A uniform rod of mass M and length a lies on a smooth horizontal plane. A particle of mass m moving at a speed v perpendicular to the length of the rod strikes it at a distance `a/4` from the centre and stops after the collision. Find a. the velocity of the cente of the rod and b. the angular velocity of the rod abut its centre just after the collision.

Explanation

To solve the problem, we will break it down into two parts: finding the velocity of the center of the rod after the collision and finding the angular velocity of the rod about its center just after the collision.

### Part (a): Finding the Velocity of the Center of the Rod

1. **Understanding the Conservation of Momentum**:
- Before the collision, the momentum of the system is due to the moving particle of mass \(m\) with velocity \(v\).
- The rod is initially at rest, so its momentum is zero.
- After the collision, the total momentum of the system must be conserved.
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