Perfectly Inelastic Collision in 1D

Collision

Collision is an event in which two or more objects exert forces on each other, resulting in a transfer of momentum, energy, and/or deformation of the objects.

All collisions conserve momentum. However, the total kinetic energy after a collision can be more than, equal to or less than the total kinetic energy before a collision.

For example, a ball fall and land on the ground. It can bounce back as high as the initial height (a), or lower (b), or higher for special reasons (c).

We can write the relationship of total kinetic energy before and after the collision in these 3 cases:

• (a) $\sum E_k$ (before) =$\sum E_k$ (after)
• (b) $\sum E_k$ (before) > $\sum E_k$ (after)
• (c) $\sum E_k$ (before) < $\sum E_k$ (after)

Those cases are respectively called the elastic collision, inelastic collision, and super-elastic collision. If the colliding objects stick to each other and share velocity, then it is called a perfectly inelastic collision.

Perfectly inelastic collision

Suppose two objects with masses $m_1$ and $m_2$ have a collision in 1 dimension. Before the collision, their velocities are $v_1$ and $v_2$; after the collision, their velocities are $u_1$ and $u_2$. If the collision is perfectly inelastic, the post-collision velocities are equal: $u_1=u_2$.

All collision conserves momentum, including the perfectly inelastic collision:

$$m_1v_1+m_2v_2=m_1u_1+m_2u_2$$

We can solve the equations and obtain that:

$$u_1=u_2=\frac{m_1v_1+m_2v_2}{m_1+m_2}$$