Question

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

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

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

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

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.

Key concepts

Newton's Second Law

In physics, Newton's Second Law of Motion is fundamental to understanding how forces interact with objects. This law states that the force acting on an object is equal to the mass of that object multiplied by its acceleration, which can be represented by the equation:

• \( F = ma \)

Force \( F \) is measured in Newtons (N), mass \( m \) in kilograms (kg), and acceleration \( a \) in meters per second squared (\( m/s^2 \)).
This principle is crucial because it describes how the velocity of an object changes when it is subjected to external forces. For example, when Venus Williams serves a tennis ball, she applies a force to it using her racquet, causing it to accelerate from rest to a high speed in a very short time.

Force Calculation

To calculate the force exerted in a situation, such as hitting a tennis ball, we need to consider both the mass of the object and its acceleration. By using Newton's Second Law, we can find the average force applied:

• The mass of the tennis ball in the exercise is given as 57 g, which is 0.057 kg after conversion.
• The acceleration of the ball for Venus's serve and her opponent's return can be found using their respective velocities and contact times.

For Venus's serve, we calculate the acceleration of the ball to be approximately 227 \( m/s^2 \), applying:

• Force for Venus = Mass x Acceleration = 0.057 kg x 227 \( m/s^2 \) = 12.94 N

For the opponent's racquet, with a different acceleration of 316 \( m/s^2 \), the required force to return the serve is:

• Force for opponent = 0.057 kg x 316 \( m/s^2 \) = 18.04 N

These calculations demonstrate the significant forces involved in high-speed tennis play.

Unit Conversion

When working on physics problems, unit conversion is often necessary to ensure calculations are consistent. It allows us to use standard units such as meters, seconds, and kilograms:

• Mass was converted from grams to kilograms by dividing by 1000: 57 g = 0.057 kg.
• Velocity conversion, from miles per hour to meters per second, used the conversion factor: 1 mph = 0.44704 m/s.

For example:

• Venus's serve: 127 mph x 0.44704 = 56.75 m/s
• Opponent's return: 50 mph x 0.44704 = 22.35 m/s

By ensuring all units are correctly converted, we can accurately apply formulas such as Newton's Second Law in our calculations, maintaining the integrity of our results.

Acceleration

Acceleration describes how quickly the velocity of an object changes. In this exercise, it is crucial for calculating the forces exerted on the tennis ball. The formula for acceleration is:

• \( a = \frac{\text{Final velocity} - \text{Initial velocity}}{\text{Time}} \)

For Venus Williams's serve, the initial velocity is 0 \( m/s \) because she starts from a standstill. So, the acceleration is:

• \( a = \frac{56.75 \, m/s - 0}{0.250 \, s} = 227 \, m/s^2 \)

Her opponent, however, must reverse the ball's direction:

• Initial velocity is negative, as it comes towards the opponent: -56.75 \( m/s \)
• Final velocity for the return is 22.35 \( m/s \)
• Hence, \( a = \frac{22.35 + 56.75}{0.250} = 316.42 \, m/s^2 \)

These accelerations are large, reflecting the high-speed nature of professional tennis serves and returns.