Interesting question for XC MTB, so I plotted a bunch of attempts (from max effort down to party pace) on a short loop that I use as a benchmark. Looks like the relationship is pretty linear, at least at my speeds.
No kidding…That is Crazy! Thank you so much for that. That answers a ton.
That’s a really narrow range of speed though. On a small enough scale, exponentials look linear, and your spread there is 3mph.
The reason why I like it so much is that it represents typical xc mtb speeds at my level. 12-17mph ave. If you look at @WindWarrior post, it aligns with his graph in that range.
So what it really tells me is that at my speeds, power and speed correlate pretty well
No. If you assume there are no aerodynamic losses, power and resulting speed are not directly proportional.
This is because the power lost to rolling resistance is itself dependent on velocity. There is a nice discussion of this on Stack Exchange here.
Note that Crr is not constant, and is in fact variable depending on speed. From the perspective of a cyclist experiencing a relatively narrow range of speeds, Crr is effectively constant for a given tire because the relationship is close to being linear… however, the best answer is still that they are not directly proportional.
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I have to go back and look… but I think there may be some disagreement on rolling resistance being more or less linear v. exponential. I probably should look into that more and see how that corresponds to the speed range I’m in as well.
You can play around with various models of your scenario here:
Again, note that at the range of speed you’re going to be going on the bike, a linear approximation is quite reasonable. It’s once you start going realllly fast that the exponential nature becomes more obvious.
But if there’s not aerodynamic drag, then maybe you would actually be going 80kph on your bike 
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I LIKE IT… Thank you! 
Maybe I should amend it to “drafting a semi”
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Makes sense. You understand the limitations of that “model” but it is useful for your purposes, so have at it! Cool discussion, though. I like being forced to recall my fluid mechanics classes from 20 years ago, lol.
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When I searched for “tire resistance for tire types with speed graph” images, I saw lots of interesting graphs showing the curves for speed x resistance for different tire types, tire pressures, and so on.
https://www.google.com/search?q=tire+resistance+for+tire+types+with+speed+graph
So, ‘tire resistance’ isn’t simple and is going to be split into many different resistance curves depending on design and then sub-divided again for different pressures for the same tire. For example, a snow-ready fat tire will increase in resistance exponentially at slower speeds than a skinny 180 PSI track tub tire.
However, the steeper portion of the exponential curve of each tire setup in these results seemed to happen at speeds faster than most cycling scenarios which has been mentioned in the posts above.
KE = 1/2mv^2
The same physics that make wind resistance exponential also requires you to exponentially increase your power input to reach increasing speeds, even if you ignore ALL losses - wind resistance, rolling resistance, wheel/bb bearing drag, drivetrain losses, etc.
This is why spacecraft have thus far been unable to achieve speeds beyond what gravitational slingshots can provide. It just takes way too much energy.
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This is only true if you are attempting to reach that higher velocity in the same amount of time.
IOW, you seem to be conflating power with energy (cf. your last sentence).
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Some energy is lost in friction/heat (entropy). May be minimal… but with mechanical energy input is always greater than the output. But how that directly relates to speed probably depends on a lot of variables, including terrain.
Going to need to ask a buddy (physics professor) this one on our next ride!
From the slowest to fastest it’s only a 28% increase in speed, but a 56% increase in effort which is quite a big spread. I did a regression for the heck of it even though there are only 7 data points, and the R^2 is 92%. I do think this shows at typical XC speeds on a well-known course, speed is more or less proportional to power.
Yes! As mentioned, on a small enough range, exponentials appear linear and the model presented above certainly seems to work well enough for these purposes. I was being an “axchally” nerd pointing out the proportion to speed cubed. 
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