David Smith is Associate Professor Emeritus of Mathematics at Duke University.
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This module was prepared with the assistance of Jer-Chin Chuang (Furman University) and John Michel (Marietta College). It is based on an article by S. R. Dunbar in the CODEE Newsletter, Spring,1994.
David Smith is Associate Professor Emeritus of Mathematics at Duke University.
Click on the button corresponding to your preferred computer algebra system (CAS). This will download a file which you may open with your CAS. You should arrange your browser and CAS windows side by side so you can see both.
Ver. 6 or higher |
Ver. 3.0 or higher |
Ver. 5.1 |
This module was prepared with the assistance of Jer-Chin Chuang (Furman University) and John Michel (Marietta College). It is based on an article by S. R. Dunbar in the CODEE Newsletter, Spring,1994.
In 1973, J. B. Keller published a theory of competitive running, in which he proposed that the speed of a sprinter (up to 300 meters) could be modeled by the differential equation
dv/dt = A - v/b,
where v(t) is the speed at time t. At the time of Keller's work, reasonable values for the constants were A = 12.2 m/sec2 and b = 0.892 sec.
We turn now to the question of appropriate values for the parameters 20 years after Keller's work. The following tables record 1993 world-class performances in the 100 meter sprint, separately for men and women. Each table shows "split times" for 30, 60, 80, and 100 meters.
Name | 30 m | 60 m | 80 m | 100 m |
Linford Christie | 3.85 | 6.45 | 8.15 | 9.87 |
Andre Carson | 3.83 | 6.43 | 8.15 | 9.92 |
Dennis Mitchell | 3.82 | 6.46 | 8.22 | 9.99 |
Carl Lewis | 3.95 | 6.59 | 8.30 | 10.02 |
Name | 30 m | 60 m | 80 m | 100 m |
Gail Devers | 4.09 | 6.95 | 8.86 | 10.82 |
Merlene Ottey | 4.13 | 6.98 | 8.87 | 10.82 |
Gwen Torrence | 4.14 | 7.00 | 8.92 | 10.89 |
Irina Privalova | 4.09 | 7.00 | 8.96 | 10.96 |
To illustrate the meanings of split times -- and the possibility that Keller's parameters are outdated -- we show the men's split times in the following figure, along with possible model distance functions. Christie's data (fastest) are shown as solid circles, and Lewis's (slowest) as open diamonds. The broken curve shows the distance function you calculated in Part 2 with Keller's parameters. The solid curve shows a possibly better fit -- with a faster terminal velocity.
Before we charge ahead to find a value of b from the displayed equation, we make a few observations about this equation and its relation to the modeling process.
(T1,D1), (T2,D2), (T3,D3), (T4,D4),
there are six different ways to pair them up so the first subscript is smaller than the second: (1,2), (1,3), (1,4), (2,3), (2,4), (3,4). Thus, we get six possibly different determinations of b from our men's data.
Before we tackle the problem of finding a numerical value for b, we will look at the problem graphically to see if it makes sense to proceed.