A body is projected vertically from earth's surface with ( 3 1 ) rd of escape velocity. The maximum height reached by the body is ( R = radius of earth)

Question

A body is projected vertically from earth's surface with ( 3 1 ​ ) rd of escape velocity. The maximum height reached by the body is ( R = radius of earth)

TestHub · Accepted Answer

Escape velocity from the surface of the earth is given by $v_{e} = \frac{2 GM}{R}$

Velocity of projection is $v = \frac{v _{e}}{3} = \frac{1}{3} \frac{2 GM}{R}$

By the principle of conservation of mechanical energy:

$\frac{1}{2} m v^{2} - \frac{GM m}{R} = - \frac{GM m}{R + h}$

Substituting the value of $v^{2}$ :

$\frac{1}{2} m (\frac{2 GM}{9 R}) - \frac{GM m}{R} = - \frac{GM m}{R + h}$

$\frac{GM m}{9 R} - \frac{GM m}{R} = - \frac{GM m}{R + h}$

Dividing by $GM m$ on both sides:

$\frac{1}{9 R} - \frac{1}{R} = - \frac{1}{R + h}$

$- \frac{8}{9 R} = - \frac{1}{R + h}$

$8 (R + h) = 9 R$

$8 R + 8 h = 9 R$

$8 h = R \Rightarrow h = \frac{R}{8}$

Answer: $\frac{R}{8}$

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