A body is projected vertically upwards from the surface of earth with a velocity equal to one third of escape velocity. The maximum height attained by the body will be
(Take radius of earth $= 6400 km$ and $g = 10 {ms}^{- 2}$ )

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TestHub · Accepted Answer

Total mechanical energy of the body at the surface of earth
$M_{i} = - \frac{G M m}{R} + \frac{1}{2} m {[\frac{1}{3} (\sqrt{\frac{2 G M}{R}})]}^{2} = - \frac{G M m}{R} + \frac{G M m}{9 R} = - \frac{8 G M m}{9 R}$
Total mechanical energy of the body at the highest point
$M_{f} = - \frac{G M m}{R + h}$
Using conservation of mechanical energy,
$M_{i} = M_{f} \Rightarrow - \frac{G M m}{R + h} = - \frac{8 G M m}{9 R}$
$\Rightarrow R + h = \frac{9 R}{8}$
$\Rightarrow h = \frac{R}{8} = \frac{6400}{8} = 800 km$

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