| | Ivars Peterson's MathTrek |
June 8, 1998
The article begins: "Canadian Donovan Bailey rocketed into the record books when he set a new world mark of 9.84 seconds for the 100-meter dash at the Atlanta Olympics." It briefly recounts Bailey's career and, toward the end, quotes Bailey: "No one has ever run as fast as I have, running 27 mph."
When Roy D. North, a computer programmer and mathematical gadfly now based in Connecticut, came across that passage, something bothered him. When he calculated Bailey's speed from the given time and distance, he obtained 22.7 miles per hour.
"What went wrong here?" North wondered. "Was Bailey misquoted? Was there a typo?"
North had to find out how that apparent mismatch had come about, and his search turned up an article in the Aug. 5, 1996, Sports Illustrated. The account mentioned that Bailey's speed at the 60-meter mark of the race was 27.1 miles per hour.
Mystery solved! One speed was the average over the entire race, and the other was the instantaneous velocity at a particular point in the race.
However, there's much more to races and world records than simple speed determinations. Jonas R. Mureika, presently a systems programmer at the University of Southern California in Los Angeles, has been developing mathematical models of sprinting to find a way to compensate for wind effects in the 100-m dash. Wind has long been a confounding factor in the 100-m and 200-m sprints. Indeed, records can be established only if the wind speed is less than 2.0 m/s. Nonetheless, sprinters in a race run with a 1.9 m/s tail wind would undoubtedly have an advantage over competitors racing in still air.
Assuming that a sprinter dissipates about 3 percent of his or her energy in overcoming drag, Mureika developed a formula that compensates for accompanying wind speeds, converting measured times into equivalent times in still air:
Interestingly, because of drag effects, a head wind has a greater effect than a tail wind of equivalent strength. Thus, a 10.00-s run with a 1.5 m/s head wind is roughly equivalent to a 9.91-s run with no wind, while a would-be world record 9.83 s with a 1.5 m/s tail wind also comes to 9.91 s in calm conditions.
The formula allowed Mureika to compare performances in all 100-m races on more or less an equal footing. Thus, Bailey's 9.84-s world-record time, aided by a tail wind of 0.7 m/s, turns into the equivalent of 9.88 s in calm air. A race run in 1996 by Frank Fredericks into a head wind of 0.4 m/s becomes 9.80 s in calm air, surpassing Bailey's world-record performance. If Fredericks had given the same performance in Atlanta, he would have crossed the finish line in roughly 9.81 s, Mureika says.
In 1997, Bailey lost his world sprint crown when he was beaten by Maurice Greene, who clocked 9.86 s at a track meet in Athens. Greene's wind-corrected time was roughly 9.88 s.
Mureika has also developed a simple model that takes into account energy losses when runners in a 200- or 400-m race must negotiate various portions of a curved track. Such efforts have put him into the prognostication game, predicting the outcome of races under various conditions and extrapolating trends to future world records.
Mathematical models, however, only suggest possibilities, based on present data and understanding. Athletes set the records.
Copyright 1998 by Ivars Peterson
References:
Guiness Media, Inc. 1998. The Guinness Book of World Records (37th ed.). New York: Bantam.
You can check out the official Donovan Bailey Web page at http://www.donovanbailey.com/.
Comments are welcome. Please send messages to Ivars Peterson at ipeterson@maa.org.
is the equivalent still-air time.