5.2 Impulse

The effect of a force on an object depends on how long it acts, as well as how great the force is. In previous examples, a very large force acting for a short time had a great effect on the momentum of the tennis ball. A small force could cause the same change in momentum    , but it would have to act for a much longer time. For example, if the ball were thrown upward, the gravitational force (which is much smaller than the tennis racquet’s force) would eventually reverse the momentum of the ball. Quantitatively, the effect we are talking about is the change in momentum $\mathrm{\Delta }\mathbf{p}$ .

By rearranging the equation ${\mathbf{F}}_{\text{net}}=\frac{\mathrm{\Delta }\mathbf{p}}{\mathrm{\Delta }t}$ to be

we can see how the change in momentum equals the average net external force multiplied by the time this force acts. The quantity ${\mathbf{F}}_{\text{net}}\mathrm{\Delta }t$ is given the name impulse    . Impulse is the same as the change in momentum.

Our definition of impulse includes an assumption that the force is constant over the time interval $\mathrm{\Delta }t$ . Forces are usually not constant . Forces vary considerably even during the brief time intervals considered. It is, however, possible to find an average effective force $F_{\text{eff}}$ that produces the same result as the corresponding time-varying force. [link] shows a graph of what an actual force looks like as a function of time for a ball bouncing off the floor. The area under the curve has units of momentum and is equal to the impulse or change in momentum between times $t_1$ and $t_2$ . That area is equal to the area inside the rectangle bounded by $F_{\text{eff}}$ , $t_1$ , and $t_2$ . Thus the impulses and their effects are the same for both the actual and effective forces.