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#12 – Pushing the Pace of Physics

5 May 2007

Last weekend I ran in the Road Ends Trail Run and despite achieving my goal, I noticed that my leg speed just wasn’t there. Thursday I hopped on a treadmill to get a feel for my stride rate (steps per minute) and sure enough, my recent lack of training had slowed my legs down a bit. Yeah, speed work is good.

While running pops between a 6:00 pace and a 10:00 pace I noticed that as the treadmill sped up, it went quickly from 10:00 to 9:00 but more slowly from 7:00 to 6:00. My brain was too tired to do math at the fast pace, but when I slowed down again I realized that this made sense – speed does not correlate with pace in a linear fashion. For example:

10:00 pace = 6.0 mph, 9:00 pace = 6.7 mph, a difference of 0.7 mph;
7:00 pace = 8.6 mph, 6:00 pace = 10.0 mph, a difference of 1.4 mph!

What this means is that the faster you run speed-wise, the harder it is to reduce your minutes-per-mile pace because speed and pace have an inverse power relationship, like you see below:

The blue line is pace relative to speed. I also added a green line representing energy to show how energy is proportional to the square of speed, meaning energy has a parabolic relationship to speed. (For simplicity I used generic “energy units” as opposed to real units – like Joules or Calories – where 100 energy units equates to a 4:00 pace.)

Since most of us runners measure ourselves using minutes-per-mile pace rather than miles-per-hour speed, I created a plot showing the relationship of energy to pace:

This shows why running fast is so hard! The faster you go, the even more energy it takes to sustain that pace. Using the paces from my first example, it takes:

4 energy units to increase pace from 10:00 to 9:00
12 energy units to increase pace from 7:00 to 6:00

And just to show how tough it is for elite runners to push themselves to a 4:00 pace:
36 energy units to increase pace from 5:00 to 4:00

Wow! So cutting your 5K time from 30 to 27 minutes isn’t nearly as difficult as trying to improve from 18 to 15 minutes. This is why elite runners measure their performance to the second while average folks like me are content to measure to the minute or half-minute. Speaking of 5Ks, here’s how my performance has improved running 5Ks over the past couple years:

24:39 overall = 7:57 pace = 25 energy units (October 2005)
22:42 overall = 7:19 pace = 30 energy units (October 2006)
21:07 overall = 6:49 pace = 35 energy units (March 2007)

Looks like I’m pretty consistent; if I can improve another 5 energy units to 40 that puts me at a 6:19 pace, which would pass my goal of running a 20:08 5K (a 6:30 pace).

One way I can see an analysis like this helping runners in training is to provide more achievable goals. As we train and improve, initial results are going to come quickly as we cut large chunks of time off our race performances. If we expect such improvements to continue at a linear pace then we’ll only become frustrated. For example, my second 5K was two minutes faster than my first; however, I should not expect my next 5K to be two minutes faster than my last one. This may seem ironic but the faster we get, the “lower” we should aim our sights for the next goal.

Try it out! How fast were you running when you first got started, and how fast are you now? How many “energy units” is that? If you maintain the intensity of your training, you could expect a similar improvement in “energy units”. Of course the human body has limits to its energy output, which makes running faster even more difficult than just the physics of energy and speed!

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