Kinetische Energie – Wikipedia

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Sie entspricht der <a href="/wiki/Arbeit_(Physik)" title="Arbeit (Physik)">Arbeit</a>, die aufgewendet werden muss, um das Objekt aus der Ruhe in die momentane Bewegung zu versetzen. Sie hängt von der <a href="/wiki/Masse_(Physik)" title="Masse (Physik)">Masse</a> und der <a href="/wiki/Geschwindigkeit" title="Geschwindigkeit">Geschwindigkeit</a> des bewegten Körpers ab. </p><p>Als <a href="/wiki/Formelzeichen" title="Formelzeichen">Formelzeichen</a> für die kinetische Energie wird häufig <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> oder <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{\mathrm {kin} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">k</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">n</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{\mathrm {kin} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/93d27518b579169b89542ab23824b416d83789ce" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.187ex; height:2.509ex;" alt="{\displaystyle E_{\mathrm {kin} }}"></span> verwendet. Die <a href="/wiki/Internationales_Einheitensystem" title="Internationales Einheitensystem">SI</a>-<a href="/wiki/Ma%C3%9Feinheit" title="Maßeinheit">Maßeinheit</a> der kinetischen Energie ist das <a href="/wiki/Joule" title="Joule">Joule</a>.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p><p>Das Konzept der kinetischen Energie als eine <a href="/wiki/Physikalische_Gr%C3%B6%C3%9Fe" title="Physikalische Größe">Größe</a>, die bei <a href="/wiki/Elastischer_Sto%C3%9F" class="mw-redirect" title="Elastischer Stoß">elastischen Stößen</a> und vielen anderen mechanischen Vorgängen erhalten bleibt, wurde als <i><a href="/wiki/Vis_viva" title="Vis viva">vis viva</a></i> (‚Lebendige Kraft‘) von <a href="/wiki/Gottfried_Wilhelm_Leibniz" title="Gottfried Wilhelm Leibniz">Gottfried Wilhelm Leibniz</a> eingeführt, der darin in Streit mit den Anhängern von <a href="/wiki/Ren%C3%A9_Descartes" title="René Descartes">René Descartes</a> die korrekte Erhaltungsgröße in der <a href="/wiki/Mechanik" title="Mechanik">Mechanik</a> sah (1686). Diese Größe war allerdings um den Faktor 2 größer als die heute gültige kinetische Energie. Der Faktor <span class="bruch template-frac" style="line-height:0"><sup style="font-size: 70%; vertical-align: 0.4em;">1</sup>⁄<sub style="font-size: 70%; vertical-align: 0em;">2</sub></span> in der Formel für die kinetische Energie findet sich schon 1726 bei <a href="/wiki/Daniel_Bernoulli" title="Daniel Bernoulli">Daniel Bernoulli</a>.<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> Das eigentliche Energiekonzept bildete sich aber erst im 19. Jahrhundert heraus, insbesondere in der Schule der angewandten Mathematik in Frankreich und mit dem Aufkommen der <a href="/wiki/Thermodynamik" title="Thermodynamik">Thermodynamik</a>. In der Mechanik des 18. Jahrhunderts, deren Hauptuntersuchungsgegenstand die <a href="/wiki/Himmelsmechanik" title="Himmelsmechanik">Himmelsmechanik</a> war, spielte es noch keine große Rolle.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> Die Ausdrücke „kinetische Energie“ und „<a href="/wiki/Potentielle_Energie" title="Potentielle Energie">potentielle Energie</a>“ wurden 1859 von dem schottischen Ingenieur <a href="/wiki/William_John_Macquorn_Rankine" title="William John Macquorn Rankine">William J. M. Rankine</a> geprägt.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p> <div id="toc" class="toc" role="navigation" aria-labelledby="mw-toc-heading"><input type="checkbox" role="button" id="toctogglecheckbox" class="toctogglecheckbox" style="display:none" /><div class="toctitle" lang="de" dir="ltr"><h2 id="mw-toc-heading">Inhaltsverzeichnis</h2><span class="toctogglespan"><label class="toctogglelabel" for="toctogglecheckbox"></label></span></div> <ul> <li class="toclevel-1 tocsection-1"><a href="#Kinetische_Energie_in_der_klassischen_Mechanik"><span class="tocnumber">1</span> <span class="toctext">Kinetische Energie in der klassischen Mechanik</span></a> <ul> <li class="toclevel-2 tocsection-2"><a href="#Massenpunkt"><span class="tocnumber">1.1</span> <span class="toctext">Massenpunkt</span></a> <ul> <li class="toclevel-3 tocsection-3"><a href="#Einfache_Herleitung"><span class="tocnumber">1.1.1</span> <span class="toctext">Einfache Herleitung</span></a></li> <li class="toclevel-3 tocsection-4"><a href="#Bewegung_in_einem_Koordinatensystem"><span class="tocnumber">1.1.2</span> <span class="toctext">Bewegung in einem Koordinatensystem</span></a></li> </ul> </li> <li class="toclevel-2 tocsection-5"><a href="#Starre_Körper"><span class="tocnumber">1.2</span> <span class="toctext">Starre Körper</span></a></li> <li class="toclevel-2 tocsection-6"><a href="#Hydrodynamik"><span class="tocnumber">1.3</span> <span class="toctext">Hydrodynamik</span></a></li> </ul> </li> <li class="toclevel-1 tocsection-7"><a href="#Kinetische_Energie_in_der_relativistischen_Mechanik"><span class="tocnumber">2</span> <span class="toctext">Kinetische Energie in der relativistischen Mechanik</span></a></li> <li class="toclevel-1 tocsection-8"><a href="#Kinetische_Energie_in_der_Quantenmechanik"><span class="tocnumber">3</span> <span class="toctext">Kinetische Energie in der Quantenmechanik</span></a></li> <li class="toclevel-1 tocsection-9"><a href="#Siehe_auch"><span class="tocnumber">4</span> <span class="toctext">Siehe auch</span></a></li> <li class="toclevel-1 tocsection-10"><a href="#Literatur"><span class="tocnumber">5</span> <span class="toctext">Literatur</span></a></li> <li class="toclevel-1 tocsection-11"><a href="#Weblinks"><span class="tocnumber">6</span> <span class="toctext">Weblinks</span></a></li> <li class="toclevel-1 tocsection-12"><a href="#Einzelnachweise"><span class="tocnumber">7</span> <span class="toctext">Einzelnachweise</span></a></li> </ul> </div> <div class="mw-heading mw-heading2"><h2 id="Kinetische_Energie_in_der_klassischen_Mechanik">Kinetische Energie in der klassischen Mechanik</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Kinetische_Energie&veaction=edit&section=1" title="Abschnitt bearbeiten: Kinetische Energie in der klassischen Mechanik" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Kinetische_Energie&action=edit&section=1" title="Quellcode des Abschnitts bearbeiten: Kinetische Energie in der klassischen Mechanik"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Massenpunkt">Massenpunkt</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Kinetische_Energie&veaction=edit&section=2" title="Abschnitt bearbeiten: Massenpunkt" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Kinetische_Energie&action=edit&section=2" title="Quellcode des Abschnitts bearbeiten: Massenpunkt"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In der <a href="/wiki/Klassische_Mechanik" title="Klassische Mechanik">klassischen Mechanik</a> ist die kinetische Energie <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}"></span> eines Massenpunktes <a href="/wiki/Proportionalit%C3%A4t" title="Proportionalität">proportional</a> zu seiner Masse <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}"></span> und dem Quadrat seiner Geschwindigkeit <span style="white-space:nowrap"><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>:</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 E_{\mathrm {kin} }={\frac {1}{2}}mv^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">k</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">n</mi> </mrow> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>m</mi> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{\mathrm {kin} }={\frac {1}{2}}mv^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cc953820bd294c5ff2ce76e218b690db64e50391" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:13.506ex; height:5.176ex;" alt="{\displaystyle E_{\mathrm {kin} }={\frac {1}{2}}mv^{2}}"></span>.</dd></dl> <p>Fährt beispielsweise ein Auto der Masse <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=1000\,\mathrm {kg} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo>=</mo> <mn>1000</mn> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">k</mi> <mi mathvariant="normal">g</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m=1000\,\mathrm {kg} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5b20bef117b51b6267d7afa97eb1eb0bdb2fb529" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.566ex; height:2.509ex;" alt="{\displaystyle m=1000\,\mathrm {kg} }"></span> mit einer Geschwindigkeit von <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=100\,\mathrm {km} /\mathrm {h} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> <mo>=</mo> <mn>100</mn> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">k</mi> <mi mathvariant="normal">m</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">h</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v=100\,\mathrm {km} /\mathrm {h} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/36e15d00d7daa554c22393ef8132c94148033b39" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.719ex; height:2.843ex;" alt="{\displaystyle v=100\,\mathrm {km} /\mathrm {h} }"></span>, hat es demzufolge eine kinetische Energie von <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle E_{\mathrm {kin} }={\frac {1}{2}}\cdot 1000\,\mathrm {kg} \cdot \left(100\,{\frac {\mathrm {km} }{\mathrm {h} }}\right)^{2}\approx {\frac {1}{2}}\cdot 1000\,\mathrm {kg} \cdot \left(27{,}78\,{\frac {\mathrm {m} }{\mathrm {s} }}\right)^{2}\approx 385.800\,\mathrm {J} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">k</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">n</mi> </mrow> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>⋅</mo> <mn>1000</mn> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">k</mi> <mi mathvariant="normal">g</mi> </mrow> <mo>⋅</mo> <msup> <mrow> <mo>(</mo> <mrow> <mn>100</mn> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">k</mi> <mi mathvariant="normal">m</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">h</mi> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>≈</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>⋅</mo> <mn>1000</mn> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">k</mi> <mi mathvariant="normal">g</mi> </mrow> <mo>⋅</mo> <msup> <mrow> <mo>(</mo> <mrow> <mn>27</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>,</mo> </mrow> <mn>78</mn> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">m</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">s</mi> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>≈</mo> <mn>385.800</mn> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">J</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle E_{\mathrm {kin} }={\frac {1}{2}}\cdot 1000\,\mathrm {kg} \cdot \left(100\,{\frac {\mathrm {km} }{\mathrm {h} }}\right)^{2}\approx {\frac {1}{2}}\cdot 1000\,\mathrm {kg} \cdot \left(27{,}78\,{\frac {\mathrm {m} }{\mathrm {s} }}\right)^{2}\approx 385.800\,\mathrm {J} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/12fa3c33ed284b2d1accd104c580ecd3bf06fee4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:69.422ex; height:5.176ex;" alt="{\textstyle E_{\mathrm {kin} }={\frac {1}{2}}\cdot 1000\,\mathrm {kg} \cdot \left(100\,{\frac {\mathrm {km} }{\mathrm {h} }}\right)^{2}\approx {\frac {1}{2}}\cdot 1000\,\mathrm {kg} \cdot \left(27{,}78\,{\frac {\mathrm {m} }{\mathrm {s} }}\right)^{2}\approx 385.800\,\mathrm {J} }"></span> (das <a href="/wiki/Joule" title="Joule">Joule</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 \mathrm {J} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">J</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {J} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ed0b3ef0cc0c96ab4e418ef3de9037d68d8a8ba0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.195ex; height:2.176ex;" alt="{\displaystyle \mathrm {J} }"></span>, ist die <a href="/wiki/Internationales_Einheitensystem" title="Internationales Einheitensystem">SI</a>-Einheit der Energie). </p><p>Wenn man den Bewegungszustand des Körpers nicht durch seine Geschwindigkeit <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>, sondern durch seinen <a href="/wiki/Impuls" title="Impuls">Impuls</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 p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span> beschreibt, wie das u. a. in der <a href="/wiki/Hamiltonsche_Mechanik" title="Hamiltonsche Mechanik">Hamiltonschen Mechanik</a> üblich ist, so gilt für die kinetische Energie (wegen <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p=mv}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>=</mo> <mi>m</mi> <mi>v</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p=mv}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2acbe7154884d4dbe30b9a0b399e43cefd8654c7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:7.525ex; height:2.009ex;" alt="{\displaystyle p=mv}"></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 E_{\mathrm {kin} }={\frac {p^{2}}{2m}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">k</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">n</mi> </mrow> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mn>2</mn> <mi>m</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{\mathrm {kin} }={\frac {p^{2}}{2m}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f7dfeffc83e0ed395942f5f70a24589ccac900b9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:11.324ex; height:5.676ex;" alt="{\displaystyle E_{\mathrm {kin} }={\frac {p^{2}}{2m}}}"></span>.