The idle ramblings of a Jack of some trades, Master of none

Somehow, I don't think Maurice Greene will take kindly to being called a thin slab of front cross-sectional area A1, creating a modified drag area Ad. But this is how mathematicians who are interested in this sort of thing model a sprinter. To the question of why bother, I can safely point to the large amounts of money being spent on biomechanical and biophysical research aiming to improve the performance of elite athletes.

One of the earliest mathematical models of competitive running was that of Keller in 1973 2, which was involved optimising a set of coupled differential equations
\dot{d}(t) = v(t)
\dot{v}(t) = f(t) - a v(t)
where a is a decay constant placing upper bounds on velocities, and with the race distance calculated as the integral
d = \int_{0}^{T}v(t)dt
Keller's original model proposed a fixed propulsion f(t) = F, because, he said, athletes need to utilise their fullest muscular effort during the race.

In reality, there are phases during the sprint during which the levels of muscular effort change. No athlete can maintain full power throughout the distance, and in any case, Keller's model could not account for the dynamical variation in each 10 metre split of a race. Indeed, there is a drive phase during which the sprinter propels himself forward from a crouching start lasting about 30 metres. The transition and maintenance phases follow.

Mureika (whose paper I cite below and which provides the meat for this post) proposes a quasi-physical equation, partly mathematical and partly physical, requiring estimation of parameters, where the equation of motion becomes
\dot{v}(t) = f_{s}+f_{m}-f_{v}-f_{d}
The drive term fs is given an inverse exponential function of t2 such that its magnitude decays rapidly during the start of the race. The maintenance term, or the muscular propulsive force that remains to the end of the race, is assumed to decay exponentially as well, although as a function of t, and hence slower than the drive term. The two remaining factors affecting the speed of the sprinter are a velocity term, a physical barrier that limits human performance (e.g. a sprinter's acceleration is less at higher speeds than at lower); and the drag term, resulting from air friction, the sprinter's mass, atmospheric density, and possible impedance (head-wind) or assistance (tail-wind). It is in this last factor that the thin slab of frontal cross-sectional area A crops up that might seem derogatory to our pal Greene.

The equations were calibrated against data obtained from various top-level athletic meets. Each sprinter would be characterised by a unique set of parameters, but for a generalised examination of the class of elite athletes, a combined set was obtained.

The drive-phase estimation proved critical in reproducing accurately the speed profiles of the athletes, especially in the first 30-40 metres. Estimating the wind-drag was important (and indeed provides a method to backtest an independent record claim to see if, indeed, the wind assist was within the IAAF tolerance of 2 metres/sec).

An interesting query that Mureika aims to answer is: what would the 100m world record have been in 1988? You may recall the blistering run by Ben Johnson, who, fifteen metres before the finish line, turned to see where his nearest competitor (Carl Lewis) was, and raised his hand in victory before he actually crossed the finish line. During those last 15 metres, clearly, he was not running at peak speed, and still he demolished the extant speed record, completing the distance in 9.79 seconds. With Murieka's estimated equations, it is possible to show that Johnson could have accomplished the race in 9.69 seconds, shaving off a tenth of a second from an already great record.

Notwithstanding Johnson's later fall from grace, this was truly a monumental moment in athletics.

UPDATE 25 Nov 2008: Now that Usain Bolt has smashed the record despite appearing to slow down to exult, one wonders if Mureika would update his paper to predict Bolt's achievable record?


1. Mureika, J.R., A Realistic Quasi-Physical Model of the 100 Metre Dash, arXiv:physics/0007052v1

2. J. B. Keller, ”A theory of competitive running”, Physics Today 43, Sept. 1973


Ros said...

The reason I never pursued areas like this was the lack of certainty.Tiny variations in the model could lead to quite large changes in the prediction.

I envy those who can put up with it. I'll stick to group theory for now!

Fëanor said...

Ros: I guess that's why it's important always to quote confidence intervals, what? These numerically unstable problems are the very devil, though, I agree.

Still, adds a bit of spice to life, no? Sort of like having to deal with the Monster Group all the time :-)

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