An outfielder throws a baseball with an initial speed of 83.2 mi/h. Just before an infielder catches the ball at the same level, the ball's speed is 110 ft/s. In foot-pounds, by how much is the mechanical energy of the ball-Earth system reduced of air drag ? (The weight of a baseball is 9.0 oz.)

Answer

To solve the problem, we need to calculate the change in mechanical energy of the ball-Earth system due to air drag. We will do this by calculating the initial and final kinetic energies of the baseball and then finding the difference.

### Step 1: Convert the initial speed from miles per hour to feet per second
The initial speed of the baseball is given as 83.2 miles per hour. We need to convert this to feet per second.

1 mile = 5280 feet
1 hour = 3600 seconds

\[ \text{Initial speed (V_i)} = 83.2 \, \text{mi/h} \times \frac{5280 \, \text{ft}}{1 \, \text{mi}} \times \frac{1 \, \text{h}}{3600 \, \text{s}} \]

Calculating this gives:

\[ V_i = 83.2 \times \frac{5280}{3600} \approx 122.0 \, \text{ft/s} \]

### Step 2: Calculate the mass of the baseball in pounds
The weight of the baseball is given as 9.0 oz. We need to convert this to pounds.

1 pound = 16 ounces

\[ \text{Weight in pounds} = \frac{9.0 \, \text{oz}}{16 \, \text{oz/lb}} = 0.5625 \, \text{lb} \]

### Step 3: Calculate the initial kinetic energy (KE_initial)
The formula for kinetic energy is:

\[ KE = \frac{1}{2} m v^2 \]

Where:
- \( m \) = mass in pounds
- \( v \) = speed in feet per second

Using the initial speed:

\[ KE_{initial} = \frac{1}{2} \times 0.5625 \, \text{lb} \times (122.0 \, \text{ft/s})^2 \]

Calculating this gives:

\[ KE_{initial} = \frac{1}{2} \times 0.5625 \times 14884 \approx 4184.25 \, \text{ft-lb} \]

### Step 4: Calculate the final kinetic energy (KE_final)
Using the final speed of 110 ft/s:

\[ KE_{final} = \frac{1}{2} \times 0.5625 \, \text{lb} \times (110.0 \, \text{ft/s})^2 \]

Calculating this gives:

\[ KE_{final} = \frac{1}{2} \times 0.5625 \times 12100 \approx 3403.125 \, \text{ft-lb} \]

### Step 5: Calculate the reduction in mechanical energy due to air drag
The reduction in mechanical energy is given by the difference between the initial and final kinetic energies:

\[ \text{Reduction in energy} = KE_{initial} - KE_{final} \]

Calculating this gives:

\[ \text{Reduction in energy} = 4184.25 \, \text{ft-lb} - 3403.125 \, \text{ft-lb} \approx 781.125 \, \text{ft-lb} \]

### Final Answer
The mechanical energy of the ball-Earth system is reduced by approximately **781.125 foot-pounds** due to air drag.

---