The ratio of the radius of the earth to that of the motion is 10. the ratio of the acceleration due to gravity on the earth to that on the moon is 6. The ratio of the escape velocity from the earth's surface to that from the moon is

Options

A. 4
B. 6
C. 12
D. None of these

Explanation

To find the ratio of the escape velocity from the Earth's surface to that from the Moon, we can use the formula for escape velocity:

\[ V = \sqrt{2gR} \]

where \( V \) is the escape velocity, \( g \) is the acceleration due to gravity, and \( R \) is the radius of the celestial body.

### Step-by-Step Solution:

1. **Identify the Ratios Given:**
- The ratio of the radius of the Earth to that of the Moon is given as:
\[ \frac{R_{Earth}}{R_{Moon}} = 10 \]
- The ratio of the acceleration due to gravity on the Earth to that on the Moon is given as:
\[ \frac{g_{Earth}}{g_{Moon}} = 6 \]

2. **Write the Escape Velocity for Earth and Moon:**
- For Earth:
\[ V_{Earth} = \sqrt{2g_{Earth}R_{Earth}} \]
- For Moon:
\[ V_{Moon} = \sqrt{2g_{Moon}R_{Moon}} \]

3. **Find the Ratio of Escape Velocities:**
- The ratio of escape velocities can be expressed as:
\[ \frac{V_{Earth}}{V_{Moon}} = \frac{\sqrt{2g_{Earth}R_{Earth}}}{\sqrt{2g_{Moon}R_{Moon}}} \]
- The \( \sqrt{2} \) cancels out:
\[ \frac{V_{Earth}}{V_{Moon}} = \sqrt{\frac{g_{Earth}R_{Earth}}{g_{Moon}R_{Moon}}} \]

4. **Substitute the Ratios:**
- Substitute the given ratios into the equation:
\[ \frac{V_{Earth}}{V_{Moon}} = \sqrt{\frac{g_{Earth}}{g_{Moon}} \cdot \frac{R_{Earth}}{R_{Moon}}} \]
- Using the ratios:
\[ \frac{V_{Earth}}{V_{Moon}} = \sqrt{6 \cdot 10} \]

5. **Calculate the Result:**
- Simplifying the expression:
\[ \frac{V_{Earth}}{V_{Moon}} = \sqrt{60} \]

6. **Final Answer:**
- Therefore, the ratio of the escape velocity from the Earth's surface to that from the Moon is:
\[ \frac{V_{Earth}}{V_{Moon}} = \sqrt{60} \]