Yesterday Om made a comment noting a sneaker with equations inscribed in it. It makes you wonder what would be appropriate. Perhaps simple Newtonian physics like F = ma, v = at, d = at2/2. You might also try to build a model of running. I scribbled the equations of a very simple running model. They wouldn’t speak to a teen and they wouldn’t speak to an expert either. It turns out that an exact model of human running don’t exist. The same is true for walking. This despite quite a bit of money from athletic shoe companies. While a good deal is understood is understood, the situation isn’t as nice as swimming was in the 70s when computational hydrodynamic models revolutionized technique as evidenced by the record books.
The basics, however, are known fairly well. To run you need to strike the ground very hard with a minimal contact time. Using the basic physics of motion you can glean a lot of information using video analysis without a detailed model of running. A pair of sports science people did this for Usain Bolt’s 9.58 second run. The specifics of his run are amazing. A few notes:
° He generated a maximum power of about 2,620 watts (about 3.5 horsepower) 0.9 seconds into the run. For a bit over a second he developed over 2,000 watts.
° He is very tall and aerodynamically inefficient - his coefficient of drag is about 1.2. A little worse than a falling piano. He’s also about 10% worse than the average world class sprinter, so he has to make it up with raw muscle power. It is good to be a freak of nature.
° During the run Usain's muscles delivered about 81.6 kilojoules of usable energy … 19.5 nutritional calories. If you assume an average metabolic efficiency of about 24%, you get to about 40 calories:-) That may describe the run, but his core has heated so much that he will be spending a lot of energy as his body attempts to cool down over the next 15 to 20 minutes. Even so the amount of energy he would have burned including the recovery period is probably less than one Snicker’s bar.
° Most of the power generated by his muscles, about 92% of it, goes into dealing with air friction. Power needed to move through the air goes up with the cube of velocity and coefficient of drag or Cd for the running human body is awful. So rather than moving forward, most of his effort is going into parting the air and creating vortices. Since his muscles are anaerobic during the run it would be interesting to set up a oxygen tent on the Tibetan plateau or in the high Andes in Chile and cut it open a second before the start. Lower air resistance would allow a faster run, but you’d want to get him back into sea level oxygen asap after the run.
° Back to that 2,600 watt burst. Assuming the 24% metabolic efficiency it means he has to dump over eight kilowatts of waste heat. For the 2,000 one second rate he has about 6 kilowatt-seconds of heat energy cooking away inside him and the amount over the run when he’s dropped back to around a kilowatt of output is still huge. It turns out the limiting factor on ten second runs like this is core body temperature which is gets to dangerous levels before the anaerobic portion of the metabolism fades. If he could keep up the rate for a 200 meter run he might not be able to recover. We have some natural limits.
° A few athletes have higher power bursts — weight lifters for extremely short periods of a few tens of milliseconds. Probably the best are indoor cyclists who can deliver 1,500 watts for about 20 seconds. It takes a very special body type.
I am so far from the limits:-) I’m really happy when I can average 200 watts to a rowing machines flywheel for an hour. (a few notes in an earlier post)
Those who probe the limits raise some interesting questions. What could Usain have done to slightly improve his time? It turns out he's a slow starter and has the Cd issue, but are there runners that have his other gifts and a slightly better number? How does wind influence these numbers (that is easy to calculate)? And the general very fascinating question of how important is body and muscles type to different types of sport?
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