A bullet of mass m moving at a speed v hits a ball of mass M kept at rest. A small part having mass m breaks from the ball and sticks to the bullet. The remaining ball is found to move at a speed v_{1} in the direction of the bullet. Find the velocity of the bullet after the collision.

#### Solution

Given:

The mass of bullet moving with speed v is m.

The mass of the ball is M and it is at rest.

m' is the fractional mass of the ball that sticks with the bullet.

The remaining mass of the ball moves with the velocity v_{1}.

Let v_{2} be the final velocity of the bullet plus fractional mass system.

On applying the law of conservation of momentum, we get:

mv + 0 = (m' + m)v_{2} + (M − m') v_{1}\[\Rightarrow v_2 = \frac{mv - (M - m') v_1}{m + m'}\]

Therefore, the velocity of the bullet after the collision is \[\frac{mv - (M - m') v_1}{m + m'}\]