An official major league baseball has a mass of 0.14 kg . A pitcher throws a 40 \( \mathrm{~m} / \mathrm{s} \) fastball which is hit by the batter straight back up the middle at a speed of \( 46 \mathrm{~m} / \mathrm{s} \). What is the impulse of the ball during the collision with the bat? Woint Numeric 1 point If this collision occurs during a time of 0.012 seconds, what is the av force exerted by the bat on the ball? Type your answer.

To solve this problem, we will follow these steps: 1. **Calculate the impulse of the ball during the collision with the bat.** 2. **Calculate the average force exerted by the bat on the ball.** ### Step 1: Calculate the Impulse Impulse (\( J \)) is defined as the change in momentum of an object. It can be calculated using the formula: \[ J = \Delta p = m(v_f - v_i) \] Where: - \( m \) is the mass of the ball (0.14 kg), - \( v_f \) is the final velocity of the ball after being hit (46 m/s), - \( v_i \) is the initial velocity of the ball before being hit (-40 m/s, negative because it is in the opposite direction). Now, we can substitute the values into the formula: \[ J = 0.14 \, \text{kg} \times (46 \, \text{m/s} - (-40 \, \text{m/s})) \] Calculating the change in velocity: \[ v_f - v_i = 46 + 40 = 86 \, \text{m/s} \] Now substituting back into the impulse formula: \[ J = 0.14 \, \text{kg} \times 86 \, \text{m/s} \] Now, let's calculate this value. Calculate the value by following steps: - step0: Calculate: \(0.14\times 86\) - step1: Multiply the numbers: \(12.04\) The impulse of the ball during the collision with the bat is: \[ J = 12.04 \, \text{kg m/s} \] ### Step 2: Calculate the Average Force The average force (\( F \)) exerted by the bat on the ball can be calculated using the formula: \[ F = \frac{J}{\Delta t} \] Where: - \( J \) is the impulse (12.04 kg m/s), - \( \Delta t \) is the time duration of the collision (0.012 seconds). Now we can substitute the values into the formula: \[ F = \frac{12.04 \, \text{kg m/s}}{0.012 \, \text{s}} \] Let's calculate this value. Calculate the value by following steps: - step0: Calculate: \(\frac{12.04}{0.012}\) - step1: Convert the expressions: \(\frac{\frac{301}{25}}{\frac{3}{250}}\) - step2: Multiply by the reciprocal: \(\frac{301}{25}\times \frac{250}{3}\) - step3: Reduce the numbers: \(301\times \frac{10}{3}\) - step4: Multiply: \(\frac{301\times 10}{3}\) - step5: Multiply: \(\frac{3010}{3}\) The average force exerted by the bat on the ball is: \[ F \approx 1003.33 \, \text{N} \] ### Summary of Results: - **Impulse of the ball:** \( 12.04 \, \text{kg m/s} \) - **Average force exerted by the bat on the ball:** \( 1003.33 \, \text{N} \)