# Cycling down a mountain

I am Dutch, so I cycle a lot, hence the following question. Say I am standing at the top of a mountain on my bicycle. I have an amount of energy $E$ available. Say that after the slope of the mountain ends, there is an infinite long, straight, smooth road ahead. How do I get the furthest on this flat road? Options:

a) using all my energy $E$ to peddle while I am coming down the mountain

b) roll gently down the mountain and start peddling precisely when I am at the end of the mountain

c) roll gently down the mountain and down the straight road and only when I am starting to get to a hold, I start cycling very slowly, keeping my acceleration and speed to a minimum

d) some kind of hybrid of the above.

In this story I would like to keep it as realistic as possible: there's friction, both from my tires and from air resistance. I once heard that air resistance goes $\propto v^2$, such that I am tempted to opt for option c). I am interested in your opinions/thoughts. Help me spend as little energy as possible while cycling!

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First, avoid using brakes, obviously. Second, your instinct is right. In the most realistic scenario possible, with a set amount of energy available for peddling, option c is the best. Note though that the speed should be kept high enough to avoid wobbling and instability, which can make you need to spend more energy per pedal.

However, when cycling, energy use is not nearly as important as power consumption. Thanks to the wonders of the human body, we are constantly replenishing our supply of energy. You get tired after spending more energy per second (power) than you are generating because your reserve is depleted. If you use less power than you generate, you can theoretically go indefinitely without tiring.

Due to friction effects though, option c is still best. Pedaling hard will quickly deplete energy reserves while pedaling at a slow but steady rate will allow you to cycle for much longer. From a physics point of view, we cannot help you spend less energy, it will inevitable take about the same amount of energy regardless of your method (some +/- due to friction, etc). But by keeping your power usage low, you can go much farther before needing a rest. It is much the same as with running and walking. Simplistic physics says both use the same amount of energy, but you won't get as far by running due to the massive power requirements.

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Option C is best. This is because air friction is non-linear. Put another way, it requires more energy for the same distance when pushing a object faster thru the air. If all friction was linear, then speed wouldn't matter and there would be a finite fixed amount of energy required per distance.

Since slower uses less energy going thru the same distance of air, don't add energy until you need the speed to stay upright and stable. The energy from the mountain is imparted where it is, and you can't do anything about it. Since air drag is the dominant power loss at high speed, tuck yourself in to make the smallest possible air crossection when rolling down the mountain. Note that you won't be pedalling then anyway. Once you slow down enough on the flat part to where you have to pedal to stay stable, you'll be going slow enough so that air drag is a minor issue. In that case, position your body for most efficient pedalling.

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