I thought it might be interesting to explore the mathematics of a common problem with a store-bought HO model train set that contains a collection of straight track segments and fixed-radius curved track segments that form a simple oval. Invariably an initial run of the train has it careening off the track when the train first meets the curved segment after running along the straight track segments.
Why is that? Well of course the train is going too fast. But even if it slows down enough not to fall off the curve, it still jerks unstably and may derail when it first reaches the beginning of the curve. What is going on?
See the Train Wreck Puzzle