## Wednesday, March 31, 2010

### running power: introduction

Kenyan runners, the paradigm of form

There is a well-established relationship between cycling power and speed. Power is proportional to retarding force multipled by speed, or equivalently, work done is proportional to retarding force multiplied by unit distance traveled. The retarding force consists of the following:
1. a rolling resistance component, proportional to weight multiplied by a "rolling resistance coefficient", which depends primarily on the tires and their air pressure
2. a wind resistance component, which is proportional to the square of the the relative wind speed, to the air mass density, to the cross-sectional area of the cyclist and bike, and to a drag coefficient which is affected by the aerodynamic efficiency of the rider and equipment
3. inertial force, which is proportional to the rate of change of speed.
In addition to the power dissipated to the retarding forces, power is lost in the drivetrain. This is typically modeled as a fraction of total power lost. As I've discussed before, this is a very crude approximation, as at higher powers a smaller fraction is lost to the drivetrain.

Anyway, what's lost in the model is the power associated with moving body parts around. With a bike, the body is relatively still, and while the legs are moving up and down, they're coupled, so one leg moving down while the other moves up allows the weight of the dropping leg push up on the rising leg, so in principal no work must be done by the muscles to sustain this motion. This is in part while cycling is so efficient.

Running is a different story. The legs are no longer coupled, so when I lift my foot at the rear end of my stride,my muscles need to actively work to make that happen. Even if my opposite foot is falling at the same time, the coupling between the two is likely fairly weak, so lifting and dropping the legs is going to take a toll. Similarly, each stride a runner's center of mass is moving up and down, much more so than with seated cycling. While a bouncing ball can move up and down a substantial number of times without external energy, humans don't bounce so well. So keeping that center of mass moving requires power from the muscles.

I'll post some equations next time...