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 step by step, we will analyze the collision between the particle and the rod using the principles of conservation of linear momentum and conservation of angular momentum.

### Step 1: Analyze the initial conditions
- The particle of mass \( m \) is moving with a velocity \( v \) perpendicular to the rod.
- The rod of mass \( M \) is stationary on a smooth horizontal plane.
- The collision occurs at a distance \( \frac{a}{4} \) from the center of the rod.

### Step 2: Apply conservation of linear momentum
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