Discussion
The escape velocities of two planets A and B are in the ratio 2 : 3. If the ratio of their radii is 3 : 4 , then the ratio of acceleration due to gravity at the surface of the planet A to that at the surface of the planet B is
Answer: C
The ratio is 16 : 27.
Escape velocity is \(v_{e} = \sqrt{2gR}\), so \(g = \frac{v_{e}^{2}}{2R}\).
\[ \begin{aligned} \frac{g_{A}}{g_{B}} &= \left(\frac{v_{A}}{v_{B}}\right)^{2}\frac{R_{B}}{R_{A}} \\ &= \frac{4}{9} \times \frac{4}{3} = \frac{16}{27} \end{aligned} \]
The option 4 : 9 squares the velocity ratio but ignores the radii. Note that the radius ratio enters inverted, because \(g\) is divided by \(R\).
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