Squishbacks

My office is built into the side of a hill, and several times while waiting for my bus, I've seen bicyclists rocketing down at tremendous speed. I was imagining they would burn through their brakes quickly, and wondered if there were a better way to limit your speed on a hill. Switchbacks used by hikers came to mind – Instead of taking a direct route up a mountain, trails wind back and forth, which reduces the slope. I was curious if a similar method could be used to slow down.

If we ignore air/rolling resistance, we can use conservation of energy to relate the drop in height to the total velocity gained:

For a road of slope m, we can set up a differential equation relating this total velocity to the individual forward, sideways, and vertical velocities:

Suppose we want to maintain a constant forward velocity, and so any extra gets put into the side-to-side motion. Then we can get the total forward travel just by multiplying the velocity by t, and we can write the sideways speed as

The trouble with this strategy is that as you go downhill, your total velocity is increasing, but your forward velocity is constant. That means proportionally, you'll be traveling far faster sideways than the direction you're planning to go. I picked a 30% slope and 10 m/s forward speed to plug into the equation above:

After only about 10 seconds, 90% of your speed is perpendicular to the direction you want to go! Now that doesn't mean you'll never get to your destination – That speed is fixed, remember. Because we ignored resistive forces, your total velocity keeps increasing. I suppose it may be better to stick with the brakes – Not only does this method still have you going at uncomfortably high velocities, you'll also take a lot longer to get where you're going.