### Gravitation Question 18

#### Question 18 - 01 February - Shift 2

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:

(1) $\frac{4}{3}$

(2) $\frac{3}{2}$

(3) $\frac{2}{3}$

(4) $\frac{3}{4}$

## Show Answer

#### Answer: (4)

#### Solution:

#### Formula: Escape velocity

$V_e=\sqrt{\frac{2 GM}{R}}=\sqrt{\frac{2 G \rho \frac{4}{3} \pi R^{3}}{R}}=C \sqrt{\rho} \cdot R$

$\frac{V _{e_1}}{V _{e_2}}=\frac{R_1}{R_2} \sqrt{\frac{\rho_1}{\rho_2}}=\frac{1}{2}$

$\frac{R_1^{2}}{R_2^{2}} \times \frac{\rho_1}{\rho_2}=\frac{1}{4}$

$\frac{R_1}{R_2}=\frac{1}{3}$

$g=\frac{GM}{R^{2}}=\frac{G \frac{4}{3} \pi R^{3} \times \rho}{R^{2}} C . \rho R$

$\frac{g_1}{g_2}=\frac{\rho_1 R_1}{\rho_2 R_2}=\frac{1}{4} \frac{R_2^{2}}{R_1^{2}} \times \frac{R_1}{R_2}$

$=\frac{1}{4} \times \frac{R_2}{R_1}=\frac{3}{4}$