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Chapter 8: Problem 6

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?

Short Answer

Expert verified
The impulse and total force exerted by Big Bill on the tennis ball during his record serve were \(4.131 kg.m/s\) and \(137.7 N\) respectively. The impulse and total force exerted by Big Bill's opponent on the ball while returning the serve were \(7.296 kg.m/s\) and \(243.2 N\) respectively.

Step by step solution

01

Calculating Impulse Exerted by Big Bill

First convert the mass of the ball into kg, because the standard unit of mass in physics is kilogram. That can be done by dividing the mass by 1000, so \(57 g = 0.057 kg\). Impulse can be calculated using the formula Impulse = mass x change in velocity. The change in velocity is equal to the final velocity, as the initial velocity is assumed to be 0. Hence, Impulse exerted by Big Bill is \(0.057 kg\) x \(73 m/s\) = \(4.131 kg.m/s\).
02

Calculating Force Exerted by Big Bill

Force can be calculated using the formula Force = Impulse/time. Convert time into seconds, because the standard unit of time in physics is second. So \(30.0 ms = 0.03 s\). Force exerted by Big Bill is then \(4.131 kg.m/s\) / \(0.03 s\) = \(137.7 N\).
03

Calculating Impulse Exerted by Big Bill's Opponent

Impulse by Big Bill's opponent can be calculated in much the same way as before. The change in velocity is given by the difference in final velocity \(V_f = +55 m/s\) and the initial velocity \(V_i = -73 m/s\) because the direction of the ball's motion reversed. Hence the change in velocity is \(+55 m/s - (-73 m/s) = 128 m/s\). Hence, Impulse exerted by Big Bill's opponent is \(0.057 kg\) x \(128 m/s\) = \(7.296 kg.m/s\).
04

Calculating Force Exerted by Big Bill's Opponent

Force exerted by Big Bill's opponent is then \(7.296 kg.m/s\) / \(0.03 s\) = \(243.2 N\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Tennis Ball Physics
Tennis ball physics involves understanding how forces and motions interact when hitting or serving the ball. A tennis ball, weighing around 57 grams, is influenced by these physics principles. In our example, the ball is served at an impressive 73 m/s. When discussing tennis ball physics, it's essential to consider how momentum and impulse play crucial roles. As the ball impacts the racket and is sent over the net, these interactions determine its movement path and speed. By calculating impulse and force, we understand the involved mechanics better.
Contact Time
Contact time refers to the duration over which the racket is in contact with the ball. In this scenario, it's 30 milliseconds.
  • This brief period is critical as it defines the time frame in which forces are applied to the ball.
  • During contact time, energy is transferred from the racket to the ball, allowing players to control the ball's speed and direction.
To convert milliseconds to seconds, crucial in physics calculations, divide by 1000. Thus, 30 ms becomes 0.03 s. This conversion ensures we can correctly calculate force using standard units, making this concept foundational to understanding tennis ball dynamics.
Force Calculation
Force calculation is crucial in analyzing a serve or return in tennis. Force indicates the interaction's strength between the racket and the ball.
  • In the example, Big Bill's serve exerts a force of 137.7 N on the tennis ball.
  • This is calculated by dividing the impulse (4.131 kg路m/s) by the contact time (0.03 s).
The opponent's return involves greater force.
  • In this case, it is 243.2 N, due to a higher impulse of 7.296 kg路m/s.
Force can vary based on factors like swing speed and contact time. Understanding these factors can improve game strategies and interpretations of players' physical performances.
Velocity Change
The change in velocity is an important component in calculating impulse.
  • For Big Bill's serve, velocity change equals his serve speed, 73 m/s, with an initial speed of 0.
  • For the return, it requires adding the initial and final speeds because the motion direction from serve to return is reversed.
Here, the return velocity change equals 128 m/s, calculated as 55 m/s - (-73 m/s). This change reflects the total adjustment the tennis ball undergoes during its travel, showcasing the force exerted by the player to send it back across the net. By meticulously calculating these changes, players can refine their understanding of game tactics and improve their performance efficiency.

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Most popular questions from this chapter

You are at the controls of a particle accelerator, sending a beam of \(1.50 \times 10^{7} \mathrm{~m} / \mathrm{s}\) protons (mass \(\mathrm{m}\) ) at a gas target of an unknown element. Your detector tells you that some protons bounce straight back after a collision with one of the nuclei of the unknown element. All such protons rebound with a speed of \(1.20 \times 10^{7} \mathrm{~m} / \mathrm{s}\). Assume that the initial speed of the target nucleus is negligible and the collision is elastic. (a) Find the mass of one nucleus of the unknown element. Express your answer in terms of the proton mass \(m\). (b) What is the speed of the unknown nucleus immediately after such a collision?

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