100 m Speed Demons		Date	Time
Donald Lippincott	USA	Sat 06 Jul 1912	10.6
Charles Paddock	USA	Sat 23 Apr 1921	10.4
Percy Williams	Can	Sat 09 Aug 1930	10.3
Jesse Owens	USA	Sat 20 Jun 1936	10.2
Willie Williams	USA	Fri 03 Aug 1956	10.1
Armin Hary	Germ	Tue 21 Jun 1960	10.0
Jim Hines	USA	Mon 14 Oct 1968	9.95
Calvin Smith	USA	Sun 03 Jul 1983	9.93
Carl Lewis	USA	Sat 24 Sep 1988	9.92
Leroy Burrell	USA	Fri 14 Jun 1991	9.90
Carl Lewis	USA	Sun 25 Aug 1991	9.86
Leroy Burrell	USA	Wed 06 Jul 1994	9.85
Donovan Bailey	Can	Sat 27 Jul 1996	9.84
Maurice Greene	USA	Wed 16 Jun 1999	9.79
Tim Montgomery	USA	Sat 14 Sep 2002	9.78

Other authors have predicted limits to performance. An ultimate limit of 9.37 sec is predicted by F. Péronnet and G. Thibault in``Mathematical analysis of running performance and world running records'', Journal of Applied Physiology, v.67, pp. 453-465 (1989) as cited in Jonas Mereika's paper "How Good Can We Get? Using mathematical models to predict the future of athletics" (published Athletics: Canada's National Track and Field / Running Magazine (April/May 1998)). Peronnet & Thibault's prediction of 9.73 s for Year 2000, however, has proven optimistic by about 0.06 secs; the record is not falling as quickly as they predicted, and the higher limit suggested above might be more accurate.

According to claims on the Charlie Francis discussion forums, Biomechanist Gideon B. Ariel Ph.D may have hypothesized in the late '70's that .: 9.60 would be the limit of the human body mechanically, that to exert the forces and limb velocites necesary to exceed this would crack bones and pull tendon from connection points.

No Limits to Performance ?

Some sports physiologists are more optimistic than this. For an intesting discussion prior to the Year 2000 Sydney Olympics, see W G Hopkins report in Sports Science Limits to Performance, and particularly the responses from Norway's Assoc Prof Stephen Seiler.

Seiler reports

Seiler's fit of a straight line uses data only since 1956, and averages top performances not the peak performance (selecting performances at particular world-class events). Unfortunately, Seiler has not reported the goodness of fit in this discussion ( maybe he has in other publications). His method may be difficult, if not impossible, to extend back in time due to timing and recording inaccuracies. Other articles by Seiler at the site suggest he may have used film or videotape of performances to correct for hand-timing inaccuracies, and finding suitable archival material more than 50 years old could be impossible.

Seiler's methodologies are impressively complex. Does this complexity, however, go beyond what Occam's Razor might caution us to seek: "Pluralitas non est ponenda sine neccesitate" or "plurality should not be posited without necessity."

One should not increase, beyond what is necessary, the number of entities required to explain anything. From the discussion of Occam's Razor on Principia Cybernetica Web:

If Seiler's model makes subjective assumptions about what is treated as a world class event for the purpose of constructing his statistics and how many performances to include (top 6, why not top 4, 8, 10 etc.,.), then it might not be minimizing the number of additional assumptions.

The discussion Hopkins reviewed includes viewpoints that there is no limit to performance. It raises such prospects as using gene therapy to insert the right muscle types into athletes, boosting their capacity. At what point ligaments, cartilage and perhaps even bones, give way and prevent a zero-second 100 m sprint is not clear - but with rewards for elite athletes sky-rocketing the temptation will obviously be there for some sportsmen to be willing guinea pigs.

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Measuring Running Speed ?

