Do you use more energy cycling on, say, a 10 km flat stretch, or on a stretch where you spend the first 5km cycling uphill and the second 5km coasting downhill? What if you spend the first 1km cycling up a pretty steep incline, and the remaining 9km coasting down a more gentle slope? What if you add the constraint that you have to complete both routes in equal amounts of time? I've considered this a bit intuitively, but I'm not sure how to model it mathematically. A thought: 1. It seems that if your bicycle is really efficient (here I just mean that it doesn't lose much energy to friction), hills might be easier. You could theoretically spend only one meter going forward, gain a little elevation, and then coast down for the remaining 10 km without using any energy. But if your bicycle is that efficient, then I guess you would also be able to go forward a long ways using little energy on flat ground. Intuitively, I have a feeling that the same amount of energy should be expended on a flat course as on a hilly one. But I don't know how to justify it mathematically. Thoughts?