I like problems that make my students write their own equations, and we've been working with position, velocity, and acceleration recently. Yesterday, I had them write an equation to model the height of a yo-yo from the ground and then differentiate to find the maximum velocity. The situation lends itself nicely to writing a sinusoidal function, which was what I wanted them to do, really. We got to practice relating position, velocity and acceleration, differentiating trig functions, and using the chain rule all at once.
The problem is, I don't think a yo-yo behaves sinusoidally. Neither did my students, so we tried to write an equation that worked better. Short on time, we wound up with the absolute value of a natural log (it moves quickly out of the hand, changes direction abruptly, and slows down on its way back up). For an impromptu activity, its going to segue nicely into finding the derivative of y = ln x... but I still want to find a good model for a yo-yo. I may head down to the Physics lab and see if there's an easy way to take some data, but I'm wondering if this has been done before (my googling reveals much more thorough models than we're ready for).
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