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

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 acceleration due to gravity, we can follow these steps:

### Step 1: Understand the formula for escape velocity
The escape velocity \( V \) from the surface of a planet is given by the formula:
\[ V = \sqrt{2gR} \]
where \( g \) is the acceleration due to gravity and \( R \) is the radius of the planet.

### Step 2: Write the escape velocities for both planets
Let the escape velocities for the two planets be \( V_1 \) and \( V_2 \). Then:
\[ V_1 = \sqrt{2g_1R_1} \]
\[ V_2 = \sqrt{2g_2R_2} \]

### Step 3: Find the ratio of escape velocities
To find the ratio \( \frac{V_1}{V_2} \), we can substitute the expressions for \( V_1 \) and \( V_2 \):
\[ \frac{V_1}{V_2} = \frac{\sqrt{2g_1R_1}}{\sqrt{2g_2R_2}} = \sqrt{\frac{g_1R_1}{g_2R_2}} \]

### Step 4: Substitute the given ratios
We are given:
- The ratio of the radii of the two planets: \( \frac{R_1}{R_2} = k \)
- The ratio of the acceleration due to gravity: \( \frac{g_1}{g_2} = g \)

Using these ratios, we can rewrite the expression:
\[ \frac{V_1}{V_2} = \sqrt{\frac{g_1}{g_2} \cdot \frac{R_1}{R_2}} = \sqrt{g \cdot k} \]

### Step 5: Final expression
Thus, the ratio of the escape velocities of the two planets is:
\[ \frac{V_1}{V_2} = \sqrt{gk} \]

### Summary
The ratio of the escape velocities of the two planets is \( \sqrt{gk} \).

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