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 1: Write the formula for escape velocity for Earth and Moon

For Earth:
\[ V_e = \sqrt{2g_e R_e} \]

For Moon:
\[ V_m = \sqrt{2g_m R_m} \]

### Step 2: Find the ratio of escape velocities

The ratio of escape velocities from Earth to Moon is given by:

\[ \frac{V_e}{V_m} = \frac{\sqrt{2g_e R_e}}{\sqrt{2g_m R_m}} = \sqrt{\frac{g_e R_e}{g_m R_m}} \]

### Step 3: Substitute the given ratios

We are given:
- The ratio of the radius of the Earth to that of the Moon:
\[ \frac{R_e}{R_m} = 10 \]

- The ratio of the acceleration due to gravity on Earth to that on the Moon:
\[ \frac{g_e}{g_m} = 6 \]

Now, substituting these ratios into our equation:

\[ \frac{V_e}{V_m} = \sqrt{\frac{g_e}{g_m} \cdot \frac{R_e}{R_m}} = \sqrt{6 \cdot 10} \]

### Step 4: Calculate the result

Now we calculate:

\[ \frac{V_e}{V_m} = \sqrt{60} \]

### Step 5: Final answer

Thus, the ratio of the escape velocity from the Earth's surface to that from the Moon is:

\[ \frac{V_e}{V_m} = \sqrt{60} \approx 7.75 \]