After the AP exam, I have my students design and build a scale model of a roller coaster. The path of the coaster must be piecewise and include at least one trig function and one polynomial function. The pieces have to joined together so they are continuous and differentiable at all points. They must also determine the velocity and acceleration functions for their paths. Perhaps this year, I will also add some other conditions so they will have to solve differential equations or determine the arc length.
The students always enjoy this and look forward to it as an end of the year project. It allows them to work together and apply many of the concepts of calculus.