First of all I would like to thank you for the nice discussion. Second, I am sorry to all the folks who are not that much into physics, my post will contain some numbers as well and this whole back and forth with
@Casual cyclist may be more suited for a physics forum, but maybe some not so technically inclined people will find these posts interesting as well.
I finally had some time today to think about what you wrote and do some analysis on my own. Below is the segment I want to analyze. This is from when the attack took place until 20 seconds after. I chose this segment because I can see from the broadcast that Healy needs 24s to ride this segment. FYI on the Discovery broadcast that lasts 2:25:24 hours the attack happens at 1:58:31 (basically when the director cuts to the frontal view of Pog) and it lasts until 1:58:51 when Pog is at point A. Healy passes point A at 1:58:55.
Now, I did some modeling and I believe I got the constants pretty spot on.
It is a shame I cannot use LaTeX, but here is my DE:
P(t) = dE/dt = m*g*sin(\theta)*v + m*v*v' + k*v + q*v^3 (for these speeds I assume that the rolling resistance (RR) is constant)
m = 72kg (Pog+bike)
g = 9.81m/s^2
k = 3.6 kg*m/s^2
q = 0.25 kg/m
\theta = 0.2 rad (we see from the Strava file that the average slope of the segment of interest is 20% and the angle is small enough so sin(\theta) = 0.2). I have to add that the model could be improved here by making the angle be a function of time, but I believe even this constant value is good enough to illustrate my ideas.
Rewritten for v (the speed) it is:
v' = P(t)/m/v - g*sin(\theta) - k/m - q/m*v^2
Let us now turn to modeling the input power P(t). I suggest we use the following model:
P(t) = IP+EP*(1-exp(-t*10)),
where the Initial Power (IP) is the steady state power before the attack and the Extra Power (EP) is the increase of power during the attack. This makes it so that the maximum power is reached after about 0.5s after the attack.
Let us try to infer what the IP should be. According to Strava, in the 14s before the attack the slope is 10.4% and his average speed is 5.36 m/s and that would require about 450W (I am including the RR and Aerodynamic Resistance (AR) at this speed), so IP = 450W. Now what should the EP be in this model so that he covers a little more than 98m in 20s? The answer is about 320W.
Let us now see what does this model predict graphically.
Let us do a sanity check. As argued before, the KE increases as the power increases but eventually the increase in KE goes to 0. The increase per unit time in PE is higher for higher total power input. The RR and AR are small but non-negligible as you correctly argued. Notice the interesting point where the AR=RR (just put of curiosity, nothing to do with the analysis). You might argue that your analysis assumed 6m/s at the beginning of the acceleration and 9 m/s at the end at a constant slope much smaller than what I have here. OK, but the reality is that Pog attacked at the steepest point and when the gradient changed so there is not such a big change of the KE of 1500J as in your model but a mere 572J in mine (I computed the integral under the d(KE)/dt curve).
Edit: You predicted accurately that it would take about 3s to reach max velocity.
If we assume a more modest EP = 180W for Healy the (equivalent) model for Healy would be something like this:
Note that it would take 4 more seconds to cover the 98m, so we are on the right track.
Now, I argue, Pog's power output of 450W + 320W = 770 W is doable whilst seated. Easily doable I would say since I managed to do about 600W seated for about 10s. Therefore, I do not see any evidence for motor doping by Pog during the FW race.
Let us now turn to your analysis. First, you argue that for some reason it is not enough to just increase the power from 450W to 770W, but you need an additional power to accelerate (I still do not understand why). Please look at the curves above and verify that everything is kosher so to speak. In any case, just to indulge you and for my understanding as well, I tried to model your assumptions as well. Here is what I get:
The parameters for this model are as follows:
P(t) = (exp(-t/4)/2+0.7).*(IP+EP*(1-exp(-t*10))),
where IP = 450W and EP = 520W.
Note that some of the KE was converted to PE near the end (lower speed), so the lower total energy at the input. However, this does not seem to be a good model (maybe I did not do justice to all of your thoughts, assumptions and considerations).
To conclude, I believe there is no evidence for motor doping stemming from Pog's performance on FW's final climb.
P.S. If you find the model interesting and want to play around with it, I can share the MATLAB code with you.
www.uaeteamemirates.com
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