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Gravitation Part 2

Section: Physics

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Question : 72 of 110

Marks: +1, -0

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:

[1-Feb-2023 Shift 2]

$\; \frac{4}{3}$
$\; \frac{3}{2}$
$\; \frac{2}{3}$
$\;{3}/{4}$

Solution:

Escape velocity is given by

v_e=\sqrt{\frac{2GM}{R}}

.

Using

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

, we get

GM=gR^2

.

Therefore,

v_e=\sqrt{2gR}

.

For planets A and B,

\frac{v_A}{v_B}=\sqrt{\frac{g_A R_A}{g_B R_B}}

.

Given

\frac{v_A}{v_B}=\frac{1}{2}

and

\frac{R_A}{R_B}=\frac{1}{3}

.

Squaring both sides gives

\frac{g_A R_A}{g_B R_B}=\frac{1}{4}

.

Hence

\frac{g_A}{g_B}=\frac{1}{4}\times \frac{R_B}{R_A}

.

Since

\frac{R_B}{R_A}=3

,

\frac{g_A}{g_B}=\frac{1}{4}\times 3=\frac{3}{4}

.

Thus the required ratio is

\frac{3}{4}

, so the correct option is option D.

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