Velocità - Wikipedia

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data-event-name="ui.sidebar-toc" class="mw-table-of-contents-container vector-toc-landmark"> <div id="vector-toc-pinned-container" class="vector-pinned-container"> <div id="vector-toc" class="vector-toc vector-pinnable-element"> <div class="vector-pinnable-header vector-toc-pinnable-header vector-pinnable-header-pinned" data-feature-name="toc-pinned" data-pinnable-element-id="vector-toc" > <h2 class="vector-pinnable-header-label">Indice</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">sposta nella barra laterale</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">nascondi</button> </div> <ul class="vector-toc-contents" id="mw-panel-toc-list"> <li id="toc-mw-content-text" class="vector-toc-list-item vector-toc-level-1"> <a href="#" class="vector-toc-link"> <div class="vector-toc-text">Inizio</div> </a> </li> <li id="toc-Velocità_media_e_istantanea" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Velocità_media_e_istantanea"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Velocità media e istantanea</span> </div> </a> <button aria-controls="toc-Velocità_media_e_istantanea-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>Attiva/disattiva la sottosezione Velocità media e istantanea</span> </button> <ul id="toc-Velocità_media_e_istantanea-sublist" class="vector-toc-list"> <li id="toc-Velocità_in_due_dimensioni" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Velocità_in_due_dimensioni"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1</span> <span>Velocità in due dimensioni</span> </div> </a> <ul id="toc-Velocità_in_due_dimensioni-sublist" class="vector-toc-list"> <li id="toc-Velocità_scalare_media" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Velocità_scalare_media"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1.1</span> <span>Velocità scalare media</span> </div> </a> <ul id="toc-Velocità_scalare_media-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </li> <li id="toc-Relazione_integrale_fra_la_posizione_e_velocità" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Relazione_integrale_fra_la_posizione_e_velocità"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Relazione integrale fra la posizione e velocità</span> </div> </a> <ul id="toc-Relazione_integrale_fra_la_posizione_e_velocità-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Composizione_delle_velocità" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Composizione_delle_velocità"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Composizione delle velocità</span> </div> </a> <button aria-controls="toc-Composizione_delle_velocità-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>Attiva/disattiva la sottosezione Composizione delle velocità</span> </button> <ul id="toc-Composizione_delle_velocità-sublist" class="vector-toc-list"> <li id="toc-Composizione_delle_velocità_in_relatività_speciale" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Composizione_delle_velocità_in_relatività_speciale"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Composizione delle velocità in relatività speciale</span> </div> </a> <ul id="toc-Composizione_delle_velocità_in_relatività_speciale-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Velocità_nei_sistemi_di_punti_materiali" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Velocità_nei_sistemi_di_punti_materiali"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Velocità nei sistemi di punti materiali</span> </div> </a> <ul id="toc-Velocità_nei_sistemi_di_punti_materiali-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Caduta_nel_campo_gravitazionale" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Caduta_nel_campo_gravitazionale"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Caduta nel campo gravitazionale</span> </div> </a> <button aria-controls="toc-Caduta_nel_campo_gravitazionale-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>Attiva/disattiva la sottosezione Caduta nel campo gravitazionale</span> </button> <ul id="toc-Caduta_nel_campo_gravitazionale-sublist" class="vector-toc-list"> <li id="toc-Velocità_terminale_di_caduta" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Velocità_terminale_di_caduta"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.1</span> <span>Velocità terminale di caduta</span> </div> </a> <ul id="toc-Velocità_terminale_di_caduta-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Velocità_limite" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Velocità_limite"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Velocità limite</span> </div> </a> <ul id="toc-Velocità_limite-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Note" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Note"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Note</span> </div> </a> <ul id="toc-Note-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Bibliografia" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Bibliografia"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Bibliografia</span> </div> </a> <ul id="toc-Bibliografia-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Voci_correlate" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Voci_correlate"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>Voci correlate</span> </div> </a> <ul id="toc-Voci_correlate-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Altri_progetti" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Altri_progetti"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>Altri progetti</span> </div> </a> <ul id="toc-Altri_progetti-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Collegamenti_esterni" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Collegamenti_esterni"> <div class="vector-toc-text"> <span class="vector-toc-numb">11</span> <span>Collegamenti esterni</span> </div> </a> <ul id="toc-Collegamenti_esterni-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="Indice" 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="Mostra/Nascondi l'indice" > <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">Mostra/Nascondi l'indice</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">Velocità</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="Vai a una voce in un'altra lingua. Disponibile in 130 lingue" > <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-130" 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">130 lingue</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/Snelheid" title="Snelheid - afrikaans" lang="af" hreflang="af" data-title="Snelheid" 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/Geschwindigkeit" title="Geschwindigkeit - tedesco svizzero" lang="gsw" hreflang="gsw" data-title="Geschwindigkeit" data-language-autonym="Alemannisch" data-language-local-name="tedesco svizzero" class="interlanguage-link-target"><span>Alemannisch</span></a></li><li class="interlanguage-link interwiki-am mw-list-item"><a href="https://am.wikipedia.org/wiki/%E1%8D%8D%E1%8C%A5%E1%8A%90%E1%89%B5" title="ፍጥነት - amarico" lang="am" hreflang="am" data-title="ፍጥነት" data-language-autonym="አማርኛ" data-language-local-name="amarico" class="interlanguage-link-target"><span>አማርኛ</span></a></li><li class="interlanguage-link interwiki-an mw-list-item"><a href="https://an.wikipedia.org/wiki/Velocidat" title="Velocidat - aragonese" lang="an" hreflang="an" data-title="Velocidat" data-language-autonym="Aragonés" data-language-local-name="aragonese" class="interlanguage-link-target"><span>Aragonés</span></a></li><li class="interlanguage-link interwiki-anp mw-list-item"><a href="https://anp.wikipedia.org/wiki/%E0%A4%B5%E0%A5%87%E0%A4%97" title="वेग - angika" lang="anp" hreflang="anp" data-title="वेग" data-language-autonym="अंगिका" data-language-local-name="angika" class="interlanguage-link-target"><span>अंगिका</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%B3%D8%B1%D8%B9%D8%A9_%D9%85%D8%AA%D8%AC%D9%87%D8%A9" title="سرعة متجهة - arabo" lang="ar" hreflang="ar" data-title="سرعة متجهة" data-language-autonym="العربية" data-language-local-name="arabo" 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%AC%E0%A7%87%E0%A6%97" title="বেগ - assamese" lang="as" hreflang="as" data-title="বেগ" 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/Velocid%C3%A1" title="Velocidá - asturiano" lang="ast" hreflang="ast" data-title="Velocidá" data-language-autonym="Asturianu" data-language-local-name="asturiano" class="interlanguage-link-target"><span>Asturianu</span></a></li><li class="interlanguage-link interwiki-azb mw-list-item"><a href="https://azb.wikipedia.org/wiki/%DB%8C%D8%A4%D9%86%D8%A6%DB%8C%D9%84%DB%8C_%D8%B3%D9%88%D8%B1%D8%B9%D8%AA" title="یؤنئیلی سورعت - South Azerbaijani" lang="azb" hreflang="azb" data-title="یؤنئیلی سورعت" data-language-autonym="تۆرکجه" data-language-local-name="South Azerbaijani" class="interlanguage-link-target"><span>تۆرکجه</span></a></li><li class="interlanguage-link interwiki-ba mw-list-item"><a href="https://ba.wikipedia.org/wiki/%D0%A2%D0%B8%D2%99%D0%BB%D0%B5%D0%BA" title="Тиҙлек - baschiro" lang="ba" hreflang="ba" data-title="Тиҙлек" data-language-autonym="Башҡортса" data-language-local-name="baschiro" class="interlanguage-link-target"><span>Башҡортса</span></a></li><li class="interlanguage-link interwiki-bar mw-list-item"><a href="https://bar.wikipedia.org/wiki/Gschwindigkeit" title="Gschwindigkeit - bavarese" lang="bar" hreflang="bar" data-title="Gschwindigkeit" data-language-autonym="Boarisch" data-language-local-name="bavarese" class="interlanguage-link-target"><span>Boarisch</span></a></li><li class="interlanguage-link interwiki-bcl mw-list-item"><a href="https://bcl.wikipedia.org/wiki/Belosidad" title="Belosidad - Central Bikol" lang="bcl" hreflang="bcl" data-title="Belosidad" 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-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%A1%D0%BA%D0%BE%D1%80%D0%B0%D1%81%D1%86%D1%8C" title="Скорасць - bielorusso" lang="be" hreflang="be" data-title="Скорасць" data-language-autonym="Беларуская" data-language-local-name="bielorusso" 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%A5%D1%83%D1%82%D0%BA%D0%B0%D1%81%D1%8C%D1%86%D1%8C" title="Хуткасьць - Belarusian (Taraškievica orthography)" lang="be-tarask" hreflang="be-tarask" data-title="Хуткасьць" data-language-autonym="Беларуская (тарашкевіца)" data-language-local-name="Belarusian (Taraškievica orthography)" class="interlanguage-link-target"><span>Беларуская (тарашкевіца)</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%A1%D0%BA%D0%BE%D1%80%D0%BE%D1%81%D1%82" title="Скорост - bulgaro" lang="bg" hreflang="bg" data-title="Скорост" data-language-autonym="Български" data-language-local-name="bulgaro" class="interlanguage-link-target"><span>Български</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" title="वेग - Bhojpuri" lang="bh" hreflang="bh" data-title="वेग" data-language-autonym="भोजपुरी" data-language-local-name="Bhojpuri" class="interlanguage-link-target"><span>भोजपुरी</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%97%E0%A6%A4%E0%A6%BF%E0%A6%AC%E0%A7%87%E0%A6%97" title="গতিবেগ - bengalese" lang="bn" hreflang="bn" data-title="গতিবেগ" data-language-autonym="বাংলা" data-language-local-name="bengalese" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Brzina" title="Brzina - bosniaco" lang="bs" hreflang="bs" data-title="Brzina" data-language-autonym="Bosanski" data-language-local-name="bosniaco" 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%A5%D1%83%D1%80%D0%B4%D0%B0%D0%BD" title="Хурдан - Russia Buriat" lang="bxr" hreflang="bxr" data-title="Хурдан" 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/Velocitat" title="Velocitat - catalano" lang="ca" hreflang="ca" data-title="Velocitat" data-language-autonym="Català" data-language-local-name="catalano" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cdo mw-list-item"><a href="https://cdo.wikipedia.org/wiki/S%C3%B3k-d%C3%B4" title="Sók-dô - Mindong" lang="cdo" hreflang="cdo" data-title="Sók-dô" data-language-autonym="閩東語 / Mìng-dĕ̤ng-ngṳ̄" data-language-local-name="Mindong" class="interlanguage-link-target"><span>閩東語 / Mìng-dĕ̤ng-ngṳ̄</span></a></li><li class="interlanguage-link interwiki-ce mw-list-item"><a href="https://ce.wikipedia.org/wiki/%D0%A1%D0%B8%D1%85%D0%B0%D0%BB%D0%BB%D0%B0" title="Сихалла - ceceno" lang="ce" hreflang="ce" data-title="Сихалла" data-language-autonym="Нохчийн" data-language-local-name="ceceno" class="interlanguage-link-target"><span>Нохчийн</span></a></li><li class="interlanguage-link interwiki-ceb mw-list-item"><a href="https://ceb.wikipedia.org/wiki/Tulin_(pisika)" title="Tulin (pisika) - cebuano" lang="ceb" hreflang="ceb" data-title="Tulin (pisika)" data-language-autonym="Cebuano" data-language-local-name="cebuano" class="interlanguage-link-target"><span>Cebuano</span></a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%D8%AE%DB%8E%D8%B1%D8%A7%DB%8C%DB%8C_(%D9%81%DB%8C%D8%B2%DB%8C%DA%A9)" title="خێرایی (فیزیک) - curdo centrale" lang="ckb" hreflang="ckb" data-title="خێرایی (فیزیک)" data-language-autonym="کوردی" data-language-local-name="curdo centrale" class="interlanguage-link-target"><span>کوردی</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Rychlost" title="Rychlost - ceco" lang="cs" hreflang="cs" data-title="Rychlost" data-language-autonym="Čeština" data-language-local-name="ceco" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%A5%C4%83%D0%B2%C4%83%D1%80%D1%82%D0%BB%C4%83%D1%85" title="Хăвăртлăх - ciuvascio" lang="cv" hreflang="cv" data-title="Хăвăртлăх" data-language-autonym="Чӑвашла" data-language-local-name="ciuvascio" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-cy mw-list-item"><a href="https://cy.wikipedia.org/wiki/Cyflymder" title="Cyflymder - gallese" lang="cy" hreflang="cy" data-title="Cyflymder" data-language-autonym="Cymraeg" data-language-local-name="gallese" 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/Hastighed" title="Hastighed - danese" lang="da" hreflang="da" data-title="Hastighed" data-language-autonym="Dansk" data-language-local-name="danese" 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/Geschwindigkeit" title="Geschwindigkeit - tedesco" lang="de" hreflang="de" data-title="Geschwindigkeit" data-language-autonym="Deutsch" data-language-local-name="tedesco" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%A4%CE%B1%CF%87%CF%8D%CF%84%CE%B7%CF%84%CE%B1" title="Ταχύτητα - greco" lang="el" hreflang="el" data-title="Ταχύτητα" data-language-autonym="Ελληνικά" data-language-local-name="greco" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Velocity" title="Velocity - inglese" lang="en" hreflang="en" data-title="Velocity" data-language-autonym="English" data-language-local-name="inglese" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Vektora_rapido" title="Vektora rapido - esperanto" lang="eo" hreflang="eo" data-title="Vektora rapido" data-language-autonym="Esperanto" data-language-local-name="esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Velocidad" title="Velocidad - spagnolo" lang="es" hreflang="es" data-title="Velocidad" data-language-autonym="Español" data-language-local-name="spagnolo" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Kiirus" title="Kiirus - estone" lang="et" hreflang="et" data-title="Kiirus" data-language-autonym="Eesti" data-language-local-name="estone" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Abiadura" title="Abiadura - basco" lang="eu" hreflang="eu" data-title="Abiadura" data-language-autonym="Euskara" data-language-local-name="basco" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-ext mw-list-item"><a href="https://ext.wikipedia.org/wiki/Velocid%C3%A1" title="Velocidá - estremegno" lang="ext" hreflang="ext" data-title="Velocidá" data-language-autonym="Estremeñu" data-language-local-name="estremegno" class="interlanguage-link-target"><span>Estremeñu</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%B3%D8%B1%D8%B9%D8%AA_%D8%A8%D8%B1%D8%AF%D8%A7%D8%B1%DB%8C" title="سرعت برداری - persiano" lang="fa" hreflang="fa" data-title="سرعت برداری" data-language-autonym="فارسی" data-language-local-name="persiano" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Nopeus" title="Nopeus - finlandese" lang="fi" hreflang="fi" data-title="Nopeus" data-language-autonym="Suomi" data-language-local-name="finlandese" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-fiu-vro mw-list-item"><a href="https://fiu-vro.wikipedia.org/wiki/Kib%C3%B5husvektor" title="Kibõhusvektor - võro" lang="vro" hreflang="vro" data-title="Kibõhusvektor" data-language-autonym="Võro" data-language-local-name="võro" class="interlanguage-link-target"><span>Võro</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Vecteur_vitesse" title="Vecteur vitesse - francese" lang="fr" hreflang="fr" data-title="Vecteur