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Chapter 7: Problem 75

A tennis ball of mass \(0.060 \mathrm{kg}\) is served. It strikes the ground with a velocity of \(54 \mathrm{m} / \mathrm{s}(120 \mathrm{mi} / \mathrm{h})\) at an angle of \(22^{\circ}\) below the horizontal. Just after the bounce it is moving at \(53 \mathrm{m} / \mathrm{s}\) at an angle of \(18^{\circ}\) above the horizontal. If the interaction with the ground lasts \(0.065 \mathrm{s}\), what average force did the ground exert on the ball?

Short Answer

Expert verified
Answer: To find the average force exerted by the ground, follow these steps: 1. Find the initial and final velocities in the x and y components. 2. Calculate the change in velocity for each component. 3. Compute the impulse in each component. 4. Determine the average force exerted in each component using the given duration. 5. Combine the force components to find the total average force. Using these steps, you will be able to find the average force exerted by the ground on the tennis ball during its interaction with the ground.

Step by step solution

01

1. Identify initial and final velocities

First, we need to find the initial velocity (before the bounce) and the final velocity (after the bounce) of the tennis ball in the x and y components separately. Before bounce: - Velocity: \(54 \mathrm{m/s}\) - Angle: \(22^\circ\) below horizontal (negative since it's going downward) After bounce: - Velocity: \(53 \mathrm{m/s}\) - Angle: \(18^\circ\) above horizontal (positive since it's going upward)
02

2. Calculate the velocity components

Next, we will calculate the x and y components of the initial and final velocities. Initial velocity components: - \(v_{ix} = 54 \cos(22^{\circ})\) - \(v_{iy} = -54 \sin(22^{\circ})\) Final velocity components: - \(v_{fx} = 53 \cos(18^{\circ})\) - \(v_{fy} = 53 \sin(18^{\circ})\)
03

3. Compute changes in velocities

Now calculate the change in velocity for the x and y components, as it will be used to calculate the impulse. \(\Delta v_{x} = v_{fx} - v_{ix}\) \(\Delta v_{y} = v_{fy} - v_{iy}\)
04

4. Calculate the impulse

The impulse experienced by the tennis ball is the change in its momentum. Since impulse is the product of force and time, we can now compute the impulse in both the x and y components. \(I_{x} = m\Delta v_{x}\) \(I_{y} = m\Delta v_{y}\)
05

5. Determine average force exerted

We are given the duration of the interaction between the ball and the ground (\(0.065 \mathrm{s}\)), so we can find the average force exerted in each component. \(F_{avg_x} = \frac{I_{x}}{0.065}\) \(F_{avg_y} = \frac{I_{y}}{0.065}\)
06

6. Combine force components

Finally, we'll find the magnitude of the total average force exerted on the ball by combining the x and y components. \(F_{avg} = \sqrt{F_{avg_x}^2 + F_{avg_y}^2}\) This will give us the average force exerted by the ground on the tennis ball during its interaction with the ground.

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