Rule-of-Thumb: 100 Cal/mile

In the last two posts, here and here, we looked at the relationships between grade, power

expenditure, and pace for running.  In those posts, we had to make an assumption about a “base” power use which we defined as the power required to run at a steady pace on flat ground.  Since we were interested mostly in the additional power associated with running up a grade, and since that “base” power is so complicated to estimate for any particular individual, we simply invoked a common rule-of-thumb and used 100 Cal per mile as our base power.  Today, we’ll explore that rule-of-thumb in a little more depth.
We know that a runner typically has some vertical displacement as they run.  That is actually a difficult thing to measure accurately since most runners are not rigid bodies.  So, for example, if you were to trace the trajectory of a person’s hand, you would find a very different path of motion than, say, their left ear.  Of course, for the purposes of energy expenditure, we would mostly be interested in the trajectory of the instantaneous center-of-mass, but with body parts flying in all directions, even that isn’t easy to find.  Researchers have sometimes used proxies for the instantaneous center of mass like the motion of the center of a runner’s chest, or the motion of a runner’s pelvis.  A single point like that is much easier to measure and probably gives a very good approximation to the motion of the center of mass.  However it is determined, we might imagine then, that the motion of a center of mass might trace out a path something like this:

This figure is just for illustrating the nomenclature and is not an actual trajectory.  Some trajectories are more sinusoidal and others have other odd irregularities.  I imagine that in real life they vary quite a bit among runners.  

In any case, we’ll call the vertical displacement “y” and the stride length “x” as shown in the figure.

Now, IF we assume that all of the energy used when running on flat ground is used to boost the center of mass of the runner up by the distance “y” on each step, and ALSO assume that there is no energy recovery when the center of mass descends by the distance “y”, then it is easy to calculate the energy use as illustrated in this figure: 

In order to compare to the rule-of-thumb (100 Cal/mile, which is an estimate of energy burned, not mechanical energy of motion) we have to invoke a conversion efficiency.  The bands (blue for x=36 inches, y=4 inches) represent a range of conversion efficiencies from 15% to 25%.  The light green band represents x=30 inches, y=2 inches, and the red band (partly obscured by overlap with the light green) represents x=36 inches, y=2 inches.  The 100 Cal/mile rule-of-thumb is also shown  for reference.  

As a rough approximation for typical runner motions and runner weights, the rule-of-thumb isn’t too bad! 

I’d like to acknowledge and thank reader KAB for suggesting this topic.