vitesse" data-language-autonym="Français" data-language-local-name="francese" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-frr mw-list-item"><a href="https://frr.wikipedia.org/wiki/Faard" title="Faard - frisone settentrionale" lang="frr" hreflang="frr" data-title="Faard" data-language-autonym="Nordfriisk" data-language-local-name="frisone settentrionale" class="interlanguage-link-target"><span>Nordfriisk</span></a></li><li class="interlanguage-link interwiki-fy mw-list-item"><a href="https://fy.wikipedia.org/wiki/Faasje" title="Faasje - frisone occidentale" lang="fy" hreflang="fy" data-title="Faasje" data-language-autonym="Frysk" data-language-local-name="frisone occidentale" 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/Treoluas" title="Treoluas - irlandese" lang="ga" hreflang="ga" data-title="Treoluas" data-language-autonym="Gaeilge" data-language-local-name="irlandese" 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/Velocidade" title="Velocidade - galiziano" lang="gl" hreflang="gl" data-title="Velocidade" data-language-autonym="Galego" data-language-local-name="galiziano" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-gv mw-list-item"><a href="https://gv.wikipedia.org/wiki/Bieauid-jeerit" title="Bieauid-jeerit - mannese" lang="gv" hreflang="gv" data-title="Bieauid-jeerit" data-language-autonym="Gaelg" data-language-local-name="mannese" class="interlanguage-link-target"><span>Gaelg</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%9E%D7%94%D7%99%D7%A8%D7%95%D7%AA" title="מהירות - ebraico" lang="he" hreflang="he" data-title="מהירות" data-language-autonym="עברית" data-language-local-name="ebraico" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%B5%E0%A5%87%E0%A4%97" title="वेग - hindi" lang="hi" hreflang="hi" data-title="वेग" data-language-autonym="हिन्दी" data-language-local-name="hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hif mw-list-item"><a href="https://hif.wikipedia.org/wiki/Raftaar" title="Raftaar - hindi figiano" lang="hif" hreflang="hif" data-title="Raftaar" data-language-autonym="Fiji Hindi" data-language-local-name="hindi figiano" class="interlanguage-link-target"><span>Fiji Hindi</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Brzina" title="Brzina - croato" lang="hr" hreflang="hr" data-title="Brzina" data-language-autonym="Hrvatski" data-language-local-name="croato" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-ht mw-list-item"><a href="https://ht.wikipedia.org/wiki/Velosite" title="Velosite - creolo haitiano" lang="ht" hreflang="ht" data-title="Velosite" data-language-autonym="Kreyòl ayisyen" data-language-local-name="creolo haitiano" class="interlanguage-link-target"><span>Kreyòl ayisyen</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Sebess%C3%A9g" title="Sebesség - ungherese" lang="hu" hreflang="hu" data-title="Sebesség" data-language-autonym="Magyar" data-language-local-name="ungherese" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D4%B1%D6%80%D5%A1%D5%A3%D5%B8%D6%82%D5%A9%D5%B5%D5%B8%D6%82%D5%B6" title="Արագություն - armeno" lang="hy" hreflang="hy" data-title="Արագություն" data-language-autonym="Հայերեն" data-language-local-name="armeno" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-ia mw-list-item"><a href="https://ia.wikipedia.org/wiki/Velocitate" title="Velocitate - interlingua" lang="ia" hreflang="ia" data-title="Velocitate" data-language-autonym="Interlingua" data-language-local-name="interlingua" class="interlanguage-link-target"><span>Interlingua</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Kecepatan" title="Kecepatan - indonesiano" lang="id" hreflang="id" data-title="Kecepatan" data-language-autonym="Bahasa Indonesia" data-language-local-name="indonesiano" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-ie mw-list-item"><a href="https://ie.wikipedia.org/wiki/Velocit%C3%A1" title="Velocitá - interlingue" lang="ie" hreflang="ie" data-title="Velocitá" data-language-autonym="Interlingue" data-language-local-name="interlingue" class="interlanguage-link-target"><span>Interlingue</span></a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/Hra%C3%B0i" title="Hraði - islandese" lang="is" hreflang="is" data-title="Hraði" data-language-autonym="Íslenska" data-language-local-name="islandese" class="interlanguage-link-target"><span>Íslenska</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E9%80%9F%E5%BA%A6" title="速度 - giapponese" lang="ja" hreflang="ja" data-title="速度" data-language-autonym="日本語" data-language-local-name="giapponese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-jv mw-list-item"><a href="https://jv.wikipedia.org/wiki/Kacepetan" title="Kacepetan - giavanese" lang="jv" hreflang="jv" data-title="Kacepetan" data-language-autonym="Jawa" data-language-local-name="giavanese" class="interlanguage-link-target"><span>Jawa</span></a></li><li class="interlanguage-link interwiki-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/%E1%83%A1%E1%83%98%E1%83%A9%E1%83%A5%E1%83%90%E1%83%A0%E1%83%94" title="სიჩქარე - georgiano" lang="ka" hreflang="ka" data-title="სიჩქარე" data-language-autonym="ქართული" data-language-local-name="georgiano" class="interlanguage-link-target"><span>ქართული</span></a></li><li class="interlanguage-link interwiki-kab mw-list-item"><a href="https://kab.wikipedia.org/wiki/Arured_amaway" title="Arured amaway - cabilo" lang="kab" hreflang="kab" data-title="Arured amaway" data-language-autonym="Taqbaylit" data-language-local-name="cabilo" class="interlanguage-link-target"><span>Taqbaylit</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%96%D1%8B%D0%BB%D0%B4%D0%B0%D0%BC%D0%B4%D1%8B%D2%9B" title="Жылдамдық - kazako" lang="kk" hreflang="kk" data-title="Жылдамдық" data-language-autonym="Қазақша" data-language-local-name="kazako" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-km mw-list-item"><a href="https://km.wikipedia.org/wiki/%E1%9E%9B%E1%9F%92%E1%9E%94%E1%9E%BF%E1%9E%93" title="ល្បឿន - khmer" lang="km" hreflang="km" data-title="ល្បឿន" data-language-autonym="ភាសាខ្មែរ" data-language-local-name="khmer" class="interlanguage-link-target"><span>ភាសាខ្មែរ</span></a></li><li class="interlanguage-link interwiki-kn mw-list-item"><a href="https://kn.wikipedia.org/wiki/%E0%B2%B5%E0%B3%87%E0%B2%97" title="ವೇಗ - kannada" lang="kn" hreflang="kn" data-title="ವೇಗ" data-language-autonym="ಕನ್ನಡ" data-language-local-name="kannada" class="interlanguage-link-target"><span>ಕನ್ನಡ</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%86%8D%EB%8F%84" title="속도 - coreano" lang="ko" hreflang="ko" data-title="속도" data-language-autonym="한국어" data-language-local-name="coreano" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-ks mw-list-item"><a href="https://ks.wikipedia.org/wiki/%D9%88%DB%8C%D9%96%DA%AF" title="ویٖگ - kashmiri" lang="ks" hreflang="ks" data-title="ویٖگ" data-language-autonym="कॉशुर / کٲشُر" data-language-local-name="kashmiri" class="interlanguage-link-target"><span>कॉशुर / کٲشُر</span></a></li><li class="interlanguage-link interwiki-kw mw-list-item"><a href="https://kw.wikipedia.org/wiki/Uskitter" title="Uskitter - cornico" lang="kw" hreflang="kw" data-title="Uskitter" data-language-autonym="Kernowek" data-language-local-name="cornico" class="interlanguage-link-target"><span>Kernowek</span></a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Velocitas" title="Velocitas - latino" lang="la" hreflang="la" data-title="Velocitas" data-language-autonym="Latina" data-language-local-name="latino" class="interlanguage-link-target"><span>Latina</span></a></li><li class="interlanguage-link interwiki-lfn mw-list-item"><a href="https://lfn.wikipedia.org/wiki/Rapidia_vetoral" title="Rapidia vetoral - Lingua Franca Nova" lang="lfn" hreflang="lfn" data-title="Rapidia vetoral" data-language-autonym="Lingua Franca Nova" data-language-local-name="Lingua Franca Nova" class="interlanguage-link-target"><span>Lingua Franca Nova</span></a></li><li class="interlanguage-link interwiki-li mw-list-item"><a href="https://li.wikipedia.org/wiki/Snelheid" title="Snelheid - limburghese" lang="li" hreflang="li" data-title="Snelheid" data-language-autonym="Limburgs" data-language-local-name="limburghese" class="interlanguage-link-target"><span>Limburgs</span></a></li><li class="interlanguage-link interwiki-lmo mw-list-item"><a href="https://lmo.wikipedia.org/wiki/Velocitaa" title="Velocitaa - lombardo" lang="lmo" hreflang="lmo" data-title="Velocitaa" data-language-autonym="Lombard" data-language-local-name="lombardo" class="interlanguage-link-target"><span>Lombard</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Vektorinis_greitis" title="Vektorinis greitis - lituano" lang="lt" hreflang="lt" data-title="Vektorinis greitis" data-language-autonym="Lietuvių" data-language-local-name="lituano" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/%C4%80trums" title="Ātrums - lettone" lang="lv" hreflang="lv" data-title="Ātrums" data-language-autonym="Latviešu" data-language-local-name="lettone" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-mg mw-list-item"><a href="https://mg.wikipedia.org/wiki/Hafainganam-pandeha" title="Hafainganam-pandeha - malgascio" lang="mg" hreflang="mg" data-title="Hafainganam-pandeha" data-language-autonym="Malagasy" data-language-local-name="malgascio" class="interlanguage-link-target"><span>Malagasy</span></a></li><li class="interlanguage-link interwiki-min mw-list-item"><a href="https://min.wikipedia.org/wiki/Laju" title="Laju - menangkabau" lang="min" hreflang="min" data-title="Laju" data-language-autonym="Minangkabau" data-language-local-name="menangkabau" class="interlanguage-link-target"><span>Minangkabau</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%91%D1%80%D0%B7%D0%B8%D0%BD%D0%B0" title="Брзина - macedone" lang="mk" hreflang="mk" data-title="Брзина" data-language-autonym="Македонски" data-language-local-name="macedone" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%AA%E0%B5%8D%E0%B4%B0%E0%B4%B5%E0%B5%87%E0%B4%97%E0%B4%82" title="പ്രവേഗം - malayalam" lang="ml" hreflang="ml" data-title="പ്രവേഗം" data-language-autonym="മലയാളം" data-language-local-name="malayalam" class="interlanguage-link-target"><span>മലയാളം</span></a></li><li class="interlanguage-link interwiki-mn mw-list-item"><a href="https://mn.wikipedia.org/wiki/%D0%A5%D1%83%D1%80%D0%B4" title="Хурд - mongolo" lang="mn" hreflang="mn" data-title="Хурд" data-language-autonym="Монгол" data-language-local-name="mongolo" class="interlanguage-link-target"><span>Монгол</span></a></li><li class="interlanguage-link interwiki-mr mw-list-item"><a href="https://mr.wikipedia.org/wiki/%E0%A4%B5%E0%A5%87%E0%A4%97" title="वेग - marathi" lang="mr" hreflang="mr" data-title="वेग" 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/Halaju" title="Halaju - malese" lang="ms" hreflang="ms" data-title="Halaju" data-language-autonym="Bahasa Melayu" data-language-local-name="malese" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-my mw-list-item"><a href="https://my.wikipedia.org/wiki/%E1%80%A1%E1%80%9C%E1%80%BB%E1%80%84%E1%80%BA" title="အလျင် - birmano" lang="my" hreflang="my" data-title="အလျင်" data-language-autonym="မြန်မာဘာသာ" data-language-local-name="birmano" class="interlanguage-link-target"><span>မြန်မာဘာသာ</span></a></li><li class="interlanguage-link interwiki-myv mw-list-item"><a href="https://myv.wikipedia.org/wiki/%D0%9C%D0%BE%D0%BB%D0%B5%D0%B2%D0%BA%D1%81" title="Молевкс - erzya" lang="myv" hreflang="myv" data-title="Молевкс" data-language-autonym="Эрзянь" data-language-local-name="erzya" class="interlanguage-link-target"><span>Эрзянь</span></a></li><li class="interlanguage-link interwiki-nds mw-list-item"><a href="https://nds.wikipedia.org/wiki/Snelligkeid" title="Snelligkeid - basso tedesco" lang="nds" hreflang="nds" data-title="Snelligkeid" data-language-autonym="Plattdüütsch" data-language-local-name="basso tedesco" class="interlanguage-link-target"><span>Plattdüütsch</span></a></li><li class="interlanguage-link interwiki-ne mw-list-item"><a href="https://ne.wikipedia.org/wiki/%E0%A4%97%E0%A4%A4%E0%A4%BF" title="गति - nepalese" lang="ne" hreflang="ne" data-title="गति" data-language-autonym="नेपाली" data-language-local-name="nepalese" class="interlanguage-link-target"><span>नेपाली</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Snelheid" title="Snelheid - olandese" lang="nl" hreflang="nl" data-title="Snelheid" data-language-autonym="Nederlands" data-language-local-name="olandese" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Hastigheit" title="Hastigheit - norvegese nynorsk" lang="nn" hreflang="nn" data-title="Hastigheit" data-language-autonym="Norsk nynorsk" data-language-local-name="norvegese nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Hastighet" title="Hastighet - norvegese bokmål" lang="nb" hreflang="nb" data-title="Hastighet" data-language-autonym="Norsk bokmål" data-language-local-name="norvegese bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-oc mw-list-item"><a href="https://oc.wikipedia.org/wiki/Velocitat" title="Velocitat - occitano" lang="oc" hreflang="oc" data-title="Velocitat" data-language-autonym="Occitan" data-language-local-name="occitano" class="interlanguage-link-target"><span>Occitan</span></a></li><li class="interlanguage-link interwiki-om mw-list-item"><a href="https://om.wikipedia.org/wiki/Ariitii" title="Ariitii - oromo" lang="om" hreflang="om" data-title="Ariitii" data-language-autonym="Oromoo" data-language-local-name="oromo" class="interlanguage-link-target"><span>Oromoo</span></a></li><li class="interlanguage-link interwiki-pa mw-list-item"><a href="https://pa.wikipedia.org/wiki/%E0%A8%B5%E0%A9%87%E0%A8%97" title="ਵੇਗ - punjabi" lang="pa" hreflang="pa" data-title="ਵੇਗ" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="punjabi" class="interlanguage-link-target"><span>ਪੰਜਾਬੀ</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Pr%C4%99dko%C5%9B%C4%87" title="Prędkość - polacco" lang="pl" hreflang="pl" data-title="Prędkość" data-language-autonym="Polski" data-language-local-name="polacco" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pms mw-list-item"><a href="https://pms.wikipedia.org/wiki/Andi" title="Andi - piemontese" lang="pms" hreflang="pms" data-title="Andi" data-language-autonym="Piemontèis" data-language-local-name="piemontese" class="interlanguage-link-target"><span>Piemontèis</span></a></li><li class="interlanguage-link interwiki-pnb mw-list-item"><a href="https://pnb.wikipedia.org/wiki/%D9%88%D9%84%D8%A7%D8%B3%D9%B9%DB%8C" title="ولاسٹی - Western Punjabi" lang="pnb" hreflang="pnb" data-title="ولاسٹی" data-language-autonym="پنجابی" data-language-local-name="Western Punjabi" class="interlanguage-link-target"><span>پنجابی</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Velocidade" title="Velocidade - portoghese" lang="pt" hreflang="pt" data-title="Velocidade" data-language-autonym="Português" data-language-local-name="portoghese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-qu mw-list-item"><a href="https://qu.wikipedia.org/wiki/Utqa_kay" title="Utqa kay - quechua" lang="qu" hreflang="qu" data-title="Utqa kay" data-language-autonym="Runa Simi" data-language-local-name="quechua" class="interlanguage-link-target"><span>Runa Simi</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Vitez%C4%83" title="Viteză - rumeno" lang="ro" hreflang="ro" data-title="Viteză" data-language-autonym="Română" data-language-local-name="rumeno" 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%A1%D0%BA%D0%BE%D1%80%D0%BE%D1%81%D1%82%D1%8C" title="Скорость - russo" lang="ru" hreflang="ru" data-title="Скорость" data-language-autonym="Русский" data-language-local-name="russo" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-rue mw-list-item"><a href="https://rue.wikipedia.org/wiki/%D0%A8%D0%B2%D1%8B%D0%B4%D0%BA%D0%BE%D1%81%D1%82%D1%8C" title="Швыдкость - ruteno" lang="rue" hreflang="rue" data-title="Швыдкость" data-language-autonym="Русиньскый" data-language-local-name="ruteno" class="interlanguage-link-target"><span>Русиньскый</span></a></li><li class="interlanguage-link interwiki-sc badge-Q17437796 badge-featuredarticle mw-list-item" title="voce in vetrina"><a href="https://sc.wikipedia.org/wiki/Velotzidade" title="Velotzidade - sardo" lang="sc" hreflang="sc" data-title="Velotzidade" data-language-autonym="Sardu" data-language-local-name="sardo" class="interlanguage-link-target"><span>Sardu</span></a></li><li class="interlanguage-link interwiki-sd mw-list-item"><a href="https://sd.wikipedia.org/wiki/%D9%88%D9%8A%D9%84%D8%A7%D8%B3%D9%BD%D9%8A" title="ويلاسٽي - sindhi" lang="sd" hreflang="sd" data-title="ويلاسٽي" data-language-autonym="سنڌي" data-language-local-name="sindhi" class="interlanguage-link-target"><span>سنڌي</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Brzina" title="Brzina - serbo-croato" lang="sh" hreflang="sh" data-title="Brzina" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="serbo-croato" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-si mw-list-item"><a href="https://si.wikipedia.org/wiki/%E0%B6%B4%E0%B7%8A%E2%80%8D%E0%B6%BB%E0%B7%80%E0%B7%9A%E0%B6%9C%E0%B6%BA" title="ප්‍රවේගය - singalese" lang="si" hreflang="si" data-title="ප්‍රවේගය" data-language-autonym="සිංහල" data-language-local-name="singalese" 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/Velocity" title="Velocity - Simple English" lang="en-simple" hreflang="en-simple" data-title="Velocity" 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/R%C3%BDchlos%C5%A5_(fyzik%C3%A1lna_veli%C4%8Dina)" title="Rýchlosť (fyzikálna veličina) - slovacco" lang="sk" hreflang="sk" data-title="Rýchlosť (fyzikálna veličina)" data-language-autonym="Slovenčina" data-language-local-name="slovacco" 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/Hitrost" title="Hitrost - sloveno" lang="sl" hreflang="sl" data-title="Hitrost" data-language-autonym="Slovenščina" data-language-local-name="sloveno" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-smn mw-list-item"><a href="https://smn.wikipedia.org/wiki/Li%C3%A4htu" title="Liähtu - sami di Inari" lang="smn" hreflang="smn" data-title="Liähtu" data-language-autonym="Anarâškielâ" data-language-local-name="sami di Inari" class="interlanguage-link-target"><span>Anarâškielâ</span></a></li><li class="interlanguage-link interwiki-sn mw-list-item"><a href="https://sn.wikipedia.org/wiki/Muchacha" title="Muchacha - shona" lang="sn" hreflang="sn" data-title="Muchacha" data-language-autonym="ChiShona" data-language-local-name="shona" class="interlanguage-link-target"><span>ChiShona</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Shpejt%C3%ABsia_(fizik%C3%AB)" title="Shpejtësia (fizikë) - albanese" lang="sq" hreflang="sq" data-title="Shpejtësia (fizikë)" data-language-autonym="Shqip" data-language-local-name="albanese" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%91%D1%80%D0%B7%D0%B8%D0%BD%D0%B0" title="Брзина - serbo" lang="sr" hreflang="sr" data-title="Брзина" data-language-autonym="Српски / srpski" data-language-local-name="serbo" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Hastighet" title="Hastighet - svedese" lang="sv" hreflang="sv" data-title="Hastighet" data-language-autonym="Svenska" data-language-local-name="svedese" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-sw mw-list-item"><a href="https://sw.wikipedia.org/wiki/Kasimwelekeo" title="Kasimwelekeo - swahili" lang="sw" hreflang="sw" data-title="Kasimwelekeo" data-language-autonym="Kiswahili" data-language-local-name="swahili" class="interlanguage-link-target"><span>Kiswahili</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%A4%E0%AE%BF%E0%AE%9A%E0%AF%88%E0%AE%B5%E0%AF%87%E0%AE%95%E0%AE%AE%E0%AF%8D" title="திசைவேகம் - tamil" lang="ta" hreflang="ta" data-title="திசைவேகம்" data-language-autonym="தமிழ்" data-language-local-name="tamil" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-te mw-list-item"><a href="https://te.wikipedia.org/wiki/%E0%B0%B5%E0%B1%87%E0%B0%97%E0%B0%82" title="వేగం - telugu" lang="te" hreflang="te" data-title="వేగం" data-language-autonym="తెలుగు" data-language-local-name="telugu" class="interlanguage-link-target"><span>తెలుగు</span></a></li><li class="interlanguage-link interwiki-tg mw-list-item"><a href="https://tg.wikipedia.org/wiki/%D0%A1%D1%83%D1%80%D1%8A%D0%B0%D1%82" title="Суръат - tagico" lang="tg" hreflang="tg" data-title="Суръат" data-language-autonym="Тоҷикӣ" data-language-local-name="tagico" class="interlanguage-link-target"><span>Тоҷикӣ</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%84%E0%B8%A7%E0%B8%B2%E0%B8%A1%E0%B9%80%E0%B8%A3%E0%B9%87%E0%B8%A7" title="ความเร็ว - thailandese" lang="th" hreflang="th" data-title="ความเร็ว" data-language-autonym="ไทย" data-language-local-name="thailandese" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tl mw-list-item"><a href="https://tl.wikipedia.org/wiki/Tulin" title="Tulin - tagalog" lang="tl" hreflang="tl" data-title="Tulin" data-language-autonym="Tagalog" data-language-local-name="tagalog" class="interlanguage-link-target"><span>Tagalog</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/H%C4%B1z" title="Hız - turco" lang="tr" hreflang="tr" data-title="Hız" data-language-autonym="Türkçe" data-language-local-name="turco" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-tt mw-list-item"><a href="https://tt.wikipedia.org/wiki/%D0%A2%D0%B8%D0%B7%D0%BB%D0%B5%D0%BA" title="Тизлек - tataro" lang="tt" hreflang="tt" data-title="Тизлек" data-language-autonym="Татарча / tatarça" data-language-local-name="tataro" class="interlanguage-link-target"><span>Татарча / tatarça</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%A8%D0%B2%D0%B8%D0%B4%D0%BA%D1%96%D1%81%D1%82%D1%8C" title="Швидкість - ucraino" lang="uk" hreflang="uk" data-title="Швидкість" data-language-autonym="Українська" data-language-local-name="ucraino" 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%88%D9%84%D8%A7%D8%B3%D9%B9%DB%8C" title="ولاسٹی - urdu" lang="ur" hreflang="ur" data-title="ولاسٹی" data-language-autonym="اردو" data-language-local-name="urdu" class="interlanguage-link-target"><span>اردو</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Tezlik" title="Tezlik - uzbeco" lang="uz" hreflang="uz" data-title="Tezlik" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="uzbeco" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></a></li><li class="interlanguage-link interwiki-vep mw-list-item"><a href="https://vep.wikipedia.org/wiki/Piguz" title="Piguz - vepso" lang="vep" hreflang="vep" data-title="Piguz" data-language-autonym="Vepsän kel’" data-language-local-name="vepso" class="interlanguage-link-target"><span>Vepsän kel’</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/V%E1%BA%ADn_t%E1%BB%91c" title="Vận tốc - vietnamita" lang="vi" hreflang="vi" data-title="Vận tốc" data-language-autonym="Tiếng Việt" data-language-local-name="vietnamita" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-war mw-list-item"><a href="https://war.wikipedia.org/wiki/Belosidad" title="Belosidad - waray" lang="war" hreflang="war" data-title="Belosidad" data-language-autonym="Winaray" data-language-local-name="waray" class="interlanguage-link-target"><span>Winaray</span></a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E9%80%9F%E5%BA%A6" title="速度 - wu" lang="wuu" hreflang="wuu" data-title="速度" 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-xh mw-list-item"><a href="https://xh.wikipedia.org/wiki/I-velocity" title="I-velocity - xhosa" lang="xh" hreflang="xh" data-title="I-velocity" data-language-autonym="IsiXhosa" data-language-local-name="xhosa" class="interlanguage-link-target"><span>IsiXhosa</span></a></li><li class="interlanguage-link interwiki-yi mw-list-item"><a href="https://yi.wikipedia.org/wiki/%D7%92%D7%99%D7%9B%D7%A7%D7%99%D7%99%D7%98" title="גיכקייט - yiddish" lang="yi" hreflang="yi" data-title="גיכקייט" data-language-autonym="ייִדיש" data-language-local-name="yiddish" class="interlanguage-link-target"><span>ייִדיש</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E9%80%9F%E5%BA%A6" title="速度 - cinese" lang="zh" hreflang="zh" data-title="速度" data-language-autonym="中文" data-language-local-name="cinese" class="interlanguage-link-target"><span>中文</span></a></li><li class="interlanguage-link interwiki-zh-classical mw-list-item"><a href="https://zh-classical.wikipedia.org/wiki/%E9%80%9F%E5%BA%A6" title="速度 - cinese classico" lang="lzh" hreflang="lzh" data-title="速度" data-language-autonym="文言" data-language-local-name="cinese classico" class="interlanguage-link-target"><span>文言</span></a></li><li class="interlanguage-link interwiki-zh-min-nan mw-list-item"><a href="https://zh-min-nan.wikipedia.org/wiki/Sok-t%C5%8D%CD%98" title="Sok-tō͘ - min nan" lang="nan" hreflang="nan" data-title="Sok-tō͘" data-language-autonym="閩南語 / Bân-lâm-gú" data-language-local-name="min nan" class="interlanguage-link-target"><span>閩南語 / Bân-lâm-gú</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E9%80%9F%E5%BA%A6" title="速度 - cantonese" lang="yue" hreflang="yue" data-title="速度" data-language-autonym="粵語" data-language-local-name="cantonese" class="interlanguage-link-target"><span>粵語</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q11465#sitelinks-wikipedia" title="Modifica collegamenti interlinguistici" class="wbc-editpage">Modifica collegamenti</a></span></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div class="vector-page-toolbar-container"> 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img{display:flex}.mw-parser-output .hatnote-text{font-style:italic}body:not(.skin-minerva) .mw-parser-output .hatnote{border:1px solid #CCC;display:flex;margin:.5em 0;padding:.2em .5em}body:not(.skin-minerva) .mw-parser-output .hatnote-text{padding-left:.5em}body.skin-minerva .mw-parser-output .hatnote-icon{padding-right:8px}body.skin-minerva .mw-parser-output .hatnote-icon img{height:auto;width:16px}body.skin--responsive .mw-parser-output .hatnote a.new{color:#d73333}body.skin--responsive .mw-parser-output .hatnote a.new:visited{color:#a55858}</style> <div class="hatnote noprint nota-disambigua"> <div class="hatnote-content"><span class="noviewer hatnote-icon" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/bc/Nota_disambigua.svg/18px-Nota_disambigua.svg.png" decoding="async" width="18" height="18" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/bc/Nota_disambigua.svg/27px-Nota_disambigua.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/bc/Nota_disambigua.svg/36px-Nota_disambigua.svg.png 2x" data-file-width="200" data-file-height="200" /></span></span> <span class="hatnote-text"><a href="/wiki/Aiuto:Disambiguazione" title="Aiuto:Disambiguazione">Disambiguazione</a> – Se stai cercando altri significati, vedi <b><a href="/wiki/Velocit%C3%A0_(disambigua)" class="mw-disambig" title="Velocità (disambigua)">Velocità (disambigua)</a></b>.</span></div> </div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r130658281"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r139142988"> <div class="hatnote noprint nota-disambigua"> <div class="hatnote-content"><span class="noviewer hatnote-icon" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/bc/Nota_disambigua.svg/18px-Nota_disambigua.svg.png" decoding="async" width="18" height="18" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/bc/Nota_disambigua.svg/27px-Nota_disambigua.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/bc/Nota_disambigua.svg/36px-Nota_disambigua.svg.png 2x" data-file-width="200" data-file-height="200" /></span></span> <span class="hatnote-text"><a href="/wiki/Aiuto:Disambiguazione" title="Aiuto:Disambiguazione">Disambiguazione</a> – "<a href="/wiki/Velocit%C3%A0_massima" class="mw-redirect" title="Velocità massima">Velocità massima</a>" rimanda qui. Se stai cercando il film del 2002 diretto da Daniele Vicari, vedi <b><a href="/wiki/Velocit%C3%A0_massima_(film)" title="Velocità massima (film)">Velocità massima (film)</a></b>.</span></div> </div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:US_Navy_040501-N-1336S-037_The_U.S._Navy_sponsored_Chevy_Monte_Carlo_NASCAR_leads_a_pack_into_turn_four_at_California_Speedway.jpg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/40/US_Navy_040501-N-1336S-037_The_U.S._Navy_sponsored_Chevy_Monte_Carlo_NASCAR_leads_a_pack_into_turn_four_at_California_Speedway.jpg/220px-US_Navy_040501-N-1336S-037_The_U.S._Navy_sponsored_Chevy_Monte_Carlo_NASCAR_leads_a_pack_into_turn_four_at_California_Speedway.jpg" decoding="async" width="220" height="147" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/40/US_Navy_040501-N-1336S-037_The_U.S._Navy_sponsored_Chevy_Monte_Carlo_NASCAR_leads_a_pack_into_turn_four_at_California_Speedway.jpg/330px-US_Navy_040501-N-1336S-037_The_U.S._Navy_sponsored_Chevy_Monte_Carlo_NASCAR_leads_a_pack_into_turn_four_at_California_Speedway.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/40/US_Navy_040501-N-1336S-037_The_U.S._Navy_sponsored_Chevy_Monte_Carlo_NASCAR_leads_a_pack_into_turn_four_at_California_Speedway.jpg/440px-US_Navy_040501-N-1336S-037_The_U.S._Navy_sponsored_Chevy_Monte_Carlo_NASCAR_leads_a_pack_into_turn_four_at_California_Speedway.jpg 2x" data-file-width="1800" data-file-height="1200" /></a><figcaption>In fisica, la velocità è un vettore che descrive la rapidità, la direzione ed il verso; pertanto la velocità delle automobili che percorrono una curva cambia ad ogni istante a causa del cambiamento di direzione, anche se la rapidità rimane costante</figcaption></figure> <p>In <a href="/wiki/Fisica" title="Fisica">fisica</a>, in primo luogo in <a href="/wiki/Cinematica" title="Cinematica">cinematica</a>, la <b>velocità</b> (dal <a href="/wiki/Lingua_latina" title="Lingua latina">latino</a> <i>vēlōcitās</i>, a sua volta derivato da <i>vēlōx</i>, cioè <i>veloce</i><sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup>) è una <a href="/wiki/Grandezza_vettoriale" title="Grandezza vettoriale">grandezza vettoriale</a> definita come la variazione della posizione di un corpo in funzione del <a href="/wiki/Tempo" title="Tempo">tempo</a>, ossia, in termini matematici, come la <a href="/wiki/Derivata" title="Derivata">derivata</a> del <a href="/wiki/Vettore_(matematica)" title="Vettore (matematica)">vettore</a> <a href="/wiki/Posizione" title="Posizione">posizione</a> rispetto al tempo.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> Nel <a href="/wiki/Sistema_internazionale_di_unit%C3%A0_di_misura" title="Sistema internazionale di unità di misura">Sistema Internazionale</a> la velocità si misura in m·s<sup>-1</sup> (<a href="/wiki/Metri_al_secondo" class="mw-redirect" title="Metri al secondo">metri al secondo</a>). </p><p>Quando non specificato, per "velocità" si intende la <i>velocità traslazionale</i>, sottintendendo che lo spostamento a cui si fa riferimento è una <a href="/wiki/Traslazione_(geometria)" title="Traslazione (geometria)">traslazione</a> nello spazio. Il termine, "velocità", infatti, può essere utilizzato con un significato più generale per indicare la variazione di una <a href="/wiki/Coordinate_generalizzate" title="Coordinate generalizzate">coordinata spaziale</a> in funzione del tempo. Ad esempio, nella descrizione del <a href="/wiki/Moto_(fisica)" title="Moto (fisica)">moto</a> <a href="/wiki/Rotazione_(matematica)" title="Rotazione (matematica)">rotatorio</a>, per definire la <i>velocità di rotazione</i> si usano la <a href="/wiki/Velocit%C3%A0_angolare" title="Velocità angolare">velocità angolare</a> e la <a href="/wiki/Velocit%C3%A0_areolare" title="Velocità areolare">velocità areolare</a>. </p><p>Si indica con <b>velocità scalare</b> il <a href="/wiki/Norma_(matematica)" title="Norma (matematica)">modulo</a> della velocità (in inglese, si usano due termini diversi, <i>speed</i> per la velocità scalare e <i>velocity</i> per la velocità in senso vettoriale). La variazione della velocità, sia in aumento che in diminuzione, è l'<a href="/wiki/Accelerazione" title="Accelerazione">accelerazione</a>, anche se nel linguaggio comune a volte si parla di "decelerazione" quando la velocità diminuisce. