Two bullets are fired simultaneously, horizontally and with different speeds from the same place. 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

Answer

To solve the problem of which bullet will hit the ground first when two bullets are fired horizontally with different speeds from the same height, we can analyze the situation step by step.

### Step-by-Step Solution:

1. **Understanding the Scenario**:
- Two bullets are fired horizontally from the same height but with different speeds. Let's denote the speed of the first bullet as \( u \) and the speed of the second bullet as \( v \) (where \( v > u \)).
- Both bullets are fired from the same height above the ground.

2. **Vertical Motion Analysis**:
- Since both bullets are fired horizontally, their initial vertical velocity (\( u_y \) and \( v_y \)) is zero. Therefore, \( u_y = 0 \) and \( v_y = 0 \).

3. **Using the Equation of Motion**:
- The vertical displacement (\( s_y \)) for both bullets can be described using the second equation of motion:
\[ s_y = u_y t - \frac{1}{2} g t^2 \]
- Since the initial vertical velocity is zero for both bullets, the equation simplifies to:
\[ s_y = -\frac{1}{2} g t^2 \]
- Here, \( s_y \) is the vertical distance fallen, and \( g \) is the acceleration due to gravity.

4. **Setting Up the Equations for Both Bullets**:
- For the first bullet (speed \( u \)):
\[ s_y = -\frac{1}{2} g t_1^2 \]
- For the second bullet (speed \( v \)):
\[ s_y = -\frac{1}{2} g t_2^2 \]

5. **Equating the Vertical Displacements**:
- Since both bullets are fired from the same height and fall the same vertical distance (\( s_y \)), we can set the equations equal to each other:
\[ -\frac{1}{2} g t_1^2 = -\frac{1}{2} g t_2^2 \]
- This simplifies to:
\[ t_1^2 = t_2^2 \]
- Taking the square root of both sides gives:
\[ t_1 = t_2 \]

6. **Conclusion**:
- Since \( t_1 = t_2 \), both bullets will hit the ground at the same time, regardless of their horizontal speeds. Therefore, the answer is that both bullets will reach the ground simultaneously.

### Final Answer:
Both bullets will hit the ground simultaneously.