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Impulsive Forces in Collision

Maximum heigh...

Question

Maximum height reached by a rocket fired with a speed equal to $50 %$ of the escape velocity from earths surface is given by $\frac{n R}{18}$ then $n$ is :

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Solution

The body reaches maximum height when the Kinetic energy of the body becomes zero .
$- \frac{G M m}{R} + \frac{1}{2} m v^{2} = - \frac{G M m}{R + H}$
$v = \frac{\sqrt{\frac{2 G M}{R}}}{2} = \sqrt{\frac{G M}{2 R}}$
Hence, $- \frac{G M m}{R} + \frac{1}{2} m \frac{G M}{2 R} = - \frac{G M m}{R + H}$
$- \frac{3 G M m}{4 R} = - \frac{G M m}{R + H}$
$⟹ H = \frac{R}{3}$
Hence, $n = 6$

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