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

Asked by Abhishika John | 1st Oct, 2013, 04:16: PM

Expert Answer:

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] 

Answered by  | 2nd Oct, 2013, 10:49: PM

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