Problem 58 - College Physics

Question

Tennis players sometimes leap into the air to return a volley. (a) If a \(57 \mathrm{~g}\) tennis ball is traveling horizontally at \(72 \mathrm{~m} / \mathrm{s}\) (which does occur), and a \(61 \mathrm{~kg}\) tennis player leaps vertically upward and hits the ball, causing it to travel at \(45 \mathrm{~m} / \mathrm{s}\) in the reverse direction, how fast will her center of mass be moving horizontally just after hitting the ball? (b) If, as is reasonable, her racket is in contact with the ball for \(30.0 \mathrm{~ms}\), what force does her racket exert on the ball? What force does the ball exert on the racket?

Step-by-Step Solution

Answer

The player's center of mass moves at approximately 0.109 m/s horizontally. The racket exerts -222.3 N on the ball, and the ball exerts 222.3 N on the racket.

Step 1: Identify Initial Momentum of the System

The initial momentum of the system consists of the momentum of the ball moving horizontally. Since the player's vertical leap does not affect horizontal momentum, her initial horizontal momentum is zero. The momentum of the ball is calculated as follows:\[ p_{ball ext{ }initial} = m_{ball} imes v_{ball ext{ }initial} = 0.057 ext{ kg} imes 72 ext{ m/s} = 4.104 ext{ kg m/s} \]

Step 2: Apply Conservation of Momentum

In the absence of external horizontal forces, the horizontal momentum of the system is conserved. Thus, the total initial momentum equals the total final momentum. The final momentum is the sum of the player's and the ball's momenta:\[ p_{total ext{ }initial} = p_{player ext{ }final} + p_{ball ext{ }final} \]Given the ball's final velocity:\[ p_{ball ext{ }final} = m_{ball} imes (-v_{ball ext{ }final}) = 0.057 ext{ kg} imes (-45 ext{ m/s}) = -2.565 ext{ kg m/s} \]

Step 3: Solve for Player's Final Velocity

Using the conservation equation:\[ p_{total ext{ }initial} = 4.104 ext{ kg m/s} = p_{player ext{ }final} + (-2.565 ext{ kg m/s}) \]Therefore, the player's final horizontal momentum:\[ p_{player ext{ }final} = 4.104 + 2.565 = 6.669 ext{ kg m/s} \]To find the player's velocity:\[ v_{player ext{ }final} = \frac{p_{player ext{ }final}}{m_{player}} = \frac{6.669 ext{ kg m/s}}{61 ext{ kg}} \approx 0.109 m/s \]

Step 4: Calculate Force Exerted by Racket

To find the force exerted by the racket on the ball, use the impulse-momentum theorem, which states that the change in momentum equals the impulse:\[ F \times \Delta t = \Delta p_{ball} \]The change in momentum of the ball is:\[ \Delta p_{ball} = p_{ball ext{ }final} - p_{ball ext{ }initial} = -2.565 - 4.104 = -6.669 ext{ kg m/s} \]Given the contact time \( \Delta t = 0.030 \text{ s} \), the force \( F \) is:\[ F = \frac{-6.669 ext{ kg m/s}}{0.030 ext{ s}} = -222.3 ext{ N} \]

Step 5: Newton's Third Law - Racket's and Ball's Reaction Forces

According to Newton's Third Law, the force exerted by the racket on the ball is equal and opposite to the force exerted by the ball on the racket. Hence, the force exerted by the racket on the ball is \(-222.3 ext{ N}\) and by the ball on the racket, it is \(222.3 ext{ N}\).

Key Concepts

Impulse-Momentum Theorem

The Impulse-Momentum Theorem is a fundamental concept in physics that connects the change in momentum of an object to the impulse applied to it. Impulse is defined as the product of the force applied to an object and the time duration over which it is applied. The theorem can be mathematically expressed as: \[ F \times \Delta t = \Delta p \] where

• \( F \) is the force applied,
• \( \Delta t \) is the time duration, and
• \( \Delta p \) is the change in momentum.

In the context of the exercise, the tennis player exerts a force on the ball, changing its momentum. Initially, the tennis ball's momentum can be calculated using its mass and velocity. As a result of the player's action, the ball reverses its direction and its velocity decreases, indicating a change in momentum. By knowing the time of contact, we can calculate the average force exerted. The large force over a short duration results in the desired change in the ball's momentum. This highlights the importance of impulse in changing the motion of objects.

Newton's Third Law

Newton's Third Law of Motion states that for every action, there is an equal and opposite reaction. This law is essential in understanding interactions between two objects. When a force is applied by one object onto another, the second object applies an equal but opposite force back onto the first object. This principle is vividly illustrated in the step-by-step solution. When the player hits the tennis ball with her racket, she exerts a force on the ball. This action, according to Newton's Third Law, results in the ball exerting an equal magnitude force back on the racket. In numerical terms from the problem, the force the racket exerts on the ball is \(-222.3 \, \text{N}\) and equally, the ball exerts \(222.3 \, \text{N}\) back on the racket. Even though they act in opposite directions, their magnitudes remain identical. This concept explains why even if the ball experiences a negative force acting against its initial motion, the force itself remains strong because of the abrupt reversal of direction, demonstrating how forces in interactions affect both involved bodies.

Physics Problem Solving

Solving physics problems often requires a disciplined approach, involving a few key steps to understand and solve the problem accurately. In this problem, we utilize a systematic method:

• Identify the known values and what you need to find. For instance, here we know the mass of the tennis ball, its initial and final velocity, and the contact time.
• Understanding the applicable physics principles. The concepts of momentum conservation, impulse, and Newton's laws are essential in this example.
• Implement mathematical equations and solve them step by step. Calculations involve using the mass, velocity, force, and time to find the resulting values such as the final velocity and forces involved.
• Check units and ensure the results are logical. The step-by-step solution confirms that momentum is conserved and Newton's Third Law is satisfied.

By breaking down the problem into these components, complex physics problems become more manageable. This structured approach helps ensure that all parts of the problem are addressed and solved effectively, encompassing everything from mathematical calculations to theoretical understanding.