Christiaan_Huygens-painting_%28cropped%29.jpeg/226px-Christiaan_Huygens-painting_%28cropped%29.jpeg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/bc/Christiaan_Huygens-painting_%28cropped%29.jpeg/301px-Christiaan_Huygens-painting_%28cropped%29.jpeg 2x" data-file-width="765" data-file-height="789" /></a><figcaption>Christiaan Huygens<br />(1629–1695)</figcaption></figure> <p>In the 1600s, <a href="/wiki/Christiaan_Huygens" title="Christiaan Huygens">Christiaan Huygens</a> concluded quite early that <a href="/wiki/Cartesian_laws_of_motion" class="mw-redirect" title="Cartesian laws of motion">Descartes's laws</a> for the elastic collision of two bodies must be wrong, and he formulated the correct laws.<sup id="cite_ref-70" class="reference"><a href="#cite_note-70"><span class="cite-bracket">[</span>70<span class="cite-bracket">]</span></a></sup> An important step was his recognition of the <a href="/wiki/Galilean_invariance" title="Galilean invariance">Galilean invariance</a> of the problems.<sup id="cite_ref-71" class="reference"><a href="#cite_note-71"><span class="cite-bracket">[</span>71<span class="cite-bracket">]</span></a></sup> His views then took many years to be circulated. He passed them on in person to <a href="/wiki/William_Brouncker,_2nd_Viscount_Brouncker" title="William Brouncker, 2nd Viscount Brouncker">William Brouncker</a> and <a href="/wiki/Christopher_Wren" title="Christopher Wren">Christopher Wren</a> in London, in 1661.<sup id="cite_ref-72" class="reference"><a href="#cite_note-72"><span class="cite-bracket">[</span>72<span class="cite-bracket">]</span></a></sup> What Spinoza wrote to <a href="/wiki/Henry_Oldenburg" title="Henry Oldenburg">Henry Oldenburg</a> about them, in 1666 during the <a href="/wiki/Second_Anglo-Dutch_War" title="Second Anglo-Dutch War">Second Anglo-Dutch War</a>, was guarded.<sup id="cite_ref-Israel2001_73-0" class="reference"><a href="#cite_note-Israel2001-73"><span class="cite-bracket">[</span>73<span class="cite-bracket">]</span></a></sup> Huygens had actually worked them out in a manuscript <span title="Latin-language text"><i lang="la">De motu corporum ex percussione</i></span> in the period 1652–1656. The war ended in 1667, and Huygens announced his results to the Royal Society in 1668. He published them in the <span title="French-language text"><i lang="fr"><a href="/wiki/Journal_des_s%C3%A7avans" title="Journal des sçavans">Journal des sçavans</a></i></span> in 1669.<sup id="cite_ref-74" class="reference"><a href="#cite_note-74"><span class="cite-bracket">[</span>74<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Momentum">Momentum</h3></div> <div class="mw-heading mw-heading4"><h4 id="John_Wallis">John Wallis</h4></div> <p>In 1670, <a href="/wiki/John_Wallis" title="John Wallis">John Wallis</a>, in <span title="Latin-language text"><i lang="la">Mechanica sive De Motu, Tractatus Geometricus</i></span>, stated the law of conservation of momentum: "the initial state of the body, either of rest or of motion, will persist" and "If the force is greater than the resistance, motion will result".<sup id="cite_ref-75" class="reference"><a href="#cite_note-75"><span class="cite-bracket">[</span>75<span class="cite-bracket">]</span></a></sup> Wallis used <i>momentum</i> for quantity of motion, and <span title="Latin-language text"><i lang="la">vis</i></span> for force. </p> <div class="mw-heading mw-heading4"><h4 id="Gottfried_Leibniz">Gottfried Leibniz</h4></div> <p>In 1686, <a href="/wiki/Gottfried_Wilhelm_Leibniz" title="Gottfried Wilhelm Leibniz">Gottfried Wilhelm Leibniz</a>, in <i><a href="/wiki/Discourse_on_Metaphysics" title="Discourse on Metaphysics">Discourse on Metaphysics</a></i>, gave an argument against Descartes' construction of the conservation of the "quantity of motion" using an example of dropping blocks of different sizes different distances. He points out that force is conserved but quantity of motion, construed as the product of size and speed of an object, is not conserved.<sup id="cite_ref-76" class="reference"><a href="#cite_note-76"><span class="cite-bracket">[</span>76<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading4"><h4 id="Isaac_Newton">Isaac Newton</h4></div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Isaac_Newton_by_James_Thronill,_after_Sir_Godfrey_Kneller.jpg" class="mw-file-description"><img alt="Portrait of Isaac Newton by James Thronill, after Sir Godfrey Kneller" src="//upload.wikimedia.org/wikipedia/commons/thumb/5/59/Isaac_Newton_by_James_Thronill%2C_after_Sir_Godfrey_Kneller.jpg/150px-Isaac_Newton_by_James_Thronill%2C_after_Sir_Godfrey_Kneller.jpg" decoding="async" width="150" height="187" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/59/Isaac_Newton_by_James_Thronill%2C_after_Sir_Godfrey_Kneller.jpg/225px-Isaac_Newton_by_James_Thronill%2C_after_Sir_Godfrey_Kneller.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/59/Isaac_Newton_by_James_Thronill%2C_after_Sir_Godfrey_Kneller.jpg/299px-Isaac_Newton_by_James_Thronill%2C_after_Sir_Godfrey_Kneller.jpg 2x" data-file-width="1162" data-file-height="1452" /></a><figcaption>Isaac Newton<br />(1642–1727)</figcaption></figure> <p>In 1687, <a href="/wiki/Isaac_Newton" title="Isaac Newton">Isaac Newton</a>, in <span title="Latin-language text"><i lang="la"><a href="/wiki/Philosophi%C3%A6_Naturalis_Principia_Mathematica" title="Philosophiæ Naturalis Principia Mathematica">Philosophiæ Naturalis Principia Mathematica</a></i></span>, just like Wallis, showed a similar casting around for words to use for the mathematical momentum. His Definition II defines <span title="Latin-language text"><i lang="la">quantitas motus</i></span>, "quantity of motion", as "arising from the velocity and quantity of matter conjointly", which identifies it as momentum.