This report is intended to be presented to high school students, with the aid of a gravity well. A simple gravity well that is familiar to most students is the coin well used by charities to collect donations. Here at UBC we had a gravity well made out of clear acrylic, by Ivor Yhap at the University of British Columbia. It was made by making a template of a 1/r curve, which was then used to guide a CMC lathe as it cut away a large block of acrylic. The well is about 30 cm in diameter, and 10 cm in height, with a 2.5 cm-diameter center hole. There is a raised ring around the outside of the top of the well, to prevent balls from rolling out. A dish is placed beneath the hole to catch balls. The well sits on four small jacks, which are adjusted until it is level. Very large balls (diameter 5 cm) "orbit" the center for the longest time before falling into the center, and it is easier to observe good ellipses with these large balls. The gravity well can be placed on an overhead projector, and since it is transparent, the three-dimensional motions of the balls seen as two-dimensional orbits. The only problem with this is finding a projector whose head can be raised enough to focus on the balls.
In the gravity well, the hole represents a large mass. Gravitational potential is proportional to 1/r, which is why the curvature of 1/r was chosen for the gravity well. One way of understanding the well is that it is a warp in space. If you take a flat plane, and particles move along this surface in a normal fashion, and you then stretch the center down, creating the gravity well, you have warped the space. Now the particles will move in an odd fashion, being pulled in by the gravity well as they try to continue along their paths. By using the third dimension to warp the motion of particles, we can look at the two-dimensional view of the particles as they are affected be gravity. By looking at the overhead, where we don't see any "depth" associated with the third dimension, we are watching the 2-D motion of particles which is warped by gravity.
In the same way, our 3-D space is "warped" by gravity. If you think about particles moving around in a normal fashion through three dimensions, and then you put a large gravitational field in this 3-D space, the field will warp the space and affect the motions of the particles. This is harder to visualize, because we cannot easily view a fourth dimension which will warp our 3-D view of the motions of particles. Because of the similar 1/r² dependence of the electric force between particles, the "gravity" well could also represent an electric potential well. Thus electric potential fields "warp" space in the same manner.
A demonstration that is easy to make, and operates like a gravity well, is as follows: Take a piece of mylar (the stuff used for insulating windows in the winter), stretch it over the top of a circular rim, and attach it to the rim. You could use an embroidery hoop or the top of a bucket as a rim. If you've cut the bottom of the bucket out, you can put this on an overhead projector for a class demonstration. Use a hair dryer on the mylar to tighten it up, so that it is taut. Then you can roll ball bearings around on the mylar. Try leaving one heavy ball in the center, and rolling a small ball around to see "orbits". Or try using two medium-weight balls, and get them to "orbit" around one another. The demonstration is re-usable, because the mylar can be tightened at any time by using a blow-dryer again. This idea came from Dr. David Hobill, at the University of Calgary.
With the gravity well, you can observe how the speed of the ball changes as it's distance from the hole changes over it's elliptical path. With the strobe-light photographs, you can show Kepler's second law. You can see for yourself that elliptical paths work for orbits in a field whose force goes as 1 / r². And with the mylar demonstration, you can also see how two particles can orbit around one another. In reality, there are many systems of binary stars (two stars that orbit around each other), which operate in a similar fashion.
The main problem with these models is friction, which causes the balls to spiral into the centre. This would happen in reality if the kinetic energy of the particle is lower than it's gravitational potential energy, but often a system arises in which the two forms of energy are approximately equal. In space, which is essentially a vacuum, offering no resistance, the particles would continue in their elliptical orbits almost indefinitely. (Of course, space is not "perfect", and nor are the orbits of "particles" (ie planets, stars, galaxies, etc.)