Calculating the Energy Required to Launch a Satellite into Orbit

[dkgojackets]

Homework Statement

Find the energy required to launch a satellite from Earth into circular orbit at radius 5Re in units of joules.

Msat = 1000 kg
Me = 5.98e24 kg
Re = 6.37e6
G = 6.67259e-11

Homework Equations

F = GMm/(r^2)

The Attempt at a Solution

I'm thinking somehow using conservation of forces, ending up with (mv^2)/r, and that v can be used to find final kinetic energy, which can be added to potential energy found using mgh, where g = GM/(r^2). I don't know how to get there though.

 

[Gyroscope]

The energy required to launch that satellite should be the change in the mechanical energy of the satellite when put in orbit. You will increase potential energy and kinetic energy. The potential energy formula you have, U=mgh, is just right for points near the Earth surface, you need the general formula, which is:

U=-GM_Em_s/r²

I hope I could help you.

 

[dobry_den]

The idea about adding up changes in potential and kinetic energies seems correct to me. Getting the change in potential energy is quite easy (using the equation mentioned by gyroscope) and finding the change in kinetic energy isn't that difficult. Just realize that the centripetal force of the circular motion around the Earth is equal to the gravitational force. Then rearrange this equation to get the velocity.

 

[dkgojackets]

The energy required to launch that satellite should be the change in the mechanical energy of the satellite when put in orbit. You will increase potential energy and kinetic energy. The potential energy formula you have, U=mgh, is just right for points near the Earth surface, you need the general formula, which is:

U=-GM_Em_s/r²

I hope I could help you.

For U as potential energy, wouldn't I multiply it by the 4R above the Earth's surface?

 

[dobry_den]

Assuming the satellite was initially situated on the surface of the Earth, you've got to calculate the difference in potential energy between the surface and its orbit, i.e. the energy needed to move the satellite on its orbit:

change in PE = U(5R_e) - U(R_e)

 

[dobry_den]

The energy also depends on whether you launch the satellite against or in the direction of the rotation of the Earth. But that's probably beyond the scope of your assignment.