<sup id="cite_ref-77" class="reference"><a href="#cite_note-77"><span class="cite-bracket">[</span>77<span class="cite-bracket">]</span></a></sup> Thus when in Law II he refers to <span title="Latin-language text"><i lang="la">mutatio motus</i></span>, "change of motion", being proportional to the force impressed, he is generally taken to mean momentum and not motion.<sup id="cite_ref-78" class="reference"><a href="#cite_note-78"><span class="cite-bracket">[</span>78<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading4"><h4 id="John_Jennings">John Jennings</h4></div> <p>In 1721, <a href="/wiki/John_Jennings_(tutor)" title="John Jennings (tutor)">John Jennings</a> published <i>Miscellanea</i>, where the momentum in its current mathematical sense is attested, five years before the final edition of Newton's <span title="Latin-language text"><i lang="la">Principia Mathematica</i></span>. <i>Momentum</i> <span class="texhtml">M</span> or "quantity of motion" was being defined for students as "a rectangle", the product of <span class="texhtml mvar" style="font-style:italic;">Q</span> and <span class="texhtml mvar" style="font-style:italic;">V</span>, where <span class="texhtml mvar" style="font-style:italic;">Q</span> is "quantity of material" and <span class="texhtml mvar" style="font-style:italic;">V</span> is "velocity", <span class="texhtml"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num"><var style="padding-right: 1px;">s</var></span><span class="sr-only">/</span><span class="den"><var style="padding-right: 1px;">t</var></span></span>⁠</span></span>.<sup id="cite_ref-79" class="reference"><a href="#cite_note-79"><span class="cite-bracket">[</span>79<span class="cite-bracket">]</span></a></sup> </p><p>In 1728, the <a href="/wiki/Cyclop%C3%A6dia,_or_an_Universal_Dictionary_of_Arts_and_Sciences" title="Cyclopædia, or an Universal Dictionary of Arts and Sciences">Cyclopedia</a> states: </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1244412712"><blockquote class="templatequote"><p>The <i>Momentum</i>, <i>Impetus</i>, or Quantity of Motion of any Body, is the <i>Factum</i> [i.e., product] of its Velocity, (or the Space it moves in a given Time, see <style data-mw-deduplicate="TemplateStyles:r920966791">.mw-parser-output span.smallcaps{font-variant:small-caps}.mw-parser-output span.smallcaps-smaller{font-size:85%}</style><span class="smallcaps smallcaps-smaller">Motion</span>) multiplied into its Mass.</p></blockquote> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2></div> <style 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class="portalbox-image"><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Stylised_atom_with_three_Bohr_model_orbits_and_stylised_nucleus.svg" class="mw-file-description"><img alt="icon" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/6f/Stylised_atom_with_three_Bohr_model_orbits_and_stylised_nucleus.svg/25px-Stylised_atom_with_three_Bohr_model_orbits_and_stylised_nucleus.svg.png" decoding="async" width="25" height="28" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/6f/Stylised_atom_with_three_Bohr_model_orbits_and_stylised_nucleus.svg/37px-Stylised_atom_with_three_Bohr_model_orbits_and_stylised_nucleus.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/6f/Stylised_atom_with_three_Bohr_model_orbits_and_stylised_nucleus.svg/49px-Stylised_atom_with_three_Bohr_model_orbits_and_stylised_nucleus.svg.png 2x" data-file-width="530" data-file-height="600" /></a></span></span><span class="portalbox-link"><a href="/wiki/Portal:Physics" title="Portal:Physics">Physics portal</a></span></li></ul> <style data-mw-deduplicate="TemplateStyles:r1184024115">.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}</style><div class="div-col"> <ul><li><a href="/wiki/Angular_momentum" title="Angular momentum">Angular momentum</a></li> <li><a href="/wiki/Crystal_momentum" title="Crystal momentum">Crystal momentum</a></li> <li><a href="/wiki/Galilean_cannon" title="Galilean cannon">Galilean cannon</a></li> <li><a href="/wiki/Momentum_compaction" title="Momentum compaction">Momentum compaction</a></li> <li><a href="/wiki/Momentum_transfer" title="Momentum transfer">Momentum transfer</a></li> <li><a href="/wiki/Newton%27s_cradle" title="Newton's cradle">Newton's cradle</a></li> <li><a href="/wiki/Position_and_momentum_space" class="mw-redirect" title="Position and momentum space">Position and momentum space</a></li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-FeynmanCh9-1"><span class="mw-cite-backlink">^ <a href="#cite_ref-FeynmanCh9_1-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-FeynmanCh9_1-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-FeynmanCh9_1-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text"><a rel="nofollow" class="external text" href="https://feynmanlectures.caltech.edu/I_09.html"><i>The Feynman Lectures on Physics</i></a> Vol. I Ch. 9: Newton's Laws of Dynamics</span> </li> <li id="cite_note-BookRags-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-BookRags_2-0">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite class="citation book cs1"><a rel="nofollow" class="external text" href="http://www.bookrags.com/research/eulers-laws-of-motion-wom/"><i>Euler's Laws of Motion</i></a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20090710162552/http://www.bookrags.com/research/eulers-laws-of-motion-wom/">Archived</a> from the original on 2009-07-10<span class="reference-accessdate">. Retrieved <span class="nowrap">2009-03-30</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Euler%27s+Laws+of+Motion&rft_id=http%3A%2F%2Fwww.bookrags.com%2Fresearch%2Feulers-laws-of-motion-wom%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMomentum" class="Z3988"></span></span> </li> <li id="cite_note-McGillKing-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-McGillKing_3-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMcGillKing1995" class="citation book cs1">McGill, David J. & King, Wilton W. (1995). <i>Engineering Mechanics: An Introduction to Dynamics</i> (3rd ed.). PWS. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-534-93399-9" title="Special:BookSources/978-0-534-93399-9"><bdi>978-0-534-93399-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Engineering+Mechanics%3A+An+Introduction+to+Dynamics&rft.edition=3rd&rft.pub=PWS&rft.date=1995&rft.isbn=978-0-534-93399-9&rft.aulast=McGill&rft.aufirst=David+J.&rft.au=King%2C+Wilton+W.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMomentum" class="Z3988"></span></span> </li> <li id="cite_note-FeynmanCh10-4"><span class="mw-cite-backlink">^ <a href="#cite_ref-FeynmanCh10_4-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-FeynmanCh10_4-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-FeynmanCh10_4-2"><sup><i><b>c</b></i></sup></a> <a href="#cite_ref-FeynmanCh10_4-3"><sup><i><b>d</b></i></sup></a></span> <span class="reference-text"><a rel="nofollow" class="external text" href="https://feynmanlectures.caltech.edu/I_10.html"><i>The Feynman Lectures on Physics</i></a> Vol. I Ch. 10: Conservation of Momentum</span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHo-KimKumarLam2004" class="citation book cs1">Ho-Kim, Quang; Kumar, Narendra; Lam, Harry C. S. (2004). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/invitationtocont00hoki"><i>Invitation to Contemporary Physics</i></a></span> (illustrated ed.). World Scientific. p. <a rel="nofollow" class="external text" href="https://archive.org/details/invitationtocont00hoki/page/19">19</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-981-238-303-7" title="Special:BookSources/978-981-238-303-7"><bdi>978-981-238-303-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Invitation+to+Contemporary+Physics&rft.pages=19&rft.edition=illustrated&rft.pub=World+Scientific&rft.date=2004&rft.isbn=978-981-238-303-7&rft.aulast=Ho-Kim&rft.aufirst=Quang&rft.au=Kumar%2C+Narendra&rft.au=Lam%2C+Harry+C.+S.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Finvitationtocont00hoki&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMomentum" class="Z3988"></span></span> </li> <li id="cite_note-Goldstein54-6"><span class="mw-cite-backlink">^ <a href="#cite_ref-Goldstein54_6-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Goldstein54_6-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-Goldstein54_6-2"><sup><i><b>c</b></i></sup></a> <a href="#cite_ref-Goldstein54_6-3"><sup><i><b>d</b></i></sup></a></span> <span class="reference-text"><a href="#CITEREFGoldstein1980">Goldstein 1980</a>, pp. 54–56</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"><a href="#CITEREFGoldstein1980">Goldstein 1980</a>, p. 276</span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text">Resnick and Halliday (1966), <i>Physics</i>, Section 10-3. Wiley Toppan, Library of Congress 66-11527</span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFNave2010" class="citation web cs1">Nave, Carl (2010). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20120818114930/http://hyperphysics.phy-astr.gsu.edu/hbase/elacol.html">"Elastic and inelastic collisions"</a>. <i>Hyperphysics</i>. Archived from <a rel="nofollow" class="external text" href="http://hyperphysics.phy-astr.gsu.edu/hbase/elacol.html">the original</a> on 18 August 2012<span class="reference-accessdate">. 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Boston, Massachusetts: Brooks/Cole, Cengage Learning. p. 245. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-133-10426-1" title="Special:BookSources/978-1-133-10426-1"><bdi>978-1-133-10426-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Principles+of+physics%3A+a+calculus-based+text&rft.place=Boston%2C+Massachusetts&rft.pages=245&rft.edition=5th&rft.pub=Brooks%2FCole%2C+Cengage+Learning&rft.date=2012&rft.isbn=978-1-133-10426-1&rft.aulast=Serway&rft.aufirst=Raymond+A.&rft.au=Jewett%2C+John+W.+Jr.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMomentum" class="Z3988"></span></span> </li> <li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFNave2010" class="citation web cs1">Nave, Carl (2010). <a rel="nofollow" class="external text" href="http://hyperphysics.phy-astr.gsu.edu/hbase/carcr.html#cc1">"Forces in car crashes"</a>. <i>Hyperphysics</i>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20120822034313/http://hyperphysics.phy-astr.gsu.edu/hbase/carcr.html#cc1">Archived</a> from the original on 22 August 2012<span class="reference-accessdate">. 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Champaign, Illinois: Human Kinetics. p. 85. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-7360-5101-9" title="Special:BookSources/978-0-7360-5101-9"><bdi>978-0-7360-5101-9</bdi></a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20160819020542/https://books.google.com/books?id=PrOKEcZXJ58C&pg=PA85&lpg=PA85&dq=coefficient+of+restitution+bounciness">Archived</a> from the original on 2016-08-19.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Biomechanics+of+sport+and+exercise&rft.place=Champaign%2C+Illinois&rft.pages=85&rft.edition=2nd&rft.pub=Human+Kinetics&rft.date=2005&rft.isbn=978-0-7360-5101-9&rft.aulast=McGinnis&rft.aufirst=Peter+M.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DPrOKEcZXJ58C%26q%3Dcoefficient%2Bof%2Brestitution%2Bbounciness%26pg%3DPA85&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMomentum" class="Z3988"></span></span> </li> <li id="cite_note-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-14">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSutton2001" class="citation book cs1">Sutton, George (2001). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=LQbDOxg3XZcC">"Chapter 1: Classification"</a>. <i>Rocket Propulsion Elements</i> (7th ed.). Chichester: John Wiley & Sons. