Two bullets are fired simultaneously, horizontally but with different speeds from the same horizontal plane. Which bullet will hit the ground first ?

Options

A. the faster one
B. the slower one
C. both will reach simultaneously
D. depends on the masses

Explanation

To solve the problem of which bullet will hit the ground first when fired horizontally from the same height but with different speeds, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Scenario**:
- We have two bullets fired horizontally from the same height (let's say from a building).
- Bullet 1 is fired with speed \( V_1 \) and Bullet 2 with speed \( V_2 \).

2. **Analyzing Horizontal Motion**:
- Both bullets are fired horizontally, which means their initial vertical velocity (\( V_{y0} \)) is 0.
- The horizontal motion does not affect the time it takes for the bullets to fall to the ground because there is no horizontal acceleration (assuming air resistance is negligible).

3. **Analyzing Vertical Motion**:
- The only force acting on the bullets in the vertical direction is gravity, which causes a downward acceleration \( g \) (approximately \( 9.81 \, \text{m/s}^2 \)).
- The vertical motion can be described using the equation of motion:
\[ s = ut + \frac{1}{2} a t^2 \]
where:
- \( s \) is the vertical displacement (height of the building),
- \( u \) is the initial vertical velocity (which is 0 for both bullets),
- \( a \) is the acceleration due to gravity (same for both bullets),
- \( t \) is the time taken to hit the ground.

4. **Setting Up the Equation**:
- For both bullets:
\[ s = 0 \cdot t + \frac{1}{2} g t^2 \]
- This simplifies to:
\[ s = \frac{1}{2} g t^2 \]

5. **Finding Time to Hit the Ground**:
- Rearranging the equation gives:
\[ t^2 = \frac{2s}{g} \]
- Taking the square root:
\[ t = \sqrt{\frac{2s}{g}} \]
- Since \( s \) (the height) and \( g \) (acceleration due to gravity) are the same for both bullets, the time \( t \) will also be the same for both bullets.

6. **Conclusion**:
- Both bullets will hit the ground at the same time regardless of their horizontal speeds \( V_1 \) and \( V_2 \).

### Final Answer:
Both bullets will hit the ground at the same time.

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