# 碰撞

## 弹性碰撞

{\displaystyle {\begin{aligned}{\frac {m_{1}\cdot |v_{1}|^{2}}{2}}+{\frac {m_{2}\cdot |v_{2}|^{2}}{2}}&={\frac {m_{1}\cdot |v_{1}'|^{2}}{2}}+{\frac {m_{2}\cdot |v_{2}'|^{2}}{2}}\\{\frac {m_{1}\cdot v_{1}^{2}}{2}}-{\frac {m_{1}\cdot v_{1}'^{2}}{2}}&={\frac {m_{2}\cdot v_{2}'^{2}}{2}}-{\frac {m_{2}\cdot v_{2}^{2}}{2}}\\{\frac {m_{1}}{2}}\cdot (v_{1}-v_{1}')(v_{1}+v_{1}')&={\frac {m_{2}}{2}}\cdot (v_{2}'-v_{2})(v_{2}'+v_{2})\\\end{aligned}}}

{\displaystyle {\begin{aligned}(m_{1}\cdot v_{1})+(m_{2}\cdot v_{2})&=(m_{1}\cdot v_{1}')+(m_{2}\cdot v_{2}')\\(m_{1}\cdot {\vec {v_{1}}})-(m_{1}\cdot {\vec {v_{1}'}})&=(m_{2}\cdot {\vec {v_{2}'}})-(m_{2}\cdot {\vec {v_{2}}})\\m_{1}\cdot ({\vec {v_{1}}}-{\vec {v_{1}'}})&=m_{2}\cdot ({\vec {v_{2}'}}-{\vec {v_{2}}})\\\end{aligned}}}

${\displaystyle v_{1}'={\frac {(m_{1}-m_{2})\cdot v_{1}+2\ m_{2}\cdot v_{2}}{m_{1}+m_{2}}}}$
${\displaystyle v_{2}'={\frac {(m_{2}-m_{1})\cdot v_{2}+2\ m_{1}\cdot v_{1}}{m_{1}+m_{2}}}}$

${\displaystyle m_{1}=m_{2}}$ 是以下方程:

${\displaystyle v_{1}'=v_{2}}$
${\displaystyle v_{2}'=v_{1}}$

## 非弹性碰撞

• 碰撞前:
${\displaystyle E_{kin}={\frac {m_{1}\cdot v_{1}^{2}}{2}}}$
${\displaystyle p=m_{1}\cdot v_{1}}$
• 碰撞后:
${\displaystyle E_{kin}={\frac {m_{1}\cdot v_{1}'^{2}+m_{2}\cdot v_{2}'^{2}}{2}}+U}$
${\displaystyle p'=m_{1}\cdot v_{1}'+m_{2}\cdot v_{2}'}$

• 碰撞后:
${\displaystyle E_{kin}={\frac {(m_{1}+m_{2})\cdot v_{2}'^{2}}{2}}+U}$
${\displaystyle p'=(m_{1}+m_{2})\cdot v_{2}'}$

${\displaystyle m_{1}\cdot v_{1}=(m_{1}+m_{2})\cdot v_{2}'-m_{2}\cdot v_{2}}$
${\displaystyle v_{2}'={\frac {m_{1}\cdot v_{1}+m_{2}\cdot v_{2}}{m_{1}+m_{2}}}}$

{\displaystyle {\begin{aligned}{\frac {m_{1}}{2}}\cdot v_{1}^{2}&={\frac {(m_{1}+m_{2})\cdot v_{2}'^{2}}{2}}+U\\{\frac {m_{1}}{2}}\cdot v_{1}^{2}&={\frac {m_{1}+m_{2}}{2}}\cdot {\frac {m_{1}^{2}}{(m_{1}+m_{2})^{2}}}\cdot v_{1}^{2}+U\\\end{aligned}}}
${\displaystyle U={\frac {1}{2}}\cdot {\frac {m_{1}\cdot m_{2}}{m_{1}+m_{2}}}\cdot v_{1}^{2}}$

## 实际碰撞

${\displaystyle k={\frac {v_{1}'-v_{2}'}{v_{2}-v_{1}}}}$
k = 0: 完全非弹性碰撞
k = 1: 完全弹性碰撞

## 参考书籍

• Christiaan Huygens, Felix Hausdorff: Christiaan Huygens' nachgelassene Abhandlungen: Über die Bewegung der Körper durch den Stoss : Über die Centrifugalkraft / Hrsg. von Felix Hausdorff. Akademische Verlagsgesellschaft, Leipzig, um 1921 （德语）