A body is projected vertically upward from the surface of the earth with a velocity equal to half the escape velocity. If R is the radius of the earth, the maximum height attained by the body is

$$\begin{gathered} Initially KE=\frac{1}{2}m{v}^2=\frac{1}{2}m{\left(\frac{v_e}{2}\right)}^2 \\ = \frac{1}{8}m{v_e}^2 \\ = \frac{GmM}{4R} \left(\therefore v_{\mathfrak{e}}=\sqrt{\frac{2MG}{R}}\right) \end{gathered}$$   

Initially PE=$-\frac{GmM}{R}$

$$\begin{gathered} \therefore Total initial energy=\frac{GmM}{4R}-\frac{GmM}{R} \\ = \frac{-3GmM}{4R} \left(i\right) \end{gathered}$$  

If the body comes to rest at a height r from the centre of the earth, its final energy will be given be 

$Final energy=\frac{-GmM}{r} \left(ii\right)$

Equating (i) and (ii) we get r = 4R / 3 . Maximum height attained $=r-R=\frac{4R}{3}-R=\frac{R}{3}.$

Hence the correct choice is (2)