</dd></dl> <div class="mw-heading mw-heading4"><h4 id="Einfache_Herleitung">Einfache Herleitung</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Kinetische_Energie&veaction=edit&section=3" title="Abschnitt bearbeiten: Einfache Herleitung" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Kinetische_Energie&action=edit&section=3" title="Quellcode des Abschnitts bearbeiten: Einfache Herleitung"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Wird ein Körper der Masse <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}"></span> aus der Ruhe heraus auf die Geschwindigkeit <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> beschleunigt, so muss man dafür die <a href="/wiki/Beschleunigungsarbeit" class="mw-redirect" title="Beschleunigungsarbeit">Beschleunigungsarbeit</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 W}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>W</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54a9c4c547f4d6111f81946cad242b18298d70b7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.435ex; height:2.176ex;" alt="{\displaystyle W}"></span> zufügen. Bei konstanter Kraft gilt </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 W=Fs}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>W</mi> <mo>=</mo> <mi>F</mi> <mi>s</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W=Fs}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b020230128c115d1b9e9cfbe6636985e98bbaf69" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.365ex; height:2.176ex;" alt="{\displaystyle W=Fs}"></span>,</dd></dl> <p>wobei <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> die in Richtung der Kraft zurückgelegte Strecke ist. Die Kraft erteilt dem Körper eine <a href="/wiki/Gleichm%C3%A4%C3%9Fig_beschleunigte_Bewegung" title="Gleichmäßig beschleunigte Bewegung">gleichmäßige Beschleunigung</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 a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span>, nach der <a href="/wiki/Grundgleichung_der_Mechanik" class="mw-redirect" title="Grundgleichung der Mechanik">Grundgleichung der Mechanik</a> ist <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 F=ma}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo>=</mo> <mi>m</mi> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F=ma}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1ca4e42b7d6d66f52294364928cb5f7c590f514c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.109ex; height:2.176ex;" alt="{\displaystyle F=ma}"></span>. Nach einer Zeit <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> ist die Geschwindigkeit <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v=at}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> <mo>=</mo> <mi>a</mi> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v=at}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ab8eed79a675337b2155d9be4004717b9e9a4c9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.296ex; height:2.009ex;" alt="{\displaystyle v=at}"></span> erreicht, und es wurde die Strecke <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={\tfrac {1}{2}}at^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mi>a</mi> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s={\tfrac {1}{2}}at^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c0f741b88f6938bb3d2a535a63c8376cdcdbdb4b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:8.971ex; height:3.509ex;" alt="{\displaystyle s={\tfrac {1}{2}}at^{2}}"></span> zurückgelegt. Alles oben eingesetzt, ergibt die Beschleunigungsarbeit </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 W=ma\cdot {\frac {1}{2}}at^{2}={\frac {1}{2}}mv^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>W</mi> <mo>=</mo> <mi>m</mi> <mi>a</mi> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>a</mi> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>m</mi> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W=ma\cdot {\frac {1}{2}}at^{2}={\frac {1}{2}}mv^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2412e05c56e236c03ae58deb05366b81521bdae7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:24.925ex; height:5.176ex;" alt="{\displaystyle W=ma\cdot {\frac {1}{2}}at^{2}={\frac {1}{2}}mv^{2}}"></span>.</dd></dl> <p>Da die kinetische Energie in Ruhe den Wert Null hat, erreicht sie nach dem Beschleunigungsvorgang genau diesen Wert <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 W}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>W</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54a9c4c547f4d6111f81946cad242b18298d70b7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.435ex; height:2.176ex;" alt="{\displaystyle W}"></span>. Folglich beträgt die kinetische Energie eines Körpers der Masse <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}"></span>, der sich mit der Geschwindigkeit <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> bewegt </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{\mathrm {kin} }={\frac {1}{2}}mv^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">k</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">n</mi> </mrow> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>m</mi> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{\mathrm {kin} }={\frac {1}{2}}mv^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cc953820bd294c5ff2ce76e218b690db64e50391" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:13.506ex; height:5.176ex;" alt="{\displaystyle E_{\mathrm {kin} }={\frac {1}{2}}mv^{2}}"></span>.</dd></dl> <div class="mw-heading mw-heading4"><h4 id="Bewegung_in_einem_Koordinatensystem">Bewegung in einem Koordinatensystem</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Kinetische_Energie&veaction=edit&section=4" title="Abschnitt bearbeiten: Bewegung in einem Koordinatensystem" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Kinetische_Energie&action=edit&section=4" title="Quellcode des Abschnitts bearbeiten: Bewegung in einem Koordinatensystem"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Beschreibt man die Bewegung eines Körpers in einem Koordinatensystem, so lässt sich die kinetische Energie je nach Wahl des <a href="/wiki/Koordinatensystem" title="Koordinatensystem">Koordinatensystems</a> wie folgt berechnen: </p> <ul><li><a href="/wiki/Kartesisches_Koordinatensystem" title="Kartesisches Koordinatensystem">Kartesische Koordinaten</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 (x,y,z)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <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 (x,y,z)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/22a8c93372e8f8b6e24d523bd5545aed3430baf4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.45ex; height:2.843ex;" alt="{\displaystyle (x,y,z)}"></span>:</li></ul> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{\mathrm {kin} }={\frac {1}{2}}m\left({\dot {x}}^{2}+{\dot {y}}^{2}+{\dot {z}}^{2}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">k</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">n</mi> </mrow> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>m</mi> <mrow> <mo>(</mo> <mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>y</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>z</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{\mathrm {kin} }={\frac {1}{2}}m\left({\dot {x}}^{2}+{\dot {y}}^{2}+{\dot {z}}^{2}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fde584a2fc87c01ebe86fd5017730fce80f5764e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:26.613ex; height:5.176ex;" alt="{\displaystyle E_{\mathrm {kin} }={\frac {1}{2}}m\left({\dot {x}}^{2}+{\dot {y}}^{2}+{\dot {z}}^{2}\right)}"></span></dd></dl> <ul><li>Ebene <a href="/wiki/Polarkoordinaten" title="Polarkoordinaten">Polarkoordinaten</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 r,\varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>,</mo> <mi>φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r,\varphi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e24a3776c244ed9013cc7b3f54687fba6d8b099e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.603ex; height:2.176ex;" alt="{\displaystyle r,\varphi }"></span>):</li></ul> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{\mathrm {kin} }={\frac {1}{2}}m\left({\dot {r}}^{2}+r^{2}{\dot {\varphi }}^{2}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">k</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">n</mi> </mrow> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>m</mi> <mrow> <mo>(</mo> <mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo>˙</mo> </mover> </mrow> </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 class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>φ</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{\mathrm {kin} }={\frac {1}{2}}m\left({\dot {r}}^{2}+r^{2}{\dot {\varphi }}^{2}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6482fe79094d525ffc427c8ccf1395063213f582" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:23.719ex; height:5.176ex;" alt="{\displaystyle E_{\mathrm {kin} }={\frac {1}{2}}m\left({\dot {r}}^{2}+r^{2}{\dot {\varphi }}^{2}\right)}"></span></dd></dl> <ul><li><a href="/wiki/Kugelkoordinaten" title="Kugelkoordinaten">Kugelkoordinaten</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 r,\varphi ,\vartheta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>,</mo> <mi>φ</mi> <mo>,</mo> <mi>ϑ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r,\varphi ,\vartheta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c183b2a3531dabe17a0ca5faafcd8ef44a7eafa4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.01ex; height:2.676ex;" alt="{\displaystyle r,\varphi ,\vartheta }"></span>):</li></ul> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{\mathrm {kin} }={\frac {1}{2}}m\left(r^{2}\left[{\dot {\vartheta }}^{2}+{\dot {\varphi }}^{2}\sin ^{2}\vartheta \right]+{\dot {r}}^{2}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">k</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">n</mi> </mrow> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>m</mi> <mrow> <mo>(</mo> <mrow> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mo>[</mo> <mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ϑ</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>φ</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>ϑ</mi> </mrow> <mo>]</mo> </mrow> <mo>+</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{\mathrm {kin} }={\frac {1}{2}}m\left(r^{2}\left[{\dot {\vartheta }}^{2}+{\dot {\varphi }}^{2}\sin ^{2}\vartheta \right]+{\dot {r}}^{2}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/147523b4fe56857363a761a057471a26678ba8f7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:38.361ex; height:5.176ex;" alt="{\displaystyle E_{\mathrm {kin} }={\frac {1}{2}}m\left(r^{2}\left[{\dot {\vartheta }}^{2}+{\dot {\varphi }}^{2}\sin ^{2}\vartheta \right]+{\dot {r}}^{2}\right)}"></span></dd></dl> <ul><li><a href="/wiki/Zylinderkoordinaten" class="mw-redirect" title="Zylinderkoordinaten">Zylinderkoordinaten</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 r,\varphi ,z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>,</mo> <mi>φ</mi> <mo>,</mo> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r,\varphi ,z}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/354e747f10244a59b6c972a026ef5444f2b39e81" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.725ex; height:2.176ex;" alt="{\displaystyle r,\varphi ,z}"></span>):</li></ul> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{\mathrm {kin} }={\frac {1}{2}}m\left({\dot {r}}^{2}+r^{2}{\dot {\varphi }}^{2}+{\dot {z}}^{2}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">k</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">n</mi> </mrow> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>m</mi> <mrow> <mo>(</mo> <mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo>˙</mo> </mover> </mrow> </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 class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>φ</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>z</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{\mathrm {kin} }={\frac {1}{2}}m\left({\dot {r}}^{2}+r^{2}{\dot {\varphi }}^{2}+{\dot {z}}^{2}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/866083df719104742170a2b7850146868d8e3b9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:28.91ex; height:5.176ex;" alt="{\displaystyle E_{\mathrm {kin} }={\frac {1}{2}}m\left({\dot {r}}^{2}+r^{2}{\dot {\varphi }}^{2}+{\dot {z}}^{2}\right)}"></span></dd></dl> <p>Dabei bedeutet der Punkt über der Koordinate ihre zeitliche Änderung, die <a href="/wiki/Differentialrechnung#Ableitungsberechnung" title="Differentialrechnung">Ableitung</a> nach der Zeit. Die Formeln berücksichtigen nicht die Energie, die möglicherweise in der Eigenrotation des Körpers steckt. </p> <div class="mw-heading mw-heading3"><h3 id="Starre_Körper"><span id="Starre_K.C3.B6rper"></span>Starre Körper</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Kinetische_Energie&veaction=edit&section=5" title="Abschnitt bearbeiten: Starre Körper" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Kinetische_Energie&action=edit&section=5" title="Quellcode des Abschnitts bearbeiten: Starre Körper"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Die kinetische Energie eines <a href="/wiki/Starrer_K%C3%B6rper" title="Starrer Körper">starren Körpers</a> mit der Gesamtmasse <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}"></span> und der Geschwindigkeit <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_{\mathrm {s} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">s</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v_{\mathrm {s} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/80cfb31bf303e4c75808e13b1bbdafd4f0ce7fd0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.008ex; height:2.009ex;" alt="{\displaystyle v_{\mathrm {s} }}"></span> seines <a href="/wiki/Massenmittelpunkt" title="Massenmittelpunkt">Schwerpunktes</a> ist die Summe der Energie aus der Bewegung des Schwerpunkts (<a href="/wiki/Translation_(Physik)" title="Translation (Physik)">Translationsenergie</a>) und der <a href="/wiki/Rotationsenergie" title="Rotationsenergie">Rotationsenergie</a> aus der Drehung um den Schwerpunkt: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{\mathrm {kin} }={\frac {1}{2}}M{v_{\mathrm {s} }}^{2}+{\frac {1}{2}}J_{\mathrm {s} }\omega ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">k</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">n</mi> </mrow> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>M</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">s</mi> </mrow> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">s</mi> </mrow> </mrow> </msub> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{\mathrm {kin} }={\frac {1}{2}}M{v_{\mathrm {s} }}^{2}+{\frac {1}{2}}J_{\mathrm {s} }\omega ^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7a87be59b6b792ce7915ab86f368f3a73078e0fa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:24.297ex; height:5.176ex;" alt="{\displaystyle E_{\mathrm {kin} }={\frac {1}{2}}M{v_{\mathrm {s} }}^{2}+{\frac {1}{2}}J_{\mathrm {s} }\omega ^{2}}"></span></dd></dl> <p>Hier ist <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 J_{\mathrm {s} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">s</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle J_{\mathrm {s} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ff9eec21765972dc93e11e1ed6f6510f96afed89" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.17ex; height:2.509ex;" alt="{\displaystyle J_{\mathrm {s} }}"></span> das <a href="/wiki/Tr%C3%A4gheitsmoment" title="Trägheitsmoment">Trägheitsmoment</a> des Körpers bezüglich seines Schwerpunktes und <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 \omega }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ω</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48eff443f9de7a985bb94ca3bde20813ea737be8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.446ex; height:1.676ex;" alt="{\displaystyle \omega }"></span> die <a href="/wiki/Winkelgeschwindigkeit" title="Winkelgeschwindigkeit">Winkelgeschwindigkeit</a> der Drehung. </p><p>Mit dem <a href="/wiki/Tr%C3%A4gheitstensor" title="Trägheitstensor">Trägheitstensor</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 I}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/535ea7fc4134a31cbe2251d9d3511374bc41be9f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.172ex; height:2.176ex;" alt="{\displaystyle I}"></span> wird dies allgemein geschrieben als: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{\mathrm {kin} }={\frac {1}{2}}M{v_{\mathrm {s} }}^{2}+{\frac {1}{2}}{\boldsymbol {\omega }}^{T}I{\boldsymbol {\omega }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">k</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">n</mi> </mrow> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>M</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">s</mi> </mrow> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ω</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msup> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ω</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{\mathrm {kin} }={\frac {1}{2}}M{v_{\mathrm {s} }}^{2}+{\frac {1}{2}}{\boldsymbol {\omega }}^{T}I{\boldsymbol {\omega }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4c4b3776889616578af5e9be2f50df77cfccaa03" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:25.526ex; height:5.176ex;" alt="{\displaystyle E_{\mathrm {kin} }={\frac {1}{2}}M{v_{\mathrm {s} }}^{2}+{\frac {1}{2}}{\boldsymbol {\omega }}^{T}I{\boldsymbol {\omega }}}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Hydrodynamik">Hydrodynamik</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Kinetische_Energie&veaction=edit&section=6" title="Abschnitt bearbeiten: Hydrodynamik" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Kinetische_Energie&action=edit&section=6" title="Quellcode des Abschnitts bearbeiten: Hydrodynamik"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In der <a href="/wiki/Hydrodynamik" class="mw-redirect" title="Hydrodynamik">Hydrodynamik</a> wird oft statt der kinetischen Energie die kinetische Energie<i>dichte</i> angegeben. Diese wird meistens durch ein kleines <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>e</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cd253103f0876afc68ebead27a5aa9867d927467" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.083ex; height:1.676ex;" alt="{\displaystyle e}"></span> oder <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 \epsilon }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϵ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \epsilon }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3837cad72483d97bcdde49c85d3b7b859fb3fd2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.944ex; height:1.676ex;" alt="{\displaystyle \epsilon }"></span> ausgedrückt: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e_{\mathrm {kin} }={\frac {E_{\mathrm {kin} }}{V}}={\frac {1}{2}}\rho v^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">k</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">n</mi> </mrow> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">k</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">n</mi> </mrow> </mrow> </msub> <mi>V</mi> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>ρ</mi> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e_{\mathrm {kin} }={\frac {E_{\mathrm {kin} }}{V}}={\frac {1}{2}}\rho v^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/689060e851d12dec36ab1a683659f4fd83a1a2d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:20.157ex; height:5.343ex;" alt="{\displaystyle e_{\mathrm {kin} }={\frac {E_{\mathrm {kin} }}{V}}={\frac {1}{2}}\rho v^{2}}"></span></dd></dl> <p>Hierbei bezeichnet <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f7d439671d1289b6a816e6af7a304be40608d64" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.202ex; height:2.176ex;" alt="{\displaystyle \rho }"></span> die <a href="/wiki/Dichte" title="Dichte">Dichte</a> und <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> das Volumen. </p> <div class="mw-heading mw-heading2"><h2 id="Kinetische_Energie_in_der_relativistischen_Mechanik">Kinetische Energie in der relativistischen Mechanik</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Kinetische_Energie&veaction=edit&section=7" title="Abschnitt bearbeiten: Kinetische Energie in der relativistischen Mechanik" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Kinetische_Energie&action=edit&section=7" title="Quellcode des Abschnitts bearbeiten: Kinetische Energie in der relativistischen Mechanik"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure typeof="mw:File/Thumb"><a href="/wiki/Datei:E_Kin.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/52/E_Kin.jpg/400px-E_Kin.jpg" decoding="async" width="400" height="256" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/52/E_Kin.jpg/600px-E_Kin.