Question

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 :

• A. $\frac{3}{2}$
• B. $\frac{2}{3}$
• C. $\frac{3}{4}$
• D. $\frac{4}{3}$

Answer 1

Answer: (C)

$V_e=\sqrt{\frac{2GM}{R}}=\sqrt{\frac{2G\rho \frac{4}{3}\pi R^3}{R}}=C\sqrt{\rho }\cdot R$
$\frac{V_{e_1}}{V_{e_2}}=\frac{R_1}{R_2}\sqrt{\frac{{\rho }_1}{{\rho }_2}}=\frac{1}{2}$
$\frac{R_1^2}{R_2^2}\times \frac{{\rho }_1}{{\rho }_2}=\frac{1}{4}$
$\frac{R_1}{R_2}=\frac{1}{3}$
$g=\frac{G{M}^2}{R^2}=\frac{G\frac{4}{3}\pi R^3\times \rho }{R^2}C\cdot \rho R$
$\frac{g_1}{g_2}=\frac{{\rho }_1{R}_1}{{\rho }_2{R}_2}=\frac{1}{4}\frac{R_2^2}{R_1^2}\times \frac{R_1}{R_2}$
$=\frac{1}{4}\times \frac{R_2}{R_1}=\frac{3}{4}$