Up until the 1968 Mexico Olympics, the 100 m sprint was hand-measured with stop-watches calibrated in tenths of seconds. So, as can be seen in Figure 1 above, the record fell in 0.1 second increments. Part of the inherent inaccuracy in this was the error due to the reaction time of the human timers. This could be between 0.1 and 0.2 seconds on starting and stopping, could vary between starting and stopping, and could vary compared to the runner's reaction to the gun. Measuring records in 0.1 second increments at least recognised the limited accuracy of the process.

For the 1968 Mexico Olympics, Jim Hines had several advantages: 0.01 second timing (allowing a new record just 0.05 seconds faster than the previous, and high altitude (equivalent to a 1.5 m/s wind assist, added to the 1.6 m/sec actual wind assistance). It's difficult to say how much of the improvement in his time may have been due simply to timing artefacts. Hines record stood for 15 years until Calvin Smith beat it in 1983 by just 0.02 seconds. That margin is so fine that if a light (eg. a LED) was flashed on for 0.02 seconds nobody would see it.

Some researchers have estimated the effect of altitude, and wind-assistance. J.R.Mereika's paper "A Matter of Time" (Published: Athletics: Canada's National Track and Field / Running Magazine (December 1999)) has estimated that the results could vary by as much as 0.18 second for the differing extremes of wind assistance (up to 2 m/sec) and altitude effects. The 0.18 seconds is nearly enough to cover all the improvement in the record for the last two decades!

The difference between no wind and 2 m/sec wind assistance could amount to around 0.1 seconds. Records that split this difference (by measuring to better than 0.1 sec resolution) could simply be measuring the varying contribution of wind. Tim Montgomery's 2002 record had the maximum legal wind assistance of 2 m/sec.

Beating the Gun or Running Fast

It is equally revealing to consider the effect of the start, and particularly the reaction time of the runners. The reaction time is the time is takes for the runner to respond to the start signal and begin leaving the starting blocks. (see Omega and Sport - Athletics for a good run-down of the sprint timing and start rules). Interestingly, it is considered that there is a limit to how fast a human can react to a start signal. As of 2002, if an athlete left the blocks sooner than 100 mSec after the start signal, he was deemed to have false-started.

The best athletes reaction times are usually in the range of 120 mSec to 160 mSec (see graphs). If you track individuals performances, there is usually a spread of 0.01 to 0.03 s in their reaction times. Until someone can consistently start in the same time (with less than 0.01 second variatin), it's fair to question what it means when events are timed in 0.01 second increments. At least some of the difference between world records and world class performances might have more to do with luck than something that can be honed by good training.

See Reaction Times and Sprint False Starts for more discussion of reaction times.

Peak Speed

Further insight can be gained by looking at the quickest 10 metre split attained by the runners - this shows the peak average speed reached. In 1988, with 1.1 m/sec wind assistance, Carl Lewis and the drug-fuelled Ben Johnson both hit a 0.83 sec split - or 12.08 m/sec ( = 43.37 km/hr). Lewis hit the same mark again in his Tokyo 91 world record run with 1.2 m/sec wind assistance. Maurice Greene's Athens 97 record, with a mere 0.2 m/sec wind assistance, could manage only a 0.85 split. In Seville 1999, Greene managed a 0.84 with a 0.2 m/sec wind assist, as did Dwain Chambers. Tim Montgomery's 2002 record run matched the 0.83 sec split, with 2m/sec wind assist (see Track And Field, Nov 2002). But over nearly 14 years, the peak speed has not been improved unless an allowance is made for wind assistance (making Greene's 99 performance perhaps the best).. This could be a strong signal that a limit has been reached.

Wind Assistance

When you get down to looking at one part in 100 changes to split times, it is wise to question what the wind-assistance figure represents. This has been researched by N.P Linthorne who reports(while with the University of Sydney School of Exercise and Sports Science):

As Linthorne suggests, gusts and swirls at fortunate parts of the track could be affecting the results. The wind guage is required to have an accuracy of only 1 part in 10 (relative to the 2 m/sec limit being measured) - nothing like the 1 part in 1000 reported timing accuracy. Inaccuracy of 0.05 seconds could account for virtually all the difference between the last three world records (9.84 to 9.78 seconds).

