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Physics Question on Gravitation

A body of mass m is projected with velocity λ v e in vertically upward direction from the surface of the earth into space. It is given that v e is escape velocity and λ< 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)

A

R1+λ2\frac{R}{1+λ^2}

B

R1−λ2\frac{R}{1-λ^2}

C

R1−λ\frac{R}{1-λ}

D

λ2R1−λ2\frac{λ^2R}{1-λ^2}

Answer

R1−λ2\frac{R}{1-λ^2}

Explanation

Solution

The correct answer is (B): R1−λ2\frac{R}{1-λ^2}
Using energy conservation
−GMemRe+12m(λ2GMeRe)2=−GMemr-\frac{GM_em}{R_e} + \frac{1}{2}m \left(λ\sqrt{\frac{2GM_e}{R_e}} \right)^2 = -\frac{GM_em}{r}
GMemr=GMemRe−GMemReλ2\frac{GM_em}{r} = \frac{GM_em}{R_e} - \frac{GM_em}{R_e} λ^2
r=Re1−λ2r = \frac{R_e}{1-λ^2}