There are two planets. The ratio of radius of two planets is `k` but radio of acceleration due to gravity of both planets is g. What will be the ratio of their escape velocity ?

Options

A. `(kg)^(1//2)`
B. `(kg)^(-1//2)`
C. `(kg)^(2)`
D. `(kg)^(-2)`

Explanation

To find the ratio of escape velocities of two planets given the ratio of their radii and the ratio of their gravitational accelerations, we can follow these steps:

### Step-by-Step Solution:

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

2. **Defining Variables**:
Let:
- \( R_1 \) and \( g_1 \) be the radius and acceleration due to gravity of the first planet.
- \( R_2 \) and \( g_2 \) be the radius and acceleration due to gravity of the second planet.

3. **Given Ratios**:
We are given:
- The ratio of the radii of the two planets:
\[ \frac{R_1}{R_2} = k \]
- The ratio of the gravitational accelerations:
\[ \frac{g_1}{g_2} = g \]

4. **Calculating Escape Velocities**:
The escape velocity for the first planet \( v_1 \) is:
\[ v_1 = \sqrt{2g_1R_1} \]
The escape velocity for the second planet \( v_2 \) is:
\[ v_2 = \sqrt{2g_2R_2} \]

5. **Finding the Ratio of Escape Velocities**:
We need to find the ratio \( \frac{v_1}{v_2} \):
\[ \frac{v_1}{v_2} = \frac{\sqrt{2g_1R_1}}{\sqrt{2g_2R_2}} = \sqrt{\frac{g_1R_1}{g_2R_2}} \]

6. **Substituting the Ratios**:
Substitute the ratios we have:
\[ \frac{v_1}{v_2} = \sqrt{\frac{g_1}{g_2} \cdot \frac{R_1}{R_2}} = \sqrt{g \cdot k} \]

7. **Final Result**:
Thus, the ratio of the escape velocities of the two planets is:
\[ \frac{v_1}{v_2} = \sqrt{kg} \]

### Conclusion:
The ratio of the escape velocities of the two planets is \( \sqrt{kg} \).

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