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I am studying an example problem, concerning the very topic mentioned in the title. In this example problem, a car has a head-on collision with the wall; the initial and final velocity are known, as well as the mass. My question is, why does the car rebound off of the wall? Why doesn't the normal force of the wall reduce the car's velocity to zero, with the car remaining there, instead of reducing the car's velocity to zero, and then giving it a velocity in the opposite direction?

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This is a partially inelastic collision, which means that while the total kinetic energy of the system is not conserved, the momentum is.

If the car was stationary after the collision, momentum would not be conserved since: ${\vec{p}} = m\vec{v}$

Imagine a rubber ball colliding with a wall. It certainly has a final velocity opposite the initial direction.

If the question was rewritten so that the car did stop against the wall, then it could be considered a perfectly inelastic collision, where the the wall and the car move forwards together for a short amount of time after the collision. Of course, the wall would likely be so massive that it really doesn't move at all.

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