The mathematics of breaking away and chasing in cycling.

In cycling stage races a small group of riders will often form a "breakaway" and establish a lead over the main group. This paper examines the factors that affect the likelihood of success for the breakaway. A mathematical approach is used, drawing on a model of cycling previously developed and validated (Olds et al. J Appl Physiol 78:1596 1611, 1995). In a breakaway group, the power required to overcome air resistance is reduced because the lead can be shared, with trailing riders sheltering or drafting behind leading riders. The benefit of drafting can be quantified as a function of the distance between riders using previously obtained data. Of course, this advantage is even greater in the (larger) chasing group, so that eventually the chasing group will catch the breakaway, assuming identical bicycles and physiological characteristics. The question addressed is: what factors determine how great a lead the breakaway must have in order for the chasing group to be unable to catch the breakaway before the finish of the race? Demand-side simulations show that the critical factors are: the distance remaining in the race; the speed of the breakaway group; the number of riders in the chasing and breakaway groups: how closely riders in each group draft one another; the grade; surface roughness; as well as head- and cross-winds. When supply-side physiological factors are incorporated, the maximum sustainable speed and maximum lead time can be calculated.