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. 

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.

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 v_x (your forward speed in m/s) and d_leg (the length of your extended leg, in metres). The first assumption I made was that R can be expressed as a function of v_x and L_leg. From the early figures in part 1, this looked to be roughly true. The second assumption was that the distance of your running step, d_step, which is the distance covered per stride was equal to L_leg 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:

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.

Running strides Part 1: ground and aerial contact times

I'm going to break apart my discussion of running into two posts. 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

Foot strike under a microscope

Foot strike plot borrowed from here 

I want to show that it's nearly impossible to perceive a running step exactly as it is happening and furthermore the idea of deliberately controlling your stride is perhaps harder than it sounds.

To begin at the end, it takes between 150 and 250 milliseconds from the moment your foot touches the ground until the tips of your toes leave during take-off. The time of your foot strike is proportional to your speed. As you run faster, your footsteps take less time. But even at a slow jog, the time it takes for a sin

gle step is short. For instance, assume you are running at 180 steps per minute. If each leg is taking 90 steps per minute then each stride cycle lasts 0.667 seconds. The time of each kick is at most half this time (because the opposite foot has to return to the start positing in the same amount of time) so that your foot landing, takeoff, and mid-air float last about 333 milliseconds.

Chess vs Running: prize money sharing

Green to play?

The 2013 world chess championships are underway, and although I do not much follow chess, I noticed how they shared their winnings is markedly different than running. By comparison, running is more of a winner take all. But there is a spectrum of distributions filling out the middle.

Below is the money distribution for Chess. The ratio from first to 8th is 115/21 = 5.47. Even more generous is the first and second-place ratio of  a mere 1.07.

• 1st place – €115,000
• 2nd place – €107,000
• 3rd place – €91,000
• 4th place – €67,000
• 5th place – €48,000
• 6th place – €34,000
• 7th place – €27,000
• 8th place – €21,000

By comparison, here's the prize structure for the 2013 Boston Marathon:

• 1st place – $150,000
• 2nd place – $75,000 
• 3rd place –  $40,000 
• 4th place –  $25,000 
• 5th place –  $15,000 
• 6th place –  $12,000 
• 7th place –  $9,000 
• 8th place –  $7,400 

The 1st/8th money ratio here is 150/7.4 = 20.2, and 1st/2nd ratio of 2. So in chess the runner up makes almost what the winner does, and 8th place make 18% of the winner, though not fantastic, compare to the runner-up for Boston who makes 50% while the 8th place makes only 5% of the winner. Boston is typical for running. For instance the ratios are similar to London, for which 2nd place makes 55% of the purse and 8th gets 7.3%.

But I wanted to see which, between chess and running, is more the outlier. That is, compared to other sports or competitions.

Title

Haven't been up to posting lately. Not for lack of things to say, but been busy. Here are cities of which I spent at least one hour in between Oct 19th and Nov 17th (i.e. Today)

London, Mumbai, Dhaka, Jaipur, Agra, Kanpur, Lucknow, Delhi, New Jersey, Fredericton, Sussex, Toronto, Hamilton, Napanee, Ottawa, Saint John, and Halifax.

Here's the total distances below. At 34,000 km total I almost, but not quite, circled the earth (40,000 km). If I ran this same distance at a rate of 20 km/day it would have taken me 4.7 years to complete.

City 1	City 2	Distance (km)
Halifax.	London	4631
London	Mumbai,	7200
Mumbai,	Dhaka,	1891
Dhaka,	Mumbai,	1891
Mumbai,	Jaipur,	928
Jaipur,	Agra,	239
Agra,	Kanpur,	278
Kanpur,	Lucknow,	83
Lucknow,	Delhi,	414
Delhi,	Newark	11768
Newark	Halifax	968
Halifax	Sussex,	344
Sussex,	Fredericton,	118
Fredericton,	Sussex,	118
Sussex,	Halifax	344
Halifax	Toronto,	1265
Toronto,	Hamilton,	70
Hamilton,	Ottawa,	478
Ottawa,	Saint John	753
Saint John	Halifax.	206
	TOTAL	33987

This was all pretty random, that's for sure. It'd take a while to explain all of it, but it include work, vacation, visitation and death. Opinions and whatnot, I'm sure there will be time eventually.