# Gravitation, orbit, escape velocity

##### Examples
###### Lessons
1. Satellite in geostationary orbit of the Earth

A 465 kg satellite is in geostationary orbit around the Earth.

1. What is the radius of its circular orbit? What is its altitude?
2. What is the satellite's speed?
3. What is the gravitational potential energy of the satellite when it is still attached to its launch rocket at the Earth's surface? What is the potential energy in orbit?
4. Concept: what is the difference between $E_{p}=mgh$ and $E_{p}=-G\frac{m_{1}m_{2}}{r}$? Why is one positive and one negative?
5. What is the total mechanical energy of the satellite once it is in orbit?
6. Concept: which of the following sets of energies could represent the kinetic energy, gravitational potential energy, and total mechanical energy of a satellite in a stable circular orbit? (Proof: for an object in stable circular orbit, $E_{p}=-2E_{k}$)

a) $E_{k} = 30000J, E_{p} = -30000J, E_{mech} = 0J$

b) $E_{k} = 30000J, E_{p} = -60000J, E_{mech} = -30000J$

c) $E_{k} = 60000J, E_{p} = -30000J, E_{mech} = 30000J$

d) $E_{k} = 60000J, E_{p} = 30000J, E_{mech} = 90000J$

7. How much work must be done on the satellite to launch it into geostationary orbit?
8. What speed does the satellite need at the surface of the Earth from its launch to reach the geostationary orbit?
9. In orbit, how much additional energy does the satellite need to escape the Earth's gravity?
10. What is the escape velocity from the surface of the Earth?