Test Your Intuition 53 by Ehud Friedgut: Which Way to Ride Your Bike

This post was kindly written by Ehud Friedgut. Shana Tova (= Happy New Jewish Year) to all our readers.

The following is a real-life question.

I recently rode my bike around a closed track.

The track consists of a flat stretch, which we shall ignore, a stretch with a (more or less constant) positive slope, and a stretch with a near-constant negative slope. When riding clockwise the climbing portion is considerably longer (distancewise, not timewise) than the descent, let’s say by a 2:1 ratio.

The question is: if I pedal with constant power output, will it be faster to ride clockwise or counterclockwise?

I tried measuring the time, but the fluctuations in my riding time between laps were larger than the differences due to direction, and I couldn’t get a clear answer from this experiment.

I thought about the extreme cases; when the total ascent is zero, the time difference between the directions is zero, and when the total ascent is large enough, the time difference becomes infinite, because I can’t cycle up a very steep slope. This would suggest that it’s easier to cycle up the mild slope and down the steep slope, but from some back-of-a-napkin calculations I did, I don’t think that for fixed distances the time-difference between directions is a monotone function of the elevation.

So, the question is: is there a rule of thumb that covers most of the range of parameters one would expect to encounter for this question in real life? For a constant power output, is it faster to cycle up the steep slope and down the gentle slope, or vice versa?

(In my case it was approximately 50 meters elevation gain, with about 2 km uphill and 1 km down.)

I’m not sure that the constant power output is a realistic assumption for a non-professional biker, but let’s assume it for simplicity’s sake.

Gil’s comment. For me, there is a positive slope α (which is rather close to zero) so that I cannot ride for more than 50 meters. This gives me a clear intuitive choice. I don’t know how common is my experience and if it is relevant to the mathematics of the question.