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/52/E_Kin.jpg/800px-E_Kin.jpg 2x" data-file-width="923" data-file-height="590" /></a><figcaption>Relativistische und klassische kinetische Energie im Vergleich, mit <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 \beta ={\frac {v}{c}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>v</mi> <mi>c</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta ={\frac {v}{c}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fa4399f8b4ceb09825943082d658a89f5497eefe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:6.394ex; height:4.676ex;" alt="{\displaystyle \beta ={\frac {v}{c}}}"></span></figcaption></figure> <p>In der <a href="/wiki/Spezielle_Relativit%C3%A4tstheorie" title="Spezielle Relativitätstheorie">relativistischen Physik</a> gilt die oben angegebene Abhängigkeit der kinetischen Energie von der Geschwindigkeit nur näherungsweise für Geschwindigkeiten deutlich kleiner als die <a href="/wiki/Lichtgeschwindigkeit" title="Lichtgeschwindigkeit">Lichtgeschwindigkeit</a>. Aus dem Ansatz, dass die kinetische Energie <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{\mathrm {kin} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">k</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">n</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{\mathrm {kin} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/93d27518b579169b89542ab23824b416d83789ce" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.187ex; height:2.509ex;" alt="{\displaystyle E_{\mathrm {kin} }}"></span> die Differenz aus Gesamtenergie und <a href="/wiki/Ruheenergie" class="mw-redirect" title="Ruheenergie">Ruheenergie</a> ist, folgt: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{\mathrm {kin} }=\gamma mc^{2}-mc^{2}=\left(\gamma -1\right)mc^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">k</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">n</mi> </mrow> </mrow> </msub> <mo>=</mo> <mi>γ</mi> <mi>m</mi> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mi>m</mi> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mi>γ</mi> <mo>−</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mi>m</mi> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{\mathrm {kin} }=\gamma mc^{2}-mc^{2}=\left(\gamma -1\right)mc^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/280a2efc820b487386e0bfe0d40c3f478ee9c07d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:34.252ex; height:3.176ex;" alt="{\displaystyle E_{\mathrm {kin} }=\gamma mc^{2}-mc^{2}=\left(\gamma -1\right)mc^{2}}"></span></dd></dl> <p>Dabei ist <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86a67b81c2de995bd608d5b2df50cd8cd7d92455" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.007ex; height:1.676ex;" alt="{\displaystyle c}"></span> die Lichtgeschwindigkeit, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}"></span> die Masse und <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> der <a href="/wiki/Lorentzfaktor" title="Lorentzfaktor">Lorentzfaktor</a> </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 \gamma ={\frac {1}{\sqrt {1-(v/c)^{2}}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>γ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mn>1</mn> <mo>−</mo> <mo stretchy="false">(</mo> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>c</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma ={\frac {1}{\sqrt {1-(v/c)^{2}}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4300e46f26fe8afab49c4e8679e21e285b62d1c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:18.331ex; height:6.509ex;" alt="{\displaystyle \gamma ={\frac {1}{\sqrt {1-(v/c)^{2}}}}.}"></span></dd></dl> <p>Aus der <a href="/wiki/Taylorreihe" title="Taylorreihe">Taylor-Entwicklung</a> nach <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/c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v/c}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bd5fbfe05e197127354e233f6c813e6b5d846672" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.297ex; height:2.843ex;" alt="{\displaystyle v/c}"></span> erhält man </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{\mathrm {kin} }={\frac {1}{2}}mv^{2}+{\frac {3}{8}}{\frac {mv^{4}}{c^{2}}}+\cdots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">k</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">n</mi> </mrow> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>m</mi> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>8</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>m</mi> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mrow> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>+</mo> <mo>⋯</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{\mathrm {kin} }={\frac {1}{2}}mv^{2}+{\frac {3}{8}}{\frac {mv^{4}}{c^{2}}}+\cdots }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6196e33bf9adcef364240daa3489dd9349fc0e74" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:28.967ex; height:6.009ex;" alt="{\displaystyle E_{\mathrm {kin} }={\frac {1}{2}}mv^{2}+{\frac {3}{8}}{\frac {mv^{4}}{c^{2}}}+\cdots }"></span>,</dd></dl> <p>also für <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\ll c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> <mo>≪</mo> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v\ll c}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eeae20cf8308560968c220b5c0da3d2b6a48e8d6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.749ex; height:1.843ex;" alt="{\displaystyle v\ll c}"></span> wieder die kinetische Energie der klassischen Mechanik. </p><p>Da die Energie über alle Grenzen wachsen müsste, wenn die Geschwindigkeit gegen die Lichtgeschwindigkeit geht, <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 \lim _{v\to c}E_{\mathrm {kin} }=\infty ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>v</mi> <mo stretchy="false">→</mo> <mi>c</mi> </mrow> </munder> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">k</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">n</mi> </mrow> </mrow> </msub> <mo>=</mo> <mi mathvariant="normal">∞</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{v\to c}E_{\mathrm {kin} }=\infty ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/217f9382099d1e132833e1b5ff3cf45f79199a9c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:13.873ex; height:3.676ex;" alt="{\displaystyle \lim _{v\to c}E_{\mathrm {kin} }=\infty ,}"></span> ist es nicht möglich, einen massebehafteten Körper auf Lichtgeschwindigkeit zu beschleunigen. </p><p>Das Diagramm rechts zeigt die relativistische kinetische Energie und die nach der klassischen Mechanik als Funktion der Geschwindigkeit (gemessen in Vielfachen der Lichtgeschwindigkeit) für einen Körper mit der Masse von <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=1\,\mathrm {kg} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo>=</mo> <mn>1</mn> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">k</mi> <mi mathvariant="normal">g</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m=1\,\mathrm {kg} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/22a2afc295060cf032b41e01bd5c648031cf4759" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.078ex; height:2.509ex;" alt="{\displaystyle m=1\,\mathrm {kg} }"></span>. </p><p>Da die Geschwindigkeit eines bewegten Körpers vom Bezugssystem abhängt, gilt dies auch für dessen kinetische Energie. Das gilt in klassischer und in relativistischer Physik. </p> <dl><dt>Anwendungsbeispiele</dt></dl> <div class="hauptartikel" role="navigation"><span class="hauptartikel-pfeil" title="siehe" aria-hidden="true" role="presentation">→ </span><i><span class="hauptartikel-text">Hauptartikel</span>: <a href="/wiki/Tests_der_relativistischen_Energie-Impuls-Beziehung" title="Tests der relativistischen Energie-Impuls-Beziehung">Tests der relativistischen Energie-Impuls-Beziehung</a></i></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Datei:Elektronengeschwindigkeit.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/80/Elektronengeschwindigkeit.png/220px-Elektronengeschwindigkeit.png" decoding="async" width="220" height="189" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/80/Elektronengeschwindigkeit.png/330px-Elektronengeschwindigkeit.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/80/Elektronengeschwindigkeit.png/440px-Elektronengeschwindigkeit.png 2x" data-file-width="510" data-file-height="439" /></a><figcaption>Relativistische Geschwindigkeit eines Elektrons nach Durchlaufen eines elektrischen Felds</figcaption></figure> <p>Im elektrischen Feld nimmt die Energie eines Elektrons der Ladung <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>e</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cd253103f0876afc68ebead27a5aa9867d927467" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.083ex; height:1.676ex;" alt="{\displaystyle e}"></span> und der Masse <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}"></span> linear mit der durchlaufenen Beschleunigungsspannung <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/458a728f53b9a0274f059cd695e067c430956025" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}"></span> zu. Die kinetische Energie ist nun die Differenz der relativistischen Gesamtenergie <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}"></span> und der Ruheenergie <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}"></span><sub>0</sub>.<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> Die kinetische Energie <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 eU}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>e</mi> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle eU}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f258ad1d5d4f10be63c34e01198d33560a6e786" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.866ex; height:2.176ex;" alt="{\displaystyle eU}"></span> ist also: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e\cdot U=E-E_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>e</mi> <mo>⋅</mo> <mi>U</mi> <mo>=</mo> <mi>E</mi> <mo>−</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e\cdot U=E-E_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b918a27f3a607386d161e5fff66dd9ee4392fce" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.029ex; height:2.509ex;" alt="{\displaystyle e\cdot U=E-E_{0}}"></span></dd></dl> <p>Beachtet man, dass für die Gesamtenergie </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E^{2}=c^{2}p^{2}+E_{0}^{2}\quad (*)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msubsup> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mspace width="1em" /> <mo stretchy="false">(</mo> <mo>∗</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E^{2}=c^{2}p^{2}+E_{0}^{2}\quad (*)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5d93245ea66eb3717cd09c31afe5b1a4d81f571" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:21.214ex; height:3.343ex;" alt="{\displaystyle E^{2}=c^{2}p^{2}+E_{0}^{2}\quad (*)}"></span></dd></dl> <p>gilt (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span>: relativistischer Impuls) und zwischen Impuls und Gesamtenergie der Zusammenhang </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 cp=E\cdot {\frac {v}{c}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> <mi>p</mi> <mo>=</mo> <mi>E</mi> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>v</mi> <mi>c</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle cp=E\cdot {\frac {v}{c}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5e7fc55f4b37f2510dfbee84d6345548aacc2ecd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:10.693ex; height:4.676ex;" alt="{\displaystyle cp=E\cdot {\frac {v}{c}}}"></span></dd></dl> <p>besteht, folgt für die Gesamtenergie aus <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 (*)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mo>∗</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (*)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fae74f24018ce1ff0dc698dbe8181555eb4cd768" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.972ex; height:2.843ex;" alt="{\displaystyle (*)}"></span> also: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E(v)={\frac {E_{0}}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo stretchy="false">(</mo> <mi>v</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msqrt> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> </msqrt> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E(v)={\frac {E_{0}}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aa28bd8e18817796568ab050549740f79af676fa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.671ex; width:17.439ex; height:8.176ex;" alt="{\displaystyle E(v)={\frac {E_{0}}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}}"></span></dd></dl> <p>Berechnet man nun die Differenz aus <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E(v)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo stretchy="false">(</mo> <mi>v</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E(v)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/716bc56eda4c3d74f0f09822f5e01ed7d0166125" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.713ex; height:2.843ex;" alt="{\displaystyle E(v)}"></span> und <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/411d268de7b1cf300d7481e3fe59f3b20887e0d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.769ex; height:2.509ex;" alt="{\displaystyle E_{0}}"></span>, setzt den Ausdruck gleich <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e\cdot U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>e</mi> <mo>⋅</mo> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e\cdot U}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/744cffba03ac902bb6c069141930da143f3eac36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.545ex; height:2.176ex;" alt="{\displaystyle e\cdot U}"></span> und löst nach <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> auf, erhält man abschließend: </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=c\cdot {\sqrt {1-{\left({\frac {1}{1+{\frac {eU}{E_{0}}}}}\right)}^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> <mo>=</mo> <mi>c</mi> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>e</mi> <mi>U</mi> </mrow> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mfrac> </mrow> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v=c\cdot {\sqrt {1-{\left({\frac {1}{1+{\frac {eU}{E_{0}}}}}\right)}^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c2d8343a8830cc6016aff05f5d99db898247160f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.838ex; width:26.209ex; height:10.009ex;" alt="{\displaystyle v=c\cdot {\sqrt {1-{\left({\frac {1}{1+{\frac {eU}{E_{0}}}}}\right)}^{2}}}}"></span> mit der Ruheenergie eines Elektrons <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{0}=0{,}51\,\mathrm {MeV} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>,</mo> </mrow> <mn>51</mn> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">M</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">V</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{0}=0{,}51\,\mathrm {MeV} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7579af87db896502e5fc0e13b415f41a3a35e88a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.296ex; height:2.509ex;" alt="{\displaystyle E_{0}=0{,}51\,\mathrm {MeV} }"></span></dd></dl> <p>Bei Beschleunigungsspannungen unterhalb 1 kV lässt sich die Geschwindigkeit aus dem klassischen Ansatz für die kinetische Energie abschätzen, bei höheren Energien muss relativistisch gerechnet werden. Bereits bei einer Spannung von 10 kV erreichen die Elektronen eine Geschwindigkeit von fast 20 % der Lichtgeschwindigkeit, bei 1 MV 94 %. </p><p>Der <a href="/wiki/Large_Hadron_Collider" title="Large Hadron Collider">Large Hadron Collider</a> führt Protonen eine kinetische Energie von 6,5 TeV zu. Diese Energie ist etwa achttausend Mal größer als die Ruheenergie eines Protons. Bei einer Kollision zwischen entgegengesetzt beschleunigten Protonen können Teilchen mit einer entsprechend hohen Ruheenergie entstehen. </p> <div class="mw-heading mw-heading2"><h2 id="Kinetische_Energie_in_der_Quantenmechanik">Kinetische Energie in der Quantenmechanik</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Kinetische_Energie&veaction=edit&section=8" title="Abschnitt bearbeiten: Kinetische Energie in der Quantenmechanik" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Kinetische_Energie&action=edit&section=8" title="Quellcode des Abschnitts bearbeiten: Kinetische Energie in der Quantenmechanik"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In der <a href="/wiki/Quantenmechanik" title="Quantenmechanik">Quantenmechanik</a> ist der <a href="/wiki/Erwartungswert" title="Erwartungswert">Erwartungswert</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 \langle {\hat {E}}_{\mathrm {kin} }\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>E</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">k</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">n</mi> </mrow> </mrow> </msub> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle {\hat {E}}_{\mathrm {kin} }\rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/764833f865a2d5c6c8f06bd298bad2950930da20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.056ex; height:3.343ex;" alt="{\displaystyle \langle {\hat {E}}_{\mathrm {kin} }\rangle }"></span> der kinetischen Energie eines Teilchens der Masse <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}"></span>, welches durch die <a href="/wiki/Wellenfunktion" title="Wellenfunktion">Wellenfunktion</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \vert \psi \rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">|</mo> <mi>ψ</mi> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \vert \psi \rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8addc466341daafbbf224730defba59505d1e760" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.065ex; height:2.843ex;" alt="{\displaystyle \vert \psi \rangle }"></span> beschrieben wird, gegeben durch </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 {\hat {E}}_{\mathrm {kin} }\rangle ={\frac {1}{2m}}\langle \psi |{\hat {P}}^{2}|\psi \rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>E</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">k</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">n</mi> </mrow> </mrow> </msub> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>m</mi> </mrow> </mfrac> </mrow> <mo fence="false" stretchy="false">⟨</mo> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>P</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>ψ</mi> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle {\hat {E}}_{\mathrm {kin} }\rangle ={\frac {1}{2m}}\langle \psi |{\hat {P}}^{2}|\psi \rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/63fde74d9e2cd277d983ceb2ee7b16a0255c8bd4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:22.189ex; height:5.176ex;" alt="{\displaystyle \langle {\hat {E}}_{\mathrm {kin} }\rangle ={\frac {1}{2m}}\langle \psi |{\hat {P}}^{2}|\psi \rangle }"></span>,</dd></dl> <p>wobei <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {P}}^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>P</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {P}}^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d4da4e507e9ad89c4de66cb0500ec8797ac17937" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.866ex; height:3.343ex;" alt="{\displaystyle {\hat {P}}^{2}}"></span> das Quadrat des <a href="/wiki/Impulsoperator" title="Impulsoperator">Impulsoperators</a> des Teilchens ist. </p><p>Im Formalismus der <a href="/wiki/Dichtefunktionaltheorie_(Quantenphysik)" title="Dichtefunktionaltheorie (Quantenphysik)">Dichtefunktionaltheorie</a> ist nur vorausgesetzt, dass die Elektronendichte bekannt ist, das heißt, dass die Wellenfunktion formal nicht bekannt sein muss. Mit der Elektronendichte <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho (\mathbf {r} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho (\mathbf {r} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/77f477411625125978c0a18946bdfae2c1f13bcb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.113ex; height:2.843ex;" alt="{\displaystyle \rho (\mathbf {r} )}"></span> ist das exakte Funktional der kinetischen Energie für <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/f5e3890c981ae85503089652feb48b191b57aae3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}"></span> Elektronen unbekannt; falls jedoch im Fall <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=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/85982022b9eb1f295b44de55023687a490db0a39" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.325ex; height:2.176ex;" alt="{\displaystyle N=1}"></span> ein einzelnes Elektron betrachtet wird, so kann die kinetische Energie als </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{\mathrm {kin} }[\rho ]=\int {\frac {1}{8}}{\frac {\nabla \rho (\mathbf {r} )\cdot \nabla \rho (\mathbf {r} )}{\rho (\mathbf {r} )}}\mathrm {d} ^{3}r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">k</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">n</mi> </mrow> </mrow> </msub> <mo stretchy="false">[</mo> <mi>ρ</mi> <mo stretchy="false">]</mo> <mo>=</mo> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>8</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∇</mi> <mi>ρ</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">)</mo> <mo>⋅</mo> <mi mathvariant="normal">∇</mi> <mi>ρ</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>ρ</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{\mathrm {kin} }[\rho ]=\int {\frac {1}{8}}{\frac {\nabla \rho (\mathbf {r} )\cdot \nabla \rho (\mathbf {r} )}{\rho (\mathbf {r} )}}\mathrm {d} ^{3}r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dd15443c620a38a60d0be2a3af972d1c14fd8a16" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:32.369ex; height:6.509ex;" alt="{\displaystyle E_{\mathrm {kin} }[\rho ]=\int {\frac {1}{8}}{\frac {\nabla \rho (\mathbf {r} )\cdot \nabla \rho (\mathbf {r} )}{\rho (\mathbf {r} )}}\mathrm {d} ^{3}r}"></span></dd></dl> <p>geschrieben werden, wobei <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{\mathrm {kin} }[\rho ]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">k</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">n</mi> </mrow> </mrow> </msub> <mo stretchy="false">[</mo> <mi>ρ</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{\mathrm {kin} }[\rho ]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3688bd08b48cb85dcc750fd148178aeca0e0034d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.682ex; height:2.843ex;" alt="{\displaystyle E_{\mathrm {kin} }[\rho ]}"></span> das <a href="/w/index.php?title=Weizs%C3%A4cker-Funktional&action=edit&redlink=1" class="new" title="Weizsäcker-Funktional (Seite nicht vorhanden)">Weizsäcker-Funktional</a> der kinetischen Energie ist. </p> <div class="mw-heading mw-heading2"><h2 id="Siehe_auch">Siehe auch</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Kinetische_Energie&veaction=edit&section=9" title="Abschnitt bearbeiten: Siehe auch" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Kinetische_Energie&action=edit&section=9" title="Quellcode des Abschnitts bearbeiten: Siehe auch"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Potentielle_Energie" title="Potentielle Energie">Potentielle Energie</a></li> <li><a href="/wiki/Energieerhaltungssatz" title="Energieerhaltungssatz">Energieerhaltungssatz</a></li> <li><a href="/wiki/Schleppkraft" title="Schleppkraft">Schleppkraft</a> (Kinetische Energie in der Geographie)</li></ul> <div class="mw-heading mw-heading2"><h2 id="Literatur">Literatur</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Kinetische_Energie&veaction=edit&section=10" title="Abschnitt bearbeiten: Literatur" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Kinetische_Energie&action=edit&section=10" title="Quellcode des Abschnitts bearbeiten: Literatur"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Wolfgang_Nolting_(Physiker)" title="Wolfgang Nolting (Physiker)">Wolfgang Nolting</a>: <i>Klassische Mechanik.</i> In: <i>Grundkurs Theoretische Physik.</i> Bd. 1, 8. Auflage. Springer, Berlin 2008, <a href="/wiki/Spezial:ISBN-Suche/9783540348320" class="internal mw-magiclink-isbn">ISBN 978-3-540-34832-0</a>.</li> <li><a href="/wiki/Richard_P._Feynman" class="mw-redirect" title="Richard P. Feynman">Richard P. Feynman</a>: <i>Feynman-Vorlesungen über Physik. Mechanik, Strahlung, Wärme</i> 5., verbesserte Auflage, definitive Edition. Oldenbourg, München / Wien 2007, <a href="/wiki/Spezial:ISBN-Suche/9783486584448" class="internal mw-magiclink-isbn">ISBN 978-3-486-58444-8</a> (= <i>The Feynman Lectures on Physics</i>, Band 1).