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-471-32642-7" title="Special:BookSources/978-0-471-32642-7"><bdi>978-0-471-32642-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Chapter+1%3A+Classification&rft.btitle=Rocket+Propulsion+Elements&rft.place=Chichester&rft.edition=7th&rft.pub=John+Wiley+%26+Sons&rft.date=2001&rft.isbn=978-0-471-32642-7&rft.aulast=Sutton&rft.aufirst=George&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DLQbDOxg3XZcC&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMomentum" class="Z3988"></span></span> </li> <li id="cite_note-FeynmanCh11-15"><span class="mw-cite-backlink">^ <a href="#cite_ref-FeynmanCh11_15-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-FeynmanCh11_15-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-FeynmanCh11_15-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text"><a rel="nofollow" class="external text" href="https://feynmanlectures.caltech.edu/I_11.html"><i>The Feynman Lectures on Physics</i></a> Vol. I Ch. 11: Vectors</span> </li> <li id="cite_note-16"><span class="mw-cite-backlink"><b><a href="#cite_ref-16">^</a></b></span> <span class="reference-text"><a href="#CITEREFRindler1986">Rindler 1986</a>, pp. 26–27</span> </li> <li id="cite_note-kleppner135-17"><span class="mw-cite-backlink">^ <a href="#cite_ref-kleppner135_17-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-kleppner135_17-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKleppnerKolenkow" class="citation book cs1">Kleppner; Kolenkow. <i>An Introduction to Mechanics</i>. pp. 135–139.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=An+Introduction+to+Mechanics&rft.pages=135-139&rft.au=Kleppner&rft.au=Kolenkow&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMomentum" class="Z3988"></span></span> </li> <li id="cite_note-18"><span class="mw-cite-backlink"><b><a href="#cite_ref-18">^</a></b></span> <span class="reference-text"><a href="#CITEREFGoldstein1980">Goldstein 1980</a>, pp. 11–13</span> </li> <li id="cite_note-19"><span class="mw-cite-backlink"><b><a href="#cite_ref-19">^</a></b></span> <span class="reference-text"><a href="#CITEREFJackson1975">Jackson 1975</a>, p. 574</span> </li> <li id="cite_note-FeynmanQM-20"><span class="mw-cite-backlink"><b><a href="#cite_ref-FeynmanQM_20-0">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="https://feynmanlectures.caltech.edu/III_21.html#Ch21-S3"><i>The Feynman Lectures on Physics</i></a> Vol. III Ch. 21-3: Two kinds of momentum</span> </li> <li id="cite_note-21"><span class="mw-cite-backlink"><b><a href="#cite_ref-21">^</a></b></span> <span class="reference-text"><a href="#CITEREFGoldstein1980">Goldstein 1980</a>, pp. 20–21</span> </li> <li id="cite_note-Lerner-22"><span class="mw-cite-backlink">^ <a href="#cite_ref-Lerner_22-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Lerner_22-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLernerTrigg2005" class="citation book cs1"><a href="/wiki/Rita_G._Lerner" title="Rita G. 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Archived from <a rel="nofollow" class="external text" href="https://www.sns.ias.edu/ckfinder/userfiles/files/%5B32%5DCMP_80_1981.pdf">the original</a> <span class="cs1-format">(PDF)</span> on 2016-11-25<span class="reference-accessdate">. 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Princeton, New Jersey: Princeton University Press. pp. <a rel="nofollow" class="external text" href="https://archive.org/details/howwhyessayonori0000park/page/139">139–141</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-691-02508-7" title="Special:BookSources/978-0-691-02508-7"><bdi>978-0-691-02508-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+how+and+the+why%3A+an+essay+on+the+origins+and+development+of+physical+theory&rft.place=Princeton%2C+New+Jersey&rft.pages=139-141&rft.edition=3rd+print&rft.pub=Princeton+University+Press&rft.date=1990&rft.isbn=978-0-691-02508-7&rft.aulast=Park&rft.aufirst=David&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fhowwhyessayonori0000park&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMomentum" class="Z3988"></span></span> </li> <li id="cite_note-:0-64"><span class="mw-cite-backlink">^ <a href="#cite_ref-:0_64-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-:0_64-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDescartes2008" class="citation book cs1">Descartes, R. 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Part II, § 36.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Principles+of+philosophy&rft.pages=Part+II%2C+%C2%A7+36.&rft.date=2008&rft.aulast=Descartes&rft.aufirst=R.&rft_id=https%3A%2F%2Fwww.earlymoderntexts.com%2Fassets%2Fpdfs%2Fdescartes1644part2.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMomentum" class="Z3988"></span></span> </li> <li id="cite_note-65"><span class="mw-cite-backlink"><b><a href="#cite_ref-65">^</a></b></span> <span class="reference-text">Alexander Afriat (2004). <a rel="nofollow" class="external text" href="http://philsci-archive.pitt.edu/1699/1/Momentum3.pdf">"Cartesian and Lagrangian Momentum"</a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20170309014638/http://philsci-archive.pitt.edu/1699/1/Momentum3.pdf">Archived</a> 2017-03-09 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a>.</span> </li> <li id="cite_note-66"><span class="mw-cite-backlink"><b><a href="#cite_ref-66">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFNewton1729" class="citation book cs1">Newton, I (1729) [Original work published 1686]. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=Tm0FAAAAQAAJ&pg=PP13"><i>The mathematical principles of natural philosophy</i></a>. 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Cambridge: Cambridge University Press. pp. 310–319. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-36696-0" title="Special:BookSources/978-0-521-36696-0"><bdi>978-0-521-36696-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Descartes%27+Physics&rft.btitle=The+Cambridge+Companion+to+Descartes&rft.place=Cambridge&rft.pages=310-319&rft.pub=Cambridge+University+Press&rft.date=1992&rft.isbn=978-0-521-36696-0&rft.aulast=Garber&rft.aufirst=Daniel&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMomentum" class="Z3988"></span></span> </li> <li id="cite_note-68"><span class="mw-cite-backlink"><b><a href="#cite_ref-68">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRothman1989" class="citation book cs1">Rothman, Milton A. 