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Velocità_media_e_istantanea"><span id="Velocit.C3.A0_media_e_istantanea"></span>Velocità media e istantanea</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Velocit%C3%A0&veaction=edit&section=1" title="Modifica la sezione Velocità media e istantanea" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Velocit%C3%A0&action=edit&section=1" title="Edit section's source code: Velocità media e istantanea"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><span id="Velocità_media"></span> La velocità è una grandezza vettoriale che descrive lo stato di moto di un corpo e, in quanto tale, è caratterizzato da una lunghezza (rapidità), una direzione e un verso. </p><p>Si definisce <i>velocità media</i> <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 {\bar {\mathbf {v} }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bar {\mathbf {v} }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7803bbf7495a6770131ba960a9f90f41d11ea5e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.411ex; height:2.009ex;" alt="{\displaystyle {\bar {\mathbf {v} }}}"></span> il rapporto tra lo <a href="/wiki/Spostamento_(cinematica)" title="Spostamento (cinematica)">spostamento</a>, inteso come la <a href="/wiki/Differenza_finita" title="Differenza finita">variazione</a> della posizione, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta \mathbf {r} =\mathbf {r_{2}} -\mathbf {r_{1}} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="bold">r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">2</mn> </mrow> </msub> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="bold">r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">1</mn> </mrow> </msub> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta \mathbf {r} =\mathbf {r_{2}} -\mathbf {r_{1}} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/21486529c8892c79ca0432712378a307ea2ba028" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.536ex; height:2.509ex;" alt="{\displaystyle \Delta \mathbf {r} =\mathbf {r_{2}} -\mathbf {r_{1}} }"></span> e l'intervallo di tempo <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\Delta t}={t_{2}-t_{1}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Δ</mi> <mi>t</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\Delta t}={t_{2}-t_{1}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cdb0742024c173c1b4f5cd1bb6c47f16c12c14f6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.502ex; height:2.509ex;" alt="{\displaystyle {\Delta t}={t_{2}-t_{1}}}"></span> impiegato a percorrerlo:<sup id="cite_ref-Maz8_3-0" class="reference"><a href="#cite_note-Maz8-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bar {\mathbf {v} }}={\frac {\mathbf {r} _{2}-\mathbf {r} _{1}}{t_{2}-t_{1}}}={\frac {\Delta \mathbf {r} }{\Delta t}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mrow> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">Δ</mi> <mi>t</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bar {\mathbf {v} }}={\frac {\mathbf {r} _{2}-\mathbf {r} _{1}}{t_{2}-t_{1}}}={\frac {\Delta \mathbf {r} }{\Delta t}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72794be759f1518f4b3dbfb70585313425bbaa69" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:19.471ex; height:5.676ex;" alt="{\displaystyle {\bar {\mathbf {v} }}={\frac {\mathbf {r} _{2}-\mathbf {r} _{1}}{t_{2}-t_{1}}}={\frac {\Delta \mathbf {r} }{\Delta t}}}"></span></dd></dl> <p>dove <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 {r_{1}} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="bold">r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">1</mn> </mrow> </msub> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {r_{1}} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3bc5c904c6ad4fe0ee92239113f6320beb682ff5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.279ex; height:2.009ex;" alt="{\displaystyle \mathbf {r_{1}} }"></span> e <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {r_{2}} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="bold">r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">2</mn> </mrow> </msub> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {r_{2}} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ce19d83d05685025879b6ed95fef920cefa641b9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.279ex; height:2.009ex;" alt="{\displaystyle \mathbf {r_{2}} }"></span> sono i <a href="/wiki/Vettore_(matematica)" title="Vettore (matematica)">vettori</a> posizione agli istanti iniziale <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cb0768c0bd659f2f84fb5ef9f4b74f336123d915" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.894ex; height:2.343ex;" alt="{\displaystyle t_{1}}"></span> e finale <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/749fee708b41e7079eabd50d61c8bf3e965db16f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.894ex; height:2.343ex;" alt="{\displaystyle t_{2}}"></span>. La velocità media può essere vista come il coefficiente angolare della <a href="/wiki/Retta" title="Retta">retta</a> secante le due posizioni in un grafico spazio-tempo. In particolare si parla di velocità positiva , se l'angolo che la retta forma con il semiasse positivo delle ascisse, nel I quadrante (del sistema spazio-tempo), è acuto e di velocità negativa , se l'angolo che la retta forma è ottuso. </p><p>Si definisce <i>velocità istantanea</i> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {v} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {v} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35c1866e359fbfd2e0f606c725ba5cc37a5195d6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.411ex; height:1.676ex;" alt="{\displaystyle \mathbf {v} }"></span> il <a href="/wiki/Limite_di_una_funzione" title="Limite di una funzione">limite</a> della velocità media per intervalli di tempo molto brevi, ovvero la <a href="/wiki/Derivata" title="Derivata">derivata</a> della posizione rispetto al tempo:<sup id="cite_ref-Maz8_3-1" class="reference"><a href="#cite_note-Maz8-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup>. In parole povere la velocità istantanea è il valore limite della velocità media nell'intorno di un determinato istante quando la variazione di tempo <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8c28867ecd34e2caed12cf38feadf6a81a7ee542" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.775ex; height:2.176ex;" alt="{\displaystyle \Delta t}"></span> considerata tende al valore 0. </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {v} =\lim _{t_{2}\to t_{1}}{\frac {\mathbf {r} (t_{2})-\mathbf {r} (t_{1})}{t_{2}-t_{1}}}=\lim _{\Delta t\to 0}{{\mathbf {r} (t+\Delta t)-\mathbf {r} (t)} \over \Delta t}={\frac {\operatorname {d} \!\mathbf {r} }{\operatorname {d} \!t}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">→</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">(</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">(</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Δ</mi> <mi>t</mi> <mo stretchy="false">→</mo> <mn>0</mn> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo>+</mo> <mi mathvariant="normal">Δ</mi> <mi>t</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="normal">Δ</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">d</mi> <mspace width="negativethinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">d</mi> <mspace width="negativethinmathspace" /> <mi>t</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {v} =\lim _{t_{2}\to t_{1}}{\frac {\mathbf {r} (t_{2})-\mathbf {r} (t_{1})}{t_{2}-t_{1}}}=\lim _{\Delta t\to 0}{{\mathbf {r} (t+\Delta t)-\mathbf {r} (t)} \over \Delta t}={\frac {\operatorname {d} \!\mathbf {r} }{\operatorname {d} \!t}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e7869a040507bd0abc1417692f023394d5ece2bb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:53.714ex; height:6.009ex;" alt="{\displaystyle \mathbf {v} =\lim _{t_{2}\to t_{1}}{\frac {\mathbf {r} (t_{2})-\mathbf {r} (t_{1})}{t_{2}-t_{1}}}=\lim _{\Delta t\to 0}{{\mathbf {r} (t+\Delta t)-\mathbf {r} (t)} \over \Delta t}={\frac {\operatorname {d} \!\mathbf {r} }{\operatorname {d} \!t}}}"></span></dd></dl> <p>Si noti che la velocità media è proprio il risultato della <a href="/wiki/Media_(statistica)" title="Media (statistica)">media</a> della velocità istantanea in un tempo finito <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8c28867ecd34e2caed12cf38feadf6a81a7ee542" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.775ex; height:2.176ex;" alt="{\displaystyle \Delta t}"></span>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle \mathbf {v} \rangle ={\frac {1}{t_{2}-t_{1}}}\int _{t_{1}}^{t_{2}}{\frac {\mathrm {d} \mathbf {r} }{\mathrm {d} t}}\,\mathrm {d} t={\frac {\mathbf {r} (t_{2})-\mathbf {r} (t_{1})}{t_{2}-t_{1}}}={\frac {\Delta \mathbf {r} }{\Delta t}}}"> <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">v</mi> </mrow> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </mfrac> </mrow> <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> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">(</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">(</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">Δ</mi> <mi>t</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle \mathbf {v} \rangle ={\frac {1}{t_{2}-t_{1}}}\int _{t_{1}}^{t_{2}}{\frac {\mathrm {d} \mathbf {r} }{\mathrm {d} t}}\,\mathrm {d} t={\frac {\mathbf {r} (t_{2})-\mathbf {r} (t_{1})}{t_{2}-t_{1}}}={\frac {\Delta \mathbf {r} }{\Delta t}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9c56afe6a37e4011225890042b0101fefeb86998" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:47.787ex; height:6.509ex;" alt="{\displaystyle \langle \mathbf {v} \rangle ={\frac {1}{t_{2}-t_{1}}}\int _{t_{1}}^{t_{2}}{\frac {\mathrm {d} \mathbf {r} }{\mathrm {d} t}}\,\mathrm {d} t={\frac {\mathbf {r} (t_{2})-\mathbf {r} (t_{1})}{t_{2}-t_{1}}}={\frac {\Delta \mathbf {r} }{\Delta t}}}"></span></dd></dl> <p>avendo usato il <a href="/wiki/Teorema_fondamentale_del_calcolo_integrale" title="Teorema fondamentale del calcolo integrale">teorema fondamentale del calcolo integrale</a>. </p><p>In un contesto più formale, sia <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 s(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s(t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c484de351ba40ccb9a5ad522c29c1aac5686c0df" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.739ex; height:2.843ex;" alt="{\displaystyle s(t)}"></span> la <a href="/wiki/Lunghezza_di_un_arco" title="Lunghezza di un arco">lunghezza di un arco</a> della <a href="/wiki/Curva_(matematica)" title="Curva (matematica)">curva</a> percorsa dall'oggetto in moto, ovvero lo spostamento dell'oggetto al tempo <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65658b7b223af9e1acc877d848888ecdb4466560" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}"></span>. La <a href="/wiki/Norma_(matematica)" title="Norma (matematica)">norma</a> della velocità istantanea nel punto <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 {r} =(x,y,z)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {r} =(x,y,z)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d47345c2e02313cffb6fdaf6c5957f66c6507472" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.651ex; height:2.843ex;" alt="{\displaystyle \mathbf {r} =(x,y,z)}"></span> è la derivata dello spostamento rispetto al tempo:<sup id="cite_ref-mathworld_4-0" class="reference"><a href="#cite_note-mathworld-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> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v={\frac {\mathrm {d} s}{\mathrm {d} t}}={\frac {\mathrm {d} }{\mathrm {d} t}}\left({\sqrt {\mathrm {d} x^{2}+\mathrm {d} y^{2}+\mathrm {d} z^{2}}}\right)={\sqrt {\left({\frac {\mathrm {d} x}{\mathrm {d} t}}\right)^{2}+\left({\frac {\mathrm {d} y}{\mathrm {d} t}}\right)^{2}+\left({\frac {\mathrm {d} z}{\mathrm {d} t}}\right)^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>s</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>y</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>z</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v={\frac {\mathrm {d} s}{\mathrm {d} t}}={\frac {\mathrm {d} }{\mathrm {d} t}}\left({\sqrt {\mathrm {d} x^{2}+\mathrm {d} y^{2}+\mathrm {d} z^{2}}}\right)={\sqrt {\left({\frac {\mathrm {d} x}{\mathrm {d} t}}\right)^{2}+\left({\frac {\mathrm {d} y}{\mathrm {d} t}}\right)^{2}+\left({\frac {\mathrm {d} z}{\mathrm {d} t}}\right)^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7558f92a928aec1d2b9b48032233db68bce51089" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:70.434ex; height:7.676ex;" alt="{\displaystyle v={\frac {\mathrm {d} s}{\mathrm {d} t}}={\frac {\mathrm {d} }{\mathrm {d} t}}\left({\sqrt {\mathrm {d} x^{2}+\mathrm {d} y^{2}+\mathrm {d} z^{2}}}\right)={\sqrt {\left({\frac {\mathrm {d} x}{\mathrm {d} t}}\right)^{2}+\left({\frac {\mathrm {d} y}{\mathrm {d} t}}\right)^{2}+\left({\frac {\mathrm {d} z}{\mathrm {d} t}}\right)^{2}}}}"></span></dd></dl> <p>ed il vettore velocità ha la direzione del moto: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {v} =v\mathbf {\hat {\mathbf {T} }} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>=</mo> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">T</mi> </mrow> <mo mathvariant="bold" stretchy="false">^</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {v} =v\mathbf {\hat {\mathbf {T} }} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f7c50fb325908e4fc3153316c4be0884755ace6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.496ex; height:2.843ex;" alt="{\displaystyle \mathbf {v} =v\mathbf {\hat {\mathbf {T} }} }"></span></dd></dl> <p>con <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {\mathbf {T} }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">T</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {\mathbf {T} }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b5bb114b459259ac4d24c8f34439b8117bc291e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.859ex; height:2.843ex;" alt="{\displaystyle {\hat {\mathbf {T} }}}"></span> il vettore unitario tangente alla curva. </p> <div class="mw-heading mw-heading3"><h3 id="Velocità_in_due_dimensioni"><span id="Velocit.C3.A0_in_due_dimensioni"></span>Velocità in due dimensioni</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Velocit%C3%A0&veaction=edit&section=2" title="Modifica la sezione Velocità in due dimensioni" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Velocit%C3%A0&action=edit&section=2" title="Edit section's source code: Velocità in due dimensioni"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Utilizzando uno spazio bidimensionale, la velocità media e quella istantanea si possono scomporre nel seguente modo: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bar {\mathbf {v} }}={\frac {\Delta x}{\Delta t}}{\hat {\mathbf {x} }}+{\frac {\Delta y}{\Delta t}}{\hat {\mathbf {y} }}\qquad \mathbf {v} ={\frac {\mathrm {d} x}{\mathrm {d} t}}{\hat {\mathbf {x} }}+{\frac {\mathrm {d} y}{\mathrm {d} t}}{\hat {\mathbf {y} }}=v_{x}{\hat {\mathbf {x} }}+v_{y}{\hat {\mathbf {y} }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">Δ</mi> <mi>x</mi> </mrow> <mrow> <mi mathvariant="normal">Δ</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">Δ</mi> <mi>y</mi> </mrow> <mrow> <mi mathvariant="normal">Δ</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">y</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mspace width="2em" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>y</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">y</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>=</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>+</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">y</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bar {\mathbf {v} }}={\frac {\Delta x}{\Delta t}}{\hat {\mathbf {x} }}+{\frac {\Delta y}{\Delta t}}{\hat {\mathbf {y} }}\qquad \mathbf {v} ={\frac {\mathrm {d} x}{\mathrm {d} t}}{\hat {\mathbf {x} }}+{\frac {\mathrm {d} y}{\mathrm {d} t}}{\hat {\mathbf {y} }}=v_{x}{\hat {\mathbf {x} }}+v_{y}{\hat {\mathbf {y} }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/80da263114b1260ba691898e953dcb124a16609f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:52.998ex; height:5.676ex;" alt="{\displaystyle {\bar {\mathbf {v} }}={\frac {\Delta x}{\Delta t}}{\hat {\mathbf {x} }}+{\frac {\Delta y}{\Delta t}}{\hat {\mathbf {y} }}\qquad \mathbf {v} ={\frac {\mathrm {d} x}{\mathrm {d} t}}{\hat {\mathbf {x} }}+{\frac {\mathrm {d} y}{\mathrm {d} t}}{\hat {\mathbf {y} }}=v_{x}{\hat {\mathbf {x} }}+v_{y}{\hat {\mathbf {y} }}}"></span></dd></dl> <p>dove <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {\mathbf {x} }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {\mathbf {x} }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8a5451354f837d5d89774e1de386b44a903d929d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.411ex; height:2.343ex;" alt="{\displaystyle {\hat {\mathbf {x} }}}"></span> e <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {\mathbf {y} }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">y</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {\mathbf {y} }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3826fc591cfe42348f58b15e4c23152f30931ef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.411ex; height:2.676ex;" alt="{\displaystyle {\hat {\mathbf {y} }}}"></span> sono due <a href="/wiki/Versore" title="Versore">versori</a> in direzione degli assi <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> e <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}"></span>. Il modulo del vettore velocità è a sua volta scomponibile nei suoi componenti: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |{\bar {\mathbf {v} }}|={\sqrt {\left({\frac {\Delta x}{\Delta t}}\right)^{2}+\left({\frac {\Delta y}{\Delta t}}\right)^{2}}}\qquad |\mathbf {v} |={\sqrt {{v_{x}}^{2}+{v_{y}}^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">Δ</mi> <mi>x</mi> </mrow> <mrow> <mi mathvariant="normal">Δ</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">Δ</mi> <mi>y</mi> </mrow> <mrow> <mi mathvariant="normal">Δ</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mspace width="2em" /> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |{\bar {\mathbf {v} }}|={\sqrt {\left({\frac {\Delta x}{\Delta t}}\right)^{2}+\left({\frac {\Delta y}{\Delta t}}\right)^{2}}}\qquad |\mathbf {v} |={\sqrt {{v_{x}}^{2}+{v_{y}}^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e6a01dd97f76ba7948c70498c99b8018cdd9198e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:50.145ex; height:7.676ex;" alt="{\displaystyle |{\bar {\mathbf {v} }}|={\sqrt {\left({\frac {\Delta x}{\Delta t}}\right)^{2}+\left({\frac {\Delta y}{\Delta t}}\right)^{2}}}\qquad |\mathbf {v} |={\sqrt {{v_{x}}^{2}+{v_{y}}^{2}}}}"></span></dd></dl> <p>mentre l'angolo <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 \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }"></span> formato dal vettore <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {v} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {v} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35c1866e359fbfd2e0f606c725ba5cc37a5195d6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.411ex; height:1.676ex;" alt="{\displaystyle \mathbf {v} }"></span> con l'asse delle ascisse è dato da: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tan \alpha ={\frac {v_{y}}{v_{x}}}\qquad v_{x}=|\mathbf {v} |\cos \alpha \qquad v_{y}=|\mathbf {v} |\sin \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>tan</mi> <mo>⁡</mo> <mi>α</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mfrac> </mrow> <mspace width="2em" /> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>cos</mi> <mo>⁡</mo> <mi>α</mi> <mspace width="2em" /> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>sin</mi> <mo>⁡</mo> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tan \alpha ={\frac {v_{y}}{v_{x}}}\qquad v_{x}=|\mathbf {v} |\cos \alpha \qquad v_{y}=|\mathbf {v} |\sin \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/016b0793d35639be88366f0c76f9c07d770b9186" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:47.333ex; height:5.343ex;" alt="{\displaystyle \tan \alpha ={\frac {v_{y}}{v_{x}}}\qquad v_{x}=|\mathbf {v} |\cos \alpha \qquad v_{y}=|\mathbf {v} |\sin \alpha }"></span></dd></dl> <p>Se si considera il vettore posizione <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 {r} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {r} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eca0f46511c4c986c48b254073732c0bd98ae0c1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.102ex; height:1.676ex;" alt="{\displaystyle \mathbf {r} }"></span>, con <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d\mathbf {r} =(dr,rd\theta )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <mi>d</mi> <mi>r</mi> <mo>,</mo> <mi>r</mi> <mi>d</mi> <mi>θ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d\mathbf {r} =(dr,rd\theta )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2585d1d738c23dd19c2eefea1d788512b5ceb191" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.879ex; height:2.843ex;" alt="{\displaystyle d\mathbf {r} =(dr,rd\theta )}"></span>, la velocità <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {v} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {v} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35c1866e359fbfd2e0f606c725ba5cc37a5195d6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.411ex; height:1.676ex;" alt="{\displaystyle \mathbf {v} }"></span> si può scomporre in direzione perpendicolare e parallela alla posizione: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {v} =v_{r}{\hat {\mathbf {r} }}+v_{\theta }{\hat {\mathbf {r} }}_{\perp }={\frac {\mathrm {d} r}{\mathrm {d} t}}{\hat {\mathbf {r} }}+r{\frac {\mathrm {d} \theta }{\mathrm {d} t}}{\hat {\mathbf {r} }}_{\perp }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>=</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>+</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>θ</mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>⊥</mo> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>r</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>+</mo> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>θ</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>⊥</mo> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {v} =v_{r}{\hat {\mathbf {r} }}+v_{\theta }{\hat {\mathbf {r} }}_{\perp }={\frac {\mathrm {d} r}{\mathrm {d} t}}{\hat {\mathbf {r} }}+r{\frac {\mathrm {d} \theta }{\mathrm {d} t}}{\hat {\mathbf {r} }}_{\perp }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/42266a9ab0bfe33f87583d7ff7b60a5082021461" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:32.638ex; height:5.509ex;" alt="{\displaystyle \mathbf {v} =v_{r}{\hat {\mathbf {r} }}+v_{\theta }{\hat {\mathbf {r} }}_{\perp }={\frac {\mathrm {d} r}{\mathrm {d} t}}{\hat {\mathbf {r} }}+r{\frac {\mathrm {d} \theta }{\mathrm {d} t}}{\hat {\mathbf {r} }}_{\perp }}"></span></dd></dl> <p>dove <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_{r}=\mathrm {d} r/\mathrm {d} t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v_{r}=\mathrm {d} r/\mathrm {d} t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b878dba55190be54188016e32f486725a5285581" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.836ex; height:2.843ex;" alt="{\displaystyle v_{r}=\mathrm {d} r/\mathrm {d} t}"></span> è il modulo della velocità in direzione di <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 {r} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {r} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eca0f46511c4c986c48b254073732c0bd98ae0c1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.102ex; height:1.676ex;" alt="{\displaystyle \mathbf {r} }"></span>, mentre <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_{\theta }=r\ \mathrm {d} \theta /\mathrm {d} t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>θ</mi> </mrow> </msub> <mo>=</mo> <mi>r</mi> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v_{\theta }=r\ \mathrm {d} \theta /\mathrm {d} t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b35e327ebd9d96466edab5b00200d3bce3bbebe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.536ex; height:2.843ex;" alt="{\displaystyle v_{\theta }=r\ \mathrm {d} \theta /\mathrm {d} t}"></span> è il modulo della velocità ortogonale a <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {r} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {r} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eca0f46511c4c986c48b254073732c0bd98ae0c1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.102ex; height:1.676ex;" alt="{\displaystyle \mathbf {r} }"></span>. </p><p>La norma del vettore è pertanto: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v={\frac {\mathrm {d} s}{\mathrm {d} t}}={\frac {\mathrm {d} }{\mathrm {d} t}}\left({\sqrt {\mathrm {d} x^{2}+\mathrm {d} y^{2}}}\right)={\sqrt {\left({\frac {\mathrm {d} r}{\mathrm {d} t}}\right)^{2}+r^{2}\left({\frac {\mathrm {d} \theta }{\mathrm {d} t}}\right)^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>s</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>r</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>θ</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v={\frac {\mathrm {d} s}{\mathrm {d} t}}={\frac {\mathrm {d} }{\mathrm {d} t}}\left({\sqrt {\mathrm {d} x^{2}+\mathrm {d} y^{2}}}\right)={\sqrt {\left({\frac {\mathrm {d} r}{\mathrm {d} t}}\right)^{2}+r^{2}\left({\frac {\mathrm {d} \theta }{\mathrm {d} t}}\right)^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/70d0ba4f49f16f665bce9672a677676b7b473382" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:55.381ex; height:7.676ex;" alt="{\displaystyle v={\frac {\mathrm {d} s}{\mathrm {d} t}}={\frac {\mathrm {d} }{\mathrm {d} t}}\left({\sqrt {\mathrm {d} x^{2}+\mathrm {d} y^{2}}}\right)={\sqrt {\left({\frac {\mathrm {d} r}{\mathrm {d} t}}\right)^{2}+r^{2}\left({\frac {\mathrm {d} \theta }{\mathrm {d} t}}\right)^{2}}}}"></span></dd></dl> <p>e la direzione è sempre tangente alla curva percorsa. Nel caso di <a href="/wiki/Moto_circolare" title="Moto circolare">moto circolare</a>, la <b>velocità tangenziale</b> è espressa come: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v={\frac {ds}{dt}}={\frac {d}{dt}}(r\cos(\omega t){\vec {i}}+r\operatorname {sen}(\omega t){\vec {j}})=-r\omega \operatorname {sen}(\omega t){\vec {i}}+r\omega \cos(\omega t){\vec {j}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>s</mi> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo stretchy="false">(</mo> <mi>r</mi> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>ω</mi> <mi>t</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>i</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>+</mo> <mi>r</mi> <mi>sen</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>ω</mi> <mi>t</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>j</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <mi>r</mi> <mi>ω</mi> <mi>sen</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>ω</mi> <mi>t</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>i</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>+</mo> <mi>r</mi> <mi>ω</mi> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>ω</mi> <mi>t</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>j</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v={\frac {ds}{dt}}={\frac {d}{dt}}(r\cos(\omega t){\vec {i}}+r\operatorname {sen}(\omega t){\vec {j}})=-r\omega \operatorname {sen}(\omega t){\vec {i}}+r\omega \cos(\omega t){\vec {j}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/242f72e22604eb8202acaaa9fa2b13d5c25c7b25" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:67.849ex; height:5.509ex;" alt="{\displaystyle v={\frac {ds}{dt}}={\frac {d}{dt}}(r\cos(\omega t){\vec {i}}+r\operatorname {sen}(\omega t){\vec {j}})=-r\omega \operatorname {sen}(\omega t){\vec {i}}+r\omega \cos(\omega t){\vec {j}}}"></span></dd></dl> <p>in quanto <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 \theta =\omega t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> <mo>=</mo> <mi>ω</mi> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta =\omega t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a312c1c22a5255e85c1372abd0d55b7768c6b1cf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.474ex; height:2.176ex;" alt="{\displaystyle \theta =\omega t}"></span> </p><p>Il modulo della velocità si ottiene ricordando che <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {sen} ^{2}\theta +\cos ^{2}\theta =1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>sen</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> <mo>+</mo> <msup> <mi>cos</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {sen} ^{2}\theta +\cos ^{2}\theta =1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b1dde79999756dce531bcc37c71a88f1db0d1c4c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:18.517ex; height:2.843ex;" alt="{\displaystyle \operatorname {sen} ^{2}\theta +\cos ^{2}\theta =1}"></span> per cui sarà: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v=\omega r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> <mo>=</mo> <mi>ω</mi> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v=\omega r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b5edc54db08559dc3e33cb3cc568126e12b46521" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.721ex; height:1.676ex;" alt="{\displaystyle v=\omega r}"></span></dd></dl> <p>Nel caso di <a href="/wiki/Moto_circolare" title="Moto circolare">moto circolare</a> uniforme la velocità tangenziale è un <a href="/wiki/Vettore_(matematica)" title="Vettore (matematica)">vettore</a> costante in modulo, ma che varia la sua direzione, infatti questa sarà sempre <a href="/wiki/Tangente_(geometria)" title="Tangente (geometria)">tangente</a> alla <a href="/wiki/Circonferenza" title="Circonferenza">circonferenza</a> e avrà verso nel senso di <a href="/wiki/Rotazione" title="Rotazione">rotazione</a>. </p> <div class="mw-heading mw-heading4"><h4 id="Velocità_scalare_media"><span id="Velocit.C3.A0_scalare_media"></span>Velocità scalare media</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Velocit%C3%A0&veaction=edit&section=3" title="Modifica la sezione Velocità scalare media" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Velocit%C3%A0&action=edit&section=3" title="Edit section's source code: Velocità scalare media"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>La velocità scalare media è una <a href="/wiki/Grandezza_scalare" title="Grandezza scalare">grandezza scalare</a> definita come lo spazio totale percorso diviso il tempo impiegato, e tale definizione è molto diversa da quella per la velocità vettoriale media. Per esempio, nel <a href="/wiki/Moto_circolare" title="Moto circolare">moto circolare</a> (il moto che avviene lungo una <a href="/wiki/Circonferenza" title="Circonferenza">circonferenza</a>) dopo un <a href="/wiki/Periodo_(fisica)" title="Periodo (fisica)">periodo</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}"></span> la velocità vettoriale media è nulla, perché il punto di arrivo e quello di partenza coincidono, ovvero <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta \mathbf {r} =0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta \mathbf {r} =0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/756caa11d7bd59db8237de91c321a93bd4b09778" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.299ex; height:2.176ex;" alt="{\displaystyle \Delta \mathbf {r} =0}"></span>, mentre la velocità scalare media è uguale a <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\pi R/T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>π</mi> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\pi R/T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e5ac98282b21e17820e2d3f4b84b5a435dfa999" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.057ex; height:2.843ex;" alt="{\displaystyle 2\pi R/T}"></span>, con <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}"></span> il raggio della circonferenza. </p><p>Data una traiettoria <a href="/wiki/Curva_(matematica)" title="Curva (matematica)">curva</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>γ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a223c880b0ce3da8f64ee33c4f0010beee400b1a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.262ex; height:2.176ex;" alt="{\displaystyle \gamma }"></span>, la velocità scalare media è definita come: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle v_{s}\rangle ={\frac {1}{\Delta t}}\int _{\gamma }\,\mathrm {d} \mathbf {r} ={\frac {1}{\Delta t}}\int _{t_{1}}^{t_{2}}\|\mathbf {v} (t)\|\mathrm {d} t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi mathvariant="normal">Δ</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>γ</mi> </mrow> </msub> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi mathvariant="normal">Δ</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <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> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle v_{s}\rangle ={\frac {1}{\Delta t}}\int _{\gamma }\,\mathrm {d} \mathbf {r} ={\frac {1}{\Delta t}}\int _{t_{1}}^{t_{2}}\|\mathbf {v} (t)\|\mathrm {d} t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32c93bf2a8c96d4d1b01d96b178100cd11ab2a09" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:36.746ex; height:6.509ex;" alt="{\displaystyle \langle v_{s}\rangle ={\frac {1}{\Delta t}}\int _{\gamma }\,\mathrm {d} \mathbf {r} ={\frac {1}{\Delta t}}\int _{t_{1}}^{t_{2}}\|\mathbf {v} (t)\|\mathrm {d} t}"></span></dd></dl> <p>dove l'integrale è la <a href="/wiki/Lunghezza_di_un_arco" title="Lunghezza di un arco">lunghezza della curva</a> che descrive la traiettoria. La velocità scalare non è quindi semplicemente la norma della velocità vettoriale media, e si può dimostrare che la prima è sempre maggiore o uguale della seconda. </p> <div class="mw-heading mw-heading2"><h2 id="Relazione_integrale_fra_la_posizione_e_velocità"><span id="Relazione_integrale_fra_la_posizione_e_velocit.C3.A0"></span>Relazione integrale fra la posizione e velocità</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Velocit%C3%A0&veaction=edit&section=4" title="Modifica la sezione Relazione integrale fra la posizione e velocità" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Velocit%C3%A0&action=edit&section=4" title="Edit section's source code: Relazione integrale fra la posizione e velocità"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Spazio_tempo_velocita.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/6b/Spazio_tempo_velocita.gif/220px-Spazio_tempo_velocita.gif" decoding="async" width="220" height="251" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/6b/Spazio_tempo_velocita.gif/330px-Spazio_tempo_velocita.