In August 2009, Usain Bolt set a men's world record for the 100 and 200 meter runs. In September, Sarah, a cheetah at the Cincinnati Zoo, set a cheetah world record for the 100 meter run. Bolt's average speed was a little over 23 miles per hour. Sarah's average speed was just over 36 miles per hour.
Average Speed vs. Top Speed
In both of these record setting runs the runner, man or cheetah, started from rest. Because they both had an initial speed of zero, their top speeds were higher than their average speeds. Computing their average speeds is relatively simple. Divide the distance run by the time needed to run the distance.
Computing the top speed from just the distance and time is more difficult, however estimating an approximate top speed is possible. What were Usain Bolt's and the cheetah's top speeds during their world record runs?
Physics of Running Races
In "A Theory of Competitive Running" (Physics Today vol. 26, p. 42, 1973) Joseph Keller analyzes human runners' speed and acceleration for races ranging from 50 to 10000 meters. Keller's mathematical analysis provides the strategy for optimum performance in these distances. Even if they are not aware of Keller's analysis, world class runners must run close to the optimum strategy to set world records.
For races less than 291 meters, Keller finds that runners get their best performances by accelerating to their top speed as fast as possible and then running at top speed for the duration of the race. Keller mathematically estimates the time needed to accelerate to top speed based on four physiological parameters. They are:
- The damping coefficient describing resistive frictional forces on the runner;
- The maximum forward force the runner can exert per kilogram of the runner's mass;
- The energy equivalent of the initial oxygen available to the runner per kilogram;
- The energy equivalent of the rate at which the runner's respiration can supply new oxygen per kilogram.
Keller's analysis indicates that human runners need about 2 seconds to accelerate to top speed. This time will of course vary for individual runners, so any estimate of the top speed of an individual runner will be an approximation rather than an accurate measurement.
Estimating the Top Speed in a 100 Meter Run
Applying the kinematic equations found in most physics textbooks, allows deriving a formula to estimate the top speed of a sprinter running at the optimum strategy in the context of Keller's theory of running. The formula is:
vmax = x/(t-(1/2)ta)
In this formula:
- vmax is the runner's top speed,
- x is the total distance of the race,
- t is the total time to run the race, and
- ta is the time spent accelerating to top speed, about 2 seconds for human runners.
Applying this formula to Usain Bolt's world record 100 meter run in 9.58 seconds gives Bolt's estimated top speed as about 12 meters per second or 26 miles per hour.
Keller's theory was intended for human runners, so it doesn't really apply to cheetahs. The largest difference for cheetahs is the time needed to accelerate to top speed. Not having this time for cheetahs, using the time for humans will give a rough estimate. With this caveat, the Cincinnati Zoo's cheetah, Sarah, ran at a top speed of about 20 meters per second or 44 miles per hour in her 100 meter run. Sarah's actual top speed could be slower or faster depending on how long her acceleration phase lasted.
The top speeds in sprint runs are considerably faster than the average speeds.
Paul A. Heckert - I have a Ph.D. in astrophysics, over 30 years experience teaching physics and astronomy, and over 60 published research articles.
