Two bodies of masses `m _(1) and m _(2)` fall from heights `h _(1) and h _(2)` respectively. The ratio of their velocities when the hit the ground is

Options

A. `(h _(1))( h _(2))`
B. ` sqrt ( h _(1))/sqrt( h _(2))`
C. `(m _(1) h _(1))/(m _(2) h _(2))`
D. ` (h _(1) ^(2) )/( h _(2) ^(2))`

Answer

To solve the problem of finding the ratio of the velocities of two bodies falling from different heights, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Problem**: We have two bodies with masses \( m_1 \) and \( m_2 \) falling from heights \( h_1 \) and \( h_2 \) respectively. We need to find the ratio of their velocities when they hit the ground.

2. **Use the Kinematic Equation**: The velocity of a body falling from a height can be determined using the equation:
\[ v^2 = u^2 + 2gh \]
where \( v \) is the final velocity, \( u \) is the initial velocity (which is 0 for free fall), \( g \) is the acceleration due to gravity, and \( h \) is the height.

3. **Apply the Equation for Each Body**:
- For the first body falling from height \( h_1 \):
\[ v_1^2 = 0 + 2gh_1 \implies v_1 = \sqrt{2gh_1} \]
- For the second body falling from height \( h_2 \):
\[ v_2^2 = 0 + 2gh_2 \implies v_2 = \sqrt{2gh_2} \]

4. **Find the Ratio of Velocities**:
- Now, we can find the ratio of the velocities \( v_1 \) and \( v_2 \):
\[ \frac{v_1}{v_2} = \frac{\sqrt{2gh_1}}{\sqrt{2gh_2}} = \frac{\sqrt{h_1}}{\sqrt{h_2}} \]

5. **Final Result**:
- Therefore, the ratio of their velocities when they hit the ground is:
\[ \frac{v_1}{v_2} = \sqrt{\frac{h_1}{h_2}} \]

### Conclusion:
The ratio of the velocities when the two bodies hit the ground is \( \sqrt{\frac{h_1}{h_2}} \).