1. Gravitational Potential Energy When we are close to the surface of the Earth we use the constant value of g. If we are at some altitude above the surface of the Earth we must use the radially dependent form of the gravitational acceleration. Assumes constant g! 0 0 We will define U i = 0 when r i = For more than two masses: Only valid for r > R E !

2. Energy Considerations for Planetary Motion This is the total energy of an object orbiting a larger stationary mass. This is a reasonable assumption if m << M. For circular orbits, the speed of the small mass is constant. For elliptical orbits the speed of the small mass is not constant! For a circular orbit we can determine v in the following way: 1 2 What happens as the planet moves from 1 to 2? What happens as the planet moves from 2 to 1? The distance from the sun decreases and hence the potential energy decreases, therefore the kinetic energy and hence the speed must increase. The distance from the sun increases and hence the potential energy increases, therefore the kinetic energy and hence the speed must decrease. We can now rewrite the energy:

3. Escape Speed The term escape speed or escape velocity if often used when discussing the launching of a rocket into space. What is escape speed? Escape speed is the minimum initial speed required for an object to approach an infinite separation from the Earth. There is no propulsion system on this object. 0