The first part of this chapter explores in more mathematical detail some of the concepts introduced in the previous sections. The mathematics we will employ are more complicated than we have used elsewhere, and not necessarily crucial to gaining a good understanding of the laws of gravitation. The aim of this chapter, though, is to show that the shape of the orbits can be deduced explicitly and precisely using the Universal Law of Gravitation and what we know about gravitational potential energy and angular momentum. Moreover, this analysis will give us greater insight into the energies associated with the various orbital shapes.

In addition to furthering our study of orbits in general, we will also explore two interesting problems related to orbital energies. First, we will calculate the escape velocity, which is the surface velocity required to completely blast a projectile out of the gravitational field of a star or planet. It will be seen that this value is independent of the mass of the projectile being launched. A black hole is a collapsed star that has such strong gravitational field that its escape velocity exceeds the speed of light--it is for this reason that no light (or anything else for that matter) can ever escape from a black hole. Second, we will consider what happens to satellites when they encounter the atmosphere in low-earth orbits. The atmosphere creates a friction on the satellites, which causes a viscous drag. Contrary to normal intuition, we will show that this drag actually causes the satellite to increase its speed!