A body starts from rest from a distance R 0 from the centre of the earth. The velocity acquired by the body when it reaches the surface of the earth will be ( R = radius of earth, M = mass of earth

Question

A body starts from rest from a distance R 0 ​ from the centre of the earth. The velocity acquired by the body when it reaches the surface of the earth will be ( R = radius of earth, M = mass of earth )

TestHub · Accepted Answer

Using the principle of conservation of mechanical energy:

Initial total energy = Final total energy

$K_{i} + U_{i} = K_{f} + U_{f}$

$0 - \frac{GM m}{R _{0}} = \frac{1}{2} m v^{2} - \frac{GM m}{R}$

$\frac{1}{2} m v^{2} = \frac{GM m}{R} - \frac{GM m}{R _{0}}$

$v^{2} = 2 GM (\frac{1}{R} - \frac{1}{R _{0}})$

$v = 2 GM (\frac{1}{R} - \frac{1}{R _{0}})$

Answer: $2 GM (\frac{1}{R} - \frac{1}{R _{0}})$

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