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 velocity for both planets
Let:
- \( V_1 \) be the escape velocity of the first planet
- \( V_2 \) be the escape velocity of the second planet

Using the formula, we can express the escape velocities as:
\[ V_1 = \sqrt{2g_1R_1} \]
\[ V_2 = \sqrt{2g_2R_2} \]

### Step 3: Establish the ratios
According to the problem:
- The ratio of the radii of the two planets is \( \frac{R_1}{R_2} = k \)
- The ratio of the acceleration due to gravity of the two planets is \( \frac{g_1}{g_2} = g \)

### Step 4: Substitute the ratios into the escape velocity formula
We can express \( R_1 \) and \( g_1 \) in terms of \( R_2 \) and \( g_2 \):
- \( R_1 = kR_2 \)
- \( g_1 = g g_2 \)

### Step 5: Substitute into the escape velocity equations
Now substituting these into the escape velocity equations:
\[ V_1 = \sqrt{2g_1R_1} = \sqrt{2(g g_2)(kR_2)} \]
\[ V_1 = \sqrt{2g g_2 k R_2} \]

For \( V_2 \):
\[ V_2 = \sqrt{2g_2R_2} \]

### Step 6: Find the ratio of escape velocities
Now we can find the ratio \( \frac{V_1}{V_2} \):
\[ \frac{V_1}{V_2} = \frac{\sqrt{2g g_2 k R_2}}{\sqrt{2g_2R_2}} \]
This simplifies to:
\[ \frac{V_1}{V_2} = \sqrt{g k} \]

### Conclusion
Thus, the ratio of the escape velocities of the two planets is:
\[ \frac{V_1}{V_2} = \sqrt{g k} \]