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Newton's Second Law; A body of mass m subject to a net force F undergoes an acceleration a that has the same direction as the force and a magnitude that is directly proportional to the force and inversely proportional to the mass, i.e., F = ma. Alternatively, the total force applied on a body is equal to the time derivative of linear momentum of the body.

To answer your question simply:

Newton's second law of motion. The force the bumper car is subject to is proportional to its decelaration, (i.e the speed at which it comes to a stop)

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