The Force-Time Graph

Few real collisions involve a constant force. For example, when a tennis ball hits a racquet, the strings stretch and the ball flattens dramatically. They are both acting like springs that obey Hooke's law, which says that the force is proportional to the amount of stretching or flattening. The force is therefore small at first, ramps up to a maximum when the ball is about to reverse directions, and ramps back down again as the ball is on its way back out. The equation F = Δp/Δt, derived under the assumption of constant acceleration, does not apply here, and the force does not even have a single well-defined numerical value that could be plugged in to the equation.

The F - t graph for a tennis racquet hitting a ball might look like this. The amount of momentum transferred equals the area under the curve.

As with similar-looking equations such as v = Δp/Δt, the equation F = Δp/Δt is correctly generalized by saying that the force is the slope of the p - t graph.

Conversely, if we wish to find Δp from a graph such as the one in figure m, one approach would be to divide the force by the mass of the ball, rescaling the F axis to create a graph of acceleration versus time. The area under the acceleration-versus-time graph gives the change in velocity, which can then be multiplied by the mass to find the change in momentum. An unnecessary complication was introduced, however, because we began by dividing by the mass and ended by multiplying by it. It would have made just as much sense to find the area under the original F - t graph, which would have given us the momentum change directly.

Discussion Question

A

A Many collisions, like the collision of a bat with a baseball, appear to be instantaneous. Most people also would not imagine the bat and ball as bending or being compressed during the collision. Consider the following possibilities: 1. The collision is instantaneous. 2. The collision takes a finite amount of time, during which the ball and bat retain their shapes and remain in contact. 3. The collision takes a finite amount of time, during which the ball and bat are bending or being compressed. How can two of these be ruled out based on energy or momentum considerations?