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Chapter 7: Problem 84

Tennis champion Venus Williams is capable of serving a tennis ball at around 127 mph. a) Assuming that her racquet is in contact with the 57.0 -g ball for \(0.250 \mathrm{~s}\), what is the average force of the racquet on the ball? b) What average force would an opponent's racquet have to exert in order to return Williams's serve at a speed of \(50.0 \mathrm{mph}\), assuming that the opponent's racquet is also in contact with the ball for 0.250 s?

Short Answer

Expert verified
Using Newton's second law of motion, the average force exerted by Venus Williams during her serve is approximately 12.94 N, while the average force exerted by her opponent to return the serve at 50 mph is approximately 18.04 N.

Step by step solution

01

Convert mass to kg and velocities to m/s

First, we need to convert the given mass of the ball from grams to kilograms, and both Venus Williams' and her opponent's velocities from mph to m/s. Mass in kg: 57.0 g = 0.057 kg Velocity conversion factor: 1 mph = 0.44704 m/s Venus Williams' velocity in m/s: 127 mph x 0.44704 = 56.75368 m/s Opponent's velocity in m/s: 50.0 mph x 0.44704 = 22.352 m/s
02

Calculate the acceleration of the ball during the serve and return

For Venus Williams' serve: Initial velocity = 0 m/s (assuming ball is at rest prior to the serve) Final velocity = 56.75368 m/s Contact time = 0.250 s Acceleration = (Final velocity - Initial velocity) / Contact time Acceleration = (56.75368 - 0) / 0.250 Acceleration = 227.01472 m/s² For opponent's return: Initial velocity = -56.75368 m/s (opposite direction to the serve) Final velocity = 22.352 m/s Contact time = 0.250 s Acceleration = (Final velocity - Initial velocity) / Contact time Acceleration = (22.352 + 56.75368) / 0.250 Acceleration = 316.42272 m/s²
03

Calculate the average force exerted by Venus Williams and her opponent

Now, we calculate the average force for both players using Newton's second law (F = ma). For Venus Williams' serve: Mass = 0.057 kg Acceleration = 227.01472 m/s² Force = Mass x Acceleration Force = 0.057 x 227.01472 Force = 12.93984 N For opponent's return: Mass = 0.057 kg Acceleration = 316.42272 m/s² Force = Mass x Acceleration Force = 0.057 x 316.42272 Force = 18.03569 N #Answer# a) The average force exerted by Venus Williams during her serve is approximately 12.94 N. b) The average force exerted by her opponent to return the serve at 50 mph is approximately 18.04 N.

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