Ratio of the radius of a planet `A` to that of planet `B` is `r`. The ratio of acceleration due to gravity for the two planets is `x`. The ratio of the escape velocities from the two planets is

Options

A. `sqrt(rx)`
B. `sqrt(r//x)`
C. `sqrt(r)`
D. `sqrt(x//r)`

Answer

To solve the problem, we need to find the ratio of the escape velocities from two planets, A and B, given the ratio of their radii and the ratio of their acceleration due to gravity.

### Step-by-Step Solution:

1. **Identify the Given Ratios:**
- Let the radius of planet A be \( R_A \) and the radius of planet B be \( R_B \).
- Given the ratio of the radii:
\[ \frac{R_A}{R_B} = r \]
- Let the acceleration due to gravity on planet A be \( g_A \) and on planet B be \( g_B \).
- Given the ratio of the accelerations:
\[ \frac{g_A}{g_B} = x \]

2. **Escape Velocity Formula:**
- The escape velocity \( v \) from a planet is given by the formula:
\[ v = \sqrt{2gR} \]
- Therefore, the escape velocity from planet A (\( v_A \)) and planet B (\( v_B \)) can be expressed as:
\[ v_A = \sqrt{2g_A R_A} \]
\[ v_B = \sqrt{2g_B R_B} \]

3. **Find the Ratio of Escape Velocities:**
- To find the ratio of the escape velocities \( \frac{v_A}{v_B} \), we substitute the expressions for \( v_A \) and \( v_B \):
\[ \frac{v_A}{v_B} = \frac{\sqrt{2g_A R_A}}{\sqrt{2g_B R_B}} = \sqrt{\frac{g_A R_A}{g_B R_B}} \]

4. **Substituting the Ratios:**
- Now, substitute the ratios we have:
\[ \frac{g_A}{g_B} = x \quad \text{and} \quad \frac{R_A}{R_B} = r \]
- Therefore, we can rewrite the ratio as:
\[ \frac{v_A}{v_B} = \sqrt{\frac{g_A}{g_B} \cdot \frac{R_A}{R_B}} = \sqrt{x \cdot r} \]

5. **Final Result:**
- Thus, the ratio of the escape velocities from the two planets is:
\[ \frac{v_A}{v_B} = \sqrt{xr} \]

### Conclusion:
The ratio of the escape velocities from planets A and B is \( \sqrt{xr} \).