Problem 31 - College Physics

Question

An incoming 0.14 -kg baseball has a speed of \(45 \mathrm{~m} / \mathrm{s}\). The batter hits the ball, giving it a speed of \(60 \mathrm{~m} / \mathrm{s}\). If the contact time is \(0.040 \mathrm{~s},\) what is the average force of the bat on the ball?

Step-by-Step Solution

Answer

The average force is 52.5 N.

Step 1: Identify known values

Firstly, gather all the values given in the problem. The mass of the baseball is \( m = 0.14 \, \text{kg} \), the initial speed \( v_i = 45 \, \text{m/s} \), the final speed \( v_f = 60 \, \text{m/s} \), and the contact time \( \Delta t = 0.040 \, \text{s} \).

Step 2: Calculate the change in velocity

Use the initial and final velocities to find the change in velocity (\( \Delta v \)) of the baseball. \( \Delta v = v_f - v_i = 60 \, \text{m/s} - 45 \, \text{m/s} = 15 \, \text{m/s} \).

Step 3: Determine the change in momentum

The change in momentum (\( \Delta p \)) is given by \( \Delta p = m \cdot \Delta v \). Substitute the known values into the equation: \( \Delta p = 0.14 \, \text{kg} \times 15 \, \text{m/s} = 2.1 \, \text{kg} \cdot \text{m/s} \).

Step 4: Calculate the average force

The average force (\( F \)) on the baseball is the change in momentum divided by the contact time: \( F = \frac{\Delta p}{\Delta t} \). Substituting the known values: \( F = \frac{2.1 \, \text{kg} \cdot \text{m/s}}{0.040 \, \text{s}} = 52.5 \, \text{N} \).

Key Concepts

Momentum

Momentum is a fundamental concept in physics, closely linked to both mass and velocity. It encapsulates the idea of how much "motion" an object possesses. Mathematically, momentum (\( p \)) is expressed as the product of an object's mass (\( m \)) and its velocity (\( v \)).

• Formula: \( p = m \, \cdot \, v \)
• Units: kg⋅m/s

A moving baseball is a great example to understand momentum in action. A heavier object moving at the same speed as a lighter one will have more momentum. Similarly, a faster object will have more momentum than a slower one, given equal mass. In our baseball scenario, momentum changes as the speed changes from 45 m/s to 60 m/s due to an external force (the bat).
The concept helps us quantify motion precisely, offering a base to calculate how forces will affect an object's state of movement.

Change in Velocity

Change in velocity (\( \Delta v \)) is essential for understanding how speed or direction alterations affect an object. In equations and analysis, it helps in measuring the impact of a force over time.

• Formula: \( \Delta v = v_f - v_i \)

Here, \( v_f \) is the final velocity, and \( v_i \) is the initial velocity. For our baseball, it moves from an initial speed of 45 m/s to a final speed of 60 m/s. Thus, \( \Delta v = 60 \, \text{m/s} - 45 \, \text{m/s} = 15 \, \text{m/s} \).
The change in velocity directly influences momentum changes, and as shown, becomes crucial while calculating the average force applied during the contact time with the bat. It signifies the overall impact made on the baseball’s speed by the external force.

Impulse

Impulse is closely tied with the concepts of force and time. It describes how much momentum changes as a result of a force acting over a certain period. Impulse is crucial for understanding collisions and impact scenarios.

• Formula: Impulse (\( J \)) = \( F \times \Delta t \)
• Also expressed as the Change in Momentum: \( J = \Delta p \)

In the baseball example, we calculated the change in momentum to be \( 2.1 \text{ kg}\cdot\text{m/s} \). This is equivalent to the impulse involved because the change resulted from the bat's action during the 0.040 s of contact.
Hence, average force is derived using impulse over the time interval, emphasizing that both magnitude and duration of force impact an object's motion significantly. Understanding impulse helps grasp how any object's motion can be altered or controlled through force application over time.