Question

Two planets $A$ and $B$ have the same material density. If the radius of $A$ is twice that of B, then the ratio of the escape velocity $\frac{V_A}{V_B}$ is

• A. $2$
• B. $\sqrt{2}$
• C. $\frac{1}{\sqrt{2}}$
• D. $\frac{1}{2}$

Answer 1

Answer: (A)

$2$

Let the density be d for both the planets. Given that $R_A=2R_B$
Now, mass of $A,M_A=\frac{4d\pi {R_A}^3}{3}=\frac{32d\pi {R_B}^3}{3}$
similarly, $M_B=\frac{4d\pi {R_B}^3}{3}$
Escape velocity for a planet is given by $V=\sqrt{\frac{2GM}{R}}$
So, $V_A=\sqrt{\frac{2G{M}_A}{3R_A}}=\sqrt{\frac{64Gd\pi {R_B}^3}{6R_B}}=\sqrt{\frac{32Gd\pi {R_B}^2}{3}}$
Similarly, $V_B=\sqrt{\frac{8Gd\pi {R_B}^2}{3}}$
Taking the ratio, $\frac{V_A}{V_B}=\sqrt{\frac{32Gd\pi {R_B}^2}{3}}\times \sqrt{\frac{3}{8Gd\pi {R_B}^2}}=2$