Question

A body is projected vertically upward from the surface of the earth with a velocity equal to half its escape velocity. If $R$ is radius of the earth, then the maximum height attained by the body is $(R / n)$. Find $n$.

Answer 1

Let the body be projected from the surface of the earth with velocity $v$ and reaches maximum height $h$.
According to law of conservation of mechanical energy, we get
$\frac{1}{2} m v^{2}-\frac{G M m}{R}=0-\frac{G M m}{(R+h)}$
or $\frac{1}{2} v^{2}-\frac{G M}{R}=-\frac{G M}{(R+h)}$
As per question,
$V =\frac{ V _{e}}{2}=\frac{1}{2} \sqrt{\frac{2 G M}{R}}$
$\left[\because v _{e}=\sqrt{\frac{2 G M}{R}}\right]$
$\Rightarrow \frac{1}{2}\left(\frac{1}{2} \frac{G M}{R}\right)-\frac{G M}{R}=-\frac{G M}{(R+h)}$
$\frac{G M}{4 R}-\frac{G M}{R}=-\frac{G M}{R+h}$
$-\frac{3}{4 R}=-\frac{1}{(R+h)}$
$3(R+h)=4 R$
or $3 R+3 h=4 R$
$3 h=R$
or $h=\frac{R}{3}$