A body starts from rest from a point distant `r_(0)` from the centre of the earth. It reaches the surface of the earth whose radius is `R`. The velocity acquired by the body is

Options

A. `2 GM ((1)/(R) - (1)/(R_(0)) ) `
B. `sqrt(2 GM ((1)/(R) - (1)/(R_(0)) ) )`
C. ` GM ((1)/(R) - (1)/(R_(0)) ) `
D. `2 GM sqrt(((1)/(R) - (1)/(R_(0))))`

Answer

B

Explanation

P.E. = `int_(R_(0))^(R) ("GMm")/(r^(2)) dr = - "GMm" [ (1)/(R) - (1)/(R_(0)) ] `
The K.E. acquired by the body at the surface = `(1)/(2) ` m`v^(2)`
`therefore (1)/(2) mv^(2) = - "GMm"[(1)/(R) - (1)/(R_(0))]`
`v = sqrt(2 GM ((1)/(R_(0)) - (1)/(R)))`