Discussion
The ratio of the escape velocities from the surface of two planets having densities and radii in the ratio 2 : 1 and 1 : 2 respectively is
Answer: C
The ratio is 1 : \(\sqrt{2}\).
Escape velocity is \(v = \sqrt{\frac{2GM}{R}}\). With \(M = \frac{4}{3}\pi R^{3} \rho\):
\[ \begin{aligned} v &= R\sqrt{\tfrac{8}{3}\pi G \rho} \propto R\sqrt{\rho} \\ \frac{v_{1}}{v_{2}} &= \frac{1}{2} \times \sqrt{\frac{2}{1}} = \frac{1}{\sqrt{2}} \end{aligned} \]
Using the density ratio without the square root gives 1 : 1, and ignoring the density altogether gives 1 : 2. The velocity depends on \(\sqrt{\rho}\) but on the first power of \(R\), which is where the other ratios go wrong.
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