David Hogg writes of teaching intro physics:
The time derivative of velocity is acceleration, both vectors of course. But I [Hogg] was reminded in office hours today of just how hard it is to get across the idea that the velocity vector and the acceleration vector can point in totally different directions. And some students have trouble seeing this when a ballistic stone is going upwards along some (parabolic) arc, some have trouble seeing it when it is going down, and some have trouble seeing it at the top. That is, different students have very different problems visualizing the differences of the vectors over time.
I wonder if it would help to use the business cycle as an example? When you’re at the top of the cycle, the position is high, the velocity is zero, and the acceleration is negative. And so on.
Would that example work or would it just confuse things further because you still have to make the transition to physical motion (in which the dimensions are positions in space, rather than the state of the economy)?
P.S. This sort of problem reminds me of why it was a good thing I left physics. I think I could figure out the answer if I worked at it, but I just don’t have the intuition, in the way that I have when thinking about statistical problems.