If two objects collide and one is initially at rest, is itpossible for both to be at rest after the collision?

Not possible

Explanation:

Suppose there's no external force, then the momentum must be conserved before and after the collision.

Since momentum is defined as the product of object mass and speed. This means that if after the collision, for both object to be at rest then the total momentum is 0.

But before the collision, only 1 is at rest, the other is not - > the total momentum is non 0 before the collision.

Therefor this is not possible with the law of momentum conservation.