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New York: Dover. pp. <a rel="nofollow" class="external text" href="https://archive.org/details/discoveringnatur0000roth/page/83">83–88</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-26178-2" title="Special:BookSources/978-0-486-26178-2"><bdi>978-0-486-26178-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Discovering+the+natural+laws%3A+the+experimental+basis+of+physics&rft.place=New+York&rft.pages=83-88&rft.edition=2nd&rft.pub=Dover&rft.date=1989&rft.isbn=978-0-486-26178-2&rft.aulast=Rothman&rft.aufirst=Milton+A.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fdiscoveringnatur0000roth%2Fpage%2F83&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMomentum" class="Z3988"></span></span> </li> <li id="cite_note-69"><span class="mw-cite-backlink"><b><a href="#cite_ref-69">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSlowik2017" class="citation encyclopaedia cs1">Slowik, Edward (Fall 2017). <a rel="nofollow" class="external text" href="https://plato.stanford.edu/archives/fall2017/entries/descartes-physics/">"Descartes' Physics"</a>. In Zalta, Edward N. (ed.). <i>The Stanford Encyclopedia of Philosophy</i><span class="reference-accessdate">. Retrieved <span class="nowrap">29 November</span> 2019</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Descartes%27+Physics&rft.btitle=The+Stanford+Encyclopedia+of+Philosophy&rft.date=2017&rft.aulast=Slowik&rft.aufirst=Edward&rft_id=https%3A%2F%2Fplato.stanford.edu%2Farchives%2Ffall2017%2Fentries%2Fdescartes-physics%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMomentum" class="Z3988"></span></span> </li> <li id="cite_note-70"><span class="mw-cite-backlink"><b><a href="#cite_ref-70">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFTaton1964" class="citation book cs1">Taton, Rene, ed. (1964) [1958]. <i>The Beginnings of Modern Science</i>. 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Israel">Israel, Jonathan I.</a> (8 February 2001). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=vMvlEweVPTsC&pg=RA3-PR62"><i>Radical Enlightenment: Philosophy and the Making of Modernity 1650–1750</i></a>. Oxford University Press. pp. lxii–lxiii. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-19-162287-8" title="Special:BookSources/978-0-19-162287-8"><bdi>978-0-19-162287-8</bdi></a><span class="reference-accessdate">. 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"Discourse on Metaphysics". In Ariew, Roger; Garber, Daniel (eds.). <i>Philosophical Essays</i>. 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New York: Dover Publications. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-65632-8" title="Special:BookSources/978-0-486-65632-8"><bdi>978-0-486-65632-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+history+of+mechanics&rft.place=New+York&rft.edition=Dover&rft.pub=Dover+Publications&rft.date=1988&rft.isbn=978-0-486-65632-8&rft.aulast=Dugas&rft.aufirst=Ren%C3%A9&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMomentum" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFeynman_Vol._1" class="citation book cs1">Feynman, Richard P.; Leighton, Robert B.; Sands, Matthew (2005). <i>The Feynman lectures on physics, Volume 1: Mainly Mechanics, Radiation, and Heat</i> (Definitive ed.). San Francisco: Pearson Addison-Wesley. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8053-9046-9" title="Special:BookSources/978-0-8053-9046-9"><bdi>978-0-8053-9046-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Feynman+lectures+on+physics%2C+Volume+1%3A+Mainly+Mechanics%2C+Radiation%2C+and+Heat&rft.place=San+Francisco&rft.edition=Definitive&rft.pub=Pearson+Addison-Wesley&rft.date=2005&rft.isbn=978-0-8053-9046-9&rft.aulast=Feynman&rft.aufirst=Richard+P.&rft.au=Leighton%2C+Robert+B.&rft.au=Sands%2C+Matthew&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMomentum" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFeynman_Vol._2" class="citation book cs1">Feynman, Richard P.; Leighton, Robert B.; Sands, Matthew (2006). <i>The Feynman lectures on physics</i> (Definitive ed.). San Francisco: Pearson Addison-Wesley. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8053-9047-6" title="Special:BookSources/978-0-8053-9047-6"><bdi>978-0-8053-9047-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Feynman+lectures+on+physics&rft.place=San+Francisco&rft.edition=Definitive&rft.pub=Pearson+Addison-Wesley&rft.date=2006&rft.isbn=978-0-8053-9047-6&rft.aulast=Feynman&rft.aufirst=Richard+P.&rft.au=Leighton%2C+Robert+B.&rft.au=Sands%2C+Matthew&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMomentum" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFeynman_Vol._3" class="citation book cs1">Feynman, Richard P.; Leighton, Robert B.; Sands, Matthew (2005). <i>The Feynman lectures on physics, Volume III: Quantum Mechanics</i> (Definitive ed.). New York: BasicBooks. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8053-9049-0" title="Special:BookSources/978-0-8053-9049-0"><bdi>978-0-8053-9049-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Feynman+lectures+on+physics%2C+Volume+III%3A+Quantum+Mechanics&rft.place=New+York&rft.edition=Definitive&rft.pub=BasicBooks&rft.date=2005&rft.isbn=978-0-8053-9049-0&rft.aulast=Feynman&rft.aufirst=Richard+P.&rft.au=Leighton%2C+Robert+B.&rft.au=Sands%2C+Matthew&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMomentum" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGoldstein1980" class="citation book cs1">Goldstein, Herbert (1980). <i>Classical mechanics</i> (2nd ed.). Reading, MA: Addison-Wesley Pub. Co. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-201-02918-5" title="Special:BookSources/978-0-201-02918-5"><bdi>978-0-201-02918-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Classical+mechanics&rft.place=Reading%2C+MA&rft.edition=2nd&rft.pub=Addison-Wesley+Pub.+Co.&rft.date=1980&rft.isbn=978-0-201-02918-5&rft.aulast=Goldstein&rft.aufirst=Herbert&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMomentum" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHandFinch" class="citation book cs1">Hand, Louis N.; Finch, Janet D. <i>Analytical Mechanics</i>. Cambridge University Press. Chapter 4.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Analytical+Mechanics&rft.pages=Chapter+4&rft.pub=Cambridge+University+Press&rft.aulast=Hand&rft.aufirst=Louis+N.&rft.au=Finch%2C+Janet+D.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMomentum" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJackson1975" class="citation book cs1">Jackson, John David (1975). <a rel="nofollow" class="external text" href="https://archive.org/details/classicalelectro00jack_0"><i>Classical electrodynamics</i></a> (2nd ed.). New York: Wiley. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-471-43132-9" title="Special:BookSources/978-0-471-43132-9"><bdi>978-0-471-43132-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Classical+electrodynamics&rft.place=New+York&rft.edition=2nd&rft.pub=Wiley&rft.date=1975&rft.isbn=978-0-471-43132-9&rft.aulast=Jackson&rft.aufirst=John+David&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fclassicalelectro00jack_0&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMomentum" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJammer1999" class="citation book cs1">Jammer, Max (1999). <i>Concepts of force: a study in the foundations of dynamics</i> (Facsim ed.). Mineola, New York: Dover Publications. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-40689-3" title="Special:BookSources/978-0-486-40689-3"><bdi>978-0-486-40689-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Concepts+of+force%3A+a+study+in+the+foundations+of+dynamics&rft.place=Mineola%2C+New+York&rft.edition=Facsim&rft.pub=Dover+Publications&rft.date=1999&rft.isbn=978-0-486-40689-3&rft.aulast=Jammer&rft.aufirst=Max&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMomentum" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLandauLifshitz2000" class="citation book cs1">Landau, L.D.; Lifshitz, E.M. (2000). <i>The classical theory of fields</i>. English edition, reprinted with corrections; translated from the Russian by Morton Hamermesh (4th ed.). Oxford: Butterworth Heinemann. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-7506-2768-9" title="Special:BookSources/978-0-7506-2768-9"><bdi>978-0-7506-2768-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+classical+theory+of+fields&rft.place=Oxford&rft.edition=4th&rft.pub=Butterworth+Heinemann&rft.date=2000&rft.isbn=978-0-7506-2768-9&rft.aulast=Landau&rft.aufirst=L.D.&rft.au=Lifshitz%2C+E.M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMomentum" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRindler1986" class="citation book cs1">Rindler, Wolfgang (1986). <i>Essential Relativity: Special, general and cosmological</i> (2nd ed.). New York: Springer. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-10090-6" title="Special:BookSources/978-0-387-10090-6"><bdi>978-0-387-10090-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Essential+Relativity%3A+Special%2C+general+and+cosmological&rft.place=New+York&rft.edition=2nd&rft.pub=Springer&rft.date=1986&rft.isbn=978-0-387-10090-6&rft.aulast=Rindler&rft.aufirst=Wolfgang&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMomentum" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSerwayJewett2003" class="citation book cs1">Serway, Raymond; Jewett, John (2003). <a rel="nofollow" class="external text" href="https://archive.org/details/physicssciengv2p00serw"><i>Physics for Scientists and Engineers</i></a> (6th ed.). Brooks Cole. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-534-40842-8" title="Special:BookSources/978-0-534-40842-8"><bdi>978-0-534-40842-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Physics+for+Scientists+and+Engineers&rft.edition=6th&rft.pub=Brooks+Cole.&rft.date=2003&rft.isbn=978-0-534-40842-8&rft.aulast=Serway&rft.aufirst=Raymond&rft.au=Jewett%2C+John&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fphysicssciengv2p00serw&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMomentum" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFStenger2000" class="citation book cs1">Stenger, Victor J. (2000). <i>Timeless Reality: Symmetry, Simplicity, and Multiple Universes</i>. Prometheus Books. pp. Chapter 12 in particular.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Timeless+Reality%3A+Symmetry%2C+Simplicity%2C+and+Multiple+Universes&rft.pages=Chapter+12+in+particular&rft.pub=Prometheus+Books.&rft.date=2000&rft.aulast=Stenger&rft.aufirst=Victor+J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMomentum" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFTipler1998" class="citation book cs1">Tipler, Paul (1998). <i>Physics for Scientists and Engineers: Vol. 1: Mechanics, Oscillations and Waves, Thermodynamics</i> (4th ed.). W.H. Freeman. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-57259-492-0" title="Special:BookSources/978-1-57259-492-0"><bdi>978-1-57259-492-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Physics+for+Scientists+and+Engineers%3A+Vol.+1%3A+Mechanics%2C+Oscillations+and+Waves%2C+Thermodynamics&rft.edition=4th&rft.pub=W.H.+Freeman&rft.date=1998&rft.isbn=978-1-57259-492-0&rft.aulast=Tipler&rft.aufirst=Paul&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMomentum" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFTritton2006" class="citation book cs1"><a href="/wiki/David_Tritton" title="David Tritton">Tritton, D.J.</a> (2006). <i>Physical fluid dynamics</i> (2nd ed.). Oxford: Clarendon Press. p. 58. