# 恢復係數

## 相關理論

${\displaystyle C_{r}=\left|{\frac {\mathbf {u} _{f}\cdot {\hat {\mathbf {n} }}}{\mathbf {u} _{i}\cdot {\hat {\mathbf {n} }}}}\right|}$

${\displaystyle \mathbf {u} _{i}=\mathbf {v} _{1i}-\mathbf {v} _{2i}}$
${\displaystyle \mathbf {u} _{f}=\mathbf {v} _{1f}-\mathbf {v} _{2f}}$

${\displaystyle C_{r}=-\ {\frac {\mathbf {u} _{f}\cdot {\hat {\mathbf {n} }}}{\mathbf {u} _{i}\cdot {\hat {\mathbf {n} }}}}}$

${\displaystyle C_{r}\ {\stackrel {def}{=}}\ {\frac {\int _{t_{1}}^{t_{2}}\mathbf {F} _{1r}\cdot {\hat {\mathbf {n} }}\ \mathrm {d} t}{\int _{t_{0}}^{t_{1}}\mathbf {F} _{1c}\cdot {\hat {\mathbf {n} }}\ \mathrm {d} t}}={\frac {\int _{t_{1}}^{t_{2}}\mathbf {F} _{2r}\cdot {\hat {\mathbf {n} }}\ \mathrm {d} t}{\int _{t_{0}}^{t_{1}}\mathbf {F} _{2c}\cdot {\hat {\mathbf {n} }}\ \mathrm {d} t}}}$

${\displaystyle C_{r}=-{\frac {u_{f}}{u_{i}}}={\frac {v_{2f}-v_{1f}}{v_{1i}-v_{2i}}}}$

${\displaystyle C_{r}={\frac {v_{f}}{v_{i}}}}$

${\displaystyle C_{r}={\sqrt {\frac {h}{H}}}}$

### 導引

${\displaystyle \int _{t_{0}}^{t_{1}}\mathbf {F} _{1c}\cdot {\hat {\mathbf {n} }}\ \mathrm {d} t=m_{1}v_{c}-m_{1}\mathbf {v} _{1i}\cdot {\hat {\mathbf {n} }}}$
${\displaystyle \int _{t_{0}}^{t_{1}}\mathbf {F} _{2c}\cdot {\hat {\mathbf {n} }}\ \mathrm {d} t=m_{2}v_{c}-m_{2}\mathbf {v} _{2i}\cdot {\hat {\mathbf {n} }}}$
${\displaystyle \int _{t_{1}}^{t_{2}}\mathbf {F} _{1r}\cdot {\hat {\mathbf {n} }}\ \mathrm {d} t=m_{1}\mathbf {v} _{1f}\cdot {\hat {\mathbf {n} }}-m_{1}v_{c}}$
${\displaystyle \int _{t_{1}}^{t_{2}}\mathbf {F} _{2r}\cdot {\hat {\mathbf {n} }}\ \mathrm {d} t=m_{2}\mathbf {v} _{2f}\cdot {\hat {\mathbf {n} }}-m_{2}v_{c}}$

${\displaystyle m_{1}\mathbf {v} _{1f}\cdot {\hat {\mathbf {n} }}-m_{1}v_{c}=C_{r}(m_{1}v_{c}-m_{1}\mathbf {v} _{1i}\cdot {\hat {\mathbf {n} }})}$
${\displaystyle m_{2}\mathbf {v} _{2f}\cdot {\hat {\mathbf {n} }}-m_{2}v_{c}=C_{r}(m_{2}v_{c}-m_{2}\mathbf {v} _{2i}\cdot {\hat {\mathbf {n} }})}$

${\displaystyle v_{c}={\frac {(\mathbf {v} _{1f}+C_{r}\mathbf {v} _{1i})\cdot {\hat {\mathbf {n} }}}{1+C_{r}}}={\frac {(\mathbf {v} _{2f}+C_{r}\mathbf {v} _{2i})\cdot {\hat {\mathbf {n} }}}{1+C_{r}}}}$

${\displaystyle C_{r}={\frac {(\mathbf {v} _{2f}-\mathbf {v} _{1f})\cdot {\hat {\mathbf {n} }}}{(\mathbf {v} _{1i}-\mathbf {v} _{2i})\cdot {\hat {\mathbf {n} }}}}=-\ {\frac {\mathbf {u} _{f}\cdot {\hat {\mathbf {n} }}}{\mathbf {u} _{i}\cdot {\hat {\mathbf {n} }}}}}$

${\displaystyle C_{r}=-\ {\frac {\mathbf {v} _{1f}\cdot {\hat {\mathbf {n} }}}{\mathbf {v} _{1i}\cdot {\hat {\mathbf {n} }}}}={\frac {v_{f}}{v_{i}}}}$

${\displaystyle m_{1}gH=m_{1}{v_{i}}^{2}/2}$
${\displaystyle m_{1}gh=m_{1}{v_{f}}^{2}/2}$

${\displaystyle C_{r}={\frac {v_{f}}{v_{i}}}={\sqrt {\frac {h}{H}}}}$

## 碰撞後的速度

${\displaystyle v_{1f}={\frac {m_{1}v_{1i}+m_{2}v_{2i}+C_{r}m_{2}(v_{2i}-v_{1i})}{m_{1}+m_{2}}}}$
${\displaystyle v_{2f}={\frac {m_{1}v_{1i}+m_{2}v_{2i}+C_{r}m_{1}(v_{1i}-v_{2i})}{m_{1}+m_{2}}}}$

### 導引

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

${\displaystyle v_{2f}=C_{r}(v_{1i}-v_{2i})+v_{1f}}$
${\displaystyle v_{1f}=(m_{1}v_{1i}+m_{2}v_{2i}-m_{2}v_{2f})/m_{1}}$

${\displaystyle v_{2f}}$ 的方程式代入 ${\displaystyle v_{1f}}$ 的方程式，可以得到

${\displaystyle v_{1f}=[m_{1}v_{1i}+m_{2}v_{2i}-C_{r}m_{2}(v_{1i}-v_{2i})-m_{2}v_{1f}]/m_{1}}$

${\displaystyle v_{1f}={\frac {m_{1}v_{1i}+m_{2}v_{2i}+C_{r}m_{2}(v_{2i}-v_{1i})}{m_{1}+m_{2}}}}$

${\displaystyle v_{2f}={\frac {m_{1}v_{1i}+m_{2}v_{2i}+C_{r}m_{1}(v_{1i}-v_{2i})}{m_{1}+m_{2}}}}$

## 參考文獻

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