Derive an expression for energy of satellite.

Explanation

Energy of an Orbiting Satellite:
The total energy of a satellite orbiting the Earth at a distance h from the surface of Earth is calculate as follow, 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))`
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)" ".....(1)`
Here v is the orbital speed of the satellite and is equal to
`v = sqrt((GM_E)/((R_E + h)))`
Substituting the value of v in (1), the kinetic energy of the satellite becomes,
K.E = `1/2(GM_E M_S)/((R_E + h))`
Therefore 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)/((2R_E + h))`
The negative sign in the total energy implies that the satellite is bound to the Earth and it cannot escape from the Earth.