Optimal Time-Trial Bicycle Racing with Headwinds and Tailwinds

Many time-trial and triathlon bicycle races take place on relatively flat, closed-circuit courses. In the absence of hills, riding-speed is limited almost solely by aerodynamic drag; consequently, winds can have a big effect on elapsed times. I analyze the special case of a straight out-and-back race in a steady wind, assuming the rider has a given total amount of energy to expend and can choose only two speeds - the aided speed with tailwind and the hindered speed into headwind. In this ideal circumstance the problem of choosing optimal riding speeds reduces to a constrained nonlinear optimization that can be solved with elementary calculus. My analysis reveals a practical rule of thumb that can be used more generally to choose optimal riding speeds for time-trial racing on closed-circuit courses in the presence of headwinds and tailwinds.