Thanks to bike racing results, we can discuss some actual data about power output over time. I previously discussed how the integral of “power” over time would diverge, but, let's look at “area under the curve” again.
Max Watts: 1232 W
Peak 1-minutes: 632 W
Peak 6-minutes: 448 W
Peak 20-minutes: 382 W
Peak 180-minutes: 251 W
library(ggplot2)
x = c(0.25, 60, 360, 1200, 10800)
y = c(1232, 632, 448, 382, 251)
decay <- function(x) {
decay = 1232 * 0.25/x
}
decay60 <- function(x) {
decay = 60 * 632/x
}
tmp = data.frame(x, y)
names(tmp) = c("time", "power")
qplot(time, power, data = tmp, ylim = c(0, 1500)) + geom_line() + stat_function(fun = decay,
color = "red") + stat_function(fun = decay60, color = "green")
This is decaying much slower than \( 1/x \). The behavior at the 180-minute tail is the most important feature for figuring out the “area under the curve”. The maximal power has almost no effect. I have helpfully added \( 1/x \) decay lines passing through the first and second points to illustrate this.
I do not think the fact that we can make this graph for a couple sports (bicycling and rowing, for instance) and that we can consider the power production of such events as the snatch, however, in any way legitimizes this sort of comparison between disparate events. The snatch is what it is, it cannot be compare to the power lifts, it cannot be compared to a 100m sprint, it cannot be compared to a marathon. Even comparing middle-distance running to long-distance running in any way is suspect. Comparing sprints to distance running is definitely suspect and if it weren't, this sort of comparison would not be the way to do it.