On optimal velocity during cycling

This paper focuses on the solution of two problems related to cycling. One is to determine the velocity as a function of distance which minimizes the cyclist's energy expenditure in covering a given distance in a set time. The other is to determine the velocity as a function of the distance which minimizes time for fixed energy expenditure. To solve these problems, an equation of motion for the cyclist riding over arbitrary terrain is written using Newton's second law. This equation is used to evaluate either energy expenditure or time, and the minimization problems are solved using an optimal control formulation in conjunction with the method of Miele [Optimization Techniques with Applications to Aerospace Systems, pp. 69-98 (1962) Academic Press, New York]. Solutions to both optimal control problems are the same. The solutions are illustrated through two examples. In one example where the relative wind velocity is zero, the optimal cruising velocity is constant regardless of terrain. In the second, where the relative wind velocity fluctuates, the optimal cruising velocity varies.