## Sunday, October 28, 2012

### The elasticity of rest

One concept which comes up over and over is what I call "the elasticity of rest". When I drop a rubber ball from a height h and it bounces off the ground, it will typically bounce up to a height αh, where α < 1. α in this context is typically called the "elasticity" of the bounce. For a fully elastic bounce, α = 1, and all kinetic energy is retained (the momentum changes from downward to upward). For α = 0, the bounce is fully inelastic, and the ball sticks to the floor (I assume the floor is at rest).

I apply (misapply?) this concept to recovery. Suppose when cycling I can average no more than 300 watts for 10 minutes. If I ride at 300 watts, I collapse with exhaustion at exactly 10 minutes. But instead of riding 300 watts for 10 minutes, half-way through I slow to 290 watts for 10 seconds. Then I do my best possible effort for the remaining 4:50. How much less work can I do during that 10 minutes than if I'd stuck to 300 watts?

Using the elasticity concept, an elasticity of zero would imply that I do 100 joules less work: all I can do is average 300 watts for the rest of the ten minutes, so the 10 less watts during the 10 seconds I was at 290 watts is lost, unrecoverable. On the other hand, an elasticity of 1 suggests that I can boost the power the rest of the 10 minutes by 100 joules / 290 seconds (0.34 watts), recovering all of the lost work (this wouldn't produce the same time in a time trial, it would be microscopically slower, but that's another matter).

You might argue a bit of rest might do me good. After all, there's no such thing as constant power. Every pedal stroke the power varies as the foot moves around the crank circle. In this case, you'd claim, elasticity > 1. The rest pays off with interest. But I'm assuming from the start there's an optimal power schedule and the rider is able to follow it. So any change in power from this schedule cannot improve on my work done. At best I can achieve the same result.

The same problem can come in the opposite order: if I go 310 watts for 10 seconds, and thus need to reduce my power for what remains of the ten minute effort, can I rest efficiently enough to still maintain a 300 watt average for the ten minutes? Or must I rest even more, losing more than the 100 joules I just gained in extra 10 watts for 10 seconds?

The first quantitative approach I've seen to addressing this question is Andrew Coggan's Normalized Power constraint. Using normalized power, you can come up with definitive answers to questions such as the one I proposed, and the result is an elasticity greater than 0 but less than 1.

The most recent instance where I've been thinking about this is in the Low-Key Hillclimbs where we've had back-to-back weeks of individual time trials. We follow triathlon passing rules for passing, although we don't have the resources to enforce them. As a result, it's basically honor system that riders pass definitively and that riders being passed allow the passing rider to do so. But it's "low-key", so honor system should be good enough. The question becomes in letting a rider pass, how much does that hurt my result? If I slow by 10% for 10 seconds, I'll cover 10% less distance during that 10 seconds. But I'll recover a bit during this time, allowing higher power the rest of the way. So how much of this do I get back?

Similarly, triathlon rules require a rider passing to close 3 bike lengths within 15 seconds. This probably requires going harder than optimal for a brief time ( this issue is complicatedly the small drafting advantage experienced by the passing rider ). The question is then can the rider recover from that effort without hurting overall power much?

Here's an example of solving the problem with the normalized power model. Neglecting the 30 second smoothing, which is unfortunately critical to the result, if power is reduced from 300 to 290 watts for 10 seconds, that reduces the 4th power of power from 8.1×109 watts4 to 7.07281×109 watts4. So for the remaining 290 seconds I can increase the 4th power of power proportionately: by 2.0963×107 watts4. Power is reduced to 300.194 watts. This increases work done during this time by 95.0 joules. But I lost 100 joules during the recovery period. So the normalized power model, neglecting smoothing, says the elasticity of this rest is 95%. When I add in the 30 second smoothing it will get considerably better.

The essence of my question in this context is: "what is the cost of courtesy?"

Now obviously I didn't invent this concept and I'm sure it already has another name. But this is just the term I've used to myself when thinking about it.