The student who wrote the original essay focused her attention on the average speed of the record setting athletes. Average speed is found by dividing the distance covered by the total time elapsed. Readers of this essay might also find it interesting to know the instantaneous speed of the athletes; that is, the speed as measured over an vanishingly small time interval. Instantaneous speed is often thought of as the "actual speed" as it varies with the actions of the athlete. In contrast, the average speed is often viewed as an artifact of calculation -- one that summarizes a complex event with a single number.

In track and field, instantaneous speed can be approximated by fitting a smooth, continuous curve to a runner's split times -- the time for the runner to reach a set of equally spaced milestone distances. Split times for the 100 m dash are taken every 10 M. A table of split times with the calculated instantaneous speeds from the 1997 World Championships in Athens, Greece is shown to the right. Note that the first and second place runners had the same instantaneous speeds (11.80 m/s) at the 60 m split. A more detailed analysis of the data (see reference 1) indicated that they also reached the same maximum instantaneous speed of 11.87 m/s just before this point.

An interesting and easy way to determine the instantaneous speed from the split times can be done using any basic data analysis program. Plot the distance against the split times and execute a curve fitting analysis. I chose to fit Maurice Greene's split times at Athens to a fourth order polynomial. The advantage of using this curve is that the software I have can determine the best fit coefficients in a fraction of a second.

Instantaneous speed can then be found by taking the derivative of this graph (the slope of a line tangent to the curve at any point).

Which when graphed looks like this…

v_max = 11.78 m/s at t = 5.53 s

There is a serious problem with this graph, however, as it shows a non-zero initial speed. Sprinters are not allowed to "jump the gun"; that is, they must be at rest when the starter's pistol is fired. Such an event would surely have been detected by race officials and a false start would have been called. A proper model would have to reflect this fact.

The value of the derivative at the origin is zero for the second and higher order terms (so they get to stay), but the derivative of the first order term is a constant (so it has to be tossed). I chose to refit the split times to a polynomial with terms of order 2 through 6. It took about an hour for the analysis software to fit this curve to the data.

Again, instantaneous speed can be found from the derivative.

Which when graphed looks like this…

v_max = 11.80 m/s at t = 7.02 s

Is this curve better than the previous one? Well, yes. The initial velocity is now zero as one would expect.

Is this the perfect curve? Well, no. Check out the slope at the origin. You should recall that the slope of a speed-time graph is the acceleration. This curve shows a non-zero acceleration at the beginning of the race, which is against our expectations. The acceleration should build up quickly starting from zero. Again, the runners can't "jump the gun".

A workaround for this problem would be to drop the second order term from our polynomial and see what happens. Maybe I will try that someday, but given the sluggish response of the data analysis software I'm using, I doubt it. A new approach is probably necessary -- one that may not be appropriate for the level of this book.

Editor's Supplement -- 2001

Records were meant to be broken. We now have a new fastest human and his name is Usain Bolt.

Editor's Supplement -- 2008, 2009