Problem 23 - College Physics

Question

When bunting, a baseball player uses the bat to change both the speed and direction of the baseball. (a) Will the magnitude of the change in momentum of the baseball before and after the bunt be (1) greater than the magnitude of the momentum of the baseball either before or after the bunt, (2) equal to the difference between the magnitudes of momenta of the baseball before and after the bunt, or (3) equal to the sum of the magnitudes of momenta of the baseball before and after the bunt? Why? (b) The baseball has a mass of \(0.16 \mathrm{~kg}\); its speeds before and after the bunt are \(15 \mathrm{~m} / \mathrm{s}\) and \(10 \mathrm{~m} / \mathrm{s}\) respectively; the bunt lasts \(0.025 \mathrm{~s}\). What is the change in momentum of the baseball? (c) What is the average force on the ball by the bat?

Step-by-Step Solution

Answer

(a) The change in momentum is equal to the difference in magnitudes. (b) The change in momentum is 0.8 kg·m/s. (c) The average force is 32 N.

Step 1: Understanding the Problem

We need to analyze the change in momentum of the baseball when a player bunts, using the given speeds and mass of the baseball. Additionally, we must determine the average force exerted by the bat during the bunt.

Step 2: Conceptual Explanation

For part (a), consider the formula for change in momentum, which is given by the difference between the final and initial momentum vectors. The magnitude of this change is not simply the sum or difference of their magnitudes because momentum is a vector quantity, and direction must be considered.

Step 3: Calculate Initial and Final Momentum

The initial momentum is determined by the formula \( p_i = m imes v_i \) and the final momentum by \( p_f = m imes v_f \). Here, \( v_i = 15 \, \text{m/s} \) and \( v_f = 10 \, \text{m/s} \). Thus, \( p_i = 0.16 \, \text{kg} \times 15 \, \text{m/s} = 2.4 \, \text{kg·m/s} \), and \( p_f = 0.16 \, \text{kg} \times 10 \, \text{m/s} = 1.6 \, \text{kg·m/s} \).

Step 4: Calculate Change in Momentum

The change in momentum is given by \( \Delta p = p_f - p_i \). Substituting the values, we get \( \Delta p = 1.6 \, \text{kg·m/s} - 2.4 \, \text{kg·m/s} = -0.8 \, \text{kg·m/s} \). Therefore, the magnitude of change in momentum is \( 0.8 \, \text{kg·m/s} \).

Step 5: Determine Average Force

The average force exerted by the bat can be found using the impulse-momentum theorem: \( F_{\text{avg}} \times \Delta t = \Delta p \). Rearranging for the force, we have \( F_{\text{avg}} = \frac{\Delta p}{\Delta t} \). Plugging in the values, \( F_{\text{avg}} = \frac{-0.8 \, \text{kg·m/s}}{0.025 \, \text{s}} = -32 \, \text{N} \). The negative sign indicates direction.

Key Concepts

Change in Momentum

Momentum is a crucial concept in physics. It essentially measures how difficult it is to stop a moving object. When you're looking at a baseball being bunted, for instance, you're essentially witnessing a change in the momentum of the ball. The change in momentum, denoted as \( \Delta p \), is calculated as the difference between the final momentum \( p_f \) and the initial momentum \( p_i \).

To find the change in momentum, you can use the formula:

• \( \Delta p = p_f - p_i \)

In the case of the baseball, with its initial speed \( v_i = 15 \, \text{m/s} \) and final speed \( v_f = 10 \, \text{m/s} \), the initial momentum \( p_i = 0.16 \, \text{kg} \times 15 \, \text{m/s} = 2.4 \, \text{kg} \cdot \text{m/s} \) and the final momentum \( p_f = 0.16 \, \text{kg} \times 10 \, \text{m/s} = 1.6 \, \text{kg} \cdot \text{m/s} \). Thus, the change in momentum \( \Delta p = 1.6 \, \text{kg} \cdot \text{m/s} - 2.4 \, \text{kg} \cdot \text{m/s} = -0.8 \, \text{kg} \cdot \text{m/s} \).

Even though the outcome is negative, it merely indicates a change in direction.

Impulse-Momentum Theorem

The impulse-momentum theorem is a helpful and powerful tool in physics. This theorem connects the concepts of force, time, and change in momentum over a specified duration. It states that the impulse exerted on an object will cause a change in the object's momentum. Essentially, this can be described with the equation:

• \( F_{\text{avg}} \times \Delta t = \Delta p \)

Where \( F_{\text{avg}} \) is the average force applied, \( \Delta t \) is the time duration, and \( \Delta p \) is the change in momentum. The impulse, \( F_{\text{avg}} \times \Delta t \), represents the total effect of a force over a period, which translates directly into momentum change.

In the baseball example, the bunt lasts \( 0.025 \, \text{s} \) and results in a momentum change \( \Delta p = -0.8 \, \text{kg} \cdot \text{m/s} \). This is how force effectively changes an object's state of motion over time.

Average Force

Average force is what you get when you spread out the total effect of a force over the time during which it acts. Determining the average force is quite straightforward once you have the values of the change in momentum and the time duration available. It is calculated using this formula derived from the impulse-momentum theorem:

• \( F_{\text{avg}} = \frac{\Delta p}{\Delta t} \)

For our baseball bunt scenario, you find this by measuring the time for which the bat contacts the ball, \( \Delta t = 0.025 \, \text{s} \). Then, take the previously calculated change in momentum, \( \Delta p = -0.8 \, \text{kg} \cdot \text{m/s} \), and compute:

\( F_{\text{avg}} = \frac{-0.8 \, \text{kg} \cdot \text{m/s}}{0.025 \, \text{s}} = -32 \, \text{N} \).

The negative sign signifies the force's direction, not its magnitude or size. This means the force exerted by the bat was in the opposite direction to the ball's initial motion. Understanding average force explains how forces can vary in magnitude and align with changes in object motion.