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The Sweet Spot for Positive Splits

As Olympic fever has started to build for the 2012 Summer Games in London, a physicist has analyzed a peculiar aspect of 400 meter and 800 meter track races. For the vast majority of world records in these races, the runner runs the first half of the race faster than the second -- a positive split.

These consistent positive splits don't happen for shorter races (the runners tend to run faster during the second half) or for longer races (world records don't seem to favor either positive or negative splits). So Jim Reardon, a physicist at the University of Wisconsin Madison and a track coach, decided to uncover the reason using physics.

The end of a women's 100m race. Image Courtesy Beat via Wikimedia Commons.

If you take a look at the table of track world records below, there's an interesting pattern. The sample of world records in the shorter events, namely the 100 m and 200 m, have been set exclusively with negative splits, meaning that the runners end the race faster than they start.

On the other hand, the middle distances -- the 400 m (one lap) and 800 m (two lap) races -- have world records set almost exclusively with positive splits. For the longer races, it's a wash.

"There was a puzzle in the data," said Reardon. "All the good runners go out faster than they can sustain," for the 400 m and 800 m races.

To uncover the reason behind this phenomenon, Reardon developed a computer model and used differential equations to describe its behavior. In fact, the model may look familiar to students who have taken a differential equations class: Reardon modeled a runner's leg as if it were a tank that is simultaneously being filled with a fluid and drained of that fluid.

After creating the model, Reardon crafted his differential equations so that they would accurately predict the optimal pacing observed in the world record data above. With optimized equations, Reardon could look back at the model to see how it matched up with the actual biology of a runner's leg.

The X Factor

With this simple model predicting the world records seen in these medium distance races, a few questions naturally arose: what could the fluid in the leg "container" represent, and how does it affect racing performance?

To answer that question, Reardon looked through some of the physiological literature to see what the fluid in his model -- which he coined as the "X-factor" -- might be. Some physiologists think that when positive hydrogen ions build up inside of our legs, they cease to function and "burn out." Others disagree, arguing that the buildup of certain chemicals may lead to the legs ceasing to function during the races.

Regardless, something builds up in the runner's legs after extreme exertion, causing the legs to stop working effectively. Regardless of what this x-factor may be, the legs need to properly diffuse the x-factor to continue working, according to Reardon.

Just like a drop of food coloring diffuses throughout a glass of clear water, muscles in the leg rid themselves of this x-factor diffusively, said Reardon. Here, Reardon's modeling of the leg as a container becomes more clear. More and more x-factor "fluid" builds up during the course of the race, and the leg has to efficiently drain it to maintain performance.

According to Reardon's research, this process plays an important role in the medium races but not the other races. First, the short races don't last long enough for the leg muscles to wear out, so the x-factor doesn't become an issue. For the longer races, the body equilibrates over a long enough period of time, and other variables, such as VO2 max, become more important for determining pacing.

"In the middle, there's this range where it matters," said Reardon.

There are certain strength training exercises that will likely make the legs more efficient at diffusing the X-factor, according to Reardon. He doesn't anticipate big changes in his training regimen in light of this research, however.

Instead, Reardon hopes that this research may inspire educators to include coursework problems that touch on this model. Differential equations students cover the same mixing problems over and over, and Reardon hopes to break the monotony.

"There's a new problem and the math isn't at a higher level," said Reardon. "And it leads to some results that may be relevant in the real world."

Reardon's research paper has been submitted to the American Journal of Physics and is currently being reviewed. You can see his paper currently on the arXiv.


  1. Is this a joke? This has got to be a joke. This is not a puzzle, the "x-factor" is called lactic acid. Conclusive research and the answer to this question has been known for more than 20 years now. This is very sloppy research.

  2. And you have hastily critiqued the article without having sufficient knowledge of chemistry.

    Do you really think a physicist wouldn't know about lactic acid?


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