Balls Rolling Down Planes

I'm a TA for a 3-week course in relativity for high school students. I've been putting together short (and sometimes not-too-short) lectures for them in the afternoons, but last weekend and this upcoming one I've been tackling a more ambitious project: teaching them calculus. (How does one conjugate a verb referring to an action you both did in the past and will do in the future, but aren't doing right now?)

It took me several hours to put together a simple example of modeling a physical problem using calculus - a ball rolling down a plane without slipping. It's in the category of simple problems used to drill students. There are hundreds of examples like it in every introductory physics book. For ten bucks you can buy a giant practice book with 3000 of them. Yet somehow it took me hours to put the notes together and they're monstrously long (7 pages with 3 diagrams).

Of course, most of the reason for this is that I do a fair amount of extraneous stuff. I solve the problem without using forces at all. I derive the expression for the kinetic energy of a rolling sphere by integrating over the kinetic energy of a differential volume element, just as an example of how to do such an integral.

The idea is that I don't want things like "moment of inertia" to be black boxes used to make quick numerical calculations, the way introductory texts invite their students to do.

On the other hand, everything I did this afternoon is pretty much worthless in terms of my original goals. Someone who's spent only three hours of their life studying calculus has no prayer of understanding the notes I wrote. Probably, the only people who would understand them are the people who already know everything they have to say.

Students complain about hard-to-follow lectures and obscure notes. But flip yourself over to the other side for a while and you'll begin to appreciate how difficult it is to give a clear explanation, even for something that's crystal clear inside your own head.