The bigger object will move some distance because the initial and final momentum of the colliding particles are not zero.
The term momentum has to do with the product of mass and velocity. We know that during a collision, momentum is conserved. This implies that the momentum before collision is equal to the momentum after collision. Thus, the total momentum of the system is constant.
Given the fact that the initial and final momentum of the colliding particles are not zero, the the big object is not going to stop immediately but must move some distance towards the smaller object.
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