</li> <li><a href="/wiki/Paul_A._Tipler" title="Paul A. Tipler">Paul A. Tipler</a>: <i>Physik.</i> 3. korrigierter Nachdruck der 1. Auflage. 1994, Spektrum Akademischer Verlag, Heidelberg / Berlin 2000, <a href="/wiki/Spezial:ISBN-Suche/3860251228" class="internal mw-magiclink-isbn">ISBN 3-86025-122-8</a>.</li> <li><a href="/wiki/Ludwig_Bergmann_(Physiker)" title="Ludwig Bergmann (Physiker)">Ludwig Bergmann</a>, <a href="/wiki/Clemens_Schaefer_(Physiker)" title="Clemens Schaefer (Physiker)">Clemens Schaefer</a>: <i>Mechanik – Akustik – Wärme.</i> In: <i>Lehrbuch der Experimentalphysik.</i> Bd. 1, 12. Auflage. Walter de Gruyter, Berlin 2008, <a href="/wiki/Spezial:ISBN-Suche/9783110193114" class="internal mw-magiclink-isbn">ISBN 978-3-11-019311-4</a>.</li> <li><span class="book"><a href="/w/index.php?title=Rainer_M%C3%BCller_(Physiker)&action=edit&redlink=1" class="new" title="Rainer Müller (Physiker) (Seite nicht vorhanden)">Rainer Müller</a>: <cite style="font-style:italic">Klassische Mechanik: Vom Weitsprung zum Marsflug</cite>. De Gruyter, 2015, <a href="/wiki/Spezial:ISBN-Suche/9783110445305" class="internal mw-magiclink-isbn">ISBN 978-3-11-044530-5</a> (<a rel="nofollow" class="external text" href="https://books.google.de/books?id=fqmnCgAAQBAJ&hl=de">google.de</a>).<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Kinetische+Energie&rft.au=Rainer+M%C3%BCller&rft.btitle=Klassische+Mechanik%3A+Vom+Weitsprung+zum+Marsflug&rft.date=2015-09-25&rft.genre=book&rft.isbn=9783110445305&rft.pub=De+Gruyter" style="display:none"> </span></span></li> <li><span class="book"><a href="/wiki/Dieter_Meschede" title="Dieter Meschede">Dieter Meschede</a>: <cite style="font-style:italic">Gerthsen Physik</cite>. Springer-Verlag, 2015, <a href="/wiki/Spezial:ISBN-Suche/9783662459775" class="internal mw-magiclink-isbn">ISBN 978-3-662-45977-5</a> (<a rel="nofollow" class="external text" href="https://books.google.de/books?id=qW7dBgAAQBAJ&hl=de">google.de</a>).<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Kinetische+Energie&rft.au=Dieter+Meschede&rft.btitle=Gerthsen+Physik&rft.date=2015-02-27&rft.genre=book&rft.isbn=9783662459775&rft.pub=Springer-Verlag" style="display:none"> </span></span></li></ul> <div class="mw-heading mw-heading2"><h2 id="Weblinks">Weblinks</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Kinetische_Energie&veaction=edit&section=11" title="Abschnitt bearbeiten: Weblinks" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Kinetische_Energie&action=edit&section=11" title="Quellcode des Abschnitts bearbeiten: Weblinks"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a rel="nofollow" class="external text" href="https://portal.dnb.de/opac.htm?method=simpleSearch&query=4163880-3">Literatur von und über Kinetische Energie</a> im Katalog der <a href="/wiki/Deutsche_Nationalbibliothek" title="Deutsche Nationalbibliothek">Deutschen Nationalbibliothek</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Einzelnachweise">Einzelnachweise</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Kinetische_Energie&veaction=edit&section=12" title="Abschnitt bearbeiten: Einzelnachweise" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Kinetische_Energie&action=edit&section=12" title="Quellcode des Abschnitts bearbeiten: Einzelnachweise"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><a href="#cite_ref-1">↑</a></span> <span class="reference-text">vergleiche 1,602·10<sup>−19</sup> J = 1 <a href="/wiki/Elektronenvolt" title="Elektronenvolt">eV</a> = 1,602·10<sup>−19</sup> <a href="/wiki/C" title="C">C</a> · <a href="/wiki/Volt" title="Volt">V</a> = 1,602·10<sup>−19</sup> <a href="/wiki/Ampere" title="Ampere">A</a>·<a href="/wiki/Sekunde" title="Sekunde">s</a>·<a href="/wiki/Volt" title="Volt">V</a> = 1,602·10<sup>−19</sup> <a href="/wiki/Watt_(Einheit)" title="Watt (Einheit)">W</a>·s = 3,827·10<sup>−23</sup> Kilokalorien <a href="/wiki/Kcal" class="mw-redirect" title="Kcal">kcal</a> (<a href="/wiki/Liste_von_Gr%C3%B6%C3%9Fenordnungen_der_Energie" title="Liste von Größenordnungen der Energie">Liste von Größenordnungen der Energie</a>).</span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><a href="#cite_ref-2">↑</a></span> <span class="reference-text"><a href="/wiki/Istv%C3%A1n_Szab%C3%B3" title="István Szabó">István Szabó</a>: <i>Geschichte der mechanischen Prinzipien.</i> Birkhäuser, S. 71.</span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><a href="#cite_ref-3">↑</a></span> <span class="reference-text"><a href="/wiki/Max_Jammer" title="Max Jammer">Max Jammer</a>: Artikel <i>Energie.</i> In: Donald Borchert (Hrsg.): <i>Encyclopedia of Philosophy.</i> Thomson Gale, 2006.</span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><a href="#cite_ref-4">↑</a></span> <span class="reference-text"><a href="/wiki/Paul_Diepgen" title="Paul Diepgen">Paul Diepgen</a>, <a href="/wiki/Heinz_Goerke" title="Heinz Goerke">Heinz Goerke</a>: <i><a href="/wiki/Ludwig_Aschoff" title="Ludwig Aschoff">Aschoff</a>/Diepgen/Goerke: Kurze Übersichtstabelle zur Geschichte der Medizin.</i> 7., neubearbeitete Auflage. Springer, Berlin/Göttingen/Heidelberg 1960, S. 40.</span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><a href="#cite_ref-5">↑</a></span> <span class="reference-text">A. P. French: <i>Die spezielle Relativitätstheorie – M.I.T. Einführungskurs Physik</i> 1968, S. 19–23.</span> </li> </ol> <div class="hintergrundfarbe1 rahmenfarbe1 navigation-not-searchable normdaten-typ-s" style="border-style: solid; border-width: 1px; clear: left; margin-bottom:1em; margin-top:1em; padding: 0.25em; overflow: hidden; word-break: break-word; word-wrap: break-word;" id="normdaten"> <div style="display: table-cell; vertical-align: middle; width: 100%;"> <div> Normdaten (Sachbegriff): <a href="/wiki/Gemeinsame_Normdatei" title="Gemeinsame Normdatei">GND</a>: <span class="plainlinks-print"><a rel="nofollow" class="external text" href="https://d-nb.info/gnd/4163880-3">4163880-3</a></span> <span class="noprint">(<a rel="nofollow" class="external text" href="https://lobid.org/gnd/4163880-3">lobid</a>, <a rel="nofollow" class="external text" href="https://swb.bsz-bw.de/DB=2.104/SET=1/TTL=1/CMD?retrace=0&trm_old=&ACT=SRCHA&IKT=2999&SRT=RLV&TRM=4163880-3">OGND</a><span class="metadata">, <a rel="nofollow" class="external text" href="https://prometheus.lmu.de/gnd/4163880-3">AKS</a></span>)</span> <span class="metadata"></span></div> </div></div></div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Abgerufen von „<a dir="ltr" href="https://de.wikipedia.org/w/index.php?title=Kinetische_Energie&oldid=248429972">https://de.wikipedia.org/w/index.php?title=Kinetische_Energie&oldid=248429972</a>“</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Wikipedia:Kategorien" title="Wikipedia:Kategorien">Kategorien</a>: <ul><li><a href="/wiki/Kategorie:Klassische_Mechanik" title="Kategorie:Klassische Mechanik">Klassische Mechanik</a></li><li><a href="/wiki/Kategorie:Spezielle_Relativit%C3%A4tstheorie" title="Kategorie:Spezielle Relativitätstheorie">Spezielle Relativitätstheorie</a></li><li><a href="/wiki/Kategorie:Energieform" title="Kategorie:Energieform">Energieform</a></li></ul></div></div> </div> </div> <div id="mw-navigation"> <h2>Navigationsmenü</h2> <div id="mw-head"> <nav id="p-personal" class="mw-portlet mw-portlet-personal vector-user-menu-legacy vector-menu" aria-labelledby="p-personal-label" > <h3 id="p-personal-label" class="vector-menu-heading " > <span class="vector-menu-heading-label">Meine Werkzeuge</span> </h3> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-anonuserpage" class="mw-list-item"><span title="Benutzerseite der IP-Adresse, von der aus du Änderungen durchführst">Nicht angemeldet</span></li><li id="pt-anontalk" class="mw-list-item"><a href="/wiki/Spezial:Meine_Diskussionsseite" title="Diskussion über Änderungen von dieser IP-Adresse [n]" accesskey="n"><span>Diskussionsseite</span></a></li><li id="pt-anoncontribs" class="mw-list-item"><a href="/wiki/Spezial:Meine_Beitr%C3%A4ge" title="Eine Liste der Bearbeitungen, die von dieser IP-Adresse gemacht wurden [y]" accesskey="y"><span>Beiträge</span></a></li><li id="pt-createaccount" class="mw-list-item"><a href="/w/index.php?title=Spezial:Benutzerkonto_anlegen&returnto=Kinetische+Energie" title="Wir ermutigen dich dazu, ein Benutzerkonto zu erstellen und dich anzumelden. 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mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%B7%D8%A7%D9%82%D8%A9_%D8%AD%D8%B1%D9%83%D9%8A%D8%A9" title="طاقة حركية – Arabisch" lang="ar" hreflang="ar" data-title="طاقة حركية" data-language-autonym="العربية" data-language-local-name="Arabisch" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-as mw-list-item"><a href="https://as.wikipedia.org/wiki/%E0%A6%97%E0%A6%A4%E0%A6%BF_%E0%A6%B6%E0%A6%95%E0%A7%8D%E0%A6%A4%E0%A6%BF" title="গতি শক্তি – Assamesisch" lang="as" hreflang="as" data-title="গতি শক্তি" data-language-autonym="অসমীয়া" data-language-local-name="Assamesisch" class="interlanguage-link-target"><span>অসমীয়া</span></a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/Enerx%C3%ADa_cin%C3%A9tico" title="Enerxía cinético – Asturisch" lang="ast" hreflang="ast" data-title="Enerxía cinético" data-language-autonym="Asturianu" data-language-local-name="Asturisch" class="interlanguage-link-target"><span>Asturianu</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/Kinetik_enerji" title="Kinetik enerji – Aserbaidschanisch" lang="az" hreflang="az" data-title="Kinetik enerji" data-language-autonym="Azərbaycanca" data-language-local-name="Aserbaidschanisch" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-bcl mw-list-item"><a href="https://bcl.wikipedia.org/wiki/Enerhiyang_kinetika" title="Enerhiyang kinetika – Zentralbikolano" lang="bcl" hreflang="bcl" data-title="Enerhiyang kinetika" data-language-autonym="Bikol Central" data-language-local-name="Zentralbikolano" 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%9A%D1%96%D0%BD%D0%B5%D1%82%D1%8B%D1%87%D0%BD%D0%B0%D1%8F_%D1%8D%D0%BD%D0%B5%D1%80%D0%B3%D1%96%D1%8F" title="Кінетычная энергія – Belarussisch" lang="be" hreflang="be" data-title="Кінетычная энергія" data-language-autonym="Беларуская" data-language-local-name="Belarussisch" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-be-x-old mw-list-item"><a href="https://be-tarask.wikipedia.org/wiki/%D0%9A%D1%96%D0%BD%D1%8D%D1%82%D1%8B%D1%87%D0%BD%D0%B0%D1%8F_%D1%8D%D0%BD%D1%8D%D1%80%D0%B3%D1%96%D1%8F" title="Кінэтычная энэргія – Weißrussisch (Taraschkewiza)" lang="be-tarask" hreflang="be-tarask" data-title="Кінэтычная энэргія" data-language-autonym="Беларуская (тарашкевіца)" data-language-local-name="Weißrussisch (Taraschkewiza)" 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%9A%D0%B8%D0%BD%D0%B5%D1%82%D0%B8%D1%87%D0%BD%D0%B0_%D0%B5%D0%BD%D0%B5%D1%80%D0%B3%D0%B8%D1%8F" title="Кинетична енергия – Bulgarisch" lang="bg" hreflang="bg" data-title="Кинетична енергия" data-language-autonym="Български" data-language-local-name="Bulgarisch" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%97%E0%A6%A4%E0%A6%BF%E0%A6%B6%E0%A6%95%E0%A7%8D%E0%A6%A4%E0%A6%BF" title="গতিশক্তি – Bengalisch" lang="bn" hreflang="bn" data-title="গতিশক্তি" data-language-autonym="বাংলা" data-language-local-name="Bengalisch" 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/Kineti%C4%8Dka_energija" title="Kinetička energija – Bosnisch" lang="bs" hreflang="bs" data-title="Kinetička energija" data-language-autonym="Bosanski" data-language-local-name="Bosnisch" class="interlanguage-link-target"><span>Bosanski</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Energia_cin%C3%A8tica" title="Energia cinètica – Katalanisch" lang="ca" hreflang="ca" data-title="Energia