The escape velocities of two planets `A `and` B` are in the ratio` 1 : 2`. If the ratio of their radii respectively is `1 : 3,` then the ratio of acceleration due to gravity of planet` A` to the acceleration of gravity of planet `B` will be :

Options

A. `3/2`
B. `2/3`
C. `3/4`
D. `4/3`

Answer

To solve the problem, we need to find the ratio of the acceleration due to gravity of planet A (g_A) to that of planet B (g_B) given the ratios of their escape velocities and radii.

### 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 and \( R \) is the radius of the planet.

2. **Setting Up the Ratios**:
We are given that the escape velocities of planets A and B are in the ratio:
\[ \frac{v_{eA}}{v_{eB}} = \frac{1}{2} \]
This implies:
\[ v_{eA} = \frac{1}{2} v_{eB} \]

3. **Expressing Escape Velocities**:
Using the escape velocity formula, we can express the escape velocities in terms of g and R:
\[ v_{eA} = \sqrt{2g_A R_A} \]
\[ v_{eB} = \sqrt{2g_B R_B} \]

4. **Substituting into the Ratio**:
Substituting these expressions into the ratio of escape velocities:
\[ \frac{\sqrt{2g_A R_A}}{\sqrt{2g_B R_B}} = \frac{1}{2} \]
This simplifies to:
\[ \frac{\sqrt{g_A R_A}}{\sqrt{g_B R_B}} = \frac{1}{2} \]

5. **Squaring Both Sides**:
Squaring both sides gives:
\[ \frac{g_A R_A}{g_B R_B} = \frac{1}{4} \]

6. **Using the Radius Ratio**:
We are given that the ratio of the radii is:
\[ \frac{R_A}{R_B} = \frac{1}{3} \]
Therefore, we can express \( R_A \) in terms of \( R_B \):
\[ R_A = \frac{1}{3} R_B \]

7. **Substituting Radius Ratio**:
Substituting \( R_A \) into the equation:
\[ \frac{g_A \left(\frac{1}{3} R_B\right)}{g_B R_B} = \frac{1}{4} \]
Simplifying this gives:
\[ \frac{g_A}{g_B} \cdot \frac{1}{3} = \frac{1}{4} \]

8. **Solving for the Gravity Ratio**:
Multiplying both sides by 3:
\[ \frac{g_A}{g_B} = \frac{3}{4} \]

### Final Result:
Thus, the ratio of the acceleration due to gravity of planet A to that of planet B is:
\[ \frac{g_A}{g_B} = \frac{3}{4} \]