To determine the final speed of the ball, we need to analyze the change in momentum due to the force applied by the bat.

Given information:

• Mass of baseball, $m = 0.145 \ \text{kg}$
• Initial speed of the baseball, $v_0 = 40.0 \ \text{m/s}$
• Force exerted by the bat, $F = 24,000 \ \text{N}$ (opposite to the ball's direction)
• Time of contact, $t = 0.500 \ \text{ms} = 0.0005 \ \text{s}$

Steps to Solution:

1. Calculate the Impulse: Impulse $J$ is given by the product of the force and the contact time: $J = F \cdot t$ Substituting in the values: $J = 24,000 \ \text{N} \times 0.0005 \ \text{s} = 12 \ \text{N}\cdot\text{s}$

2. Determine the Change in Momentum: The impulse is equal to the change in momentum ($\Delta p$): $\Delta p = J = 12 \ \text{kg} \cdot \text{m/s}$ Since the force is opposite to the ball’s initial direction, this impulse will reduce its momentum and then reverse it.

3. Calculate Initial Momentum: The initial momentum $p_0$ is: $p_0 = m \cdot v_0 = 0.145 \ \text{kg} \times 40.0 \ \text{m/s} = 5.8 \ \text{kg} \cdot \text{m/s}$

4. Determine Final Momentum: Since the impulse is in the opposite direction, the final momentum $p_f$ will be: $p_f = p_0 - \Delta p = 5.8 \ \text{kg} \cdot \text{m/s} - 12 \ \text{kg} \cdot \text{m/s} = -6.2 \ \text{kg} \cdot \text{m/s}$ The negative sign indicates the ball’s direction has reversed.

5. Calculate Final Speed: Using $p_f = m \cdot v_f$, we can solve for $v_f$: $v_f = \frac{p_f}{m} = \frac{-6.2 \ \text{kg} \cdot \text{m/s}}{0.145 \ \text{kg}} = -42.8 \ \text{m/s}$

   The magnitude of the final speed is therefore: $v_f = 42.8 \ \text{m/s}$

Answer:

The final speed of the ball is 42.8 m/s.

Would you like further details on any of the steps or concepts?

Related Questions:

1. How would the final speed change if the contact time were doubled?
2. What would be the effect of a lighter baseball on the final speed?
3. If the ball were initially at rest, what would the final speed be after the impact?
4. How would air resistance affect the speed if included in calculations?
5. What other forces might act on the baseball during a real game scenario?

Tip:

Impulse provides a convenient way to calculate momentum change when force acts over a very short time, especially in collision scenarios.