Problem Analysis:
- A uniform rod of mass M and length a is lying on a smooth horizontal plane.
- A particle of mass m is moving perpendicular to the length of the rod with a speed V.
- The particle strikes the rod at a distance of A/4 from the center and comes to rest after the collision.

Initial Conditions:
- The rod is at rest, so the initial velocity of the center of the rod is zero (Vc = 0).
- The angular velocity of the rod about its center is also zero (ω = 0) initially.

Conservation of Linear Momentum:
- Before the collision, the linear momentum of the particle is given by mv.
- After the collision, the particle comes to rest, so the linear momentum is zero.
- By conservation of linear momentum, mv = 0.
- This implies that the initial velocity of the particle is equal to its final velocity, which is zero.

Conservation of Angular Momentum:
- Before the collision, the angular momentum of the particle with respect to the center of the rod is given by mvr, where r is the distance of the point of impact from the center.
- After the collision, the particle comes to rest, so the angular momentum is zero.
- By conservation of angular momentum, mvr = 0.
- Since the mass, m, and velocity, v, are not zero, this implies that r = 0.

Implication:
- The point of impact is at a distance of A/4 from the center of the rod.
- Since r = 0, this means that the point of impact is at the center of the rod.

Velocity of the Center of the Rod:
- The initial velocity of the center of the rod is zero (Vc = 0).
- After the collision, the particle comes to rest, so the final velocity of the center of the rod is also zero.
- Therefore, the velocity of the center of the rod remains zero throughout the process.

Angular Velocity of the Rod:
- The initial angular velocity of the rod about its center is zero (ω = 0).
- After the collision, the particle comes to rest, so the final angular velocity of the rod is also zero.
- Therefore, the angular velocity of the rod about its center remains zero throughout the process.

Summary:
- After the collision, the velocity of the center of the rod and the angular velocity of the rod about its center both remain zero.
- This is because the particle comes to rest after the collision, and the point of impact is at the center of the rod.
- The collision does not cause any motion of the rod, as the linear and angular momenta are conserved.

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