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:

The escape velocities of planets A and B are in ratio 1:2, and their radii are in ratio 1:3. We need the ratio of their gravitational accelerations.

To begin,

$$ v_e = \sqrt{2gR} $$

Squaring: $$v_e^2 = 2gR$$, which gives $$g = \frac{v_e^2}{2R}$$.

Next,

$$ \frac{v_{eA}^2}{v_{eB}^2} = \frac{g_A R_A}{g_B R_B} $$

Given $$\frac{v_{eA}}{v_{eB}} = \frac{1}{2}$$ and $$\frac{R_A}{R_B} = \frac{1}{3}$$:

$$ \left(\frac{1}{2}\right)^2 = \frac{g_A}{g_B} \times \frac{1}{3} $$

$$ \frac{1}{4} = \frac{g_A}{3g_B} $$

$$ \frac{g_A}{g_B} = \frac{3}{4} $$

The ratio of acceleration due to gravity is $$\dfrac{3}{4}$$.

The correct answer is Option 4: $$\dfrac{3}{4}$$.