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-19-854493-7" title="Special:BookSources/978-0-19-854493-7"><bdi>978-0-19-854493-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Physical+fluid+dynamics&rft.place=Oxford&rft.pages=58&rft.edition=2nd&rft.pub=Clarendon+Press&rft.date=2006&rft.isbn=978-0-19-854493-7&rft.aulast=Tritton&rft.aufirst=D.J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMomentum" class="Z3988"></span></li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div> <style data-mw-deduplicate="TemplateStyles:r1235681985">.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 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title="Acoustics">Acoustics</a></li> <li><a href="/wiki/Vibrations" class="mw-redirect" title="Vibrations">Vibrations</a></li> <li><a href="/wiki/Rigid_body_dynamics" title="Rigid body dynamics">Rigid body dynamics</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Laws and Definitions</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <dl><dt>Laws</dt></dl> <ul><li><a href="/wiki/Conservation_of_mass" title="Conservation of mass">Conservation of mass</a></li> <li><a class="mw-selflink selflink">Conservation of momentum</a> <ul><li><a href="/wiki/Navier%E2%80%93Stokes_equations" title="Navier–Stokes equations">Navier-Stokes</a></li> <li><a href="/wiki/Bernoulli%27s_principle" title="Bernoulli's principle">Bernoulli</a></li> <li><a href="/wiki/Poiseuille_equation" class="mw-redirect" title="Poiseuille equation">Poiseuille</a></li> <li><a href="/wiki/Archimedes%27_principle" title="Archimedes' principle">Archimedes</a></li> <li><a href="/wiki/Pascal%27s_law" title="Pascal's law">Pascal</a></li></ul></li> <li><a href="/wiki/Conservation_of_energy" title="Conservation of energy">Conservation of energy</a></li> <li><a href="/wiki/Clausius%E2%80%93Duhem_inequality" title="Clausius–Duhem inequality">Entropy inequality</a></li></ul> <dl><dt>Definitions</dt></dl> <ul><li><a href="/wiki/Stress_(mechanics)" title="Stress (mechanics)">Stress</a> <ul><li><a href="/wiki/Cauchy_stress_tensor" title="Cauchy stress tensor">Cauchy stress</a></li> <li><a href="/wiki/Stress_measures" class="mw-redirect" title="Stress measures">Stress measures</a></li></ul></li> <li><a href="/wiki/Deformation_(mechanics)" class="mw-redirect" title="Deformation (mechanics)">Deformation</a></li> <li><a href="/wiki/Infinitesimal_strain_theory" title="Infinitesimal strain theory">Small strain</a> <ul><li><a href="/wiki/Antiplane_shear" title="Antiplane shear">Antiplane shear</a></li></ul></li> <li><a href="/wiki/Finite_strain_theory" title="Finite strain theory">Large strain</a></li> <li><a href="/wiki/Compatibility_(mechanics)" title="Compatibility (mechanics)">Compatibility</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Solid_mechanics" title="Solid mechanics">Solid</a> and <a href="/wiki/Structural_mechanics" title="Structural mechanics">Structural mechanics</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <dl><dt><a href="/wiki/Solid" title="Solid">Solids</a></dt></dl> <ul><li><a href="/wiki/Elasticity_(physics)" title="Elasticity (physics)">Elasticity</a> <ul><li><a href="/wiki/Linear_elasticity" title="Linear elasticity">linear</a></li> <li><a href="/wiki/Hooke%27s_law" title="Hooke's law">Hooke's law</a></li> <li><a href="/wiki/Transverse_isotropy" title="Transverse isotropy">Transverse isotropy</a></li> <li><a href="/wiki/Orthotropic_material" title="Orthotropic material">Orthotropy</a></li> <li><a href="/wiki/Hyperelastic_material" title="Hyperelastic material">hyperelasticity</a> <ul><li><a href="/wiki/Elasticity_of_cell_membranes" title="Elasticity of cell membranes">Membrane elasticity</a></li></ul></li> <li><a href="/wiki/Equation_of_state" title="Equation of state">Equation of state</a> <ul><li><a href="/wiki/Hugoniot_elastic_limit" class="mw-redirect" title="Hugoniot elastic limit">Hugoniot</a></li> <li><a href="/wiki/Equation_of_state#Jones–Wilkins–Lee_equation_of_state_for_explosives_(JWL_equation)" title="Equation of state">JWL</a></li></ul></li> <li><a href="/wiki/Hypoelastic_material" title="Hypoelastic material">hypoelasticity</a></li> <li><a href="/wiki/Cauchy_elastic_material" title="Cauchy elastic material">Cauchy elasticity</a></li> <li><a href="/wiki/Viscoelasticity" title="Viscoelasticity">Viscoelasticity</a> <ul><li><a href="/wiki/Creep_(deformation)" title="Creep (deformation)">Creep</a></li> <li><a href="/wiki/Creep_and_shrinkage_of_concrete" title="Creep and shrinkage of concrete">Concrete creep</a></li></ul></li></ul></li> <li><a href="/wiki/Plasticity_(physics)" title="Plasticity (physics)">Plasticity</a> <ul><li><a href="/wiki/Rock_mass_plasticity" title="Rock mass plasticity">Rock mass plasticity</a></li> <li><a href="/wiki/Viscoplasticity" title="Viscoplasticity">Viscoplasticity</a></li> <li><a href="/wiki/Yield_surface" title="Yield surface">Yield criterion</a> <ul><li><a href="/wiki/Bresler%E2%80%93Pister_yield_criterion" title="Bresler–Pister yield criterion">Bresler-Pister</a></li></ul></li></ul></li> <li><a href="/wiki/Contact_mechanics" title="Contact mechanics">Contact mechanics</a> <ul><li><a href="/wiki/Contact_mechanics" title="Contact mechanics">Frictionless</a></li> <li><a href="/wiki/Frictional_contact_mechanics" title="Frictional contact mechanics">Frictional</a></li></ul></li></ul> <dl><dt><a href="/wiki/Failure" title="Failure">Material failure theory</a></dt></dl> <ul><li><a href="/wiki/Drucker_stability" title="Drucker stability">Drucker stability</a></li> <li><a href="/wiki/Material_failure_theory" title="Material failure theory">Material failure theory</a></li> <li><a href="/wiki/Fatigue_(material)" title="Fatigue (material)">Fatigue</a></li> <li><a href="/wiki/Fracture_mechanics" title="Fracture mechanics">Fracture mechanics</a> <ul><li><a href="/wiki/J-integral" title="J-integral">J-integral</a></li> <li><a href="/wiki/Compact_tension_specimen" title="Compact