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/6/6b/Spazio_tempo_velocita.gif 2x" data-file-width="350" data-file-height="400" /></a><figcaption>La figura mostra il grafico di uno <a href="/wiki/Spostamento_(cinematica)" title="Spostamento (cinematica)">spostamento</a> unidimensionale. Per studiare dal punto di vista geometrico la velocità è comodo ricorrere a due tipi di grafici: quello spazio-tempo e quello velocità-tempo.<br />Il grafico dello spostamento presenta concavità verso il basso: questo corrisponde al fatto che il grafico della velocità è decrescente.<br />Al tempo <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cb0768c0bd659f2f84fb5ef9f4b74f336123d915" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.894ex; height:2.343ex;" alt="{\displaystyle t_{1}}"></span> il grafico di <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x(t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d54c275db3a1e620737b58e143b0818107fa5f5c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.979ex; height:2.843ex;" alt="{\displaystyle x(t)}"></span> ha pendenza positiva, per cui <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(t_{1})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> <mo stretchy="false">(</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v(t_{1})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f41b5e0381c86d96f786d889d0818ecda75720ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.831ex; height:2.843ex;" alt="{\displaystyle v(t_{1})}"></span> è maggiore di zero.<br />Al tempo <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/749fee708b41e7079eabd50d61c8bf3e965db16f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.894ex; height:2.343ex;" alt="{\displaystyle t_{2}}"></span> il grafico di <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x(t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d54c275db3a1e620737b58e143b0818107fa5f5c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.979ex; height:2.843ex;" alt="{\displaystyle x(t)}"></span> ha pendenza nulla, per cui <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(t_{2})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> <mo stretchy="false">(</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v(t_{2})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d370e3d9702b9c8e136ffc93d91e7353d45f9020" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.831ex; height:2.843ex;" alt="{\displaystyle v(t_{2})}"></span> è nulla.<br />Al tempo <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t_{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t_{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c62226a0dd66a3f9ad5706a365227331d9acb3a5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.894ex; height:2.343ex;" alt="{\displaystyle t_{3}}"></span> il grafico di <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x(t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d54c275db3a1e620737b58e143b0818107fa5f5c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.979ex; height:2.843ex;" alt="{\displaystyle x(t)}"></span> ha pendenza negativa, per cui <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(t_{3})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> <mo stretchy="false">(</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v(t_{3})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c15734702ac1c233f6086f8a487efb9cffe228b7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.831ex; height:2.843ex;" alt="{\displaystyle v(t_{3})}"></span> è minore di zero.</figcaption></figure><p>Tramite l'<a href="/wiki/Integrale" title="Integrale">integrazione</a> è possibile conoscere la variazione della posizione ricavandola dalla velocità. Dalla definizione di velocità: </p><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {v} (t)={\frac {\mathrm {d} \mathbf {r} (t)}{\mathrm {d} t}}{\partial ^{2} \over \partial x_{1}\partial x_{2}}y\backprime }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi mathvariant="normal">∂</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </mfrac> </mrow> <mi>y</mi> <mi class="MJX-variant">‵</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {v} (t)={\frac {\mathrm {d} \mathbf {r} (t)}{\mathrm {d} t}}{\partial ^{2} \over \partial x_{1}\partial x_{2}}y\backprime }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/09d9b963786282eb35f149ebf59ab9b33087cbda" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:23.073ex; height:6.176ex;" alt="{\displaystyle \mathbf {v} (t)={\frac {\mathrm {d} \mathbf {r} (t)}{\mathrm {d} t}}{\partial ^{2} \over \partial x_{1}\partial x_{2}}y\backprime }"></span></dd></dl> <p>Si può effettuare una <a href="/wiki/Separazione_delle_variabili" title="Separazione delle variabili">separazione delle variabili</a> portando a primo membro <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 {r} (t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {r} (t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7216e69d28fdb11c29c859f015a284c1ec6f06ac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.751ex; height:2.843ex;" alt="{\displaystyle \mathbf {r} (t)}"></span> e al secondo membro il resto dell'equazione: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {d} \mathbf {r} (t)=\mathbf {v} (t)\mathrm {d} t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {d} \mathbf {r} (t)=\mathbf {v} (t)\mathrm {d} t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca51e9328e9f0c265d452be30aa422416084ca4b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.334ex; height:2.843ex;" alt="{\displaystyle \mathrm {d} \mathbf {r} (t)=\mathbf {v} (t)\mathrm {d} t}"></span></dd></dl> <p>in modo che sia possibile integrare entrambi i membri: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{\mathbf {r} (t_{2})}^{\mathbf {r} (t_{1})}\mathrm {d} \mathbf {r} (t)=\int _{t_{2}}^{t_{1}}\mathbf {v} (t)\mathrm {d} t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">(</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">(</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{\mathbf {r} (t_{2})}^{\mathbf {r} (t_{1})}\mathrm {d} \mathbf {r} (t)=\int _{t_{2}}^{t_{1}}\mathbf {v} (t)\mathrm {d} t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5192669a339c450f50ecc6e1290ea2b0b070bbc9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:25.41ex; height:6.676ex;" alt="{\displaystyle \int _{\mathbf {r} (t_{2})}^{\mathbf {r} (t_{1})}\mathrm {d} \mathbf {r} (t)=\int _{t_{2}}^{t_{1}}\mathbf {v} (t)\mathrm {d} t}"></span></dd></dl> <p>e determinare così la variazione di <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 {r} (t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {r} (t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7216e69d28fdb11c29c859f015a284c1ec6f06ac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.751ex; height:2.843ex;" alt="{\displaystyle \mathbf {r} (t)}"></span>. Se, ad esempio, la velocità è costante l'integrale si riduce alla legge del moto rettilineo uniforme: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {r} (t_{2})-\mathbf {r} (t_{1})=\mathbf {r} (t_{2}-t_{1})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">(</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">(</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">(</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {r} (t_{2})-\mathbf {r} (t_{1})=\mathbf {r} (t_{2}-t_{1})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b37a541eb10e2b59573ec2f7ad1d7c1947f0748b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.089ex; height:2.843ex;" alt="{\displaystyle \mathbf {r} (t_{2})-\mathbf {r} (t_{1})=\mathbf {r} (t_{2}-t_{1})}"></span>.</dd></dl> <div class="mw-heading mw-heading2"><h2 id="Composizione_delle_velocità"><span id="Composizione_delle_velocit.C3.A0"></span>Composizione delle velocità</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Velocit%C3%A0&veaction=edit&section=5" title="Modifica la sezione Composizione delle velocità" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Velocit%C3%A0&action=edit&section=5" title="Edit section's source code: Composizione delle velocità"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r130657691">body:not(.skin-minerva) .mw-parser-output .vedi-anche{font-size:95%}</style><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r139142988"> <div class="hatnote noprint vedi-anche"> <div class="hatnote-content"><span class="noviewer hatnote-icon" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/87/Magnifying_glass_icon_mgx2.svg/18px-Magnifying_glass_icon_mgx2.svg.png" decoding="async" width="18" height="18" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/87/Magnifying_glass_icon_mgx2.svg/27px-Magnifying_glass_icon_mgx2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/87/Magnifying_glass_icon_mgx2.svg/36px-Magnifying_glass_icon_mgx2.svg.png 2x" data-file-width="286" data-file-height="280" /></span></span> <span class="hatnote-text">Lo stesso argomento in dettaglio: <b><a href="/wiki/Teorema_di_Coriolis" title="Teorema di Coriolis">Teorema di Coriolis</a></b>.</span></div> </div> <p>Considerando ad esempio una barca che si muove con una velocità <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e07b00e7fc0847fbd16391c778d65bc25c452597" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle v}"></span> rispetto all'acqua di un canale, che a sua volta si muove con una velocità <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"></span> rispetto alla riva, si prenda un osservatore <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 O}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>O</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle O}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d70e1d0d87e2ef1092ea1ffe2923d9933ff18fc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.773ex; height:2.176ex;" alt="{\displaystyle O}"></span> solidale con la riva e un osservatore <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 O'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>O</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle O'}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/470d46b4d626e35d82cd0d9f9aadc65ac47ab568" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.458ex; height:2.509ex;" alt="{\displaystyle O'}"></span> solidale con la barca. Si ha che: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v=v_{O'}+V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> <mo>=</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>O</mi> <mo>′</mo> </msup> </mrow> </msub> <mo>+</mo> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v=v_{O'}+V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2e9a9ecfc719ca2079c21dcad9791a6961611803" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.999ex; height:2.676ex;" alt="{\displaystyle v=v_{O'}+V}"></span></dd></dl> <p>Quindi, per l'osservatore fisso le velocità della corrente e della barca si compongono sommandosi quando la barca va nel verso della corrente e sottraendosi quando va controcorrente. Va sottolineato che <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 O'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>O</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle O'}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/470d46b4d626e35d82cd0d9f9aadc65ac47ab568" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.458ex; height:2.509ex;" alt="{\displaystyle O'}"></span> con i suoi strumenti misura sempre la velocità <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e07b00e7fc0847fbd16391c778d65bc25c452597" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle v}"></span> della barca rispetto all'acqua, e può anche misurare la velocità con la quale l'acqua scorre davanti <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 O}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>O</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle O}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d70e1d0d87e2ef1092ea1ffe2923d9933ff18fc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.773ex; height:2.176ex;" alt="{\displaystyle O}"></span>. Questo misura anch'esso la velocità con la quale si muove l'acqua e, a differenza di <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 O'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>O</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle O'}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/470d46b4d626e35d82cd0d9f9aadc65ac47ab568" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.458ex; height:2.509ex;" alt="{\displaystyle O'}"></span>, misura pure la velocità di <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 O'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>O</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle O'}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/470d46b4d626e35d82cd0d9f9aadc65ac47ab568" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.458ex; height:2.509ex;" alt="{\displaystyle O'}"></span> rispetto alla sponda del canale. Una situazione del tutto analoga si verifica pure quando la barca si muove trasversalmente alla corrente. </p><p>Questo tipo di composizione delle velocità, introdotta da Galilei nella teoria della relatività galileiana, era già nota a Leonardo da Vinci che fa l'esempio di un arciere che lancia una freccia dal centro della Terra verso la superficie. L'esempio è ripreso in maniera più formale da Galilei: qui un osservatore esterno alla Terra vede comporsi il moto rettilineo della freccia lungo un raggio e il moto rotatorio della Terra. Il moto risultante è una spirale di Archimede. La freccia si muove con il moto rettilineo uniforme, e lo spazio percorso risulta allora: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s=vt}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> <mo>=</mo> <mi>v</mi> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s=vt}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c1749ca68553b93a3976ddf713733d540ed8934" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.156ex; height:2.009ex;" alt="{\displaystyle s=vt}"></span></dd></dl> <p>Le proiezioni di <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 s}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01d131dfd7673938b947072a13a9744fe997e632" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:1.676ex;" alt="{\displaystyle s}"></span> sui due assi è quindi: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=vt\cos(\omega t)\qquad y=vt\,\mathrm {sen} (\omega t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mi>v</mi> <mi>t</mi> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>ω</mi> <mi>t</mi> <mo stretchy="false">)</mo> <mspace width="2em" /> <mi>y</mi> <mo>=</mo> <mi>v</mi> <mi>t</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">s</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">n</mi> </mrow> <mo stretchy="false">(</mo> <mi>ω</mi> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=vt\cos(\omega t)\qquad y=vt\,\mathrm {sen} (\omega t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ac1d4457a42a44742d9589e5cf054c7f27a1f1e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:32.578ex; height:2.843ex;" alt="{\displaystyle x=vt\cos(\omega t)\qquad y=vt\,\mathrm {sen} (\omega t)}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Composizione_delle_velocità_in_relatività_speciale"><span id="Composizione_delle_velocit.C3.A0_in_relativit.C3.A0_speciale"></span>Composizione delle velocità in relatività speciale</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Velocit%C3%A0&veaction=edit&section=6" title="Modifica la sezione Composizione delle velocità in relatività speciale" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Velocit%C3%A0&action=edit&section=6" title="Edit section's source code: Composizione delle velocità in relatività speciale"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r130657691"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r139142988"> <div class="hatnote noprint vedi-anche"> <div class="hatnote-content"><span class="noviewer hatnote-icon" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/87/Magnifying_glass_icon_mgx2.svg/18px-Magnifying_glass_icon_mgx2.svg.png" decoding="async" width="18" height="18" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/87/Magnifying_glass_icon_mgx2.svg/27px-Magnifying_glass_icon_mgx2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/87/Magnifying_glass_icon_mgx2.svg/36px-Magnifying_glass_icon_mgx2.svg.png 2x" data-file-width="286" data-file-height="280" /></span></span> <span class="hatnote-text">Lo stesso argomento in dettaglio: <b><a href="/wiki/Composizione_delle_velocit%C3%A0" title="Composizione delle velocità">Composizione delle velocità</a></b>.