2.14 Impulse

The effect of a force on an object depends on how long it acts, as well as how great the force is. In [link] , 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.

Calculating magnitudes of impulses: two billiard balls striking a rigid wall

Two identical billiard balls strike a rigid wall with the same speed, and are reflected without any change of speed. The first ball strikes perpendicular to the wall. The second ball strikes the wall at an angle of $\text{30º}$ from the perpendicular, and bounces off at an angle of $\text{30º}$ from perpendicular to the wall.

(a) Determine the direction of the force on the wall due to each ball.

(b) Calculate the ratio of the magnitudes of impulses on the two balls by the wall.

Strategy for (a)

In order to determine the force on the wall, consider the force on the ball due to the wall using Newton’s second law and then apply Newton’s third law to determine the direction. Assume the $x$ -axis to be normal to the wall and to be positive in the initial direction of motion. Choose the $y$ -axis to be along the wall in the plane of the second ball’s motion. The momentum direction and the velocity direction are the same.