tension specimen">Compact tension specimen</a></li></ul></li> <li><a href="/wiki/Damage_mechanics" title="Damage mechanics">Damage mechanics</a> <ul><li><a href="/wiki/Johnson-Holmquist_damage_model" class="mw-redirect" title="Johnson-Holmquist damage model">Johnson-Holmquist</a></li></ul></li></ul> <dl><dt><a href="/wiki/Structural_mechanics" title="Structural mechanics">Structures</a></dt></dl> <ul><li><a href="/wiki/Bending" title="Bending">Bending</a> <ul><li><a href="/wiki/Bending_moment" title="Bending moment">Bending moment</a></li> <li><a href="/wiki/Bending_of_plates" title="Bending of plates">Bending of plates</a></li> <li><a href="/wiki/Sandwich_theory" title="Sandwich theory">Sandwich theory</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Fluid_mechanics" title="Fluid mechanics">Fluid mechanics</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <dl><dt><a href="/wiki/Fluid" title="Fluid">Fluids</a></dt></dl> <ul><li><a href="/wiki/Fluid_statics" class="mw-redirect" title="Fluid statics">Fluid statics</a></li> <li><a href="/wiki/Fluid_dynamics" title="Fluid dynamics">Fluid dynamics</a></li> <li><a href="/wiki/Navier%E2%80%93Stokes_equations" title="Navier–Stokes equations">Navier–Stokes equations</a></li> <li><a href="/wiki/Bernoulli%27s_principle" title="Bernoulli's principle">Bernoulli's principle</a></li> <li><a href="/wiki/Poiseuille_equation" class="mw-redirect" title="Poiseuille equation">Poiseuille equation</a></li> <li><a href="/wiki/Buoyancy" title="Buoyancy">Buoyancy</a></li> <li><a href="/wiki/Viscosity" title="Viscosity">Viscosity</a> <ul><li><a href="/wiki/Newtonian_fluid" title="Newtonian fluid">Newtonian</a></li> <li><a href="/wiki/Non-Newtonian_fluid" title="Non-Newtonian fluid">Non-Newtonian</a></li></ul></li> <li><a href="/wiki/Archimedes%27_principle" title="Archimedes' principle">Archimedes' principle</a></li> <li><a href="/wiki/Pascal%27s_law" title="Pascal's law">Pascal's law</a></li> <li><a href="/wiki/Pressure" title="Pressure">Pressure</a></li> <li><a href="/wiki/Liquid" title="Liquid">Liquids</a></li> <li><a href="/wiki/Surface_tension" title="Surface tension">Surface tension</a></li> <li><a href="/wiki/Capillary_action" title="Capillary action">Capillary action</a></li></ul> <dl><dt><a href="/wiki/Gas" title="Gas">Gases</a></dt></dl> <ul><li><a href="/wiki/Atmosphere" title="Atmosphere">Atmosphere</a></li> <li><a href="/wiki/Boyle%27s_law" title="Boyle's law">Boyle's law</a></li> <li><a href="/wiki/Charles%27s_law" title="Charles's law">Charles's law</a></li> <li><a href="/wiki/Gay-Lussac%27s_law" title="Gay-Lussac's law">Gay-Lussac's law</a></li> <li><a href="/wiki/Combined_gas_law" class="mw-redirect" title="Combined gas law">Combined gas law</a></li></ul> <dl><dt><a href="/wiki/Plasma_(physics)" title="Plasma (physics)">Plasma</a></dt></dl> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Acoustics" title="Acoustics">Acoustics</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Acoustic_theory" title="Acoustic theory">Acoustic theory</a></li> <li><a href="/wiki/Aeroacoustics" title="Aeroacoustics">Aeroacoustics</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Rheology" title="Rheology">Rheology</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Viscoelasticity" title="Viscoelasticity">Viscoelasticity</a></li> <li><a href="/wiki/Smart_fluid" title="Smart fluid">Smart fluids</a> <ul><li><a href="/wiki/Magnetorheological_fluid" title="Magnetorheological fluid">Magnetorheological</a></li> <li><a href="/wiki/Electrorheological_fluid" title="Electrorheological fluid">Electrorheological</a></li> <li><a href="/wiki/Ferrofluid" title="Ferrofluid">Ferrofluids</a></li></ul></li> <li><a href="/wiki/Rheometry" title="Rheometry">Rheometry</a></li> <li><a href="/wiki/Rheometer" title="Rheometer">Rheometer</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Scientist" title="Scientist">Scientists</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Daniel_Bernoulli" title="Daniel Bernoulli">Bernoulli</a></li> <li><a href="/wiki/Robert_Boyle" title="Robert Boyle">Boyle</a></li> <li><a href="/wiki/Augustin-Louis_Cauchy" title="Augustin-Louis Cauchy">Cauchy</a></li> <li><a href="/wiki/Jacques_Charles" title="Jacques Charles">Charles</a></li> <li><a href="/wiki/Leonhard_Euler" title="Leonhard Euler">Euler</a></li> <li><a href="/wiki/Joseph_Louis_Gay-Lussac" title="Joseph Louis Gay-Lussac">Gay-Lussac</a></li> <li><a href="/wiki/Robert_Hooke" title="Robert Hooke">Hooke</a></li> <li><a href="/wiki/Blaise_Pascal" title="Blaise Pascal">Pascal</a></li> <li><a href="/wiki/Isaac_Newton" title="Isaac Newton">Newton</a></li> <li><a href="/wiki/Claude-Louis_Navier" title="Claude-Louis Navier">Navier</a></li> <li><a href="/wiki/Sir_George_Stokes,_1st_Baronet" title="Sir George Stokes, 1st 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1,\n [\"CITEREFGriffiths2017\"] = 1,\n [\"CITEREFGrimsehl1932\"] = 1,\n [\"CITEREFGrossman2012\"] = 1,\n [\"CITEREFGubbins1992\"] = 1,\n [\"CITEREFHallidayResnick2013\"] = 1,\n [\"CITEREFHandFinch\"] = 1,\n [\"CITEREFHandFinch1998\"] = 1,\n [\"CITEREFHo-KimKumarLam2004\"] = 1,\n [\"CITEREFIsrael2001\"] = 1,\n [\"CITEREFJackson1975\"] = 1,\n [\"CITEREFJammer1999\"] = 1,\n [\"CITEREFJennings1721\"] = 1,\n [\"CITEREFKleppnerKolenkow\"] = 1,\n [\"CITEREFLandauLifshitz2000\"] = 1,\n [\"CITEREFLeBlondMysak1980\"] = 1,\n [\"CITEREFLeibniz1989\"] = 1,\n [\"CITEREFLernerTrigg2005\"] = 1,\n [\"CITEREFMcGillKing1995\"] = 1,\n [\"CITEREFMcGinnis2005\"] = 1,\n [\"CITEREFMcIntyre1981\"] = 1,\n [\"CITEREFMisnerThorneWheeler1973\"] = 1,\n [\"CITEREFNasrRazavi1996\"] = 1,\n [\"CITEREFNave2010\"] = 3,\n [\"CITEREFNewton1729\"] = 1,\n [\"CITEREFPark1990\"] = 1,\n [\"CITEREFPowell2013\"] = 1,\n [\"CITEREFRescigno2003\"] = 1,\n [\"CITEREFRindler1986\"] = 1,\n [\"CITEREFRindler1991\"] = 1,\n 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Momentum - Wikipedia