A body is released from a point of distance `R'` from the centre of earth. Its velocity at the time of striking the earth will be `(R’gtR_(e))`

Options

A. `sqrt(gR)`
B. `sqrt(2gR)`
C. `sqrt( (2g Rr)/(r - R))`
D. `sqrt((2gR (r - R))/(r ))`

Answer

D

Explanation

Let v be the velocity with which the body strikes the surface of earth. Then by the law of conservation of energy,
`1/2 mv^2 - (GMm)/( R) = - (GMm)/(r )`
`(v^2)/(2) = (GM)/( R) = GM (( r - R)/(rR)) = g R ((r - R))/(r ) ( because g = (GM)/(R^2))`
` v = sqrt((2gR (r - R) )/(r ))`