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!