Problem 34 - 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 Ns; (b) Average force is 261 N.

Step 1: Identify the Formula for Impulse

Impulse is defined as the change in momentum of an object. Mathematically, it is given by the product of mass and change in velocity: \[ J = m \cdot \Delta v \] where \( J \) is the impulse, \( m \) is mass, and \( \Delta v \) is the change in velocity.

Step 2: Convert Mass to Kilograms

To use standard units in calculations, convert the mass of the baseball from grams to kilograms: \[ 145 \text{ g} = 0.145 \text{ kg} \].

Step 3: Calculate Change in Velocity

Since the baseball comes to a stop, its initial velocity is 36.0 m/s and final velocity is 0 m/s. Therefore, the change in velocity \( \Delta v \) is: \[ \Delta v = 0 - 36.0 \text{ m/s} = -36.0 \text{ m/s} \].

Step 4: Calculate Impulse to the Catcher

Substitute the mass and change in velocity into the impulse formula: \[ J = 0.145 \text{ kg} \times (-36.0 \text{ m/s}) = -5.22 \text{ Ns} \]. The impulse is 5.22 Ns in magnitude (direction is implicit as stopping).

Step 5: Identify Time and Formula for Average Force

Average force is calculated using the impulse-momentum theorem, which states \[ F_{avg} = \frac{J}{\Delta t} \] where \( F_{avg} \) is the average force and \( \Delta t \) is the time duration.

Step 6: Convert Time to Seconds

Convert the given time from milliseconds to seconds: \[ 20 \text{ ms} = 0.020 \text{ s} \].

Step 7: Calculate Average Force

Substitute the impulse and time duration into the formula: \[ F_{avg} = \frac{-5.22 \text{ Ns}}{0.020 \text{ s}} = -261 \text{ N} \]. The average force exerted by the ball on the catcher is 261 N (the direction is implicit as opposing motion).

Key Concepts

Average Force

When we talk about force in physics, one key concept is that of the "average force". This measure gives us an idea of the consistent force exerted over a period of time during a certain event, such as when a baseball hits a catcher's mitt.

The formula to calculate average force uses the impulse and the time over which this impulse acts. The impulse-momentum theorem provides this equation:

• The formula is given by \[ F_{avg} = \frac{J}{\Delta t} \]
• Here, \( J \) represents the impulse, and \( \Delta t \) is the time duration over which the force acts.

In simpler terms, average force is the force that would produce the same change in momentum if it acted constantly over the time interval \( \Delta t \). It helps simplify calculations when dealing with varying forces over time.

Impulse-Momentum Theorem

The Impulse-Momentum Theorem is a fundamental principle in physics. It reveals the relationship between impulse and the change in momentum of an object.

Momentum, which is the product of mass and velocity, changes when a force is applied over time. Impulse is basically this force applied over a time period. Therefore, impulse can effectively change an object's momentum.

• The theorem states: \[ J = \Delta p \]
• Where \( J \) is the impulse and \( \Delta p \) is the change in momentum, calculated as the product of mass \( m \) and change in velocity \( \Delta v \).

This theorem helps us connect the dots between forces acting on an object and how it affects motion, which is crucial in analyzing real-world physics problems.

Change in Velocity

The change in velocity is a pivotal concept in physics, especially when dealing with motion and momentum. Velocity measures speed in a specific direction. So, a change in velocity can occur due to an alteration in speed, direction, or both.To calculate change in velocity, consider:

• Initial velocity, \( v_i \), at which an object begins.
• Final velocity, \( v_f \), which is the speed after the event (in this case, when the baseball stops).

The change in velocity is then given by:\[ \Delta v = v_f - v_i \]Here, since the baseball comes to a stop, \( v_f = 0 \), resulting in a negative change reflecting a stop or deceleration.

Physics Problem Solving

Physics problem solving is an art form that involves logical reasoning, mathematical equations, and a systematic approach to uncover solutions to complex problems. It breaks down to understanding the problem, identifying what is known and unknown, and applying the right principles and formulas.

Key steps include:

• Understanding concepts: Grasp the core ideas involved—like impulse, force, and momentum in the current problem.
• Converting units: Ensure all quantities are in compatible units (e.g., mass in kilograms, time in seconds).
• Applying formulas: Use the correct equations (like impulse and average force formulas) to transition from known to unknown quantities.
• Checking results: Evaluate if the solution is logical and fits the context of the problem.

Through such careful steps, students can gain confidence and proficiency in tackling physics challenges effectively.