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