The Revolution Blog

Let’s nerd out on brachistochrone curves

Monday, Aug 29, 2022

We engineers always abide by certain empirical truths: Water is wet. Rock beats scissors. And the shortest path between two points is always a straight line.

But what’s the fastest path between two points? That’s what’s called the Brachistochrone question, and the answer isn’t necessarily a linear one. But it does involve how things roll—and since rolling is our middle name here at Hamilton, we thought it might be fun to geek out on physics for a moment.

To understand the answer to the Brachistochrone question, let’s look at the tracks from this video. Which track is the fastest? If we put a rolling ball on each track, you might think the straight track (#1) would be the fastest. But you would be wrong. It turns out the #2 track, which is curved and longer, is actually faster than the straighter, shorter track. Why? Because it creates a steeper path at inception, which picks up more speed. This added speed more than makes up for the extra distance that you the ball has to travel with the curved path.

This is a cycloid generated by a rolling circle.

This curved path of fastest descent under uniform gravity is known as the brachistochrone curve, which is a type of cycloid. In geometry, a cycloid is the curve traced by a point on a circle as it rolls along a straight line without slipping. See?

The #2 track is the Brachistrocrone curve. But all curved paths are not Brachistocrone. The horizontal and vertical components of gravity provide velocity in such a ratio that this curve is regarded as the path of quickest descent. It’s something to consider when we design our casters and carts to move material from one point to a lower point, such as down a ramp in a warehouse.

And it’s something to think about the next time we’re racing our hot wheels with our kids. Now don’t you feel smarter?

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