JEE Main 2023 — Gravitation Question with Solution

The orbital speed of an object moving in a circular orbit around Earth (or any other celestial body) is given by the formula:

$v=\sqrt{\frac{GM}{r}}$

where (v) is the orbital speed, (G) is the gravitational constant, (M) is the mass of the central body (Earth, in this case), and (r) is the radius of the orbit.

For the two satellites of masses (m) and (3m) in orbits of radii (r) and (3r) respectively, the ratio of their orbital speeds ($v_1/v_2$) is:

$\frac{v_1}{v_2}=\sqrt{\frac{\frac{GM}{r}}{\frac{GM}{3r}}}=\sqrt{\frac{3r}{r}}=\sqrt{3}$

So, the ratio of the orbital speeds of the satellites is ($\sqrt{3}:1$), which corresponds to Option B.

The inclusion of two different masses for the satellites, $m$ and $3m$, in the problem might initially seem to suggest that the masses would influence their orbital speeds. However, when it comes to circular orbital motion, especially around a much larger body like the Earth, the mass of the orbiting satellite does not directly affect its orbital speed. This is because the orbital speed equation:

$v=\sqrt{\frac{GM}{r}}$

only takes into account the mass of the central body (in this case, Earth's mass $M$), and the radius of the orbit $(r)$, where $G$ is the gravitational constant.