A 68.0-kg tennis player (including racket) leaps vertically...

A $68.0-\mathrm{kg}$ tennis player (including racket) leaps vertically to return a serve. The $57.0-\mathrm{g}$ ball was initially moving horizontally at $50.0 \mathrm{~m} / \mathrm{s} .$ Due to the ball's height, the player can't swing her racket, but merely allows the ball to bounce off of it, which reverses the ball's direction and reduces its speed by $75.0 \%$. If the ball/racket contact lasts for $35.0 \mathrm{~ms},$ find (a) the average force the racket exerts on the ball and (b) the racket's recoil speed.

Key Concepts

Impulse-Momentum Theorem

This principle states that the impulse acting on an object (the product of the average force and the time interval over which it acts) is equal to the change in momentum of that object. It is fundamental for analyzing collisions where forces act over short time intervals, allowing one to compute the average force when the momentum change and duration of contact are known.

Change in Momentum

The concept of momentum, defined as the product of an object's mass and velocity, is critical when analyzing collisions. By computing the difference between an object’s final and initial momentum, one quantifies the impulse delivered during the interaction. This change in momentum is central to understanding how forces affect motion during impacts.

Newton's Third Law

Newton's Third Law, which states that every action has an equal and opposite reaction, is key when studying collisions. This law explains that the force exerted on one body during a collision is met with an equal and opposite force on the other body. Consequently, the recoil or motion of a more massive object can be deduced by applying this principle in conjunction with the momentum change of the lighter object.

Transcript

00:01 We're talking about impulse.

00:03 So we have a tennis player who jumps vertically to return a serve.

00:07 The ball just simply bounces off of her racket without her swinging it.

00:13 And the ball will return with only one quarter of its velocity.

00:19 And we're told the mass of the player, which includes the mass of the racket, the mass of the ball, the time that the racket and ball were in contact with one another, and also the initial velocity of the tennis ball.

00:32 Question a, we want to find the average force that was exerted by the tennis racket on the ball.

00:41 So let's start with our impulse equation, which says that force times the amount of time over which the impulse occurs is equal to the change in momentum, which is equal to mass times the final velocity minus the initial velocity.

01:02 All right, so we're looking for this force, so we'll divide delta t over to the other side.

01:07 So our force is going to be equal to, here we have the mass of the ball since all the momentum was contained in the ball since the racket was not moving initially, times the ball's initial velocity, which was 50 meters per second, minus its, or excuse me, its final velocity, which is 12 .5 meters per second minus its initial velocity which was 50 minutes per second and we're dividing by 35 milliseconds.

01:40 So we'll have to make sure that all of our units are correct.

01:44 So we'll have to convert the mass of the ball from grams to kilograms and the time of the interaction to seconds rather than milliseconds.

01:58 But this will give us our force, rather average force over the impulse, which is equal to 61 .1 mediums.

02:13 All right, in question b, now we want to find the recoil speed of the racket, which i called vip r...

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