</span></div> </div> <p>Nella teoria della <a href="/wiki/Relativit%C3%A0_speciale" class="mw-redirect" title="Relatività speciale">relatività speciale</a>, passando da un sistema di riferimento <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 S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}"></span> a un sistema di riferimento <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 S'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>S</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S'}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf9961844d1f539adee019e432dc18aa2a7ede59" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.206ex; height:2.509ex;" alt="{\displaystyle S'}"></span>, la velocità di una particella si trasforma nel modo seguente: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v'_{x}={\frac {v_{x}-V}{1-v_{x}V/c^{2}}}\qquad v'_{y}={\frac {v_{y}}{\gamma (1-v_{x}V/c^{2})}}\qquad v'_{z}={\frac {v_{z}}{\gamma (1-v_{x}V/c^{2})}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> <mo>′</mo> </msubsup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>−</mo> <mi>V</mi> </mrow> <mrow> <mn>1</mn> <mo>−</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mspace width="2em" /> <msubsup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> <mo>′</mo> </msubsup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mrow> <mi>γ</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mspace width="2em" /> <msubsup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> <mo>′</mo> </msubsup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mrow> <mi>γ</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v'_{x}={\frac {v_{x}-V}{1-v_{x}V/c^{2}}}\qquad v'_{y}={\frac {v_{y}}{\gamma (1-v_{x}V/c^{2})}}\qquad v'_{z}={\frac {v_{z}}{\gamma (1-v_{x}V/c^{2})}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2984a907c132bf308d78e28eac19bc844213a5e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:67.785ex; height:6.009ex;" alt="{\displaystyle v'_{x}={\frac {v_{x}-V}{1-v_{x}V/c^{2}}}\qquad v'_{y}={\frac {v_{y}}{\gamma (1-v_{x}V/c^{2})}}\qquad v'_{z}={\frac {v_{z}}{\gamma (1-v_{x}V/c^{2})}}}"></span></dd></dl> <p>dove <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"></span> è la velocità (diretta lungo l'asse <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span>) del sistema <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 S'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>S</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S'}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf9961844d1f539adee019e432dc18aa2a7ede59" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.206ex; height:2.509ex;" alt="{\displaystyle S'}"></span> rispetto al sistema di riferimento <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 S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}"></span>, e <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma =\gamma (V)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>γ</mi> <mo>=</mo> <mi>γ</mi> <mo stretchy="false">(</mo> <mi>V</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma =\gamma (V)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3237f9c4544ee7125554b73a24f6ecf7cc233517" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.22ex; height:2.843ex;" alt="{\displaystyle \gamma =\gamma (V)}"></span> è il <a href="/wiki/Fattore_di_Lorentz" title="Fattore di Lorentz">fattore di Lorentz</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Velocità_nei_sistemi_di_punti_materiali"><span id="Velocit.C3.A0_nei_sistemi_di_punti_materiali"></span>Velocità nei sistemi di punti materiali</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Velocit%C3%A0&veaction=edit&section=7" title="Modifica la sezione Velocità nei sistemi di punti materiali" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Velocit%C3%A0&action=edit&section=7" title="Edit section's source code: Velocità nei sistemi di punti materiali"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Se gli <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 n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> punti materiali di un sistema sono in movimento, solitamente, la posizione del <a href="/wiki/Centro_di_massa" title="Centro di massa">centro di massa</a> varia. Pertanto, nell'ipotesi in cui la massa totale <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=\sum _{i=1}^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m=\sum _{i=1}^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b3685dc653e25c6cd522f58e5e4b9b5c8c1dfd27" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:8.494ex; height:6.843ex;" alt="{\displaystyle m=\sum _{i=1}^{n}}"></span> sia costante, la velocità del centro di massa sarà: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {v} _{G}={\frac {\mathrm {d} \mathbf {r} _{G}}{\mathrm {d} t}}={\frac {\mathrm {d} }{\mathrm {d} t}}{\frac {S_{r}^{*}}{m}}={\frac {\displaystyle {{\cancel {m}}\left({\cancel {\frac {\mathrm {d} m}{\mathrm {d} t}}}\sum _{i=1}^{n}\mathbf {r} _{i}+m\sum _{i=1}^{n}{\frac {\mathrm {d} \mathbf {r} _{i}}{\mathrm {d} t}}\right)+S_{r}^{*}{\cancel {\frac {\mathrm {d} m}{\mathrm {d} t}}}}}{m^{\cancel {2}}}}={\frac {\mathbf {p} }{m}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>G</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>G</mi> </mrow> </msub> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msubsup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> <mi>m</mi> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <menclose notation="updiagonalstrike"> <mi>m</mi> </menclose> </mrow> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <menclose notation="updiagonalstrike"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>m</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </menclose> </mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>+</mo> <mi>m</mi> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <msubsup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <menclose notation="updiagonalstrike"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>m</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </menclose> </mrow> </mrow> </mstyle> <msup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <menclose notation="updiagonalstrike"> <mn>2</mn> </menclose> </mrow> </msup> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> <mi>m</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {v} _{G}={\frac {\mathrm {d} \mathbf {r} _{G}}{\mathrm {d} t}}={\frac {\mathrm {d} }{\mathrm {d} t}}{\frac {S_{r}^{*}}{m}}={\frac {\displaystyle {{\cancel {m}}\left({\cancel {\frac {\mathrm {d} m}{\mathrm {d} t}}}\sum _{i=1}^{n}\mathbf {r} _{i}+m\sum _{i=1}^{n}{\frac {\mathrm {d} \mathbf {r} _{i}}{\mathrm {d} t}}\right)+S_{r}^{*}{\cancel {\frac {\mathrm {d} m}{\mathrm {d} t}}}}}{m^{\cancel {2}}}}={\frac {\mathbf {p} }{m}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2ac8c4c1abfb27ad4771ba2eed6690251907b929" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:72.943ex; height:11.176ex;" alt="{\displaystyle \mathbf {v} _{G}={\frac {\mathrm {d} \mathbf {r} _{G}}{\mathrm {d} t}}={\frac {\mathrm {d} }{\mathrm {d} t}}{\frac {S_{r}^{*}}{m}}={\frac {\displaystyle {{\cancel {m}}\left({\cancel {\frac {\mathrm {d} m}{\mathrm {d} t}}}\sum _{i=1}^{n}\mathbf {r} _{i}+m\sum _{i=1}^{n}{\frac {\mathrm {d} \mathbf {r} _{i}}{\mathrm {d} t}}\right)+S_{r}^{*}{\cancel {\frac {\mathrm {d} m}{\mathrm {d} t}}}}}{m^{\cancel {2}}}}={\frac {\mathbf {p} }{m}}}"></span></dd></dl> <p>dove <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 S_{r}^{*}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{r}^{*}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a0c56ca799f8f701915618e27dfefe03c5471ed6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.576ex; height:2.509ex;" alt="{\displaystyle S_{r}^{*}}"></span> è il <a href="/wiki/Momento_statico" title="Momento statico">momento statico</a> e <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {p} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {p} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dd73e3862cb92b016721b8c492eadb4e8a577527" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.485ex; height:2.009ex;" alt="{\displaystyle \mathbf {p} }"></span> la <a href="/wiki/Quantit%C3%A0_di_moto" title="Quantità di moto">quantità di moto</a> totale del sistema. </p> <div class="mw-heading mw-heading2"><h2 id="Caduta_nel_campo_gravitazionale">Caduta nel campo gravitazionale</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Velocit%C3%A0&veaction=edit&section=8" title="Modifica la sezione Caduta nel campo gravitazionale" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Velocit%C3%A0&action=edit&section=8" title="Edit section's source code: Caduta nel campo gravitazionale"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In caso di caduta di un oggetto immerso in un <a href="/wiki/Campo_gravitazionale" title="Campo gravitazionale">campo gravitazionale</a>, la velocità finale dell'oggetto può essere determinata utilizzando la <a href="/wiki/Conservazione_dell%27energia" class="mw-redirect" title="Conservazione dell'energia">conservazione dell'energia</a>, ottenendo così una semplice espressione:<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v={\sqrt {2gh}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> <mi>g</mi> <mi>h</mi> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v={\sqrt {2gh}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/377de578bc045c27043fd5209195c970b7b09bc9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:10.167ex; height:3.509ex;" alt="{\displaystyle v={\sqrt {2gh}}}"></span></dd></dl> <p>dove <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 h}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b26be3e694314bc90c3215047e4a2010c6ee184a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:2.176ex;" alt="{\displaystyle h}"></span> è la differenza di quota tra il punto di caduta e quello in cui l'oggetto si ferma. </p><p>In quest'ultimo caso si parla di <a href="/wiki/Velocit%C3%A0_di_impatto" class="mw-redirect" title="Velocità di impatto">velocità di impatto</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Velocità_terminale_di_caduta"><span id="Velocit.C3.A0_terminale_di_caduta"></span>Velocità terminale di caduta</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Velocit%C3%A0&veaction=edit&section=9" title="Modifica la sezione Velocità terminale di caduta" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Velocit%C3%A0&action=edit&section=9" title="Edit section's source code: Velocità terminale di caduta"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Per <a href="/wiki/Velocit%C3%A0_terminale_di_caduta" class="mw-redirect" title="Velocità terminale di caduta">velocità terminale di caduta</a>, o <i>velocità limite</i>, si intende la velocità massima che raggiunge un corpo in caduta. Cadendo attraverso un fluido infatti il corpo incontra una crescente resistenza all'aumentare della velocità e quando l'attrito eguaglia la forza di attrazione gravitazionale la velocità si stabilizza. </p> <div class="mw-heading mw-heading2"><h2 id="Velocità_limite"><span id="Velocit.C3.A0_limite"></span>Velocità limite</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Velocit%C3%A0&veaction=edit&section=10" title="Modifica la sezione Velocità limite" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Velocit%C3%A0&action=edit&section=10" title="Edit section's source code: Velocità limite"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>La <a href="/wiki/Velocit%C3%A0_della_luce" title="Velocità della luce">velocità della luce</a>, o di qualsiasi altra onda elettromagnetica, è identica nel <a href="/wiki/Vuoto_(fisica)" title="Vuoto (fisica)">vuoto</a> per tutti i <a href="/wiki/Sistema_di_riferimento" title="Sistema di riferimento">sistemi di riferimento</a>. Questa invarianza, implicita nelle simmetrie delle <a href="/wiki/Equazioni_di_Maxwell" title="Equazioni di Maxwell">equazioni di Maxwell</a> per la propagazione delle <a href="/wiki/Onda_elettromagnetica" class="mw-redirect" title="Onda elettromagnetica">onde elettromagnetiche</a> e verificata sperimentalmente alla fine del 1800 con l'<a href="/wiki/Esperimento_di_Michelson-Morley" title="Esperimento di Michelson-Morley">esperimento di Michelson-Morley</a>, ha portato alla necessità di modificare le equazioni del moto e della dinamica. Una delle conseguenze della <a href="/wiki/Teoria_della_relativit%C3%A0_ristretta" class="mw-redirect" title="Teoria della relatività ristretta">teoria della relatività ristretta</a> di <a href="/wiki/Albert_Einstein" title="Albert Einstein">Albert Einstein</a> è che la velocità massima raggiungibile al limite da un qualunque oggetto fisico è quella della luce nel vuoto.<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> <div class="mw-heading mw-heading2"><h2 id="Note">Note</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Velocit%C3%A0&veaction=edit&section=11" title="Modifica la sezione Note" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Velocit%C3%A0&action=edit&section=11" title="Edit section's source code: Note"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><a href="#cite_ref-1"><b>^</b></a> <span class="reference-text"><cite class="citation libro" style="font-style:normal"> <a rel="nofollow" class="external text" href="http://www.treccani.it/enciclopedia/velocita_(Dizionario-delle-Scienze-Fisiche)/"><span style="font-style:italic;">velocità</span></a>, in <span style="font-style:italic;">Dizionario delle Scienze Fisiche</span>, Roma, Istituto dell'Enciclopedia Italiana, 1996. <small>URL consultato il 2 marzo 2016</small>.</cite></span> </li> <li id="cite_note-2"><a href="#cite_ref-2"><b>^</b></a> <span class="reference-text"><cite class="citation web" style="font-style:normal">(<span style="font-weight:bolder; font-size:80%"><abbr title="inglese">EN</abbr></span>) <a rel="nofollow" class="external text" href="http://goldbook.iupac.org/V06607.html"><span style="font-style:italic;">"velocity"</span></a>, su <span style="font-style:italic;">goldbook.iupac.org</span>, IUPAC Gold Book. <small>URL consultato il 26 marzo 2016</small>.</cite></span> </li> <li id="cite_note-Maz8-3"><span class="mw-cite-backlink"><b>^</b> <sup><i><a href="#cite_ref-Maz8_3-0">a</a></i></sup> <sup><i><a href="#cite_ref-Maz8_3-1">b</a></i></sup></span> <span class="reference-text"><cite class="citation cita" style="font-style:normal"><a href="#CITEREFMazzoldi">Mazzoldi</a>, p. 8</cite>.</span> </li> <li id="cite_note-mathworld-4"><a href="#cite_ref-mathworld_4-0"><b>^</b></a> <span class="reference-text">Weisstein, Eric W. <i><a rel="nofollow" class="external text" href="http://mathworld.wolfram.com/Acceleration.html">Acceleration</a>.</i> From MathWorld.</span> </li> <li id="cite_note-5"><a href="#cite_ref-5"><b>^</b></a> <span class="reference-text">Weisstein, Eric W. <i><a rel="nofollow" class="external text" href="http://mathworld.wolfram.com/Velocity.html">Velocity</a>.</i> From MathWorld.</span> </li> <li id="cite_note-6"><a href="#cite_ref-6"><b>^</b></a> <span class="reference-text"><cite class="citation cita" style="font-style:normal"><a href="#CITEREFMazzoldi">Mazzoldi</a>, p. 16</cite>.</span> </li> <li id="cite_note-7"><a href="#cite_ref-7"><b>^</b></a> <span class="reference-text">Da non confondere con la <a href="/wiki/Velocit%C3%A0_terminale_di_caduta" class="mw-redirect" title="Velocità terminale di caduta">velocità terminale di caduta</a>, a volte detta anche velocità limite.</span> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="Bibliografia">Bibliografia</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Velocit%C3%A0&veaction=edit&section=12" title="Modifica la sezione Bibliografia" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Velocit%C3%A0&action=edit&section=12" title="Edit section's source code: Bibliografia"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><cite id="CITEREFMazzoldi" class="citation libro" style="font-style:normal"> Paolo Mazzoldi, Massimo Nigro, Cesare Voci, <a rel="nofollow" class="external text" href="http://books.google.it/books?id=O6s7AAAACAAJ"><span style="font-style:italic;">Fisica</span></a>, vol. 1, 2ª ed., Edises, 2000, <a href="/wiki/ISBN" title="ISBN">ISBN</a> <a href="/wiki/Speciale:RicercaISBN/88-7959-137-1" title="Speciale:RicercaISBN/88-7959-137-1">88-7959-137-1</a>.</cite></li> <li><cite id="CITEREFLandau" class="citation libro" style="font-style:normal"> Lev D. Landau, Evgenij M. Lifšic, <span style="font-style:italic;">Fisica teorica 1 - Meccanica</span>, Roma, Editori Riuniti Edizioni Mir, 1976, <a href="/wiki/ISBN" title="ISBN">ISBN</a> <a href="/wiki/Speciale:RicercaISBN/88-6473-202-0" title="Speciale:RicercaISBN/88-6473-202-0">88-6473-202-0</a>.</cite></li> <li><cite id="CITEREFmencuccini" class="citation libro" style="font-style:normal"> Corrado Mencuccini, Vittorio Silvestrini, <span style="font-style:italic;">Fisica I - Meccanica e Termodinamica</span>, 3rd Edition, Napoli, Liguori Editore, 1996, <a href="/wiki/ISBN" title="ISBN">ISBN</a> <a href="/wiki/Speciale:RicercaISBN/88-207-1493-0" title="Speciale:RicercaISBN/88-207-1493-0">88-207-1493-0</a>.