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A body is projected vertically from the surface of the earth of radius $R$ with a velocity equal to half of the escape velocity. The maximum height reached by the body is
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$\frac{R}{3}$
$\begin{aligned} & \text { } K_f+P_f=K_i+P_i \\ & \mathrm{O}-\frac{G M m}{R+h}=\frac{1}{2} m\left(\frac{V_e}{2}\right)^2-\frac{G M m}{R} \\ & -\frac{G M m}{1}\left[-\frac{1}{R}+\frac{1}{R+h}\right]=\frac{1}{8} m V_e^2\end{aligned}$
$\begin{aligned} & -G M m\left[\frac{-h}{R(R+h)}\right]=\frac{1}{8} m V_e^2 \\ & -G M\left[-\frac{}{R(R \quad h)}\right]=-\times\left(\sqrt{2 R . \frac{G M}{}}\right) \\ & \frac{h}{R^2+R h}=\frac{1}{4 R} \Rightarrow 4 R h=R^2+R h \\ & \Rightarrow \quad R^2=3 R h \\ & \Rightarrow \quad h=R / 3\end{aligned}$
$\begin{aligned} & -G M m\left[\frac{-h}{R(R+h)}\right]=\frac{1}{8} m V_e^2 \\ & -G M\left[-\frac{}{R(R \quad h)}\right]=-\times\left(\sqrt{2 R . \frac{G M}{}}\right) \\ & \frac{h}{R^2+R h}=\frac{1}{4 R} \Rightarrow 4 R h=R^2+R h \\ & \Rightarrow \quad R^2=3 R h \\ & \Rightarrow \quad h=R / 3\end{aligned}$
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