Hmm.. the acceleration part.

E=(mv^2)/2 - so we put more energy accelerating the bike from 10-20m/s than 0-10m/s, no?

Yet a=F/m - which suggests the acceleration is proportional to force, which would suggest that applying force F for time t should speed you up 0-10m/s the same way as 10-20m/s?

I suspect the force applied to the pedals is not the force which is acting on the bike (counter-force of the ground-bike system) and this second force is somehow relatable to the current speed of the bike, no?