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(C) Using the principle of conservation of mechanical energy between the surface of the Earth and the maximum height $h$ from the surface:Initial ener

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$\frac{R}{1 - k^2}$

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(C) Using the principle of conservation of mechanical energy between the surface of the Earth and the maximum height $h$ from the surface:Initial energy at the surface $= \frac{1}{2} m(k v_e)^2 - \frac{GMm}{R}$Final energy at maximum height $h = - \frac{GMm}{R+h}$Since $v_e = \sqrt{\frac{2GM}{R}}$,we have $v_e^2 = \frac{2GM}{R}$.Equating initial and final energy:$\frac{1}{2} m k^2 (\frac{2GM}{R}) - \frac{GMm}{R} = - \frac{GMm}{R+h}$$k^2 \frac{GMm}{R} - \frac{GMm}{R} = - \frac{GMm}{R+h}$$k^2 - 1 = - \frac{R}{R+h}$$1 - k^2 = \frac{R}{R+h}$$R+h = \frac{R}{1 - k^2}$Since the distance from the centre of the Earth is $r = R + h$,we get $r = \frac{R}{1 - k^2}$.

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