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{2 GM }{ R }}=\sqrt{\frac{2 G \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{ GM ^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}$