Kinetic energy in an orbit

The 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








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




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]


Note: this is very much simplified calculation. The mass used is just that of the shuttle orbiter. To raise the orbiter into its orbit requires rockets and fuel and these in turn need energy to lift them off the ground. The actual total energy required for the mission is therefore much larger than the result quoted above.