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 $V_A/V_B$ is

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

Answer 1

Answer: (A)

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, $\frac{V_A}{V_B}=\sqrt{\frac{32Gd\pi R_B^2}{3}}\times \sqrt{\frac{3}{8Gd\pi R_B{}^2}}$
$=2$