Two bodies of masses `(m_(1))` and `(m_(2))` are droppded from heithts `h_(1)` and `h_(2)`, respectively. They reach the ground after time `t_(1)` and `t_(2)` and strike the ground with `v_(1)` and `v_(2)`, respectively Choose the correct relations from the following.

Options

A. `(t_(1))/(t_(2)) =sqrt((h_(1))/(h_(2))`
B. `(t_(1))/(t_(2)) =sqrt((h_(2))/(h_(1))`
C. `(v_(1))/(v_(2)) =sqrt((h_(1))/(h_(2))`
D. `(v_(1))/(v_(2)) =(h_(2))/(h_(1))`

Explanation

To solve the problem, we will analyze the motion of the two bodies dropped from heights \( h_1 \) and \( h_2 \). We will use the equations of motion under uniform acceleration due to gravity.

### Step-by-Step Solution:

1. **Identify the Initial Conditions**:
- Both bodies are dropped from rest, so their initial velocities \( u_1 \) and \( u_2 \) are 0.
- The acceleration \( g \) is constant for both bodies.

2. **Use the Equation of Motion for Height**:
- The equation of motion for an object in free fall is given by:
\[ h = ut + \frac{1}{2}gt^2 \]
- For the first body:
\[ h_1 = 0 \cdot t_1 + \frac{1}{2} g t_1^2 \Rightarrow h_1 = \frac{1}{2} g t_1^2 \]
- For the second body:
\[ h_2 = 0 \cdot t_2 + \frac{1}{2} g t_2^2 \Rightarrow h_2 = \frac{1}{2} g t_2^2 \]

3. **Establish the Ratio of Heights**:
- Taking the ratio of \( h_1 \) and \( h_2 \):
\[ \frac{h_1}{h_2} = \frac{\frac{1}{2} g t_1^2}{\frac{1}{2} g t_2^2} = \frac{t_1^2}{t_2^2} \]
- This implies:
\[ \frac{t_1}{t_2} = \sqrt{\frac{h_1}{h_2}} \]

4. **Use the Equation of Motion for Velocity**:
- The final velocity of an object in free fall is given by:
\[ v = u + gt \]
- For the first body:
\[ v_1 = 0 + g t_1 \Rightarrow v_1 = g t_1 \]
- For the second body:
\[ v_2 = 0 + g t_2 \Rightarrow v_2 = g t_2 \]

5. **Establish the Ratio of Velocities**:
- Taking the ratio of \( v_1 \) and \( v_2 \):
\[ \frac{v_1}{v_2} = \frac{g t_1}{g t_2} = \frac{t_1}{t_2} \]
- This implies:
\[ \frac{v_1}{v_2} = \sqrt{\frac{h_1}{h_2}} \]

6. **Conclusion**:
- From the above calculations, we have established the following relationships:
- \( \frac{h_1}{h_2} = \left(\frac{t_1}{t_2}\right)^2 \)
- \( \frac{v_1}{v_2} = \sqrt{\frac{h_1}{h_2}} \)
- Therefore, the correct options are:
- Option 1: \( \frac{h_1}{h_2} = \left(\frac{t_1}{t_2}\right)^2 \)
- Option 3: \( \frac{v_1}{v_2} = \sqrt{\frac{h_1}{h_2}} \)