Originally Posted by
cplager
I like your idea of drawing a free body diagram.
If we're ignoring rolling resistance, then there is no force that depends on speed relative to the ground (consider a hovercraft instead if you prefer).
According to your diagram, what is the difference of the forces acting on a bicycle between (A) a bicycle moving at 30 mph with no wind and (B) a bicycle moving at 10 mph heading into a 20 MPH wind? In both cases, there is a single resistive force of the aerodynamic drag of 30mph acting on a bicycle. In both cases, it takes the same exact effort of the rider.
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Yes, the motive force required is the same in both the still air and 20mph wind examples. But the distance is different. Work = force x distance, so the issue is what distance you use. If you can't decide whether it's air distance or ground disance, ignor both and consider the force and distance that the rider's legs move.
A bicycle is a machine in the classic sense, so the rider can set his pedaling force to what ever he wants. However the work, force X distance will be the same as the bike's. (not adjusting for frictional losses) So for the same driving force, regardless of gearing the rider's legs will produce the same force through 1/3rd the distance, or he can gear down and produce 1/3rd the force for the same distance. Either way he's producing less work.