Two particles collide, one of which was initially moving and the other initially at rest.

a. Is it possible for both particles to be at rest after the collision? Give an example in which this happens, or explain why it can’t happen.

b. Is it possible for one particle to be at rest after the collision? Give an example in which this happens, or explain why it can’t happen.

We need to find that Is it possible for both particles to be at rest after the collision?

No external forces such as friction is exerted on the system that affects the result which consider the system is isolated. after the collision, the two objects become at rest, the final momentum will be zero and it couldn't occur due to the conservation law of momentum. so the initial momentum of the system is not zero. Maybe there is a loss in the energy but not in the momentum. Hence, the answer will be "No, it is not possible" .

We need to find that Is it possible for one particles to be at rest after the collision?

One of the objects is at rest before the collision occurs, so the initial momentum of the system coincides with that of the moving object

$p_i=m_1{v}_1$

This momentum could be totally transferred to the other object after the collision or makes the first object moves in the opposite direction such as a rubber ball when it rebounds from the wall. so the answer is "Yes, it is possible"