Q 11-07-205JEE MainJEE Main 2025 (4 Apr, Shift 2)Easy
An object is kept at rest at a distance of $3R$ above the earth's surface, where $R$ is the earth's radius. The minimum speed with which it must be projected so that it does not return to earth is: (Assume $M$ = mass of earth, $G$ = universal gravitational constant)
Answer: (A) $\sqrt{\dfrac{GM}{2R}}$
The object is at distance $R + 3R = 4R$ from the earth's centre. To escape, its total energy must be at least zero:
$$\tfrac12mv^2 - \frac{GMm}{4R} = 0 \Rightarrow v = \sqrt{\frac{2GM}{4R}} = \sqrt{\frac{GM}{2R}}$$
Solution by Sreeraj P, M.Sc Physics