Question

Two planets of same density, have radii $R_1$ and $R_2$ such that $R_1=2R_2$. If the escape velocity on the planets are $v_1$ and $v_2$ respectively. Then ratio $\frac{v_2}{v_1}$ is equal to

• A. $1$
• B. $4$
• C. $\frac{1}{2}$
• D. $2$

Answer 1

The correct option is C $\frac{1}{2}$

Gravitational Field$g=\frac{GM}{R^2}=\frac{G}{R^2}\times \rho \times \frac{4}{3}\pi R^3=\frac{4}{3}\pi \rho GR\Rightarrow g\propto R$
Escape Velocity$v=\sqrt{2gR}$
$v\propto \sqrt{R\times R}\propto R$
Ratio of escape velocities
$\frac{v_2}{v_1}=\frac{R_2}{R_1}=\frac{1}{2}$