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Ch 11: Impulse and Momentum
Knight Calc - Physics for Scientists and Engineers 5th Edition
Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook
All textbooksKnight Calc 5th EditionCh 11: Impulse and MomentumProblem 39b
Chapter 11, Problem 39b

A tennis player swings her 1000g racket with a speed of 10 m/s. She hits a 60g tennis ball that was approaching her at a speed of 20 m/s. The ball rebounds at 40 m/s. If the tennis ball and racket are in contact for 10 ms, what is the average force that the racket exerts on the ball? How does this compare to the gravitational force on the ball?

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Step 1: Identify the given values and convert them into standard units if necessary. The mass of the tennis ball is \( m_b = 60 \, \text{g} = 0.060 \, \text{kg} \), the initial velocity of the ball is \( v_{b,i} = -20 \, \text{m/s} \) (negative because it is approaching the racket), and the final velocity of the ball is \( v_{b,f} = 40 \, \text{m/s} \). The time of contact is \( \Delta t = 10 \, \text{ms} = 0.010 \, \text{s} \).
Step 2: Use the impulse-momentum theorem to calculate the average force. The impulse-momentum theorem states that \( F_{\text{avg}} \Delta t = \Delta p \), where \( \Delta p \) is the change in momentum of the ball. First, calculate the initial momentum of the ball: \( p_i = m_b v_{b,i} \). Then calculate the final momentum of the ball: \( p_f = m_b v_{b,f} \). The change in momentum is \( \Delta p = p_f - p_i \).
Step 3: Rearrange the impulse-momentum equation to solve for the average force: \( F_{\text{avg}} = \frac{\Delta p}{\Delta t} \). Substitute the values of \( \Delta p \) and \( \Delta t \) into this equation to find the average force exerted by the racket on the ball.
Step 4: Calculate the gravitational force on the ball for comparison. The gravitational force is given by \( F_g = m_b g \), where \( g = 9.8 \, \text{m/s}^2 \) is the acceleration due to gravity. Substitute the mass of the ball into this equation to find \( F_g \).
Step 5: Compare the average force exerted by the racket to the gravitational force on the ball. This can be done by calculating the ratio \( \frac{F_{\text{avg}}}{F_g} \) to determine how many times greater the racket's force is compared to the gravitational force.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Momentum

Momentum is the product of an object's mass and its velocity, represented by the equation p = mv. In collisions, the total momentum before the event equals the total momentum after, according to the law of conservation of momentum. This principle is crucial for analyzing the interaction between the tennis racket and the ball, as it helps determine the changes in their velocities during the impact.
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Impulse

Impulse is defined as the change in momentum of an object when a force is applied over a period of time, expressed as J = FΔt. It is directly related to the average force exerted during a collision. In this scenario, calculating the impulse experienced by the tennis ball will allow us to find the average force exerted by the racket during the brief contact time.
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Gravitational Force

Gravitational force is the attractive force between two masses, calculated using Newton's law of universal gravitation, F = mg, where m is mass and g is the acceleration due to gravity (approximately 9.81 m/s² on Earth). Comparing the average force exerted by the racket on the ball to the gravitational force helps to understand the relative magnitudes of these forces and their effects on the ball's motion.
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Related Practice
Textbook Question

A 60 g tennis ball with an initial speed of 32 m/s hits a wall and rebounds with the same speed. FIGURE P11.40 shows the force of the wall on the ball during the collision. What is the value of Fmax , the maximum value of the contact force during the collision?

Textbook Question

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Textbook Question

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Textbook Question

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