Over at Uncertain Principles, Chad is talking football. There’s this pesky problem of spotting the ball at the end of the play. In a game where fractions of an inch can make or break the end result, too often the issue is determined by a more or less random guess by the referee of where the ball stopped. Instant replay has helped the issue, but not come anywhere close to fixing it. It’s too imprecise and often made less useful because there’s enormous football players diving for the ball and thus obscuring it from the cameras.

There’s good suggestions. DGPS, radar, optical tracking, and lots of other things have been proposed. I threw in dead reckoning by accelerometer as a left field suggestion, though it’s probably among the least practical. The idea is that you put an accelerometer in the ball and integrate the acceleration twice with respect to time to determine the final position. Submarines (and I think maybe some aircraft) do this, though uncertainty builds up and the position becomes more and more uncertain with time.

How uncertain? Well, let’s take a look at the equation which gives you distance traveled if you’re accelerating with some uniform acceleration *a* for a time *t*. (In football the acceleration is far from constant, but you can treat it as the limiting case of many very small accelerations over the course of the play.) The equation is:

There’s two sources of error. The first is error involving uncertainty in the measured acceleration. If your measured value is a little lower than the real value, your calculated distance will be shorter than the real distance. Same thing with time. If you get the acceleration perfectly right but incorrectly measure the time the same kind of error will result. We can put approximate boundaries on this error by calculating how fast the distance function varies with slightly different accelerations and times, and multiplying by the error in those measurements. We do this with a little calculus. Calling the distance error delta d, we get:

That’s more formal than we need. In our football application it will be very easy to measure the time of the motion very precisely. The accelerations involved are large, but not so large that a modern clock circuit can’t easily measure intervals small enough so that the acceleration is constant for all practical purposes. That means the second term under the square root is zero. And that means the square and square root undo each other, leading to:

We would like delta d to be small, maybe something close to 1cm. The accelerometer need only function over very short time intervals, because so long as the ball is not covered by the bodies of the players it can be tracked by optical means or something similar. Only during the final hit that stops the play is the position really at issue. The time t should be in the single second range or so. I propose that adequate accuracy might be reachable even with an accelerometer that’s only good to within an error of 0.01 m/s^2 or so. Maybe even looser tolerances would do. That’s still pretty precise, especially when the hits are so hard and produce such large accelerations. And you don’t want the sensor to distort the handling characteristics of the ball in any way.

As such my impractical suggestion probably remains impractical. But it might at least be an interesting thing for a project for a clever undergraduate engineer looking for a project for class!