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Coefficient of friction doesn't matter as long as the tires don't slip...?
I'd think that at higher speeds the tire actually becomes stiffer since it experiences strong centripetal forces...
I think the reason why it apparently takes extra power at high speeds could be the fact that C_d changes a bit as speed changes. I'm not sure about this but for example a fastback whose rear windshield is angled at 20 degrees could have attached flow at 40mph, but at 120mph the flow will have significant separation and a much larger wake. With FlowIllustrator (a 2d fluid dynamics thingy) turning up the reynolds number definitely has this effect.
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