A `400 kg` satellite is in a circular orbit of radius `2 R_(E)` around the Earth. How much energy is required to transfer it to a circular orbit of radius `4 R_(E)`? What are the changes in the kinetic and potential energies?
Given `g = 9.81 m^(-2) , R_(E) = 6.37 xx 10^(6) m`.

Explanation

Initially.
`E_(t)=(GM_(E)m)/(4R_(E))`
while finally
`E_(f)=-(GM_(E)m)/(8R_(E))`
The change in the total energy is
`DeltaE=E_(f)-E_(i)`
`=(GM_(E)m)/(8R_(E))=((GM_(E))/(R_(E)^(2)))(mR_(E))/(8)`
`DeltaE=(gmR_(E))/(8)=(9.81xx400xx6.37xx10^(6))/(8)=3.13xx10^(9)J`
The kinetic energy is reduced and it minics
`DeltaE`, namely, `DeltaK=k_(f)-k_(i)=-3.13xx10^(9)J`.
The change in potential energy is twice the chagne in the total energy. namely
`DeltaV=V_(f)-v_(i)=-6.25xx10^(9)J`.