Most of the recent improvement in the record has been due to improved reaction times (as shown for the Carl Lewis case above).. Is it really a record run, or just a record start ? If the improvement is purely due to luck or flukes in beating the gun, then what's the record worth ? What if it's just random wind gusts ?

The change in the starting rules may make gambling on fast-starts less likely and beating the records could be difficult without this advantage. If the 100 mSec false-start limit is reduced, and new records arise, will they be due to a genuine improvement in performance, or just good-luck in beating the gun ? How long will it be before anyone runs faster than Carl Lewis's 0.83 s split from 1988 ?

What effect will the April 2003 revelations of Carl Lewis and other American athletes positive drug tests have, if any ? Lewis was found to have tested positive to the stimulant ephedrine in the US trials in 1988. Is that how sprinters improve reaction times ?

The Comparing 100 m Sprinting - Wind & Altitude Correction page shows how correction for wind and altitude might affect rankings, and can also be useful in eliminating some of the variability in trends in average top performances.

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Racehorse Limits

Thoroughbred Race Horses are the product of over 300 years of selective breeding. Today, racehorse records remain much the same as they were 100 years ago. As an example, Erica Goode reports (New York Times 8th May 2001, "Thoroughbreds Face Physical Limits to Improvement"):

Lest you gather from this that Thoroughbreds have reached perfection, Loren Bolinger's 1998 article On the Manipulation of the Gene Pool What's Best for the Thoroughbred? -A Human Conceit, (once at "http://www.sunshow.com/pool.html"), gone by 2004)" raises a number of shortfalls in the history of thoroughbred breeding. This include a reluctance to incorporate female bloodlines until recent times, and disastrous destructive inbreeding. These mistakes and shortfalls lead to Loren's question on the wisdom of genetic manipulation:

It does not seem reasonable to expect human performance to improve forever and ever, when it's clear that human efforts - including selective breeding over hundreds of years - has not been able to achieve that for racehorses - does it ? Have we learned enough from the mistakes in thoroughbred selective breeding to not make even more disastrous mistakes with human athletic inbreeding ? Today's panacea, Genetic manipulation, may turn out to be hi-tech snake oil.

Interestingly, thoroughbreds continue to be raced while records no longer fall. Financial support for genetic manipulation of horses was so little that researchers have had great difficulty in funding the Horse Genome Project. The horse racing industry is not convinced of its potential benefits.

Racing is about competition - which is beating the other horse, not necessarily winning world records. Thoroughbread racing is also a lot about gambling, and that doesn't require record-breaking.

Eventually, the human 100 m sprint will get to that !

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Mathematics of Logistic Equation

For there to be a performance limit, we need the Performance P(t) function to reach a limit at some future time - with no further change. The objective function is then:

dP/dt = 0                                       ...(1)

There clearly has to be a limit - even if it is the no-time-flat value of P_limit(t®¥) = 0.

We don't know what sort of function to use for dP/dt. Let's postulate that it might be some function of the current performance level (so that we can use current performance to predict future performance):

dP/dt = f(P) = 0                                ...(2)

The Taylor series expansion of any function f(P) gives:

dP/dt = bP + dP2 + gP3 + ....                    ...(3)

Let's take to equation 3 with Occam's Razor. Using one term of the taylor series expansion for f(P) gives an exponentially increasing performance - and can't satisfy the objective (1) of reaching a limit (except for the trivial case of b = 0, which implies no change at all)

Taking the first two terms of the taylor series expansion is the simplest approximation to f(P) that will permit a non-trivial solution to (1).

dP/dt = bP + dP2                                       ...(4)

Solving (2) and (4) gives

 0      = Plimit ( b + dPlimit )

hence

b + dPlimit= 0
 
 Plimit = - b /d                       ...(5a)

( 
Since 

 Plimit >= 0
 
 we would expect that b  >= 0 and d <= 0  in equation (9).
 )

The solution to the differential equation in (4) is given is

Summary: The logistic equation is the simplest mathematical model that captures a process of change that reaches a limit.

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Applications of Logistic Equation

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