The acceleration due to gravity on the surface of the moon is `(1)/(6)th` of that on the surface of earth and the diameter of the moon is one-fourth that of earth. The ratio of escape velocities on earth and moon will be

Options

A. `(sqrt(6))/(2)`
B. `sqrt(24)`
C. 3
D. `(sqrt(3))/(2)`

Explanation

To solve the problem, we need to find the ratio of escape velocities on Earth and the Moon using the given information about gravity and diameter.

### Step-by-Step Solution:

1. **Understand the formula for escape velocity**:
The escape velocity \( v_e \) from the surface of a celestial body is given by the formula:
\[ v_e = \sqrt{\frac{2GM}{R}} \]
where \( G \) is the gravitational constant, \( M \) is the mass of the body, and \( R \) is its radius.

2. **Relate escape velocity to gravity**:
We can also express escape velocity in terms of gravitational acceleration \( g \):
\[ v_e = \sqrt{2gR} \]

3. **Set up the ratio of escape velocities**:
We want to find the ratio of escape velocities on Earth (\( v_{e, \text{Earth}} \)) and the Moon (\( v_{e, \text{Moon}} \)):
\[ \frac{v_{e, \text{Earth}}}{v_{e, \text{Moon}}} = \sqrt{\frac{g_{\text{Earth}} R_{\text{Earth}}}{g_{\text{Moon}} R_{\text{Moon}}}} \]

4. **Substitute the given values**:
We know:
- \( g_{\text{Moon}} = \frac{1}{6} g_{\text{Earth}} \)
- The diameter of the Moon is \( \frac{1}{4} \) that of Earth, which means the radius of the Moon \( R_{\text{Moon}} = \frac{1}{4} R_{\text{Earth}} \).

5. **Calculate the ratio**:
Substitute the values into the ratio:
\[ \frac{v_{e, \text{Earth}}}{v_{e, \text{Moon}}} = \sqrt{\frac{g_{\text{Earth}} R_{\text{Earth}}}{\left(\frac{1}{6} g_{\text{Earth}}\right) \left(\frac{1}{4} R_{\text{Earth}}\right)}} \]
Simplifying this gives:
\[ = \sqrt{\frac{g_{\text{Earth}} R_{\text{Earth}} \cdot 6 \cdot 4}{g_{\text{Earth}}}} = \sqrt{24} \]

6. **Final result**:
Thus, the ratio of escape velocities on Earth to that on the Moon is:
\[ \frac{v_{e, \text{Earth}}}{v_{e, \text{Moon}}} = \sqrt{24} \]

### Conclusion:
The ratio of escape velocities on Earth and the Moon is \( \sqrt{24} \).