The escape velocities of two planets $\mathrm{A}$ and $\mathrm{B}$ 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 velocity of a planet is given by the formula: <br/><br/>$\mathrm{v}_{\text{escape}} = \sqrt{\frac{2GM}{R}}$ <br/><br/>where G is the gravitational constant, M is the mass of the planet, and R is its radius. <br/><br/>If the escape velocity of planet A is v_A and the escape velocity of planet B is v_B, then we can write the following relationship: <br/><br/>$${{{v_B}} \over {{v_A}}} = {{\sqrt {{{2G{M_B}} \over {{R_B}}}} } \over {\sqrt {{{2G{M_A}} \over {{R_A}}}} }} = \sqrt {{{{R_A}{M_B}} \over {{M_A}{R_B}}}} $$ <br/><br/>$\Rightarrow$ $\sqrt {{{{R_A}{M_B}} \over {{M_A}{R_B}}}} = 2$ <br/><br/>$\Rightarrow$ ${{{M_B}} \over {{M_A}}} \times {1 \over 3} = 4$ <br/><br/>$\Rightarrow$ ${{{M_B}} \over {{M_B}}} = 12$ <br/><br/>We know, The acceleration due to gravity on a planet can be calculated using the formula: <br/><br/>$\mathrm{g} = \frac{\mathrm{G} \mathrm{M}}{\mathrm{R}^2}$ <br/><br/>where G is the gravitational constant, M is the mass of the planet, and R is its radius. <br/><br/>$${{{g_A}} \over {{g_B}}} = {{{M_A}R_B^2} \over {{M_B}R_A^2}} = {1 \over {12}} \times {\left( {{3 \over 1}} \right)^2} = {9 \over {12}} = {3 \over 4}$$