The escape velocity from a spherical planet $A$ is $10 \mathrm{~km} / \mathrm{s}$. The escape veloci

Question

The escape velocity from a spherical planet $A$ is $10 \mathrm{~km} / \mathrm{s}$. The escape velocity from another planet $B$ whose density and radius are $10 \%$ of those of planet $A$, is $\_\_\_\_$ $\mathrm{m} / \mathrm{s}$.

Professor PC Thomas Classes · Answer

The escape velocity is given by $v = \sqrt{\frac{2GM}{R}}$. In terms of density $ho$, $v = \sqrt{\frac{2G}{R} \left(\frac{4}{3}\pi R^3 hoight)} = R \sqrt{\frac{8}{3}\pi G ho}$. Thus, $v \propto R\sqrt{ho}$. For planet B, $R_B = 0.1 R_A$ and $ho_B = 0.1 ho_A$. The ratio $\frac{v_B}{v_A} = \frac{R_B}{R_A} \sqrt{\frac{ho_B}{ho_A}} = 0.1 	imes \sqrt{0.1} = \frac{1}{10\sqrt{10}}$. Given $v_A = 10 	ext{ km/s} = 10000 	ext{ m/s}$, $v_B = \frac{10000}{10\sqrt{10}} = \frac{1000}{\sqrt{10}} = 100\sqrt{10} 	ext{ m/s}$.