The ratio of the radii of the planets `P_(1) and P_(2)` is `K_(1)`. the ratio of the acceleration due to gravity is `K_(2)`. the ratio of the escape velocities from them will be

Options

A. `sqrt(K_1K_2)`
B. `sqrt(2K_1K_2)`
C. `sqrt((K_1)/(K_2))`
D. `sqrt((K_2)/(K_1))`

Answer

To find the ratio of escape velocities from two planets \( P_1 \) and \( P_2 \) given the ratio of their radii \( K_1 \) and the ratio of their acceleration due to gravity \( K_2 \), we can follow these steps:

### Step-by-Step Solution

1. **Understanding Escape Velocity**:
The escape velocity \( V_e \) from the surface of a planet is given by the formula:
\[ V_e = \sqrt{2gR} \]
where \( g \) is the acceleration due to gravity at the surface of the planet and \( R \) is the radius of the planet.

2. **Expressing Escape Velocity for Both Planets**:
For planet \( P_1 \):
\[ V_{e1} = \sqrt{2g_1R_1} \]
For planet \( P_2 \):
\[ V_{e2} = \sqrt{2g_2R_2} \]

3. **Finding the Ratio of Escape Velocities**:
The ratio of the escape velocities from planets \( P_1 \) and \( P_2 \) can be expressed as:
\[ \frac{V_{e1}}{V_{e2}} = \frac{\sqrt{2g_1R_1}}{\sqrt{2g_2R_2}} = \sqrt{\frac{g_1R_1}{g_2R_2}} \]

4. **Using Given Ratios**:
We know:
- The ratio of the radii \( \frac{R_1}{R_2} = K_1 \)
- The ratio of the accelerations due to gravity \( \frac{g_1}{g_2} = K_2 \)

Substituting these ratios into the escape velocity ratio:
\[ \frac{V_{e1}}{V_{e2}} = \sqrt{\frac{g_1}{g_2} \cdot \frac{R_1}{R_2}} = \sqrt{K_2 \cdot K_1} \]

5. **Final Result**:
Therefore, the ratio of the escape velocities from planets \( P_1 \) and \( P_2 \) is:
\[ \frac{V_{e1}}{V_{e2}} = \sqrt{K_1 K_2} \]