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Chapter 9: Problem 10

A tennis player receives a shot with the ball \((0.0600 \mathrm{kg})\) traveling horizontally at \(50.0 \mathrm{m} / \mathrm{s}\) and returns the shot with the ball traveling horizontally at \(40.0 \mathrm{m} / \mathrm{s}\) in the opposite direction. (a) What is the impulse delivered to the ball by the racquet? (b) What work does the racquet do on the ball?

Short Answer

Expert verified
The racquet delivers an impulse of \(-5.40 \mathrm{kg}\cdot\mathrm{m} / \mathrm{s}\), and does \( -27.0 \mathrm{J}\) of work on the ball.

Step by step solution

01

Calculate the initial and final momenta

The momentum of an object can be found using the formula \( p = mv \), where \( m \) is mass and \( v \) is velocity. The initial momentum \( p_i \) of the ball is \( (0.0600 \mathrm{kg})(50.0 \mathrm{m} / \mathrm{s}) = 3.00 \mathrm{kg}\cdot\mathrm{m} / \mathrm{s} \). The final momentum \( p_f \) is \( (0.0600 \mathrm{kg})(-40.0 \mathrm{m} / \mathrm{s}) = -2.40 \mathrm{kg}\cdot\mathrm{m} / \mathrm{s} \), wherein the negative sign indicates the direction opposite to initial momentum.
02

Calculate the impulse

Impulse is simply the change in momentum of an object. So, impulse \( J \) would be \( p_f - p_i \) which equals \( -2.40 \mathrm{kg}\cdot\mathrm{m} / \mathrm{s} - 3.00 \mathrm{kg}\cdot\mathrm{m} / \mathrm{s} = -5.40 \mathrm{kg}\cdot\mathrm{m} / \mathrm{s} \). The negative sign shows that the impulse was in the opposite direction of the initial momentum.
03

Calculate the kinetic energy before and after the impact

The kinetic energy of the object can be found by using the formula \( KE = \frac{1}{2}mv^2 \). So, the initial kinetic energy \( KE_{i} \) of the ball is \( \frac{1}{2}(0.0600 \mathrm{kg})(50.0 \mathrm{m} / \mathrm{s})^2 = 75.0 \mathrm{J} \) and the final kinetic energy \( KE_{f} \) is \( \frac{1}{2}(0.0600 \mathrm{kg})(40.0 \mathrm{m} / \mathrm{s})^2 = 48.0 \mathrm{J} \).
04

Calculate the work done

The work done by the racquet on the ball can be calculated as the change in its kinetic energy, based on the work-energy theorem. So, work \(W\) would be \( KE_{f} - KE_{i} \) which equals \( 48.0 \mathrm{J} - 75.0 \mathrm{J} = -27.0 \mathrm{J} \). The negative sign indicates that the work was done in the direction opposite to the initial motion of the ball.

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Key Concepts

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

Work-Energy Theorem
The work-energy theorem is a fundamental concept in physics that relates the work done by forces acting on an object to the change in its kinetic energy. In simple terms, when you apply a force over a distance to an object, you are doing work on it, which causes a change in its kinetic energy.

In the context of our tennis ball example, the theorem allows us to understand how the racquet does work on the ball to change its motion. When the tennis racquet hits the ball, it applies a force over the time during which they are in contact. This force alters the ball鈥檚 velocity and consequently, its kinetic energy.

If the initial kinetic energy is greater than the final kinetic energy, it means the work done is negative, indicating that the ball is losing energy. The negative work done by the racquet occurs because it acts in the opposite direction of the ball's motion. This results in the ball slowing down, thus reducing its kinetic energy.
Kinetic Energy
Kinetic energy, often abbreviated as KE, is the energy an object possesses due to its motion. It is calculated using the formula: \[ KE = \frac{1}{2}mv^2 \] where \( m \) is the mass and \( v \) is the velocity of the object.

In the tennis ball problem, we initially calculated the kinetic energies both before and after it was hit by the racquet. Initially, with a velocity of 50.0 m/s, the ball had a kinetic energy of 75.0 J. After the ball was returned at a velocity of 40.0 m/s, its kinetic energy dropped to 48.0 J.

This decrease in kinetic energy illustrates that 27.0 J of energy was transferred out of the ball. That energy is related to the work done by the racquet. Understanding kinetic energy fluctuations can give insights into how energy is transformed and conserved in different situations.
Momentum Change
Momentum is a measure of the motion of an object and is the product of its mass and velocity. The formula is: \( p = mv \).

When dealing with momentum, it is crucial to consider both magnitude and direction, as momentum is a vector quantity. In the scenario involving the tennis ball, the initial momentum was calculated as 3.00 kg路m/s. The final momentum was -2.40 kg路m/s, with the negative indicating a change in direction.

The change in the ball's momentum \( \Delta p \) is found by subtracting the initial momentum from the final momentum. This gives us an impulse value of -5.40 kg路m/s, which signifies the total momentum transferred to the ball during its interaction with the racquet. This implies the racquet exerted a force that significantly changed the ball's state of motion, highlighting the relationship between force, time, and momentum change.

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Most popular questions from this chapter

A rocket has total mass \(M_{i}=360 \mathrm{kg},\) including \(330 \mathrm{kg}\) of fuel and oxidizer. In interstellar space it starts from rest. Its engine is turned on at time \(t=0,\) and it puts out exhaust with relative speed \(v_{e}=1500 \mathrm{m} / \mathrm{s}\) at the constant rate \(2.50 \mathrm{kg} / \mathrm{s} .\) The burn lasts until the fuel runs out, at time \(330 \mathrm{kg} /(2.5 \mathrm{kg} / \mathrm{s})=132 \mathrm{s} .\) Set up and carry out a computer analysis of the motion according to Euler's method. Find (a) the final velocity of the rocket and (b) the distance it travels during the burn.

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A railroad car of mass \(2.50 \times 10^{4} \mathrm{kg}\) is moving with a speed of \(4.00 \mathrm{m} / \mathrm{s} .\) It collides and couples with three other coupled railroad cars, each of the same mass as the single car and moving in the same direction with an initial speed of \(2.00 \mathrm{m} / \mathrm{s} .\) (a) What is the speed of the four cars after the collision? (b) How much mechanical energy is lost in the collision?

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