A body of mass m is projected with velocity ${λv}_{e}$ 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 the negligible, then the maximum height from the centre of earth, to which the body can go, will be (R : radius of earth)

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$\frac{R}{1 + λ^{2}}$

$\frac{R}{1 - λ^{2}}$

$\frac{R}{1 - λ}$

$\frac{λ^{2} R}{1 - λ^{2}}$

answer is B.

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Detailed Solution

$\begin{array}{l} - \frac{GMm}{R} + \frac{1}{2} {mλ}^{2} V_{e}^{2} = - \frac{GMm}{h} \\ - \frac{GMm}{R} + \frac{1}{2} λ^{2} \frac{2 GMm}{R} = - \frac{GMm}{h} \\ \frac{λ^{2}}{R} - \frac{1}{R} = \frac{- 1}{h} \\ \frac{1}{h} = \frac{1 - λ^{2}}{R} \\ h = \frac{R}{1 - λ^{2}} \end{array}$