(a) It is not possible for both particles to be at rest after the collision. (b) It is possible for one particle to be at rest after the collision.
Work Step by Step
(a) If one particle is moving and the other particle is at rest, then the initial momentum of the system is non-zero. If both particles are at rest after the collision, then the final momentum of the system would be zero. Then momentum would not be conserved and this would violate the conservation of momentum which is a fundamental law of physics. Therefore, it is not possible for both particles to be at rest after the collision. (b) Let's suppose that two particles have the same mass, one particle is moving with velocity $v_0$, and the other particle is at rest. Let's suppose that they have a head on collision which is perfectly elastic. In this situation, the initial velocities of the two particles are exchanged. The particle which was initially moving will be at rest after the collision, while the other particle will be moving with velocity $v_0$.