What would be the maximum height reached by the body if the body is projected vertically from the surface of the earth of radius R with a velocity equal to half of the escape velocity?

R/3

Conservation of Energy : \(E_i= E_f\)

... where E : total energy ; E_i = total energy initial ; E_f = total energy final

\(\Rightarrow T_i + U_i = T_f + U_f\)

... where T : kinetic energy ; T_i = kinetic energy initial ; T_f = kinetic energy final, and 

... where U : potential energy ; U_i = potential energy initial ; U_f = potential energy final

\(\frac{1}{2}m \frac{v^2_e}{4} + [-\frac{GMm}{R}] = -\frac{GMm}{R+H}\)

\(\frac{2GMm}{8R} - \frac{GMm}{R} = -\frac{GMm}{R+H}\) ... (\(because v^2_e = \frac{2GM}{R}\))

\(-\frac{3GMm}{4R} = -\frac{GMm}{R+H}\)

\(R+H = \frac{4}{3}R\)

\(\Rightarrow H = \frac{R}{3}\)