Total energy in orbit
In the example of the previous section, we found how much work it took to raise an object to a certain altitude. To put a satellite in orbit, we can’t follow that same method because the satellite would just fall down. For a satellite to remain in (circular) orbit, we need to have . The satellite will have a certain amount of total energy at all times:
For a stable orbit, we must have
What amount of work does it take to put a 745 kg satellite into orbit 1108 km above the surface of the Earth?
The satellite begins on Earth, so . When it reaches the altitude of 1108 km, it will be at a radius of
Now since , we should find the change in energy. For the initial value we simply use because the satellite begins at rest, but for the final value we use the special formula for stable orbit energy:
Substituting all the known quantities and evaluating gives us the answer: it would take 2.68 × 1010 J of work to put the satellite in orbit.