A tennis ball of mass 57 grams was moving horizontally at 68.4 meters per second when it collided with a vertical racket that was a rest, after which it rebounded at 18.8 meters per second. Given that the contact time between the ball and racket was one 25th of a second, find the magnitude of the average impact force.
We’ll call the mass of the tennis ball, given as 57 grams, 𝑚. The speed it had before it hit the racket, 68.4 meters per second, we’ll call 𝑣 sub 𝑖. And the speed it had after hitting the racket, 18.8 meters per second, we’ll call 𝑣 sub 𝑓.
We’re also told that the contact time between the ball and the racket was one 25th of a second, what we’ll call Δ𝑡. We want to solve for the magnitude of the average impact force of the ball on the racket. We’ll call that 𝐹.
Let’s start by drawing out this interaction between the tennis ball and the racket. We’re told that a tennis ball is incident on a racket with a speed we’ve called 𝑣 sub 𝑖 then bounces off at a speed we’ve called 𝑣 sub 𝑓. And the whole interaction with the racket takes a time of one 25th of a second.
To solve for the average force exerted by the racket on the ball during this time, we can recall the impulse momentum theorem. This theorem says that the average force exerted on an object multiplied by the time over which that force is exerted, with this quantity being known as the impulse, is equal to the object’s mass times its change in velocity, which is its change in momentum, 𝑚 times 𝑣.
Returning to our diagram, if we define motion to the right to be motion in the positive direction, then that means 𝑣 sub 𝑓, the velocity of the tennis ball after it interacts with the racket, is negative 18.8 meters per second.
Applying the impulse momentum theorem to our scenario, we can write that the average force exerted on the tennis ball 𝐹 multiplied by Δ𝑡 is equal to the ball’s mass multiplied by its initial speed minus its final speed Δ𝑣.
Dividing both sides of the equation by Δ𝑡, we find a form that will let us solve for the variable we’re interested in. Since we’re given the ball’s mass, its initial and final speeds, as well as Δ𝑡, we’re ready to plug in and solve for 𝐹. When we do, we’re careful to convert our mass to units of kilograms.
Entering these values on our calculator, we find that 𝐹 is equal to 124.26 newtons. That’s the average force magnitude exerted on the tennis ball.