The bicycle provides a context-rich problem accessible to students in a first-year physics course, encircling several core physics principles such as conservation of total energy and angular momentum, dissipative forces, and vectors. In this article, I develop a simple numerical model that can be used by any first-year physics student to investigate the maximum speed of a road bicycle under various real-world conditions. The model calculates speed as a function of time for a given set of input parameters, such as the rider's input power (delivered through the drive train to the rear wheel) and frontal surface area. (Contains 2 footnotes and 5 figures.)