To solve this problem, we need to apply the impulse-momentum theorem which states that:

$J = \Delta p$

where $J$ is the impulse, and $\Delta p$ is the change in momentum of the ball.

Impulse can be calculated as:

$J = F_{\text{avg}} \cdot \Delta t$

where:

• $F_{\text{avg}}$ is the average force applied by the batter, which is $57500 \, \text{N}$,
• $\Delta t$ is the time over which the force acts, which is $3.00 \, \text{ms} = 3.00 \times 10^{-3} \, \text{s}$.

Now, the change in momentum $\Delta p$ is given by:

$\Delta p = m(v_f - v_i)$

where:

• $m$ is the mass of the softball, which is $2.50 \, \text{kg}$,
• $v_i$ is the initial velocity of the ball (before the batter hits it), which is $20.0 \, \text{m/s}$,
• $v_f$ is the final velocity of the ball (after the batter hits it).

Step 1: Calculate Impulse

First, calculate the impulse $J$:

$J = F_{\text{avg}} \cdot \Delta t = 57500 \, \text{N} \times 3.00 \times 10^{-3} \, \text{s} = 172.5 \, \text{N·s}$

Step 2: Calculate Change in Momentum

Now, we know that:

$J = \Delta p = m(v_f - v_i)$

Substitute the known values:

$172.5 = 2.50 \cdot (v_f - 20.0)$

Step 3: Solve for $v_f$

Solve for $v_f$:

$172.5 = 2.50(v_f - 20.0)$

$\frac{172.5}{2.50} = v_f - 20.0$

$69.0 = v_f - 20.0$

$v_f = 69.0 + 20.0 = 89.0 \, \text{m/s}$

Thus, the velocity of the softball after being hit by the batter is 89.0 m/s in the opposite direction (since the ball is hit straight back to the pitcher).

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Would you like further explanation or more examples? Here are some related questions for further study:

1. What would be the velocity if the force applied by the batter was half of the original value?
2. How does the time duration of contact affect the final velocity of the ball?
3. What would happen if the batter applied a force in the same direction as the ball's initial motion?
4. If the mass of the ball was doubled, how would that affect the final velocity?
5. What is the relationship between impulse and work done by the batter?

Tip: Impulse is related to the change in momentum. If you know the force applied and the time interval, you can calculate the final velocity of an object by considering the momentum change.