动碰静
设 \(m_1\) 创物, \(m_2\) 被创物, \(v_1\) 创物初速度, \(v_1^{'}\) 创物末速度, \(v_2^{'}\) 被创物末速度。
联立以下方程组:
\[ \left\{
\begin{aligned}
m_1v_1 & = m_1v_1^{'}+m_2v_2^{'} \\
\frac{1}{2}m_1v_1^2&= \frac{1}{2}m_1{v_1^{'} }^2+\frac{1}{2}m_2{v_2^{'}}^2
\end{aligned}
\right.
\]
易得解:
\[ \left\{
\begin{aligned}
v_1^{'}&=\frac{m_1-m_2}{m_1+m_2}v_1 \\
v_2^{'}&=\frac{2m_1}{m_1+m_2}v_1
\end{aligned}
\right.
\]
特殊情况:\(m_1>>m_2\) 大创小
代入得解:
\[ \left\{
\begin{aligned}
v_1^{'}&=v_1 \\
v_2^{'}&=2v_1
\end{aligned}
\right.
\]
最大二倍也
特殊情况:\(m_1<<m_2\) 小创大
代入得解:
\[ \left\{
\begin{aligned}
v_1^{'}&=-v_1\\
v_2^{'}&=0
\end{aligned}
\right.
\]
反激之力也
特殊情况:\(m_1==m_2\) 同创同
代入得解:
\[ \left\{
\begin{aligned}
v_1^{'}&=0 \\
v_2^{'}&=v_1
\end{aligned}
\right.
\]
速度传递也
动碰动
设 \(m_1\) 创物1, \(m_2\) 创物2, \(v_1\) 创物1初速度, \(v_2\) 创物2初速度, \(v_1^{'}\) 创物1末速度, \(v_2^{'}\) 创物2末速度。
\[ \left\{
\begin{aligned}
m_1v_1+m_2v_2 & = m_1v_1^{'}+m_2v_2^{'} \\
\frac{1}{2}m_1v_1^2+\frac{1}{2}m_2v_2^2&= \frac{1}{2}m_1{v_1^{'} }^2+\frac{1}{2}m_2{v_2^{'}}^2
\end{aligned}
\right.
\]
易得解:
\[ \left\{
\begin{aligned}
v_1^{'}&=\frac{m_1-m_2}{m_1+m_2}v_1 + \frac{2m_2}{m_1+m_2}v_2 \\
v_2^{'}&=\frac{2m_1}{m_1+m_2}v_1 + \frac{m_2-m_1}{m_1+m_2}v_2
\end{aligned}
\right.
\]
动碰静公式本质上是动碰动公式的特殊情况