A ruler that is right, in the wrong room
Throw a ball spinning about a vertical axis and it curves. Everybody knows that much. What is less known is that if you are willing to measure the flight in distance travelled rather than in seconds, the curving becomes almost absurdly simple: the ball’s heading swings by the same angle for every metre of path it covers, and the rate depends on nothing except the ball. Not on how hard you threw it. Not on the angle. Not on gravity, which drops out of the horizontal problem entirely, and not on the drag, which slows the ball without bending it. A baseball turning a half circle needs 884 metres of path, and it needs exactly that whether you lob it or fire it.
Nobody seems to have written this down until last week. The pieces are all old, one of them three hundred years old, and somebody finally set them side by side.
There is a bonus hidden in it. As the heading swings, the horizontal speed falls off exponentially, and the rate of that fall is the ratio of the ball’s drag coefficient to its lift coefficient. So take one tracked flight, measure how far the heading swung and how much horizontal speed was lost, divide one logarithm by one angle, and out comes a number that ordinarily requires a wind tunnel. You do not need to know the ball’s mass. You do not need its radius, or the air density, or even which sport it came from. You do not need gravity. Divide, and read.
I spent an evening trying to break that.
My guess going in was the usual one. Formulas like this tend to be exact and useless at the same time, because the thing you divide by is small and the thing you divide is a difference between two nearly equal measurements, and a small difference over a small angle is how you manufacture an enormous error bar out of perfectly good data. I have written about that before. So I built the trajectory and sprinkled position noise over every frame. Three millimetres: a plausible figure for optical ball tracking, though I went looking for a published one and came back without it. Then I ran the estimator a few thousand times to see how wide the answer came out.
It came out to a tenth of a percent.
That is a good result by any standard, and I had been so sure of the opposite that I checked the arithmetic by hand before I would believe my own program. The pitch, which swings its heading less than four degrees, still lands inside two percent. On any reasonable reading, the measurement works.
So I went looking for the catch somewhere else, and found it immediately, sitting in plain sight in a word I had not thought about. The whole derivation is a statement about the ball’s motion through the air. The measurement is a statement about the ball’s motion across the ground. In a sealed room those are the same sentence. Outdoors they are not.
Put a breeze in. Half a metre per second, the sort of air you would call still, the sort that does not move a flag, and the answer comes out four and a half percent wrong. A metre per second costs about nine. There is no averaging that away, because it is not noise: for a given wind it is the same error every time, in the same direction.
The shape of it turns out to be simple enough to write on a napkin. Call the wind speed divided by the ball’s horizontal speed the small number you are stuck with. The fractional error in the answer is that small number, times the sine of the wind’s bearing, times a factor that is the drag-to-lift ratio added to its own reciprocal.
Two things fall out of that, and they are the reason I am writing this down.
The first is that the last factor has a floor. A number plus its reciprocal is never less than two. It does not matter what ball you choose or how you throw it; the best case is a ball whose drag and lift coefficients happen to be equal, and even that case pays double the wind-to-speed ratio. There is no clever selection of equipment that makes this small. Every ball is bad, some are worse.
The second is the one that stopped me. Every other quantity in this problem gets better with a longer flight. A longer flight swings the heading further, which makes the angle you divide by bigger. It sheds more speed, which makes the logarithm bigger. It gives you more camera frames, which shrinks the noise. If you had asked me how to improve this measurement I would have said, without hesitating, throw it further. But the flight length cancels out of the wind term exactly. It is not that a longer throw helps a little less than you hoped. It is that a longer throw does not help at all, and when I stopped arguing and measured it, the error grew.
I want to be careful about what kind of failure this is, because I have catalogued two others recently and this is neither.
An instrument can be broken: it gives the wrong answer to a question it was perfectly capable of answering, and the cure is to check it against something whose answer you already know. An instrument can also be sound and aimed wrong: it works exactly as built, and the quantity you care about simply cannot show up in what it reports, the way a photograph of a wheel cannot tell you which way the wheel is turning. That one is nastier, because every calibration passes.
This is a third thing. The instrument is exact. It is precise. It is pointed squarely at the quantity you want, and it reports that quantity faithfully — in a frame of reference that is not the one you are standing in. The error is not in the mathematics and not in the aim. It is in the unstated assumption that the room is still.
And the signature of this third kind, the thing I would now look for first, is that the error refuses to respond to effort. Noise falls as you gather more. A bias from bad aim can at least be characterised and subtracted. But when a term survives every improvement you can think to make — more frames, longer baseline, better camera, more throws — that is the frame error talking, and no amount of care inside the room will reach it. You have to go and change the room.
Which, for this one, means exactly what it sounds like. The formula is beautiful and it is exact and it will tell you the truth about a spinning ball. Take it indoors.
