Suppose a player hits several baseballs. Which baseball will be in the air for the longest time ?

Options

A. The one with the farthest range
B. The one which reaches maximum height
C. The one with the greatest initial velocity
D. The one leaving the bat at `45^(@)` with respect to the ground

Answer

To determine which baseball will be in the air for the longest time after being hit by a player, we need to analyze the projectile motion of the baseballs based on the angle of projection and the initial velocity.

### Step-by-Step Solution:

1. **Understanding Projectile Motion**:
- When a baseball is hit, it follows a projectile motion path. The time of flight, maximum height, and range depend on the angle of projection and the initial velocity.

2. **Time of Flight Formula**:
- The time of flight \( T \) for a projectile launched with an initial velocity \( u \) at an angle \( \theta \) is given by:
\[ T = \frac{2u \sin \theta}{g} \]
where \( g \) is the acceleration due to gravity.

3. **Maximum Time of Flight**:
- To maximize the time of flight \( T \), we need to maximize \( \sin \theta \). The maximum value of \( \sin \theta \) is 1, which occurs when \( \theta = 90^\circ \). Thus, the baseball will be in the air for the longest time when it is hit straight up.

4. **Analyzing the Options**:
- Let's analyze the options provided:
- **Option 1**: Farthest range (achieved at \( \theta = 45^\circ \)) - This will not give the longest time.
- **Option 2**: Reaches maximum height (also achieved at \( \theta = 90^\circ \)) - This will give the longest time.
- **Option 3**: Greatest initial velocity - While a greater initial velocity increases time, it is not the only factor; the angle of projection is crucial.
- **Option 4**: Leaving the bat at an angle of \( 45^\circ \) - This is the angle for maximum range, not maximum time.

5. **Conclusion**:
- The baseball that is hit straight up (at \( \theta = 90^\circ \)) will be in the air for the longest time. Therefore, **Option 2** is the correct answer.

### Summary:
The baseball that is hit at an angle of \( 90^\circ \) will be in the air for the longest time.