JEE Main · 2023 · 01 Feb 2023 · Shift 2 · Gravitation

The escape velocities of two planets A and B are in the ratio

1 : 2

. If the ratio of their radii respectively is

1 : 3

, then the ratio of acceleration due to gravity of planet A to the acceleration of gravity of planet B will be:

(A) $\frac{4}{3}$
(B) $\frac{3}{2}$
(C) $\frac{2}{3}$
(D) $\frac{3}{4}$ ✓ Correct

Answer: (D)

Solution

Escape velocity is

v_{e} = \frac{2 GM}{R}

, and since

g = \frac{GM}{R ^{2}}

this can be written

v_{e} = 2 g R

, so

g = \frac{v _{e}^{2}}{2 R}

. Taking the ratio for the two planets,

\frac{g _{A}}{g _{B}} = \frac{v _{e A}^{2}}{v _{e B}^{2}} \cdot \frac{R _{B}}{R _{A}} = (\frac{1}{2})^{2} \times \frac{3}{1} = \frac{3}{4}

. The ratio of the accelerations due to gravity is

\frac{3}{4}

JEE Main · 2023 · 01 Feb 2023 · Shift 2 · Single correct option

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