# 恢復係數

## 相關理論

$C_r=\left|\frac{\mathbf{u}_f\cdot\hat{\mathbf{n}}}{\mathbf{u}_i\cdot\hat{\mathbf{n}}}\right|$

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

$C_r=-\ \frac{\mathbf{u}_f\cdot\hat{\mathbf{n}}}{\mathbf{u}_i\cdot\hat{\mathbf{n}}}$

$C_r\ \stackrel{def}{=}\ \frac{\int_{t_1}^{t_2} \mathbf{F}_{1r}\cdot\hat{\mathbf{n}}\ \mathrm{d}t}{\int_{t_0}^{t_1} \mathbf{F}_{1c}\cdot\hat{\mathbf{n}}\ \mathrm{d}t} =\frac{\int_{t_1}^{t_2} \mathbf{F}_{2r}\cdot\hat{\mathbf{n}}\ \mathrm{d}t}{\int_{t_0}^{t_1} \mathbf{F}_{2c}\cdot\hat{\mathbf{n}}\ \mathrm{d}t}$

$C_r= -\frac{u_f}{u_i}=\frac{v_{2f} - v_{1f}}{v_{1i} - v_{2i}}$

$C_r = \frac{v_{f}}{v_{i}}$

$C_r = \sqrt{\frac{h}{H}}$

### 导引

$\int_{t_0}^{t_1} \mathbf{F}_{1c}\cdot\hat{\mathbf{n}}\ \mathrm{d}t =m_1 v_{c}-m_1 \mathbf{v}_{1i}\cdot\hat{\mathbf{n}}$
$\int_{t_0}^{t_1} \mathbf{F}_{2c}\cdot\hat{\mathbf{n}}\ \mathrm{d}t =m_2 v_{c}-m_2\mathbf{v}_{2i}\cdot\hat{\mathbf{n}}$
$\int_{t_1}^{t_2} \mathbf{F}_{1r}\cdot\hat{\mathbf{n}}\ \mathrm{d}t =m_1\mathbf{v}_{1f}\cdot\hat{\mathbf{n}}-m_1 v_{c}$
$\int_{t_1}^{t_2} \mathbf{F}_{2r}\cdot\hat{\mathbf{n}}\ \mathrm{d}t =m_2\mathbf{v}_{2f}\cdot\hat{\mathbf{n}}-m_2 v_{c}$

$m_1\mathbf{v}_{1f}\cdot\hat{\mathbf{n}}-m_1 v_{c} =C_r(m_1 v_{c}-m_1 \mathbf{v}_{1i}\cdot\hat{\mathbf{n}})$
$m_2\mathbf{v}_{2f}\cdot\hat{\mathbf{n}}-m_2 v_{c} =C_r(m_2 v_{c}-m_2\mathbf{v}_{2i}\cdot\hat{\mathbf{n}})$

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

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

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

$m_1 g H=m_1 {v_i}^2/2$
$m_1 g h=m_1 {v_f}^2/2$

$C_r=\frac{v_f}{v_i}=\sqrt{\frac{h}{H}}$

## 碰撞後的速度

$v_{1f}=\frac{m_1 v_{1i}+m_2 v_{2i}+C_r m_2( v_{2i}-v_{1i})}{m_1 + m_2}$
$v_{2f}=\frac{m_1 v_{1i}+m_2 v_{2i}+C_r m_1( v_{1i}-v_{2i})}{m_1 + m_2}$

### 导引

$C_r=\frac{v_{2f} - v_{1f}}{v_{1i} - v_{2i}}$
$m_1 v_{1i}+ m_2 v_{2i} = m_{1}v_{1f} + m_{2}v_{2f}$

$v_{2f}=C_r(v_{1i} - v_{2i})+v_{1f}$
$v_{1f}=(m_1 v_{1i}+ m_2 v_{2i}- m_{2}v_{2f})/m_{1}$

$v_{2f}$ 的方程式代入 $v_{1f}$ 的方程式，可以得到

$v_{1f}=[m_1 v_{1i}+ m_2 v_{2i}-C_r m_{2}(v_{1i} - v_{2i})-m_{2}v_{1f}]/m_{1}$

$v_{1f}=\frac{m_1 v_{1i}+m_2 v_{2i}+C_r m_2( v_{2i}-v_{1i})}{m_1 + m_2}$

$v_{2f}=\frac{m_1 v_{1i}+m_2 v_{2i}+C_r m_1( v_{1i}-v_{2i})}{m_1 + m_2}$

## 参考文献

1. ^ 硬圆球与彈塑性圆盘斜碰撞後，正常恢复运动的不規則行為。
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