Derive an expression for energy of satellite.

Explanation

The total energy of the satellite is the sum of its kinetic energy and the gravitational potential energy. The potential energy of the satellite is,
`U=-(GM_(S)M_(E))/((R_(E)+h))" "...(1)`
Here `M_(S)` - Mass of the satellite, `M_(E)` - Mass of the Earth, `R_(E)` - Radius of the Earth.
The Kinetic energy of the satellite is
`K.E=(1)/(2)M_(S)v^(2)" "...(2)`
Here v is the orbital speed of the satellite and is equal to
`v=sqrt((GM_E)/((R_(E)+h)))" "...(3)`
Substituting the value of v in (2), the kinetic energy of the satellite becomes,
`K.E=(1)/(2)(GM_(E)M_(S))/((R_(E)+h))`
Hence the total energy of the satellite is
`E=(1)/(2)(GM_(E)M_(S))/(R_(E)+h))-(GM_(S)M_(E))/((R_(E)+h))`
`E=-(GM_(S)M_(E))/(2(R_(E)+h))`