# 动量

## 古典力学中的动量

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

${\displaystyle {\frac {\mathrm {d} \mathbf {p} }{\mathrm {d} t}}={\frac {\mathrm {d} (m\mathbf {v} )}{\mathrm {d} t}}=m{\frac {\mathrm {d} \mathbf {v} }{\mathrm {d} t}}+v{\frac {\mathrm {d} m}{\mathrm {d} t}}}$

${\displaystyle {\frac {\mathrm {d} \mathbf {p} }{\mathrm {d} t}}=m{\frac {\mathrm {d} \mathbf {v} }{\mathrm {d} t}}}$

### 定理

${\displaystyle \sum \mathbf {I} =\Delta \mathbf {p} }$

## 碰撞中的动量守恒

${\displaystyle m_{1}\mathbf {v} _{1{\text{i}}}+m_{2}\mathbf {v} _{2{\text{i}}}=m_{1}\mathbf {v} _{1{\text{f}}}+m_{2}\mathbf {v} _{2{\text{f}}}}$

## 弹性碰撞

${\displaystyle {\frac {1}{2}}m_{1}v_{1{\text{i}}}^{2}+{\frac {1}{2}}m_{2}v_{2{\text{i}}}^{2}={\frac {1}{2}}m_{1}v_{1{\text{f}}}^{2}+{\frac {1}{2}}m_{2}v_{2{\text{f}}}^{2}}$

### 正向碰撞（一维）

${\displaystyle m_{1}v_{1{\text{i}}}+m_{2}v_{2{\text{i}}}=m_{1}v_{1{\text{f}}}+m_{2}v_{2{\text{f}}}}$
${\displaystyle {\frac {1}{2}}m_{1}v_{1{\text{i}}}^{2}+{\frac {1}{2}}m_{2}v_{2{\text{i}}}^{2}={\frac {1}{2}}m_{1}v_{1{\text{f}}}^{2}+{\frac {1}{2}}m_{2}v_{2{\text{f}}}^{2}}$

${\displaystyle v_{1{\text{f}}}={\frac {m_{1}-m_{2}}{m_{1}+m_{2}}}v_{1{\text{i}}}+{\frac {2m_{2}}{m_{1}+m_{2}}}v_{2{\text{i}}}}$
${\displaystyle v_{2{\text{f}}}={\frac {2m_{1}}{m_{1}+m_{2}}}v_{1{\text{i}}}+{\frac {m_{2}-m_{1}}{m_{1}+m_{2}}}v_{2{\text{i}}}}$

## 动量的现代定义

### 相对论力学中的动量

${\displaystyle \mathbf {p} =\gamma m\mathbf {u} }$

• ${\displaystyle m}$ 表示运动物体的静止质量；
• ${\displaystyle \gamma ={\frac {1}{\sqrt {1-u^{2}/c^{2}}}}}$
• u表示物体与观察者之间的相对速度；
• c表示光速

${\displaystyle \left({E \over c},p_{x},p_{y},p_{z}\right)}$

${\displaystyle E=\gamma mc^{2}\;}$

${\displaystyle \mathbf {p} \cdot \mathbf {p} -E^{2}/c^{2}}$

#### 无静止质量物体的动量

${\displaystyle p={\frac {h}{\lambda }}={\frac {E}{c}}}$

${\displaystyle h}$ 表示普朗克常量
${\displaystyle \lambda }$ 表示光子的波长；
${\displaystyle E}$ 表示光子的能量
${\displaystyle c}$ 表示光速

### 量子力学中的动量

${\displaystyle \mathbf {\hat {p}} ={\hbar \over i}\nabla =-i\hbar \nabla }$

## 参考文献

1. ^ Daniel Garber. Descartes' Physics. John Cottingham (编). The Cambridge Companion to Descartes. Cambridge: Cambridge University Press. 1992: 310–319. ISBN 0-521-36696-8.
2. ^ Rothman, Milton A. Discovering the natural laws : the experimental basis of physics 2nd. New York: Dover Publications. 1989: 83–88. ISBN 9780486261782.
3. ^ 人民教育出版社物理室《全日制普通高级中学教科书物理》第二册ISBN 978-7-107-16500-9