An artifical satellite of mass `m` is moving in a circular orbit at a height equal to the radius `R` of the earth. Suddenly due to intensity explosion the satellite breakes into two parts of equal pieces. One part of the satellite stops just after the explosion. The increase in the mechanical energy of the system due to explosion will be
(Given, acceleration due to gravity on the surface of earth is g)

Options

A. `mgR`
B. `(mgR)/(2)`
C. `(mgR)/(4)`
D. `(3mgR)/(2)`

Answer

C

Explanation

Just before collision,
`nu_(0) = sqrt((GM)/(r )) = sqrt((GM)/(R + R)) = sqrt((GM)/(2R ))`
From conservation of linear momentum,
`m nu_(0) = (m)/(2) xx 0 + (m)/(2) nu`
`:.nu = 2nu_(0) = sqrt((2GM)/(R ))`
Increase in mechanical energy
`= K_(f) - K_(i)`
`= (1)/(2) (m)/(nu) nu^(2) - (1)/(2) m nu_(0)^(2)`
`= (1)/(4) m (2nu_(0))^(2) - (1)/(2) m nu_(0)^(2)`
`= (1)/(2) m nu_(0)^(2) = (1)/(2)m ((GM)/(2R )) = (1)/(4) (GMm)/(R )`
`= (1)/(4) mgR` (as `g = (GM)/(R^(2)))`