Two stones are falling at a place from heights in the ratio 2:3. Their velocities on reaching the ground are in the ratio

Options

A. `sqrt3:sqrt2`
B. `sqrt2:sqrt3`
C. `2:3`
D. `3:2`

Answer

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

### Step-by-Step Solution:

1. **Define the Heights**:
Let the heights from which the two stones are dropped be in the ratio of 2:3. We can express these heights as:
- Height of stone 1, \( h_1 = 2x \)
- Height of stone 2, \( h_2 = 3x \)
where \( x \) is a common factor.

2. **Use the Formula for Final Velocity**:
The final velocity \( V \) of an object falling freely from a height \( h \) is given by the formula:
\[ V = \sqrt{2gh} \]
where \( g \) is the acceleration due to gravity.

3. **Calculate the Velocities**:
- For stone 1 (falling from height \( h_1 = 2x \)):
\[ V_1 = \sqrt{2g(2x)} = \sqrt{4gx} \]
- For stone 2 (falling from height \( h_2 = 3x \)):
\[ V_2 = \sqrt{2g(3x)} = \sqrt{6gx} \]

4. **Find the Ratio of the Velocities**:
Now, we can find the ratio of the velocities \( V_1 \) and \( V_2 \):
\[ \frac{V_1}{V_2} = \frac{\sqrt{4gx}}{\sqrt{6gx}} = \frac{\sqrt{4}}{\sqrt{6}} = \frac{2}{\sqrt{6}} = \frac{2}{\sqrt{6}} \cdot \frac{\sqrt{6}}{\sqrt{6}} = \frac{2\sqrt{6}}{6} = \frac{\sqrt{6}}{3} \]

5. **Simplify the Ratio**:
The ratio of the velocities can also be expressed in terms of square roots:
\[ V_1 : V_2 = \sqrt{4} : \sqrt{6} = 2 : \sqrt{6} \]

### Final Answer:
Thus, the ratio of the velocities of the two stones on reaching the ground is:
\[ V_1 : V_2 = 2 : \sqrt{6} \]