Parametric Playground

I'm playing around with the idea of letting kids design and create an animation via parametric equations in GeoGebra. To play around with the idea myself, I tried to come up with some possible motions. Basically, I think that any motion that the kids can dream up, as long as it's a physically sensible pattern, we can create an animation via parametric equation.

Check these examples out: http://www.geogebratube.org/student/mnl0rUuxl . In this, I played around with a bouncing ball, a rotating circle, a rolling wheel, a dampened bouncing ball, and a ball that flies through air with downwards acceleration. In each case, there is an additional rotating point that stays with the circle as it moves around.

If they can dream it, they can create it! My thought is that they would design something, create it in GeoGebra via parametric equations, explain every part of their parametric equation, and plot x(t) and y(t) functions in terms of t (not by hand by via technology), and analyze some critical points along the graph.

Exciiiting!! I love projects like this, because as a baseline, it's reinforcing everyone's understanding of parametric equations, but the upper end is limitless to allow the creative and mathematically confident students to challenge the limits of their knowledge. For example, in order to create the dampened bouncing ball, I had to use the form y1(t) = a/(t + 1)*|sin(bt)| + c just to get the center of the ball to move/bounce in a dampened way, which made the height of the rotating point around the circle more complex:
y2(t) = csin(dt) + a/(t + 1)*|sin(bt)| + c .... In creating even something that looks simple, I incorporated rational, absolute-value, and sine functions. I would be very happy if some of my kids approached this level of complexity in their own projects.