A rocket of mass M is launched vertically from the surface of the earth with an initial speed V. Assuming the radius of the earth to be R and negligible air resistance, the maximum height attained by the rocket above the surface of the earth is :

\(\frac{R}{\frac{2gR}{V^2}-1}\)

Kinetic energy given to rocket at the surface of earth = Change in potential energy of the rocket in reaching from ground to highest point : 

\(\Rightarrow \frac{1}{2} mv^2 = \frac{mgh}{1 + \frac{h}{R}}\)

\(\Rightarrow \frac{v^2}{2} = \frac{g}{\frac{1}{h} + \frac{1}{R}}\)

\(\frac{1}{h} + \frac{1}{R} = \frac{2g}{v^2}\)

\(\Rightarrow \frac{1}{h} = \frac{2g}{v^2} - \frac{1}{R}\)

\(\Rightarrow \frac{1}{h} = \frac{2gR - v^2}{v^2 R} \Rightarrow h = \frac{v^2 R }{2gR - v^2}\)

\(\Rightarrow h = \frac{R}{\frac{2g4}{v^2}-1}\)