A stone thrown vertically upwards with a speed of 5 m/sec attains a height `H_(1)`. Another stone thrown upwards from the same point with a speed of 10m/sec attains a height `H_(2)`. The correct relation between `H_(1) and H_(2)` is

Options

A. `H_(2) = 4H_(1)`
B. `H_(2) = 3H_(1)`
C. `H_(1) = 2H_(2)`
D. `H_(1) = H_(2)`

Explanation

To solve the problem, we need to determine the heights \( H_1 \) and \( H_2 \) attained by two stones thrown vertically upwards with different initial velocities. We will use the kinematic equation that relates initial velocity, final velocity, acceleration, and displacement.

### Step-by-Step Solution:

1. **Identify the variables:**
- For the first stone:
- Initial velocity \( u_1 = 5 \, \text{m/s} \)
- Final velocity \( v_1 = 0 \, \text{m/s} \) (at the maximum height)
- Acceleration \( a = -g \) (where \( g \approx 9.81 \, \text{m/s}^2 \))
- Displacement \( H_1 \) (height attained)

- For the second stone:
- Initial velocity \( u_2 = 10 \, \text{m/s} \)
- Final velocity \( v_2 = 0 \, \text{m/s} \) (at the maximum height)
- Acceleration \( a = -g \)
- Displacement \( H_2 \) (height attained)

2. **Use the kinematic equation:**
The kinematic equation we will use is:
\[ v^2 = u^2 + 2a s \]
Rearranging this for displacement \( s \):
\[ s = \frac{v^2 - u^2}{2a} \]

3. **Calculate \( H_1 \) for the first stone:**
Plugging in the values for the first stone:
\[ H_1 = \frac{0^2 - (5)^2}{2(-g)} = \frac{-25}{-2g} = \frac{25}{2g} \]

4. **Calculate \( H_2 \) for the second stone:**
Plugging in the values for the second stone:
\[ H_2 = \frac{0^2 - (10)^2}{2(-g)} = \frac{-100}{-2g} = \frac{100}{2g} = \frac{50}{g} \]

5. **Relate \( H_1 \) and \( H_2 \):**
Now we can express \( H_2 \) in terms of \( H_1 \):
\[ H_1 = \frac{25}{2g} \quad \text{and} \quad H_2 = \frac{100}{2g} = 4 \cdot \frac{25}{2g} = 4H_1 \]

6. **Conclusion:**
The relationship between the heights attained by the two stones is:
\[ H_2 = 4H_1 \]

### Final Answer:
The correct relation between \( H_1 \) and \( H_2 \) is:
\[ H_2 = 4H_1 \]