# A new toy problem

First, a short math question from Peter:

Question: What is the coefficient of $x^{12}$ in the simplified expression of $(a-x)(b-x) \dots (z-x)$?

I often hate these questions, but this one gave me a laugh. Perhaps it was just at the right time.

A police car passed me the other day with sirens wailing and I became reminded full on about the Doppler Effect. But the siren happened to agree with a song I was whistling to myself at the time, and this made me wonder – suppose we had a piece of music (or to start, a scale) that we wanted to hear, and we stood in the middle of a perfectly straight train track. Now suppose the train had on it a very loud (so that we could hear it no matter how far away it was) siren that always held the same pitch. If the train moved so that via the Doppler effect, we heard the song (or scale), what would its possible movements look like? How far away would it have to be to not run us over?

Some annoying things come up like the continuity of the velocity and pitch, so we might further specify that we have some sort of time interval. So we have a scale, and the note changes every second. And perhaps we want the train to have the exact right pitch at the start of every second (so that it would have constant acceleration, I believe – not so exciting). Or perhaps we are a bit looser, and demand only that the train hit the correct pitch each second. Or perhaps we let it have instantaneous acceleration – I haven’t played with the problem yet, so I don’t really know. I’m just throwing out the idea because I liked it, and perhaps I’ll play with it sometime soon.

Now, the reason I like it is because we can go up a level. Suppose we have a car instead, and we’re in an infinitely large, empty, parking lot (or perhaps not empty – that’d be interesting). Suppose the car had a siren that wailed a constant pitch, too. What do the possible paths of the car look like? How does one minimize the distance the car travels, or how far from us it gets, or how fast it must go (ok – this one isn’t as hard as the previous 2)? It’s more interesting, because there’s this whole other dimension thing going on.

And even better: what about a plane? I sit on the ground and a plane flies overhead. What do its paths look like?

All together, this sounds like there could be a reasonable approach to some aspect of this problem. Under the name, “Planes, trains, and automobiles” – or perhaps in order – “Trains, automobiles, and planes,” this could be a humorous and fun article for something like Mathematics Magazine or AMMonthly. Or it might be really hard. I don’t know – I haven’t played with it yet. I can only play with so many things at a time, after all.

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