cinètica" data-language-autonym="Català" data-language-local-name="Katalanisch" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Kinetick%C3%A1_energie" title="Kinetická energie – Tschechisch" lang="cs" hreflang="cs" data-title="Kinetická energie" data-language-autonym="Čeština" data-language-local-name="Tschechisch" 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%9A%D0%B8%D0%BD%D0%B5%D1%82%D0%B8%D0%BA%C4%83%D0%BB%D0%BB%D0%B0_%D1%8D%D0%BD%D0%B5%D1%80%D0%B3%D0%B8" title="Кинетикăлла энерги – Tschuwaschisch" lang="cv" hreflang="cv" data-title="Кинетикăлла энерги" data-language-autonym="Чӑвашла" data-language-local-name="Tschuwaschisch" 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/Egni_cinetig" title="Egni cinetig – Walisisch" lang="cy" hreflang="cy" data-title="Egni cinetig" data-language-autonym="Cymraeg" data-language-local-name="Walisisch" class="interlanguage-link-target"><span>Cymraeg</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Kinetisk_energi" title="Kinetisk energi – Dänisch" lang="da" hreflang="da" data-title="Kinetisk energi" data-language-autonym="Dansk" data-language-local-name="Dänisch" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%9A%CE%B9%CE%BD%CE%B7%CF%84%CE%B9%CE%BA%CE%AE_%CE%B5%CE%BD%CE%AD%CF%81%CE%B3%CE%B5%CE%B9%CE%B1" title="Κινητική ενέργεια – Griechisch" lang="el" hreflang="el" data-title="Κινητική ενέργεια" data-language-autonym="Ελληνικά" data-language-local-name="Griechisch" 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/Kinetic_energy" title="Kinetic energy – Englisch" lang="en" hreflang="en" data-title="Kinetic energy" data-language-autonym="English" data-language-local-name="Englisch" 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/Kineta_energio" title="Kineta energio – Esperanto" lang="eo" hreflang="eo" data-title="Kineta energio" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Energ%C3%ADa_cin%C3%A9tica" title="Energía cinética – Spanisch" lang="es" hreflang="es" data-title="Energía cinética" data-language-autonym="Español" data-language-local-name="Spanisch" 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/Kineetiline_energia" title="Kineetiline energia – Estnisch" lang="et" hreflang="et" data-title="Kineetiline energia" data-language-autonym="Eesti" data-language-local-name="Estnisch" 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/Energia_zinetiko" title="Energia zinetiko – Baskisch" lang="eu" hreflang="eu" data-title="Energia zinetiko" data-language-autonym="Euskara" data-language-local-name="Baskisch" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%A7%D9%86%D8%B1%DA%98%DB%8C_%D8%AC%D9%86%D8%A8%D8%B4%DB%8C" title="انرژی جنبشی – Persisch" lang="fa" hreflang="fa" data-title="انرژی جنبشی" data-language-autonym="فارسی" data-language-local-name="Persisch" 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/Liike-energia" title="Liike-energia – Finnisch" lang="fi" hreflang="fi" data-title="Liike-energia" data-language-autonym="Suomi" data-language-local-name="Finnisch" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/%C3%89nergie_cin%C3%A9tique" title="Énergie cinétique – Französisch" lang="fr" hreflang="fr" data-title="Énergie cinétique" data-language-autonym="Français" data-language-local-name="Französisch" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-fy mw-list-item"><a href="https://fy.wikipedia.org/wiki/Kinetyske_energy" title="Kinetyske energy – Westfriesisch" lang="fy" hreflang="fy" data-title="Kinetyske energy" data-language-autonym="Frysk" data-language-local-name="Westfriesisch" class="interlanguage-link-target"><span>Frysk</span></a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/Fuinneamh_cin%C3%A9iteach" title="Fuinneamh cinéiteach – Irisch" lang="ga" hreflang="ga" data-title="Fuinneamh cinéiteach" data-language-autonym="Gaeilge" data-language-local-name="Irisch" class="interlanguage-link-target"><span>Gaeilge</span></a></li><li class="interlanguage-link interwiki-gcr mw-list-item"><a href="https://gcr.wikipedia.org/wiki/%C3%89nerji_sin%C3%A9tik" title="Énerji sinétik – Französisch-Guayana Kreolisch" lang="gcr" hreflang="gcr" data-title="Énerji sinétik" data-language-autonym="Kriyòl gwiyannen" data-language-local-name="Französisch-Guayana Kreolisch" class="interlanguage-link-target"><span>Kriyòl gwiyannen</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Enerx%C3%ADa_cin%C3%A9tica" title="Enerxía cinética – Galicisch" lang="gl" hreflang="gl" data-title="Enerxía cinética" data-language-autonym="Galego" data-language-local-name="Galicisch" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%90%D7%A0%D7%A8%D7%92%D7%99%D7%94_%D7%A7%D7%99%D7%A0%D7%98%D7%99%D7%AA" title="אנרגיה קינטית – Hebräisch" lang="he" hreflang="he" data-title="אנרגיה קינטית" data-language-autonym="עברית" data-language-local-name="Hebräisch" 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%97%E0%A4%A4%E0%A4%BF%E0%A4%9C_%E0%A4%8A%E0%A4%B0%E0%A5%8D%E0%A4%9C%E0%A4%BE" title="गतिज ऊर्जा – Hindi" lang="hi" hreflang="hi" data-title="गतिज ऊर्जा" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Kineti%C4%8Dka_energija" title="Kinetička energija – Kroatisch" lang="hr" hreflang="hr" data-title="Kinetička energija" data-language-autonym="Hrvatski" data-language-local-name="Kroatisch" 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/En%C3%A8ji_sinetik" title="Enèji sinetik – Haiti-Kreolisch" lang="ht" hreflang="ht" data-title="Enèji sinetik" data-language-autonym="Kreyòl ayisyen" data-language-local-name="Haiti-Kreolisch" 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/Mozg%C3%A1si_energia" title="Mozgási energia – Ungarisch" lang="hu" hreflang="hu" data-title="Mozgási energia" data-language-autonym="Magyar" data-language-local-name="Ungarisch" 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%BF%D5%AB%D5%B6%D5%A5%D5%BF%D5%AB%D5%AF_%D5%A7%D5%B6%D5%A5%D6%80%D5%A3%D5%AB%D5%A1" title="Կինետիկ էներգիա – Armenisch" lang="hy" hreflang="hy" data-title="Կինետիկ էներգիա" data-language-autonym="Հայերեն" data-language-local-name="Armenisch" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Energi_kinetik" title="Energi kinetik – Indonesisch" lang="id" hreflang="id" data-title="Energi kinetik" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesisch" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-io mw-list-item"><a href="https://io.wikipedia.org/wiki/Cinetik_energio" title="Cinetik energio – Ido" lang="io" hreflang="io" data-title="Cinetik energio" data-language-autonym="Ido" data-language-local-name="Ido" class="interlanguage-link-target"><span>Ido</span></a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/Hreyfiorka" title="Hreyfiorka – Isländisch" lang="is" hreflang="is" data-title="Hreyfiorka" data-language-autonym="Íslenska" data-language-local-name="Isländisch" class="interlanguage-link-target"><span>Íslenska</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Energia_cinetica" title="Energia cinetica – Italienisch" lang="it" hreflang="it" data-title="Energia cinetica" data-language-autonym="Italiano" data-language-local-name="Italienisch" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E9%81%8B%E5%8B%95%E3%82%A8%E3%83%8D%E3%83%AB%E3%82%AE%E3%83%BC" title="運動エネルギー – Japanisch" lang="ja" hreflang="ja" data-title="運動エネルギー" data-language-autonym="日本語" data-language-local-name="Japanisch" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-jam mw-list-item"><a href="https://jam.wikipedia.org/wiki/Kinetik_enaji" title="Kinetik enaji – Jamaikanisch-Kreolisch" lang="jam" hreflang="jam" data-title="Kinetik enaji" data-language-autonym="Patois" data-language-local-name="Jamaikanisch-Kreolisch" class="interlanguage-link-target"><span>Patois</span></a></li><li class="interlanguage-link interwiki-jv mw-list-item"><a href="https://jv.wikipedia.org/wiki/Energi_kinetis" title="Energi kinetis – Javanisch" lang="jv" hreflang="jv" data-title="Energi kinetis" data-language-autonym="Jawa" data-language-local-name="Javanisch" 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%99%E1%83%98%E1%83%9C%E1%83%94%E1%83%A2%E1%83%98%E1%83%99%E1%83%A3%E1%83%A0%E1%83%98_%E1%83%94%E1%83%9C%E1%83%94%E1%83%A0%E1%83%92%E1%83%98%E1%83%90" title="კინეტიკური ენერგია – Georgisch" lang="ka" hreflang="ka" data-title="კინეტიკური ენერგია" data-language-autonym="ქართული" data-language-local-name="Georgisch" class="interlanguage-link-target"><span>ქართული</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%9A%D0%B8%D0%BD%D0%B5%D1%82%D0%B8%D0%BA%D0%B0%D0%BB%D1%8B%D2%9B_%D1%8D%D0%BD%D0%B5%D1%80%D0%B3%D0%B8%D1%8F" title="Кинетикалық энергия – Kasachisch" lang="kk" hreflang="kk" data-title="Кинетикалық энергия" data-language-autonym="Қазақша" data-language-local-name="Kasachisch" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-kn mw-list-item"><a href="https://kn.wikipedia.org/wiki/%E0%B2%9A%E0%B2%B2%E0%B2%A8%E0%B2%B6%E0%B2%95%E0%B3%8D%E0%B2%A4%E0%B2%BF" title="ಚಲನಶಕ್ತಿ – Kannada" lang="kn" hreflang="kn" data-title="ಚಲನಶಕ್ತಿ" data-language-autonym="ಕನ್ನಡ" data-language-local-name="Kannada" class="interlanguage-link-target"><span>ಕನ್ನಡ</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%9A%B4%EB%8F%99_%EC%97%90%EB%84%88%EC%A7%80" title="운동 에너지 – Koreanisch" lang="ko" hreflang="ko" data-title="운동 에너지" data-language-autonym="한국어" data-language-local-name="Koreanisch" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Energia_cinetica" title="Energia cinetica – Latein" lang="la" hreflang="la" data-title="Energia cinetica" data-language-autonym="Latina" data-language-local-name="Latein" 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/Enerjia_cinetica" title="Enerjia cinetica – Lingua Franca Nova" lang="lfn" hreflang="lfn" data-title="Enerjia cinetica" data-language-autonym="Lingua Franca Nova" data-language-local-name="Lingua Franca Nova" class="interlanguage-link-target"><span>Lingua Franca Nova</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Kinetin%C4%97_energija" title="Kinetinė energija – Litauisch" lang="lt" hreflang="lt" data-title="Kinetinė energija" data-language-autonym="Lietuvių" data-language-local-name="Litauisch" 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/Kin%C4%93tisk%C4%81_ener%C4%A3ija" title="Kinētiskā enerģija – Lettisch" lang="lv" hreflang="lv" data-title="Kinētiskā enerģija" data-language-autonym="Latviešu" data-language-local-name="Lettisch" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%9A%D0%B8%D0%BD%D0%B5%D1%82%D0%B8%D1%87%D0%BA%D0%B0_%D0%B5%D0%BD%D0%B5%D1%80%D0%B3%D0%B8%D1%98%D0%B0" title="Кинетичка енергија – Mazedonisch" lang="mk" hreflang="mk" data-title="Кинетичка енергија" data-language-autonym="Македонски" data-language-local-name="Mazedonisch" 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%97%E0%B4%A4%E0%B4%BF%E0%B4%95%E0%B5%8B%E0%B5%BC%E0%B4%9C%E0%B5%8D%E0%B4%9C%E0%B4%82" title="ഗതികോർജ്ജം – Malayalam" lang="ml" hreflang="ml" data-title="ഗതികോർജ്ജം" data-language-autonym="മലയാളം" data-language-local-name="Malayalam" class="interlanguage-link-target"><span>മലയാളം</span></a></li><li class="interlanguage-link interwiki-mr mw-list-item"><a href="https://mr.wikipedia.org/wiki/%E0%A4%97%E0%A4%A4%E0%A4%BF%E0%A4%9C_%E0%A4%8A%E0%A4%B0%E0%A5%8D%E0%A4%9C%E0%A4%BE" title="गतिज ऊर्जा – Marathi" lang="mr" hreflang="mr" data-title="गतिज ऊर्जा" data-language-autonym="मराठी" data-language-local-name="Marathi" class="interlanguage-link-target"><span>मराठी</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Tenaga_kinetik" title="Tenaga kinetik – Malaiisch" lang="ms" hreflang="ms" data-title="Tenaga kinetik" data-language-autonym="Bahasa Melayu" data-language-local-name="Malaiisch" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-my mw-list-item"><a href="https://my.wikipedia.org/wiki/%E1%80%A1%E1%80%9B%E1%80%BD%E1%80%B1%E1%80%B7%E1%80%85%E1%80%BD%E1%80%99%E1%80%BA%E1%80%B8%E1%80%A1%E1%80%84%E1%80%BA" title="အရွေ့စွမ်းအင် – Birmanisch" lang="my" hreflang="my" data-title="အရွေ့စွမ်းအင်" data-language-autonym="မြန်မာဘာသာ" data-language-local-name="Birmanisch" class="interlanguage-link-target"><span>မြန်မာဘာသာ</span></a></li><li