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My question is quite simple "Does the friction between the tires of a bicycle and the ground increase as speed increases?" I know that drag does increase as velocity increases, but what happens to the friction between the tire and the ground?

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There may be data out there for bike tires, but most of what I've seen has been for car tires. Most rolling resistance is due to hysteresis in the rubber, i.e., losses because energy is dissipated into heat as the tire flexes. The sources of information that I was able to find gave seemingly contradictory claims about the variation of rolling resistance with velocity.

https://www.machinedesign.com/automotive/understanding-rolling-resistance-car-tires --

Between 5 and 55 mph, rolling resistance only goes up by about 1 lb. But above those speeds, tires contribute more to rolling resistance.

Andersen, "Rolling Resistance Modelling: From Functional Data Analysis to Asset Management System" (PhD thesis), http://milne.ruc.dk/imfufatekster/pdf/503.pdf --

Graphs on p. 105 (p. 117 of the pdf) show rolling resistance decreasing with speed (green curve).

Pascoa et al, https://www.researchgate.net/figure/Values-of-rolling-resistance-coefficient-as-a-function-of-velocity-for-typical_fig2_257774870 --

Graph shows rolling resistance increasing with speed.

I would tend to believe the Andersen thesis, since he did some very elaborate regression analyses to try to disentangle different variables. Or it may be that there is no universal behavior here, and the results just depend on complicated factors like the type of rubber, hence the seemingly contradictory claims.

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