We know that,
escape velocity ve = \(\sqrt{\frac{2GM}{R}}\)
conservation of energy
Ei = Ef = 0
Ei - Ef = 0
Ei = \(\frac{1}{2}\)m(kve)2 - \(\frac{GMm}{R}\) .........(1)
Ef = \(-\frac{GMm}{R}\) ............(2)
Equal equation (1) and (2)
\(\frac{1}{2}\)m(k)2\(\frac{2GM}{R}\)- \(\frac{GMm}{R}\) = \(-\frac{GMm}{R+h}\)
k2\(\frac{GMm}{R}\) - \(\frac{GMm}{R}\) = \(-\frac{GMm}{R+h}\)
k2\(\frac{GMm}{R}\) = \(\frac{GMm}{R}\) \(-\frac{GMm}{R+h}\)
k2\(\frac{GMm}{R}\) = GMm (\(\frac{1}{R}\) - \(\frac{1}{R+h}\))
\(\frac{k^2}{R}\) = \(\frac{R+h-R}{R(R+h)}\)
\(\frac{k^2}{R}\) = \(\frac{h}{R(R+h)}\)
k2 = \(\frac{h}{R+h}\)
\(\frac{R+h}{h}\) = \(\frac{1}{k^2}\)
⇒ \(\frac{R}{h}\) + 1 = \(\frac{1}{k^2}\)
⇒ \(\frac{R}{h}\) = \(\frac{1}{k^2}\) - 1
h = \(\frac{R\,k^2}{1-k^2}\)