Q 11-07-131JEE MainJEE Main 2022 (27 Jul, Shift 2)Medium
A body of mass $m$ is projected with velocity $\lambda v_e$ in vertically upward direction from the surface of the earth into space. It is given that $v_e$ is escape velocity and $\lambda < 1$. If air resistance is considered to be negligible, then the maximum height from the centre of earth, to which the body can go, will be ($R$ : radius of earth)
Answer: (B) $\dfrac{R}{1 - \lambda^2}$
Energy conservation, with $v_e^2 = \dfrac{2GM}{R}$:
$$\frac12m\lambda^2\frac{2GM}{R} - \frac{GMm}{R} = -\frac{GMm}{r}$$
$$\frac{\lambda^2 - 1}{R} = -\frac1r \Rightarrow r = \frac{R}{1 - \lambda^2}$$
Solution by Sreeraj P, M.Sc Physics