The question provides details on a 0.25 kg softball that initially has a velocity of $12 \, \text{m/s}$ at an angle of $44^\circ$ below the horizontal, just before it hits the bat. After contact with the bat, the ball's final velocities are given in two parts:

1. (a) $16 \, \text{m/s}$ vertically downward.
2. (b) $16 \, \text{m/s}$ horizontally back toward the pitcher.

We need to determine the magnitude of the change in momentum for each case.

Step-by-Step Solution

1. Initial Velocity Components (Before Contact)

   • The initial velocity, $v_i = 12 \, \text{m/s}$ at an angle of $44^\circ$ below the horizontal.
   • Horizontal component of $v_i$: $v_{i_x} = 12 \cos(44^\circ)$
   • Vertical component of $v_i$: $v_{i_y} = -12 \sin(44^\circ)$ (negative because it is downward).

2. Calculating Momentum Components

   • Momentum before contact $(p_i)$: $p_{i_x} = m \cdot v_{i_x} \quad \text{and} \quad p_{i_y} = m \cdot v_{i_y}$ where $m = 0.25 \, \text{kg}$.

3. Final Momentum (for Each Case)

   • Case (a): Final velocity is $16 \, \text{m/s}$ vertically downward.

     • Horizontal final momentum $p_{f_x} = 0$.
     • Vertical final momentum $p_{f_y} = m \cdot (-16)$.

   • Case (b): Final velocity is $16 \, \text{m/s}$ horizontally back toward the pitcher.

     • Horizontal final momentum $p_{f_x} = m \cdot (-16)$.
     • Vertical final momentum $p_{f_y} = 0$.

4. Calculating Change in Momentum (Δp)

   • Change in momentum in each component: $\Delta p_x = p_{f_x} - p_{i_x}$ and $\Delta p_y = p_{f_y} - p_{i_y}$.
   • The magnitude of the change in momentum is given by: $|\Delta p| = \sqrt{(\Delta p_x)^2 + (\Delta p_y)^2}$

Would you like the detailed calculations to find the final answer for each case?

Related Questions

1. How would the answer change if the ball’s mass were different?
2. What effect does the angle of the initial velocity have on the momentum change?
3. Why do we consider the direction (sign) of momentum components in calculations?
4. How would you calculate the change in kinetic energy instead of momentum?
5. What are common applications of momentum principles in sports like baseball?

Tip: Always resolve vectors into components to simplify physics problems involving angles and directions.