"All of these issues considered togethersuggests that a new world record of less than 9.5 s is withinreach by Usain Bolt in the near future."
This paper was just published in the March 2009 edition of The American Journal of Physics (I have no idea if the above links will work, but the e-print is available here. The authors speculate on the possible winning time of Usain Bolt in the 2008 Olympics if he had not celebrated prematurely at the 80m mark.
It's an easily-readable article, and from what I can tell they did about as good a job as is possible with the data they had to work with. An outline of their process is:
- Use frozen frames of the race to estimate Usain Bolt and Richard Thompson's positions as functions of time.
- Estimate error in these positions as well as error in the time of the frame
- Fit splines to the data to extract the runners' velocity and accelerations. Use Monte Carlo methods to obtain error on these figures.
- Make assumptions about Bolt's acceleration if he had not celebrated.
- Extrapolate from these assumptions to a new finish time.
When Bolt began celebrating, his acceleration dropped below that of Thompson. The two assumptions the authors test are:
- Instead of Bolt's acceleration dropping below Thompson's, the two accelerations remain equal. In this case, Bolt's 100m time is estimated to improve from 9.69 to 9.61 +/- .04s (95%)
- Bolt's acceleration remains higher than Thompsons through to the finish line. In this case, the authors estimate Bolt's 100m time would be 9.55 +/- .04s.
The authors also comment that because Bolt ran with zero wind, if there had been the maximum allowable tailwind, he would get another 0.1s for free. Also, he could cut about .02 or .03s with a faster reaction time (he had the slowest reaction time in the entire field). This is how they stretch things all the way out to sub-9.5.
I thought the paper was quite sensible and occasionally clever, but there were a couple of things I was curious about. First, it was unclear to me how the authors estimated the error on the runners' positions. Their only comment on this was
"We then assigned an uncertainty to each position
measurement by estimating how many ticks we believed
we were off in a given frame. For later frames, when the
camera angle is almost orthogonal to the track, this uncertainty
is smaller than in the beginning of the race because of
the camera perspective."
Perhaps a well-funded effort could have involved specialized filming of sprinters to get very accurate readings, and simultaneously using the track-bolt-calibration methods the authors describe. Then the more accurate results could be compared with the ones used to analyze Bolt's run to estimate uncertainty. That would hardly be reasonable given that this paper was clearly the extra-curricular effort of a couple of guys who know some stuff about data and video analysis and just downloaded the race videos from online and went at it.
A more realistic possibility might have been for the investigators to estimate positions independently and compare the spread in their results.
Second, it seems unusual to me that the investigators focused solely on Bolt and Thompson, ignoring the data available from the six other runners in the race. Maybe Thompson gave up a little when he realized Bolt had him beat, or maybe he made a slight technical error in the final stages of his race. Then the estimate for Bolt would be too conservative, because his acceleration profile was being projected relative to Thompson's. On the other hand, maybe Thompson is an exceptionally-strong late runner and had a fantastic last 30m to take the silver, in which case the estimate of Bolt's potential is too generous. Comparing his run to the rest of the field seems more robust and more likely to minimize random error inherent in trying to figure out where someone's shoulders are relative to a track two meters below them.
One part of the analysis that may be unfamiliar is the idea of fitting data points with splines. I've never used this technique and don't have the source referenced in the text on hand, but somebody explained these things to me in conversation once.
If you have a set of data points and want to fit them to a curve, one very simple thing to do is draw a straight line from point 1 to point 2, another from point 2 to point 3, etc. Using this method, you will always fit the data exactly, and always get a unique solution. However, the resulting curve is not differentiable.
A more sophisticated approach is to fit a quadratic (quadratic A) through points 1 and 2. Then fit another one (quadratic B) through points 2 and 3. Because you're fitting a curve with three degrees of freedom through only two constraints, each of these quadratics has a leftover degree of freedom. There are two quadratics, so two total degrees of freedom just hanging out. You can remove a degree of freedom by requiring that at point 2 quadratics A and B have the same derivative. You then add a third quadratic (C) through points 3 and 4, requiring that quadratics B and C have the same derivative at point 3. In this way, you continue through all your points, and obtain a curve that fits all the points and has a first derivative everywhere. To get a curve with two derivatives, fit a bunch of cubics through the points. This is, I suppose, how the authors obtained velocity and acceleration of Thompson and Bolt from their position measurements. (Ryan, I know astronomers frequently use these things when looking at time series of luminosity and whatnot. Do you have experience you can lend here?)
Here is a picture showing the real Usain Bolt beating the Olympic field, and the projected Usain Bolt beating them by even more:
Of course, in order for Bolt to run 9.5 or anything near it, he'll have to have one thing occur that the authors did not mention: he'll have to return to the absolutely most incredible form ever exhibited by a sprinter in the history of the human race.
