Problem 37 - College Physics

Question

A catcher catches a 145 g baseball traveling horizontally at 36.0 \(\mathrm{m} / \mathrm{s}\) . (a) How large an impulse does the ball give to the catcher? (b) If the ball takes 20 \(\mathrm{ms}\) to stop once it is in contact with the catcher's glove, what average force did the ball exert on the catcher?

Step-by-Step Solution

Answer

(a) Impulse is -5.22 kg m/s. (b) Average force is -261 N.

Step 1: Understanding Impulse

Impulse is defined as the change in momentum of an object when a force is applied over a period of time. The formula for impulse \( J \) is given by \( J = \Delta p \), where \( \Delta p \) is the change in momentum.

Step 2: Calculate Initial Momentum

The initial momentum \( p_i \) of the baseball can be calculated using the formula \( p = m \, v \), where \( m \) is mass and \( v \) is velocity. Here, \( m = 145 \, \text{g} = 0.145 \, \text{kg} \) and \( v = 36.0 \, \text{m/s} \). Thus, \( p_i = 0.145 \, \text{kg} \times 36.0 \, \text{m/s} = 5.22 \, \text{kg} \, \text{m/s} \).

Step 3: Determine Final Momentum

Since the ball is caught and brought to a stop, its final velocity is 0 \( \text{m/s} \), making the final momentum \( p_f = 0 \text{kg m/s} \).

Step 4: Calculate Impulse

The impulse can be calculated as the change in momentum, which is \( J = \Delta p = p_f - p_i = 0 - 5.22 = -5.22 \, \text{kg m/s} \). The negative sign indicates direction change.

Step 5: Understanding Average Force

Average force \( F_{\text{avg}} \) exerted by the ball is related to impulse by the equation \( J = F_{\text{avg}} \, \Delta t \), where \( \Delta t \) is the time duration over which the force acts.

Step 6: Convert Time Units and Calculate Average Force

Convert the time \( \Delta t = 20 \, \text{ms} = 0.020 \, \text{s} \). The average force is then given by \( F_{\text{avg}} = \frac{J}{\Delta t} = \frac{-5.22 \, \text{kg m/s}}{0.020 \, \text{s}} = -261 \, \text{N} \). The negative sign indicates direction.

Key Concepts

Momentum

Momentum is a fundamental concept in physics that helps describe the motion of objects. It is the product of an object's mass and its velocity. Mathematically, momentum is expressed as \( p = m \times v \), where \( p \) is momentum, \( m \) is mass, and \( v \) is velocity.
Momentum is a vector quantity, which means it has both magnitude and direction. This makes it important to consider both speed and direction when calculating momentum.

• Example: A 145 g baseball moving at 36.0 m/s has a momentum of 5.22 kg m/s because \( p = 0.145 \times 36.0 \).
• If the velocity direction changes, the momentum direction changes too, while its magnitude might remain the same.

Understanding momentum is key to solving many physics problems related to motion, collisions, and impulse.

Average Force

The average force exerted on an object during a time interval is critical in understanding how an object accelerates or decelerates. The average force can be calculated when impulse is known, using the formula \( F_{\text{avg}} = \frac{J}{\Delta t} \). Here, \( J \) is the impulse, and \( \Delta t \) is the time duration over which the force acts.
In the case of our baseball scenario, the impulse was calculated as \( -5.22 \; \text{kg m/s} \), and the time duration was 20 milliseconds, or 0.020 seconds.

• This leads to an average force calculation: \( F_{\text{avg}} = \frac{-5.22}{0.020} = -261 \; \text{N} \).
• The negative sign indicates that the force direction is opposing the initial direction of the baseball.

Average force provides insight into how strongly or gently an object has been acted upon in a given scenario.

Impulse-Momentum Theorem

The Impulse-Momentum Theorem is a powerful tool in physics that relates impulse to the change in an object's momentum. It states that the impulse \( J \) exerted on an object is equal to the change in its momentum \( \Delta p \).
This can be expressed mathematically as \( J = \Delta p \), where \( \Delta p = p_f - p_i \), with \( p_f \) as final momentum and \( p_i \) as initial momentum.

• In the example of the baseball, the impulse calculated was \( J = -5.22 \; \text{kg m/s} \), indicating that the momentum of the baseball reduced to zero when it was caught.
• The theorem illustrates that the force applied over time results in a momentum change, crucial for understanding scenarios like catching, colliding, or stopping a moving object.

This theorem simplifies analyzing situations in which forces act over time, making it easier to understand motion and force relationships.