</cite></li> <li><cite class="citation libro" style="font-style:normal">(<span style="font-weight:bolder; font-size:80%"><abbr title="inglese">EN</abbr></span>) Edwin Bidwell Wilson, <a rel="nofollow" class="external text" href="https://hdl.handle.net/2027/mdp.39015000962285?urlappend=%3Bseq=149"><span style="font-style:italic;">Vector analysis: a text-book for the use of students of mathematics and physics, founded upon the lectures of J. Willard Gibbs</span></a>, 1901, p. 125.</cite></li></ul> <div class="mw-heading mw-heading2"><h2 id="Voci_correlate">Voci correlate</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Velocit%C3%A0&veaction=edit&section=13" title="Modifica la sezione Voci correlate" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Velocit%C3%A0&action=edit&section=13" title="Edit section's source code: Voci correlate"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Posizione" title="Posizione">Posizione</a></li> <li><a href="/wiki/Accelerazione" title="Accelerazione">Accelerazione</a></li> <li><a href="/wiki/Strappo_(cinematica)" class="mw-redirect" title="Strappo (cinematica)">Strappo</a></li> <li><a href="/wiki/Composizione_delle_velocit%C3%A0" title="Composizione delle velocità">Composizione delle velocità</a></li> <li><a href="/wiki/Velocit%C3%A0_commerciale" title="Velocità commerciale">Velocità commerciale</a></li> <li><a href="/wiki/Velocit%C3%A0_operativa" title="Velocità operativa">Velocità operativa</a></li> <li><a href="/wiki/Velocit%C3%A0_di_fuga" title="Velocità di fuga">Velocità di fuga</a></li> <li><a href="/wiki/Velocit%C3%A0_di_deriva" title="Velocità di deriva">Velocità di deriva</a></li> <li><a href="/wiki/Velocit%C3%A0_della_luce" title="Velocità della luce">Velocità della luce</a></li> <li><a href="/wiki/Velocit%C3%A0_del_suono" title="Velocità del suono">Velocità del suono</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Altri_progetti">Altri progetti</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Velocit%C3%A0&veaction=edit&section=14" title="Modifica la sezione Altri progetti" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Velocit%C3%A0&action=edit&section=14" title="Edit section's source code: Altri progetti"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <div id="interProject" class="toccolours" style="display: none; clear: both; margin-top: 2em"><p id="sisterProjects" style="background-color: #efefef; color: black; font-weight: bold; margin: 0"><span>Altri progetti</span></p><ul title="Collegamenti verso gli altri progetti Wikimedia"> <li class="" title=""><a href="https://it.wikiquote.org/wiki/Velocit%C3%A0" class="extiw" title="q:Velocità">Wikiquote</a></li> <li class="" title=""><a href="https://it.wiktionary.org/wiki/velocit%C3%A0" class="extiw" title="wikt:velocità">Wikizionario</a></li> <li class="" title=""><span class="plainlinks" title="commons:Category:Velocity"><a class="external text" href="https://commons.wikimedia.org/wiki/Category:Velocity?uselang=it">Wikimedia Commons</a></span></li></ul></div> <ul><li><span typeof="mw:File"><a href="https://it.wikiquote.org/wiki/" title="Collabora a Wikiquote"><img alt="Collabora a Wikiquote" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Wikiquote-logo.svg/18px-Wikiquote-logo.svg.png" decoding="async" width="18" height="21" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Wikiquote-logo.svg/27px-Wikiquote-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Wikiquote-logo.svg/36px-Wikiquote-logo.svg.png 2x" data-file-width="300" data-file-height="355" /></a></span> <a href="https://it.wikiquote.org/wiki/" class="extiw" title="q:">Wikiquote</a> contiene citazioni sulla <b><a href="https://it.wikiquote.org/wiki/Velocit%C3%A0" class="extiw" title="q:Velocità">velocità</a></b></li> <li><span typeof="mw:File"><a href="https://it.wiktionary.org/wiki/" title="Collabora a Wikizionario"><img alt="Collabora a Wikizionario" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f9/Wiktionary_small.svg/18px-Wiktionary_small.svg.png" decoding="async" width="18" height="18" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f9/Wiktionary_small.svg/27px-Wiktionary_small.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f9/Wiktionary_small.svg/36px-Wiktionary_small.svg.png 2x" data-file-width="350" data-file-height="350" /></a></span> <a href="https://it.wiktionary.org/wiki/" class="extiw" title="wikt:">Wikizionario</a> contiene il lemma di dizionario «<b><a href="https://it.wiktionary.org/wiki/velocit%C3%A0" class="extiw" title="wikt:velocità">velocità</a></b>»</li> <li><span typeof="mw:File"><a href="https://commons.wikimedia.org/wiki/?uselang=it" title="Collabora a Wikimedia Commons"><img alt="Collabora a Wikimedia Commons" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/18px-Commons-logo.svg.png" decoding="async" width="18" height="24" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/27px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/36px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /></a></span> <span class="plainlinks"><a class="external text" href="https://commons.wikimedia.org/wiki/?uselang=it">Wikimedia Commons</a></span> contiene immagini o altri file sulla <b><span class="plainlinks"><a class="external text" href="https://commons.wikimedia.org/wiki/Category:Velocity?uselang=it">velocità</a></span></b></li></ul> <div class="mw-heading mw-heading2"><h2 id="Collegamenti_esterni">Collegamenti esterni</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Velocit%C3%A0&veaction=edit&section=15" title="Modifica la sezione Collegamenti esterni" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Velocit%C3%A0&action=edit&section=15" title="Edit section's source code: Collegamenti esterni"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li class="mw-empty-elt"></li> <li><cite id="CITEREFTreccani.it" class="citation web" style="font-style:normal"> <a rel="nofollow" class="external text" href="https://www.treccani.it/enciclopedia/velocita"><span style="font-style:italic;">velocità</span></a>, su <span style="font-style:italic;">Treccani.it – Enciclopedie on line</span>, <a href="/wiki/Istituto_dell%27Enciclopedia_Italiana" title="Istituto dell'Enciclopedia Italiana">Istituto dell'Enciclopedia Italiana</a>.</cite> <span class="mw-valign-text-top noprint" typeof="mw:File/Frameless"><a href="https://www.wikidata.org/wiki/Q11465#P3365" title="Modifica su Wikidata"><img alt="Modifica su Wikidata" src="//upload.wikimedia.org/wikipedia/commons/thumb/7/73/Blue_pencil.svg/10px-Blue_pencil.svg.png" decoding="async" width="10" height="10" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/73/Blue_pencil.svg/15px-Blue_pencil.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/73/Blue_pencil.svg/20px-Blue_pencil.svg.png 2x" data-file-width="600" data-file-height="600" /></a></span></li> <li><cite id="CITEREFBritannica.com" class="citation web" style="font-style:normal">(<span style="font-weight:bolder; font-size:80%"><abbr title="inglese">EN</abbr></span>) <a rel="nofollow" class="external text" href="https://www.britannica.com/science/velocity"><span style="font-style:italic;">velocity</span></a> / <a rel="nofollow" class="external text" href="https://www.britannica.com/science/average-velocity"><span style="font-style:italic;">average velocity</span></a> / <a rel="nofollow" class="external text" href="https://www.britannica.com/science/instantaneous-velocity"><span style="font-style:italic;">instantaneous velocity</span></a>, su <span style="font-style:italic;"><a href="/wiki/Enciclopedia_Britannica" title="Enciclopedia Britannica">Enciclopedia Britannica</a></span>, Encyclopædia Britannica, Inc.</cite> <span class="mw-valign-text-top noprint" typeof="mw:File/Frameless"><a href="https://www.wikidata.org/wiki/Q11465#P1417" title="Modifica su Wikidata"><img alt="Modifica su Wikidata" src="//upload.wikimedia.org/wikipedia/commons/thumb/7/73/Blue_pencil.svg/10px-Blue_pencil.svg.png" decoding="async" width="10" height="10" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/73/Blue_pencil.svg/15px-Blue_pencil.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/73/Blue_pencil.svg/20px-Blue_pencil.svg.png 2x" data-file-width="600" data-file-height="600" /></a></span></li> <li><cite id="CITEREFMathWorld" class="citation web" style="font-style:normal">(<span style="font-weight:bolder; font-size:80%"><abbr title="inglese">EN</abbr></span>) Eric W. Weisstein, <a rel="nofollow" class="external text" href="http://mathworld.wolfram.com/Velocity.html"><span style="font-style:italic;">Velocità</span></a>, su <span style="font-style:italic;"><a href="/wiki/MathWorld" title="MathWorld">MathWorld</a></span>, Wolfram Research.</cite> <span class="mw-valign-text-top noprint" typeof="mw:File/Frameless"><a href="https://www.wikidata.org/wiki/Q11465#P2812" title="Modifica su Wikidata"><img alt="Modifica su Wikidata" src="//upload.wikimedia.org/wikipedia/commons/thumb/7/73/Blue_pencil.svg/10px-Blue_pencil.svg.png" decoding="async" width="10" height="10" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/73/Blue_pencil.svg/15px-Blue_pencil.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/73/Blue_pencil.svg/20px-Blue_pencil.svg.png 2x" data-file-width="600" data-file-height="600" /></a></span></li> <li><cite id="CITEREFTreccani_velocita" class="citation testo" style="font-style:normal"> <a rel="nofollow" class="external text" href="http://www.treccani.it/enciclopedia/velocita"><span style="font-style:italic;">Velocità</span></a>, in <span style="font-style:italic;">Treccani.it – Enciclopedie on line</span>, Roma, Istituto dell'Enciclopedia Italiana. <small>URL consultato il 2 marzo 2016</small>.</cite></li></ul> <style data-mw-deduplicate="TemplateStyles:r140554510">.mw-parser-output .CdA{border:1px solid #aaa;width:100%;margin:auto;font-size:90%;padding:2px}.mw-parser-output .CdA th{background-color:#f2f2f2;font-weight:bold;width:20%}@media screen{html.skin-theme-clientpref-night .mw-parser-output .CdA{border-color:#54595D}html.skin-theme-clientpref-night .mw-parser-output .CdA th{background-color:#202122}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .CdA{border-color:#54595D}html.skin-theme-clientpref-os .mw-parser-output .CdA th{background-color:#202122}}</style><table class="CdA"><tbody><tr><th><a href="/wiki/Aiuto:Controllo_di_autorit%C3%A0" title="Aiuto:Controllo di autorità">Controllo di autorità</a></th><td><a href="/wiki/Nuovo_soggettario" title="Nuovo soggettario">Thesaurus BNCF</a> <span class="uid"><a rel="nofollow" class="external text" href="https://thes.bncf.firenze.sbn.it/termine.php?id=12335">12335</a></span><span style="font-weight:bold;"> ·</span> <a href="/wiki/Library_of_Congress_Control_Number" title="Library of Congress Control Number">LCCN</a> <span class="uid">(<span style="font-weight:bolder; font-size:80%"><abbr title="inglese">EN</abbr></span>) <a rel="nofollow" class="external text" href="http://id.loc.gov/authorities/subjects/sh85126486">sh85126486</a></span><span style="font-weight:bold;"> ·</span> <a href="/wiki/Gemeinsame_Normdatei" title="Gemeinsame Normdatei">GND</a> <span class="uid">(<span style="font-weight:bolder; font-size:80%"><abbr title="tedesco">DE</abbr></span>) <a rel="nofollow" class="external text" href="https://d-nb.info/gnd/4020574-5">4020574-5</a></span><span style="font-weight:bold;"> ·</span> <a href="/wiki/Biblioteca_nazionale_di_Francia" title="Biblioteca nazionale di Francia">BNF</a> <span class="uid">(<span style="font-weight:bolder; font-size:80%"><abbr title="francese">FR</abbr></span>) <a rel="nofollow" class="external text" href="https://catalogue.bnf.fr/ark:/12148/cb119762787">cb119762787</a> <a rel="nofollow" class="external text" href="https://data.bnf.fr/ark:/12148/cb119762787">(data)</a></span><span style="font-weight:bold;"> ·</span> <a href="/wiki/Biblioteca_nazionale_di_Israele" title="Biblioteca nazionale di Israele">J9U</a> <span class="uid">(<span style="font-weight:bolder; font-size:80%"><abbr title="inglese">EN</abbr>, <abbr title="ebraico">HE</abbr></span>) <a rel="nofollow" class="external text" href="http://olduli.nli.org.il/F/?func=find-b&local_base=NLX10&find_code=UID&request=987007565830505171">987007565830505171</a></span></td></tr></tbody></table> <div class="noprint" style="width:100%; padding: 3px 0; display: flex; flex-wrap: wrap; row-gap: 4px; column-gap: 8px; box-sizing: border-box;"><div style="flex-grow: 1"><style data-mw-deduplicate="TemplateStyles:r140555418">.mw-parser-output .itwiki-template-occhiello{width:100%;line-height:25px;border:1px solid #CCF;background-color:#F0EEFF;box-sizing:border-box}.mw-parser-output .itwiki-template-occhiello-progetto{background-color:#FAFAFA}@media screen{html.skin-theme-clientpref-night .mw-parser-output .itwiki-template-occhiello{background-color:#202122;border-color:#54595D}html.skin-theme-clientpref-night .mw-parser-output .itwiki-template-occhiello-progetto{background-color:#282929}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .itwiki-template-occhiello{background-color:#202122;border-color:#54595D}html.skin-theme-clientpref-os .mw-parser-output .itwiki-template-occhiello-progetto{background-color:#282929}}</style><div class="itwiki-template-occhiello"><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Nuvola_apps_katomic.svg" class="mw-file-description" title="Fisica"><img alt=" " src="//upload.wikimedia.org/wikipedia/commons/thumb/5/55/Nuvola_apps_katomic.svg/25px-Nuvola_apps_katomic.svg.png" decoding="async" width="25" height="25" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/55/Nuvola_apps_katomic.svg/38px-Nuvola_apps_katomic.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/55/Nuvola_apps_katomic.svg/50px-Nuvola_apps_katomic.svg.png 2x" data-file-width="128" data-file-height="128" /></a></span> <b><a href="/wiki/Portale:Fisica" title="Portale:Fisica">Portale Fisica</a></b>: accedi alle voci di Wikipedia che trattano di fisica</div></div></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1&useformat=desktop" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Estratto da "<a dir="ltr" href="https://it.wikipedia.org/w/index.php?title=Velocità&oldid=142098432">https://it.wikipedia.org/w/index.php?title=Velocità&oldid=142098432</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Categoria:Categorie" title="Categoria:Categorie">Categoria</a>: <ul><li><a href="/wiki/Categoria:Grandezze_cinematiche" title="Categoria:Grandezze cinematiche">Grandezze cinematiche</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Categorie nascoste: <ul><li><a href="/wiki/Categoria:Voci_con_modulo_citazione_e_parametro_coautori" title="Categoria:Voci con modulo citazione e parametro coautori">Voci con modulo citazione e parametro coautori</a></li><li><a href="/wiki/Categoria:Voci_con_modulo_citazione_e_parametro_pagine" title="Categoria:Voci con modulo citazione e parametro pagine">Voci con modulo citazione e parametro pagine</a></li><li><a href="/wiki/Categoria:P3365_letta_da_Wikidata" title="Categoria:P3365 letta da Wikidata">P3365 letta da Wikidata</a></li><li><a href="/wiki/Categoria:P1417_letta_da_Wikidata" title="Categoria:P1417 letta da Wikidata">P1417 letta da Wikidata</a></li><li><a href="/wiki/Categoria:P2812_letta_da_Wikidata" title="Categoria:P2812 letta da Wikidata">P2812 letta da Wikidata</a></li><li><a href="/wiki/Categoria:Voci_con_codice_Thesaurus_BNCF" title="Categoria:Voci con codice Thesaurus BNCF">Voci con codice Thesaurus BNCF</a></li><li><a href="/wiki/Categoria:Voci_con_codice_LCCN" title="Categoria:Voci con codice LCCN">Voci con codice LCCN</a></li><li><a href="/wiki/Categoria:Voci_con_codice_GND" title="Categoria:Voci con codice GND">Voci con codice GND</a></li><li><a href="/wiki/Categoria:Voci_con_codice_BNF" title="Categoria:Voci con codice BNF">Voci con codice BNF</a></li><li><a href="/wiki/Categoria:Voci_con_codice_J9U" title="Categoria:Voci con codice J9U">Voci con codice J9U</a></li><li><a href="/wiki/Categoria:Voci_non_biografiche_con_codici_di_controllo_di_autorit%C3%A0" title="Categoria:Voci non biografiche con codici di controllo di autorità">Voci non biografiche con codici di controllo di autorità</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> Questa pagina è stata modificata per l'ultima volta il 10 nov 2024 alle 05:58.</li> <li id="footer-info-copyright">Il testo è disponibile secondo la <a rel="nofollow" class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.it">licenza Creative Commons Attribuzione-Condividi allo stesso modo</a>; 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