To solve this problem, let's go through the steps carefully.

Step 1: Understand Given Data and the Equation for Coefficient of Restitution (COR)

The coefficient of restitution $e$ between two objects is given by:

$e = \sqrt{\frac{KE_{\text{final}}}{KE_{\text{initial}}}}$

In this scenario:

• Mass of the ball, $m = 0.600 \ \text{kg}$
• Initial height, $h_{\text{initial}} = 3.65 \ \text{m}$
• COR, $e = 0.750$
• Contact time with the floor, $\Delta t = 0.0150 \ \text{s}$

Step 2: Calculate Initial Kinetic Energy before Impact

The ball is dropped from a height, so initially it has potential energy, which converts to kinetic energy just before impact.

Using the gravitational potential energy equation:

$KE_{\text{initial}} = m \cdot g \cdot h_{\text{initial}}$

where:

• $g = 9.81 \ \text{m/s}^2$ (acceleration due to gravity).

$KE_{\text{initial}} = 0.600 \times 9.81 \times 3.65$

Calculating this gives:

$KE_{\text{initial}} \approx 21.441 \ \text{J}$

Step 3: Calculate Final Kinetic Energy after Rebound

Using the coefficient of restitution:

$e = \sqrt{\frac{KE_{\text{final}}}{KE_{\text{initial}}}}$

Squaring both sides:

$e^2 = \frac{KE_{\text{final}}}{KE_{\text{initial}}}$

Solving for $KE_{\text{final}}$:

$KE_{\text{final}} = e^2 \cdot KE_{\text{initial}}$

Substitute $e = 0.750$ and $KE_{\text{initial}} = 21.441 \ \text{J}$:

$KE_{\text{final}} = (0.750)^2 \times 21.441$ $KE_{\text{final}} \approx 12.082 \ \text{J}$

Step 4: Calculate Change in Kinetic Energy (Impulse)

The change in kinetic energy, which corresponds to the change in momentum, is:

$\Delta KE = KE_{\text{initial}} - KE_{\text{final}}$

$\Delta KE = 21.441 - 12.082$

$\Delta KE \approx 9.359 \ \text{J}$

Step 5: Calculate the Impulse and Average Force

Impulse $J$ is defined as the change in momentum, which is also equal to the average force $F$ multiplied by the contact time $\Delta t$:

$J = F \cdot \Delta t$

The impulse can also be expressed as the change in velocity:

1. Find initial velocity ( v_{\text{initial}} \