2.9 Inelastic Collision in One Dimension

2.9 Inelastic Collision in One Dimension

Consider two bodies having masses $m_{1}$ and $m_{2}$, moving with velocities $v_{1}$ and $v_{2}$, along the same line such that $v_{1} > v_{2}$. In such a case $m_{1}$ is regarded as projectile and $m_{2}$ as target. After time $t$ both the bodies make inelastic collision and stick together. Let their combined mass become $m_{1} + m_{2}$, which moves with final velocity $v_{r}$ after collision.

Since the collision is perfectly inelastic, the total momentum of balls is conserved. Using law of conservation of momentum:

Total momentum of system before collision = Total momentum of the system after collision

Which gives the common velocity of the body after inelastic collision.

In a special case when the target $m_{2}$ is at rest, $v_{2} = 0$, the above equation becomes:

It shows that velocity of $m_{1}$ is reduced by the mass ratio i.e., $m_{1}/(m_{1} + m_{2})$.