Derive an expression for total energy of a satellite orbiting the earth.

he kinetic energy of an object in orbit can easily be found from the following equations:
Centripetal force on a satellite of mass m moving at velocity v in an orbit of radius r = mv²/r 

But this is equal to the gravitational force (F) between the planet (mass M) and the satellite: 

F =GMm/r² and so mv² = GMm/r

But kinetic energy = ½mv² and so:
kinetic energy of the satellite = ½ GMm/r

Kinetic energy in orbit = ½ mv² = + ½GMm/r

All satellites have to be given a tangential velocity (v) to maintain their orbit position and this process is called orbit injection. 

 

Total energy of satellite in orbit = -GMm/2r

However the total energy INPUT required to put a satellite into an orbit of radius r around a planet of mass M and radius R is therefore the sum of the gravitational potential energy (GMm[1/R-1/r]) and the kinetic energy of the satellite ( ½GMm/r).


Energy of launch = GMm[1/R – 1/2r]