The mass of a regulation tennis ball is 57 g (although it...

The mass of a regulation tennis ball is 57 g (although it can vary slighaly), and tests have shown that the ball is in conthet with the tennis racket for $30 \mathrm{~ms}$. (This number can also vary, depending on the racket and swing.) We shall assume a $30.0 \mathrm{~ms}$ contact time. One of the fastest-known served tennis balls was served by "Big Bill" Tilden in $1931,$ and its speed was measured to be $73 \mathrm{~m} / \mathrm{s}$. (a) What impulse and what total force did Big Bill exert on the tennis ball in his record serve? (b) If Big Bill's opponcnt returned his serve with a speed of $55 \mathrm{~m} / \mathrm{s},$ what total force and what impulse did he exert on the ball, assuming only horizontal motion?

The mass of a regulation tennis ball is 57 g (although it can vary slighaly), and tests have shown that the ball is in conthet with the tennis racket for $30 \mathrm{~ms}$. (This number can also vary, depending on the racket and swing.) We shall assume a $30.0 \mathrm{~ms}$ contact time. One of the fastest-known served tennis balls was served by "Big Bill" Tilden in $1931,$ and its speed was measured to be $73 \mathrm{~m} / \mathrm{s}$. (a) What impulse and what total force did Big Bill exert on the tennis ball in his record serve?
(b) If Big Bill's opponcnt returned his serve with a speed of $55 \mathrm{~m} / \mathrm{s},$ what total force and what impulse did he exert on the ball, assuming only horizontal motion?

Key Concepts

Impulse

Impulse is a measure of the effect of a force applied over a time interval. It is defined as the product of the force and the time during which the force acts, and it directly relates to the change in momentum of an object. In physics, calculating impulse helps in understanding how and by how much an object's momentum changes during an interaction.

Momentum

Momentum is the vector quantity defined as the product of an object's mass and its velocity. It represents the amount of motion an object has and is a central concept in mechanics, especially in understanding collisions and interactions between objects. The conservation of momentum principle is often applied in analyzing collision problems.

Impulse-Momentum Theorem

The impulse-momentum theorem states that the impulse delivered to an object is equal to the change in its momentum. This theorem provides a clear relationship between the force applied over a time interval and the resulting acceleration or deceleration of the object. It is a direct consequence of Newton's second law of motion.

Average Force and Contact Time

When a force is applied over a specific time period, the average force can be calculated by dividing the impulse by the contact time. This concept is essential in scenarios where forces vary over time but an average value is needed to characterize the overall effect during the interaction. It simplifies the analysis of collisions and impacts, such as those encountered in sports or engineering applications.

Transcript

00:02 Hello, for this problem, we have a tennis ball.

00:07 Given a mass of 57 grams, we're also provided with the time of contact between the tennis ball and the rocket ball, which is 30 milliseconds.

00:22 Over here, you can see that i already converted units, 0 .03 seconds and 0 .057 kilograms.

00:29 Now, the question of this problem is asking you to find the impulse on the first contact when the initial velocity of the serve is 73 meters per second.

00:47 Before starting, we also need to consider that before hitting the ball, it has an initial velocity of zero meters per second in the horizontal direction.

00:59 Also, we need to use the formula of impulse provided on your left, which is equal to the change of momentum, and also is associated to the force times the time.

01:17 All right, let's begin.

01:19 To find the impulse, we can find it as the change of momentum.

01:26 Given that we have two velocities, we can find momentum zero and momentum.

01:34 1.

01:36 Therefore, the change of momentum will be equal to p1 minus p0.

01:45 Remember your formula of momentum, it's equal to mass times velocity.

01:50 In this case, the first momentum is zero, given that the initial velocity was zero.

01:59 For the second one, we're going to multiply the mass times the serving velocity which is 73 meters per second we multiply and we get 4 .161 kilograms meters over second so now change of momentum is equal to the second one 4 .161 minus 0 therefore this one gives me a value of 4 .161 and that's how we find the as the change of momentum.

02:44 Therefore, impulse is equal to 4 .161 kilograms meters over second.

02:54 This is the impulse.

02:57 Second, we are asked to find the force exerted by the tennis player, which in this case, his name is bill.

03:08 Remember that impulse is associated to force with the following formula.

03:15 So given the time of contact, we can find the force...

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