At large distance potential energy \(=0\)

Conserving mechanical energy at surface of earth and large distance from earth,

\(\frac{1}{2} m\left(n v_e\right)^2-\frac{G M m}{R}=\frac{1}{2} m v^2\)

Also, \(v_\theta=\sqrt{\frac{2 G M}{R}}\)

\(\Rightarrow \frac{2 G M m}{R}\left(n^2-1\right)=m v^2\)

\(\Rightarrow v=\sqrt{\frac{2 G M}{R}}\left(n^2-1\right)^{1 / 2}\)

\(\Rightarrow v=v_e \sqrt{n^2-1}\)