class="interlanguage-link interwiki-ne mw-list-item"><a href="https://ne.wikipedia.org/wiki/%E0%A4%9A%E0%A4%BE%E0%A4%B2_%E0%A4%B6%E0%A4%95%E0%A5%8D%E0%A4%A4%E0%A4%BF" title="चाल शक्ति – Nepalesisch" lang="ne" hreflang="ne" data-title="चाल शक्ति" data-language-autonym="नेपाली" data-language-local-name="Nepalesisch" 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/Kinetische_energie" title="Kinetische energie – Niederländisch" lang="nl" hreflang="nl" data-title="Kinetische energie" data-language-autonym="Nederlands" data-language-local-name="Niederländisch" 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/Kinetisk_energi" title="Kinetisk energi – Norwegisch (Nynorsk)" lang="nn" hreflang="nn" data-title="Kinetisk energi" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegisch (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/Kinetisk_energi" title="Kinetisk energi – Norwegisch (Bokmål)" lang="nb" hreflang="nb" data-title="Kinetisk energi" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegisch (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/Energia_cinetica" title="Energia cinetica – Okzitanisch" lang="oc" hreflang="oc" data-title="Energia cinetica" data-language-autonym="Occitan" data-language-local-name="Okzitanisch" class="interlanguage-link-target"><span>Occitan</span></a></li><li class="interlanguage-link interwiki-om mw-list-item"><a href="https://om.wikipedia.org/wiki/Anniisaa_sochii" title="Anniisaa sochii – Oromo" lang="om" hreflang="om" data-title="Anniisaa sochii" data-language-autonym="Oromoo" data-language-local-name="Oromo" class="interlanguage-link-target"><span>Oromoo</span></a></li><li class="interlanguage-link interwiki-pa mw-list-item"><a href="https://pa.wikipedia.org/wiki/%E0%A8%97%E0%A8%A4%E0%A8%BF%E0%A8%9C_%E0%A8%8A%E0%A8%B0%E0%A8%9C%E0%A8%BE" title="ਗਤਿਜ ਊਰਜਾ – Punjabi" lang="pa" hreflang="pa" data-title="ਗਤਿਜ ਊਰਜਾ" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="Punjabi" class="interlanguage-link-target"><span>ਪੰਜਾਬੀ</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Energia_kinetyczna" title="Energia kinetyczna – Polnisch" lang="pl" hreflang="pl" data-title="Energia kinetyczna" data-language-autonym="Polski" data-language-local-name="Polnisch" 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/Energ%C3%ACa_cin%C3%A9tica" title="Energìa cinética – Piemontesisch" lang="pms" hreflang="pms" data-title="Energìa cinética" data-language-autonym="Piemontèis" data-language-local-name="Piemontesisch" 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/%DA%A9%D8%A7%D8%A6%DB%8C%D9%86%DB%8C%D9%B9%DA%A9_%D8%AA%D9%88%D8%A7%D9%86%D8%A7%D8%A6%DB%8C" title="کائینیٹک توانائی – Westliches Panjabi" lang="pnb" hreflang="pnb" data-title="کائینیٹک توانائی" data-language-autonym="پنجابی" data-language-local-name="Westliches Panjabi" 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/Energia_cin%C3%A9tica" title="Energia cinética – Portugiesisch" lang="pt" hreflang="pt" data-title="Energia cinética" data-language-autonym="Português" data-language-local-name="Portugiesisch" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Energie_cinetic%C4%83" title="Energie cinetică – Rumänisch" lang="ro" hreflang="ro" data-title="Energie cinetică" data-language-autonym="Română" data-language-local-name="Rumänisch" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9A%D0%B8%D0%BD%D0%B5%D1%82%D0%B8%D1%87%D0%B5%D1%81%D0%BA%D0%B0%D1%8F_%D1%8D%D0%BD%D0%B5%D1%80%D0%B3%D0%B8%D1%8F" title="Кинетическая энергия – Russisch" lang="ru" hreflang="ru" data-title="Кинетическая энергия" data-language-autonym="Русский" data-language-local-name="Russisch" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sco mw-list-item"><a href="https://sco.wikipedia.org/wiki/Kinetic_energy" title="Kinetic energy – Schottisch" lang="sco" hreflang="sco" data-title="Kinetic energy" data-language-autonym="Scots" data-language-local-name="Schottisch" class="interlanguage-link-target"><span>Scots</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Kineti%C4%8Dka_energija" title="Kinetička energija – Serbokroatisch" lang="sh" hreflang="sh" data-title="Kinetička energija" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="Serbokroatisch" 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%A0%E0%B7%8F%E0%B6%BD%E0%B6%9A_%E0%B7%81%E0%B6%9A%E0%B7%8A%E0%B6%AD%E0%B7%92%E0%B6%BA" title="චාලක ශක්තිය – Singhalesisch" lang="si" hreflang="si" data-title="චාලක ශක්තිය" data-language-autonym="සිංහල" data-language-local-name="Singhalesisch" class="interlanguage-link-target"><span>සිංහල</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Kinetic_energy" title="Kinetic energy – einfaches Englisch" lang="en-simple" hreflang="en-simple" data-title="Kinetic energy" data-language-autonym="Simple English" data-language-local-name="einfaches Englisch" 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/Kinetick%C3%A1_energia" title="Kinetická energia – Slowakisch" lang="sk" hreflang="sk" data-title="Kinetická energia" data-language-autonym="Slovenčina" data-language-local-name="Slowakisch" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Kineti%C4%8Dna_energija" title="Kinetična energija – Slowenisch" lang="sl" hreflang="sl" data-title="Kinetična energija" data-language-autonym="Slovenščina" data-language-local-name="Slowenisch" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sn mw-list-item"><a href="https://sn.wikipedia.org/wiki/Simba_reNhekairo" title="Simba reNhekairo – Shona" lang="sn" hreflang="sn" data-title="Simba reNhekairo" data-language-autonym="ChiShona" data-language-local-name="Shona" class="interlanguage-link-target"><span>ChiShona</span></a></li><li class="interlanguage-link interwiki-so mw-list-item"><a href="https://so.wikipedia.org/wiki/Awood_Socota" title="Awood Socota – Somali" lang="so" hreflang="so" data-title="Awood Socota" data-language-autonym="Soomaaliga" data-language-local-name="Somali" class="interlanguage-link-target"><span>Soomaaliga</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Energjia_kinetike" title="Energjia kinetike – Albanisch" lang="sq" hreflang="sq" data-title="Energjia kinetike" data-language-autonym="Shqip" data-language-local-name="Albanisch" 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%9A%D0%B8%D0%BD%D0%B5%D1%82%D0%B8%D1%87%D0%BA%D0%B0_%D0%B5%D0%BD%D0%B5%D1%80%D0%B3%D0%B8%D1%98%D0%B0" title="Кинетичка енергија – Serbisch" lang="sr" hreflang="sr" data-title="Кинетичка енергија" data-language-autonym="Српски / srpski" data-language-local-name="Serbisch" 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/Kinetisk_energi" title="Kinetisk energi – Schwedisch" lang="sv" hreflang="sv" data-title="Kinetisk energi" data-language-autonym="Svenska" data-language-local-name="Schwedisch" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%87%E0%AE%AF%E0%AE%95%E0%AF%8D%E0%AE%95_%E0%AE%86%E0%AE%B1%E0%AF%8D%E0%AE%B1%E0%AE%B2%E0%AF%8D" title="இயக்க ஆற்றல் – Tamil" lang="ta" hreflang="ta" data-title="இயக்க ஆற்றல்" data-language-autonym="தமிழ்" data-language-local-name="Tamil" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-te mw-list-item"><a href="https://te.wikipedia.org/wiki/%E0%B0%97%E0%B0%A4%E0%B0%BF_%E0%B0%B6%E0%B0%95%E0%B1%8D%E0%B0%A4%E0%B0%BF" title="గతి శక్తి – Telugu" lang="te" hreflang="te" data-title="గతి శక్తి" data-language-autonym="తెలుగు" data-language-local-name="Telugu" class="interlanguage-link-target"><span>తెలుగు</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%9E%E0%B8%A5%E0%B8%B1%E0%B8%87%E0%B8%87%E0%B8%B2%E0%B8%99%E0%B8%88%E0%B8%A5%E0%B8%99%E0%B9%8C" title="พลังงานจลน์ – Thailändisch" lang="th" hreflang="th" data-title="พลังงานจลน์" data-language-autonym="ไทย" data-language-local-name="Thailändisch" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Kinetik_enerji" title="Kinetik enerji – Türkisch" lang="tr" hreflang="tr" data-title="Kinetik enerji" data-language-autonym="Türkçe" data-language-local-name="Türkisch" 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/Kinetik_energi%C3%A4" title="Kinetik energiä – Tatarisch" lang="tt" hreflang="tt" data-title="Kinetik energiä" data-language-autonym="Татарча / tatarça" data-language-local-name="Tatarisch" 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%9A%D1%96%D0%BD%D0%B5%D1%82%D0%B8%D1%87%D0%BD%D0%B0_%D0%B5%D0%BD%D0%B5%D1%80%D0%B3%D1%96%D1%8F" title="Кінетична енергія – Ukrainisch" lang="uk" hreflang="uk" data-title="Кінетична енергія" data-language-autonym="Українська" data-language-local-name="Ukrainisch" 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/%D8%AD%D8%B1%DA%A9%DB%8C_%D8%AA%D9%88%D8%A7%D9%86%D8%A7%D8%A6%DB%8C" title="حرکی توانائی – Urdu" lang="ur" hreflang="ur" data-title="حرکی توانائی" 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/Kinetik_energiya" title="Kinetik energiya – Usbekisch" lang="uz" hreflang="uz" data-title="Kinetik energiya" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="Usbekisch" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/%C4%90%E1%BB%99ng_n%C4%83ng" title="Động năng – Vietnamesisch" lang="vi" hreflang="vi" data-title="Động năng" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamesisch" 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/Enerhiya_kinetic" title="Enerhiya kinetic – Waray" lang="war" hreflang="war" data-title="Enerhiya kinetic" data-language-autonym="Winaray" data-language-local-name="Waray" class="interlanguage-link-target"><span>Winaray</span></a></li><li class="interlanguage-link interwiki-wo mw-list-item"><a href="https://wo.wikipedia.org/wiki/K%C3%A0ttanug_imbiku" title="Kàttanug imbiku – Wolof" lang="wo" hreflang="wo" data-title="Kàttanug imbiku" data-language-autonym="Wolof" data-language-local-name="Wolof" class="interlanguage-link-target"><span>Wolof</span></a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E5%8A%A8%E8%83%BD" title="动能 – Wu" lang="wuu" hreflang="wuu" data-title="动能" data-language-autonym="吴语" data-language-local-name="Wu" class="interlanguage-link-target"><span>吴语</span></a></li><li class="interlanguage-link interwiki-yi mw-list-item"><a href="https://yi.wikipedia.org/wiki/%D7%A7%D7%99%D7%A0%D7%A2%D7%98%D7%99%D7%A9%D7%A2_%D7%A2%D7%A0%D7%A2%D7%A8%D7%92%D7%99%D7%A2" title="קינעטישע ענערגיע – Jiddisch" lang="yi" hreflang="yi" data-title="קינעטישע ענערגיע" data-language-autonym="ייִדיש" data-language-local-name="Jiddisch" 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/%E5%8A%A8%E8%83%BD" title="动能 – Chinesisch" lang="zh" hreflang="zh" data-title="动能" data-language-autonym="中文" data-language-local-name="Chinesisch" 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/%E5%8B%95%E8%83%BD" title="動能 – Klassisches Chinesisch" lang="lzh" hreflang="lzh" data-title="動能" data-language-autonym="文言" data-language-local-name="Klassisches Chinesisch" class="interlanguage-link-target"><span>文言</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E5%8B%95%E8%83%BD" title="動能 – Kantonesisch" lang="yue" hreflang="yue" data-title="動能" data-language-autonym="粵語" data-language-local-name="Kantonesisch" class="interlanguage-link-target"><span>粵語</span></a></li><li class="interlanguage-link interwiki-zu mw-list-item"><a href="https://zu.wikipedia.org/wiki/Impekumpeku" title="Impekumpeku – Zulu" lang="zu" hreflang="zu" data-title="Impekumpeku" data-language-autonym="IsiZulu" data-language-local-name="Zulu" class="interlanguage-link-target"><span>IsiZulu</span></a></li> </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/Q46276#sitelinks-wikipedia" title="Links auf Artikel in anderen Sprachen bearbeiten" class="wbc-editpage">Links bearbeiten</a></span></div> </div> 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Kinetische Energie – Wikipedia