Perfectly Elastic Collision in 1D

Elastic collision

Suppose two objects with masses $m_1$ and $m_2$ have an elastic collision in 1 dimension. Before the collision, their velocities are $v_1$ and $v_2$; after the collision, their velocities are $u_1$ and $u_2$.

All collision conserves momentum, including the elastic collision:

$$m_1v_1+m_2v_2=m_1u_1+m_2u_2$$

For elastic collision, kinetic energy is also conserved:

$$\frac{1}{2}m_1v_1^2+\frac{1}{2}m_2v_2^2=\frac{1}{2}m_1u_1^2+\frac{1}{2}m_2u_2^2$$

We can solve the equations and obtain that:

$$\begin{gather} u_1=\frac{m_1-m_2}{m_1+m_2}v_1+\frac{2m_2}{m_1+m_2}v_2 \\ u_2=\frac{2m_1}{m_1+m_2}v_1+\frac{m_2-m_1}{m_1+m_2}v_2 \end{gather}$$