A satallite of mass `m`, initally at rest on the earth, is launched into a circular orbit at a height equal to the the radius of the earth. The minimum energy required is

Options

A. `(sqrt(3))/(4)mgR`
B. `(1)/(2)mgR`
C. `(1)/(4)mgR`
D. `(3)/(4)mgR`

Answer

D

Explanation

We know
`V_(0) = sqrt((GM)/(r )) & g = (GM)/(R^(2))`
From energy conservation
`U_(i)+K_(i)=U_(f)+K_(f)`
`-(GMm)/(R )+K_(f)=-(GMm)/(2R)+(1)/(2)mv_(0)^(1)`
`K_(i)=(GMm)/(2R)+(1)/(2)m(sqrt((GM)/(2R)))^(2) rArr K_(i) = (3GMm)/(4R) rarr K_(i)=(3)/(4) mgR`