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## 经典力学

${\displaystyle E_{k}={\frac {1}{2}}mv^{2}}$

${\displaystyle E_{k}={\frac {p^{2}}{2m}}}$

### 推导与定义

${\displaystyle W=\int {\vec {F}}\cdot d{\vec {s}}}$

${\displaystyle {\vec {F}}={\frac {d{\vec {p}}}{dt}}}$

${\displaystyle {\vec {p}}=m{\vec {v}}}$

${\displaystyle W=\int {\frac {d{\vec {p}}}{dt}}\cdot d{\vec {s}}=\int m{\frac {d{\vec {v}}}{dt}}\cdot d{\vec {s}}=\int m{\vec {v}}\cdot d{\vec {v}}={\frac {1}{2}}\int md({\vec {v}}\cdot {\vec {v}})={\frac {1}{2}}mv^{2}+C_{0}}$

${\displaystyle E_{k}={\frac {1}{2}}mv^{2}}$

### 自转的物体

${\displaystyle E_{r}={\frac {1}{2}}\int v^{2}dm={\frac {1}{2}}\int r^{2}\omega ^{2}dm={\frac {1}{2}}\omega ^{2}\int r^{2}dm={\frac {1}{2}}I\omega ^{2}}$

## 相对论

${\displaystyle E_{\text{k}}=\int \mathbf {v} \cdot d\mathbf {p} =\int \mathbf {v} \cdot d(m\gamma \mathbf {v} )=m\gamma \mathbf {v} \cdot \mathbf {v} -\int m\gamma \mathbf {v} \cdot d\mathbf {v} =m\gamma v^{2}-{\frac {m}{2}}\int \gamma d(v^{2})}$

{\displaystyle {\begin{aligned}E_{\text{k}}&=m\gamma v^{2}-{\frac {-mc^{2}}{2}}\int \gamma d(1-v^{2}/c^{2})\\&=m\gamma v^{2}+mc^{2}(1-v^{2}/c^{2})^{1/2}-E_{0}\end{aligned}}}

{\displaystyle {\begin{aligned}E_{\text{k}}&=m\gamma (v^{2}+c^{2}(1-v^{2}/c^{2}))-E_{0}\\&=m\gamma (v^{2}+c^{2}-v^{2})-E_{0}\\&=m\gamma c^{2}-E_{0}\end{aligned}}}

${\displaystyle E_{0}=mc^{2}\,}$

${\displaystyle E_{\text{k}}=m\gamma c^{2}-mc^{2}={\frac {mc^{2}}{\sqrt {1-v^{2}/c^{2}}}}-mc^{2}}$

### 極限

${\displaystyle \lim _{v\rightarrow c}E_{\text{k}}=\infty }$

{\displaystyle {\begin{aligned}E_{\text{k}}&={\frac {mc^{2}}{\sqrt {1-(v/c)^{2}}}}-mc^{2}\\&=mc^{2}(1+{\frac {1}{2}}v^{2}/c^{2}+{\frac {3}{8}}v^{4}/c^{4}+\cdots )-mc^{2}\\&=mc^{2}+{\frac {mv^{2}}{2}}+{\frac {3}{8}}{mv^{4}/c^{2}}+\cdots -mc^{2}\\&\approx {\frac {1}{2}}mv^{2}\end{aligned}}}

## 参考文献

1. 赵志敏. 高中物理竞赛教程.基础篇. 复旦大学出版社. 2011年10月: P139. ISBN 978-7-309-08251-7.

## 參見

This article uses material from the Wikipedia article "动能", which is released under the Creative Commons Attribution-Share-Alike License 3.0. There is a list of all authors in Wikipedia

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