Saturday, 22 March 2014

Arches: a new conceptual model for running



I don't like pyramid models. Although a well-built pyramid certainly looks nice, they rarely provide a good model for a healthy system. Think Ponzi schemes: a strict hierarchy where only the topmost members benefit. Pyramids are a great analogy for dictatorships, kingdoms, or the catholic church. None of these things are something you'd aspire to mimic for a system that benefits most through cooperation.


Before I get to running, consider one other bad pyramid model: food. For some reason pyramids are used in nutrition. The food pyramid clearly makes no sense. How is it that vegetables are supporting fish and oils, and not the other way around? Why are eggs and sweets near the top? What if you are vegetarian? Do you want to place the most important food items on top, or the least? Why are calorie-rich foods scattered so randomly? For any practical purposes the food pyramid is confusing. More to the point it's just a bad model for something so intricate.

Sunday, 23 February 2014

Which country 'won' Sochi?

There's always some debate about how to rank the medal tally of all the countries. Some news outlets  rank by number of gold. The second option is to tally the total bronze, silver, and gold. CBC chose to rank by pure gold, which puts Canada in third and USA fourth, while NBC took the total count, which places USA second and Canada fourth. Hmmm. How about we try a points system, where Gold = 3 points, Silver = 2, and Bronze = 1. This takes a middle ground where runner-up performances still count while admitting gold ought to be worth more than bronze. In this case here's the Sochi 2014 medal table:


Russia is the clear winner with 70 points, while Canada's 55 points edges out Norway and the United States who are tied third with 53. The overall rankings are only tweaked a little, which is good not to upset the apple cart entirely.  One modification I could suggest is to count team sports for more points since it's impossible for a given country to sweep the medals (i.e. Canada's men's hockey team can win at most one gold while the Netherlands can, and have, won multiple medals per event). Then again sweeping the podium is an equal opportunity event hence I'm not going to change the table.

I forgot to mention a third way people rank the olympics, which is dividing that countries' population by the metal count to 'normalize' the rankings. You could do that here too, but with points instead of using the (oversimplified) medal count. Here's the rankings again with a points per capita:


No surprise that Norway is the clear winner with 10.3 points per million people; on average every 100,000, or a tiny city in Norway, generates an Olympic point. Slovenia and Switzerland rank in second and third, which I would not have considered intuitive choices. Meanwhile Canada and Russia slip all the way down to ninth and 14th, respectively (I kept the original rank numeration so you can see how much shifting there is). No surprise that China sits in dead last for winter, but maybe if we tried doing this with London 2012 something interesting could emerge. But that's for another post.

Sunday, 16 February 2014

I came across an info graphic from CBC about which countries pay money for winning Olympic medals.

Figure 1. From CBC website 
My first reaction was there wasn't much correlation between either how many medals the country actually won and how much the athlete got paid for gold. Nor is it clear how much these athletes get paid, if anything, when not winning gold. Most likely the wealthier countries pay their athletes some stipend when training. Then again, poorer countries may have elite training programs, assuming you qualify for one such as Russia's Red Army.

Nevertheless, since winning gold is rarely, if ever, a reliable source of income I figured these cash prizes were a form of saving face for the countries themselves and less so for rewarding athletes. It's as if to say "look, we don't shortchange our athletes, at least if they are winning".  I wondered if there was a correlation between the prize money and the general wealth of these countries per capita. Hence I took the figure 1 prize values and plotted them against GDP.

Figure 2: Olympic prize money compared with GDP per capita wealth per country

The correlation is not perfect, but the results are relatively clear: poorer countries give relatively more money to the winning athletes than richer ones. My guess is that these athletes make little money and to avoid the embarrassment of having a gold medalist living in poverty, it would be wiser from a publicity point of view to reward them enough to live comfortably in future years, or at least until they are forgotten.

There is a deeper significance to the negative correlation in figure 2. Wealthier countries can potentially afford larger medal prizes than poorer ones. And poorer countries don't win enough medals for these payouts to be a significant cash drain. Hence the absolute money given out to athletes is rather arbitrary.  Canada pays out $20,000 per gold medal. This is a pittance when you consider the years of effort required to earn one. An annual graduate stipend in a canadian science program is more than 20 grand, which is also small, and there are a lot more graduate students than Olympics athletes in Canada. The majority of legitimate money comes from sponsorship deals like commercials and public appearances for talks. Is this a good point in favour of capitalism in amateur sports? I need to look into this a little deeper.

Thursday, 30 January 2014

Trail running and brain training

Trail running. Image from here.

I was asked to maybe write something about "trail running and the brain", so I gave it a shot. Here's the article, more or less how it may appear in a few months time.

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Some runners might avoid trail running due to a (legitimate) concern over twisted ankles. Others stay on the roads because this is where GPS watches -and similar devices- are useful for pace and distance. By contrast, there is no way to interpret what several miles of rugged terrain mean either in terms of speed or mileage. Perhaps one mile up a mountain is worth five on a treadmill. What does running an "even pace" mean on trails? My first mountain race was two laps around a ski hill, and despite never doing such a thing before I kept a constant effort for most of the race. Somehow the body knows. Growing up I had never given much thought to precise pacing in cross country (be it skiing or running); whatever the speed, it always felt instinctively right. 

