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

Mass of satellite `m=400 kg`
Initial radius of circular orbit `=2R_(E)`
Final radius of circular orbit `=4R_(E)`
Initial total energy `E_(i)=(GM_(E)m)/(4R_(E))`
Final total energy `E_(f)=(GM_(E)m)/(8R_(E))`
The change in the total energy is `DeltaE=E_(f)-E_(i)`
`DeltaE=-(GM_(E)m)/(8R_(E))+(GM_(E)m)/(4R_(E))`
`DeltaE=-(GM_(E)m)/(8R_(E))=((GM_(E))/(R_(E)^(2)))(mR_(E))/8`
`:' (GM_(E))/(R_(E)^(2))=g=9.81 m//s^(2)`
`DeltaE=(gmR_(E))/8(R_(E)=6.37xx10^(6)m)`
`:. DeltaE=(9.81xx400xx6.37xx10^(6))/8=3.13xx10^(9) J`
Kinetic energy is reduced
`DeltaK=K_(f)-K_(i)=-3.13xx10^(9)J`
The change in potential energy is twice the change in total energy
`:. DeltaU=U_(f)-U_(i)=-6.25xx10^(9)J`