The escape velocities of two planets AandB are in the ratio 1: 2 ....

The formula to calculate the mass $\left(M\right)$ of any planet of radius $R$ and density $ρ$, considering it to be a sphere, is given by

$M = \dfrac{4}{3}\pi{R}^{3}ρ . . . . . . . . . . . . . . . . . . . . . ( 1 )$

The escape velocity \left({v}_{A}\right)$for planet A can be expressed as</p><p>$\begin{cases}{V}_{A} & = & \sqrt{\dfrac{2G{M}_{A}}{{R}_{A}}} \\ & = & \sqrt{\dfrac{2G{ρ}_{A}\dfrac{4}{3}\pi{{R}_{A}}^{3}}{{R}_{A}}} \\ & = & C\sqrt{{ρ}_{A} .}{R}_{A} . . . . . . . . . . . . . . . . . . . . . . . . . . . . ( 2 )\end{cases}$</p><p>where,$G$is Gravitational constant and$C = \sqrt{\dfrac{8\piG}{3}}$is a constant.</p><p>Similarly, the escape velocity \left({v}_{B}\right)$ of planet B can be expressed as

${v}_{B} = C\sqrt{{ρ}_{B}}{R}_{B} . . . . . . . . . . . . . . . . . . . . . . ( 3 )$

Divide equation (2) by equation (3) and solve with the known parameters to obtain the ratio of the densities of the planets.

$\begin{cases}\dfrac{{V}_{A}}{{V}_{B}} & = & \dfrac{C\sqrt{{ρ}_{A}}{R}_{A}}{C\sqrt{{ρ}_{B}}{R}_{B}} \\ \sqrt{\dfrac{{ρ}_{A}}{{ρ}_{B}}} & = & \dfrac{{V}_{A}{R}_{B}}{{V}_{B}{R}_{A}} \\ \dfrac{{ρ}_{A}}{{ρ}_{B}} & = & {\left(\dfrac{{V}_{A}{R}_{B}}{{V}_{B}{R}_{A}}\right)}^{2} \\ & = & {\left(\dfrac{1}{2} \times \dfrac{3}{1}\right)}^{2} \\ & = & \dfrac{9}{4} . . . . . . . . . . . . . . . . . . . . . . . . . . . . ( 4 )\end{cases}$

The acceleration due to gravity \left({g}_{A}\right)$for planet A is given by</p><p>$\begin{cases}{g}_{A} & = & \dfrac{G{M}_{A}}{{{R}_{A}}^{2}} \\ = & \dfrac{G\dfrac{4}{3}\pi{{R}_{A}}^{3} \times {ρ}_{A}}{{{R}_{A}}^{2}} \\ = & \dfrac{{C}^{2}}{2}{ρ}_{A}{R}_{A} . . . . . . . . . . . . . . . . . . . . . . . . . . ( 5 )\end{cases}$</p><p>Similarly, the acceleration due to gravity \left({g}_{B}\right)$ can be expressed as

${g}_{B} = \dfrac{{C}^{2}}{2}{ρ}_{B}{R}_{B} . . . . . . . . . . . . . . . . . . . . . . . . . . . ( 6 )$

Divide equation (5) by equation (6) and solve with the help of known parameters to calculate the required ratio of the accelerations due to gravity in both the planets.

$\begin{cases}\dfrac{{g}_{A}}{{g}_{B}} & = & \dfrac{\dfrac{{C}^{2}}{2}{ρ}_{A}{R}_{A}}{\dfrac{{C}^{2}}{2}{ρ}_{B}{R}_{B}} \\ = & \dfrac{{ρ}_{A}}{{ρ}_{B}}\dfrac{{R}_{A}}{{R}_{B}} \\ = & \dfrac{9}{4} \times \dfrac{1}{3} \\ = & \dfrac{3}{4}\end{cases}$

$= \dfrac{1}{4} \times \dfrac{{R}_{2}}{{R}_{1}} = \dfrac{3}{4}$