Problem 23 - Applied Physics

Question

A standard \(5.0\) -oz baseball is thrown and reaches a batter with a velocity of \(85 \mathrm{mi} / \mathrm{h}\) when it is struck with a bat that causes it to reverse direction with a velocity of \(95 \mathrm{mi} / \mathrm{h}\). Find (a) the impulse and (b) the force exerted on the baseball if the bat is in contact with the ball for \(7.0 \mathrm{~ms}\).

Step-by-Step Solution

Answer

The impulse is -11.403 Ns, and the force exerted is -1629.0 N.

Step 1: Convert Units from Ounces to Kilograms

First, convert the weight of the baseball from ounces to kilograms. Given that there are 16 ounces in a pound and 1 pound is approximately 0.453592 kg, we can calculate the weight in kilograms. \[ 5.0 \text{ oz} \times \frac{1 \text{ lb}}{16 \text{ oz}} \times 0.453592 \text{ kg/lb} = 0.1417 \text{ kg} \].

Step 2: Convert Units from Miles per Hour to Meters per Second

Next, convert the velocities from miles per hour to meters per second. Use the conversion factor that 1 mile is 1609.34 meters and 1 hour is 3600 seconds. Therefore, \[ 85 \text{ mi/h} = 85 \times \frac{1609.34}{3600} \approx 37.998 \text{ m/s } \] and \[ 95 \text{ mi/h} = 95 \times \frac{1609.34}{3600} \approx 42.467 \text{ m/s} \].

Step 3: Calculate the Change in Velocity

Determine the change in velocity of the baseball. The initial velocity \(v_i\) is 85 mi/h (37.998 m/s), and the final velocity \(v_f\) is -95 mi/h (-42.467 m/s), as the direction is reversed.\[ \Delta v = v_f - v_i = -42.467 \text{ m/s} - 37.998 \text{ m/s} = -80.465 \text{ m/s} \].

Step 4: Compute the Impulse

Impulse \(J\) is the change in momentum, which is given by \(J = m \Delta v\). Use the baseball's mass and the change in velocity to find the impulse.\[ J = 0.1417 \text{ kg} \times -80.465 \text{ m/s} = -11.403 \text{ Ns} \].

Step 5: Calculate the Average Force

To find the force \(F\), use the equation for impulse: \(J = F \Delta t\), where \(\Delta t\) is the time in seconds (7.0 ms = 0.007 s).\[ F = \frac{J}{\Delta t} = \frac{-11.403 \text{ Ns}}{0.007 \text{ s}} \approx -1629.0 \text{ N} \].

Key Concepts

Unit Conversion

When solving physics problems, unit conversion is an essential skill. It's common to encounter different measurement units, especially when dealing with problems from various fields. In the baseball problem, several units need conversion to ensure calculation consistency. Start by converting the baseball weight from ounces to kilograms. Remember:

• 1 pound (lb) = 16 ounces (oz)
• 1 pound = 0.453592 kilograms (kg)

Using these conversion factors, we calculate the baseball's weight as follows:\[5.0 ext{ oz} = 5.0 imes \frac{1 ext{ lb}}{16 ext{ oz}} \times 0.453592 ext{ kg/lb} \approx 0.1417 ext{ kg}\] Next is velocity conversion from miles per hour (mi/h) to meters per second (m/s).

• 1 mile = 1609.34 meters
• 1 hour = 3600 seconds

For example:\[85 ext{ mi/h} = 85 \times \frac{1609.34}{3600} \approx 37.998 ext{ m/s}\]By understanding and applying these conversions, calculations become accurate and coordinated with SI units.

Momentum Change

Momentum in physics is the product of an object's mass and velocity. The change in momentum is crucial when a force acts on an object. To find this change, determine the initial and final velocities and use the mass from the previous conversion. In this problem:

• Initial velocity of the baseball = 85 mi/h (converted to 37.998 m/s)
• Final velocity of the baseball, after being hit, = -95 mi/h (converted to -42.467 m/s, with the negative sign indicating direction reversal)

The change in velocity \(\Delta v\) says how fast and in what direction the ball's velocity changes:\[\Delta v = v_f - v_i = -42.467 ext{ m/s} - 37.998 ext{ m/s} = -80.465 ext{ m/s}\] This change is crucial because it directly influences the impulse and force in the scenario.

Physics Problem Solving

Solving physics problems often involves breaking down complex situations into manageable steps. With the baseball scenario, identifying which physics principles to apply is important. We primarily use the concept of impulse and momentum. The solution path involves:1. **Unit Conversion**: Begin with converting all given data to SI units for consistency.2. **Identify The Variables**: Determine pertinent values like mass, initial, and final velocities.3. **Calculate Momentum Change**: Use the formula \(\Delta p = m \Delta v\) to assess changes in momentum.This systematic approach helps solve diverse problems in physics. Clarifying the method ensures clarity and accuracy.

Average Force Calculation

Average force tells us how much force, effectively, acted on an object over a specific time. This problem requires using the impulse formula:Impulse \( J \) is related to force \( F \) by the equation \( J = F \Delta t \). Rearrange to solve for force:\[ F = \frac{J}{\Delta t} \]Using the previously calculated impulse \(-11.403 \text{ Ns}\) and time \(\Delta t = 7.0 ext{ ms} = 0.007 ext{ s}\), the force is:\[ F = \frac{-11.403 \text{ Ns}}{0.007 ext{ s}} \approx -1629.0 ext{ N} \]Understanding the negative sign is important—it shows the force's direction opposes the original ball movement. Calculating average force helps us understand motion changes accurately.