The Bicyclist's Paradox

Abstract

It's a situation every avid cyclist knows only too well. If you cycle up a hill and then back down with no net change in elevation, it seems as if your slower uphill speed and faster downhill speed should offset each other. But they don't. Your average speed is less than it would have been had you cycled the same distance on a level road. Similarly, cycling into a headwind for half your trip and returning home with a tailwind yields an average speed less than you would have achieved on a windless day. The faster part of the ride doesn't compensate for the slower part. It seems unjust. Most cyclists expect the uphill and downhill, or the headwind and tailwind, to more or less cancel and are surprised (and frustrated!) when they don't. The purpose of this paper is to resolve this paradox. Doing so involves some nice real-world applications of Newton's laws, numerical problem solving, and exercise physiology. There's a lot to learn from analyzing this problem, and it's readily accessible to introductory physics students.