I remember learning about orbits early in high school after seeing Newton's sketch for making a cannonball fall all the way around the Earth. See [url]http://en.wikipedia.org/wiki/Newton's_cannonball[/url] if you are unfamiliar with the image.
I understood that the path went from looking like the parabola to an ellipse. This I could handle. I just didn't understand the transition from one to another. Even after taking preCalc and physics, knowing more of the theoretical background for why it happened still didn't let me understand what it looked like for an object to just barely stop being a parabola and start being an ellipse.
I put together this simulation in Geogebra to give my physics students the chance to play with this idea themselves and see how Newton's original idea could really work. I realize now there are other tools to do this built in to Geogebra, and haven't tried to rebuild it to use the ODE solution functions (yet), but that may come come time in the future.
