A body moving towards another body of finite mass at rest collides with it. It is possible that :
(i) both bodies come to rest
(ii) both bodies move after collision
(iii) the moving body stops and body at rest start moving
(iv) the stationary body remains stationary and the moving rebounds

Both bodies cannot come to rest as it violates conservation of momentum. Both of them starts moving is possible but total momentum of the system must be equal to momentum of the colliding body. For perfectly elastic collision of equal masses, they exchange their velocities If stationary body rebounds, then the momentum by direction is not conserved as the heavier body is till at rest.