A particle (A) is dropped from a height and another particles (B) is thrown into horizontal direction with speed of 5m/s sec from the same height. The correct statement is

Options

A. both particles will reach at ground simultaneously
B. both particles will reach at ground with same speed
C. particle (A) will reach at ground first with respect to particle (B)
D. particle (B) will reach at ground first with respect to particle (A)

Explanation

To solve the problem, we need to analyze the motion of both particles A and B.

### Step-by-Step Solution:

1. **Understanding the Problem:**
- Particle A is dropped from a height \( h \).
- Particle B is thrown horizontally from the same height \( h \) with a speed of \( 5 \, \text{m/s} \).

2. **Analyzing Particle A:**
- Since particle A is dropped, its initial vertical velocity \( u_A = 0 \).
- The only force acting on it is gravity, with acceleration \( g \).
- Using the equation of motion for vertical displacement:
\[ h = u_A t + \frac{1}{2} g t^2 \]
Substituting \( u_A = 0 \):
\[ h = 0 + \frac{1}{2} g t^2 \]
Rearranging gives:
\[ h = \frac{1}{2} g t^2 \implies t^2 = \frac{2h}{g} \implies t_A = \sqrt{\frac{2h}{g}} \]

3. **Analyzing Particle B:**
- Particle B is thrown horizontally, so its initial vertical velocity \( u_B = 0 \) in the vertical direction.
- The vertical motion is still influenced by gravity \( g \).
- Using the same equation of motion:
\[ h = u_B t + \frac{1}{2} g t^2 \]
Substituting \( u_B = 0 \):
\[ h = 0 + \frac{1}{2} g t^2 \]
Rearranging gives:
\[ h = \frac{1}{2} g t^2 \implies t^2 = \frac{2h}{g} \implies t_B = \sqrt{\frac{2h}{g}} \]

4. **Comparing Times of Flight:**
- From the calculations, we find:
\[ t_A = \sqrt{\frac{2h}{g}} \quad \text{and} \quad t_B = \sqrt{\frac{2h}{g}} \]
- This shows that both particles A and B take the same time \( t \) to reach the ground.

5. **Conclusion:**
- Since both particles A and B reach the ground simultaneously, the correct statement is:
- **Both particles will reach the ground simultaneously.**

### Final Answer:
The correct statement is that both particles A and B will reach the ground simultaneously.

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