A golfer imparts a speed of 30.3 m/s to a ball, and it travels the maximum possible distance before landing on the green. The tee and the green are at the same elevation. How much time does the ball spend in the air?

Options

A. 4.37 s
B. 2.68 s
C. 3.51 s
D. 1.84 s

Answer

To solve the problem of how much time a golf ball spends in the air after being hit with an initial speed of 30.3 m/s at an angle that maximizes its range, we can follow these steps:

### Step 1: Understand the Problem
The golfer hits the ball with a speed of 30.3 m/s, and we need to determine the time the ball spends in the air. Since the tee and the green are at the same elevation, we can treat this as a projectile motion problem.

**Hint:** Remember that the time of flight for a projectile depends on the initial velocity and the angle of projection.

### Step 2: Determine the Angle for Maximum Range
For a projectile to achieve maximum range, it should be launched at an angle of 45 degrees. This angle maximizes the horizontal distance traveled.

**Hint:** Recall that the formula for range is maximized when the angle of projection is 45 degrees.

### Step 3: Use the Time of Flight Formula
The time of flight \( T \) for a projectile launched at an angle \( \theta \) with an initial velocity \( U \) is given by the formula:
\[ T = \frac{2U \sin \theta}{g} \]
where:
- \( U = 30.3 \, \text{m/s} \) (initial speed)
- \( g = 9.8 \, \text{m/s}^2 \) (acceleration due to gravity)
- \( \theta = 45^\circ \)

**Hint:** Remember that \( \sin(45^\circ) = \frac{\sqrt{2}}{2} \).

### Step 4: Calculate the Time of Flight
Substituting the values into the formula:
\[ T = \frac{2 \times 30.3 \times \sin(45^\circ)}{9.8} \]
\[ T = \frac{2 \times 30.3 \times \frac{\sqrt{2}}{2}}{9.8} \]
\[ T = \frac{30.3 \times \sqrt{2}}{9.8} \]

Calculating \( \sqrt{2} \approx 1.414 \):
\[ T \approx \frac{30.3 \times 1.414}{9.8} \]
\[ T \approx \frac{42.9}{9.8} \]
\[ T \approx 4.38 \, \text{seconds} \]

**Hint:** Make sure to perform the calculations step by step to avoid mistakes.

### Step 5: Conclusion
The time the ball spends in the air is approximately 4.38 seconds.

**Final Answer:** The ball spends approximately **4.38 seconds** in the air.