Sunday, 19 January 2014

Optimal speed on an indoor track

Consider yourself running on an indoor running track surface. The turn has a banked incline, which ostensibly helps making fast turns easier. How fast does one need to run around a banked curve to balance the forces of gravity and centripetal motion?

When running slow, say a light jog, a banked turn can feel awkward (you feel yourself being pulled off the track). Run too fast and you'll feel a pull outwards (though less than if there was no bank at all.)

The following diagram outlines the physics involved.
Wikipedia diagram of a banked turn; balancing the forces.
The specific forces we wish to balance are gravity and centripetal, at angle theta:


Assuming no friction, the velocity v for angle theta and radius r is


The IAAF gives a detailed plan for building an indoor 200m track here. Curves are banked at 10 degrees (18% grade) and the internal radius r = 17.5 m. Each additional lane (e.g. 2, 3..6) adds an extra 0.9 meters to the total curved radius.

The optimal speed v for the first lane of a track is therefore 5.5 m/s, or 19.8 km/h. The optical speed for the 6th lane is 22.2 km/h. One should feel the most "balanced" racing a banked turn at approximately 20 km/h. To place this in perspective, a 10 minute 3000m race is 18 km/h; at 9 minute it is run at 20 km/h, and 8 minutes is 22.5 km/h. 20 km/h is around the speed of a competitive women's 1500m-3000m university race.

Of course one can run much faster than 20-something km/h on an actual course since cleated footwear is high in fractional forces. Friction, mu, prevents sideways movement hence adds to the maximum speed.


Assuming a static friction coefficient of mu = 1, you'd stay glued to a 10-degree banked surface up 15.6 m/s or 56 km/h, enough for even the fastest sprinter. What if we increased the banking? If mu = 0 (no friction) but banked at 45 degree like a Velodrome, you could run up to 13 m/s. The optimal speed for a given banking is given the table below:


If there is no banking at all (theta = 0) then all centripetal motion is countered with frictional forces. Assuming mu = 1, the maximum speed around a non-banked curve is 13.1 m/s, still fast enough to support any runner. Even a tiny flat track of radius r = 5 m could support a speed of 7 m/s (25 km/h), fast enough for a 3:36 1500m run. This is why we never see people fly off the outside of a track.

Returning to our official track of r = 17.5 and theta = 10 degree, then running the inside lane at 20 km/h, or 36 seconds a lap, is ideal if interested in minimizing sideways forces.

Wednesday, 15 January 2014

Running strides part 2: Takeoff angles and other predictions


In Part 1 I showed that some basic predictions can be made about running contact times knowing only  one's turnover rate R (cadence; steps per second) as a function of vx (your forward speed in m/s) and dleg (the length of your extended leg, in metres). The first assumption I made was that R can be expressed as a function of vx and Lleg. From the early figures in part 1, this looked to be roughly true. The second assumption was that the distance of your running step, dstep, which is the distance covered per stride was equal to Lleg and that this is true regardless of your running speed.  To verify if these assumptions were indeed correct, I calculated the ground contact time (time spent on ground; GCT) and aerial time (time spent in air; AT) and compared them to actual empirical values.

Again, here are the two equations for GCT and AT, respectively:

$GCT = d_{step}/v_x$

$AT = 1/R - d_{step}/v_x$

I found my predictions were, within measurement error and natural human variation, equivalent to several published values. The next step is to then expand our horizons and make bolder predictions.

Sunday, 12 January 2014

Running strides Part 1: ground and aerial contact times


Be warned, this is a long topic with an unforgiving amount of math. Nevertheless, I think some people might find it interesting. If not, that's ok too. This is my first time using LaTeX in a post, so if there are issue with the equations let me know!

I'm going to break apart my discussion of running into two posts (with the possibility of a third). This first post lays out the foundations of how to calculate certain elementary running metrics including time spend in the air per running stride and time spent on the ground. Consider this an expansion of my previous discussion on the short life of a foot strike. The second part will use the foundations here to predict some perhaps more original calculations for running, including stride angles and ground forces experienced by a runner. I am not claiming I've discovered any novel numbers per se, but maybe a new approach to obtaining them. That is to come later, however. For now let's consider some basic calculations, for even these results may surprise the curious (scientist) runner.

Turnover rates, R

Let's begin with running turnover rates, which are complex entities but with lots of empirical data to back them up. They are easy to measure but hard to predict. Empirically you just measure a runner's speed and count their strides per unit time. Theoretical turnover rates, which require knowledge of a runner's muscle system, surface stiffness, etc, are by contrast very difficult-to-predict entities.

Stride rate is defined here by R in steps per second. Running at 180 steps per minute translates to 3 steps per second. Alex Hutchinson has already disused in some detail  the degree to which step rates increase with speed. Here is the a plot of his combining several running studies (and some self-reported numbers) on turnover rate. Note his plot show steps per second of a single leg, not both, hence the numbers are half of the total turnover (180 strides per minute is equivalent to 1.5 strides per second per leg). Here is Alex's combined data set for turnovers for various runners:

Figure 1: Turnover rates at variable speeds