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The Brachistochrone Curve

January 21, 2017

We’ve always been told that the fastest way from point A to point B is a straight line, right? Well, as it turns out, this isn’t always true! If the concept is considered in three dimensions – such as when a ball is rolling down a hill – the fastest way is actually a curve.

Say what?

I’d never heard of a brachistochrone curve before, and to be honest I can barely pronounce it. The name comes from the Greek words for “shortest time,” referring to a really cool property of this shape. Namely, that a ball rolled down this type of curve will reach the end faster than any other type of slope… including a straight line.

brachistochrone

In fact, the straight line is the SLOWEST path from point A to point B

 

I realize that computer generated models are not the most convincing – so let’s check it out in real life courtesy of Vsauce and Adam Savage, of MythBusters.

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Full video here

It’s not even a close race – the brachistochrone curve CLEARLY wins! So what the heck is happening? First, let’s define the geometry of the shape. A brachistochrone curve is drawn by tracing the rim of a rolling circle, like so:

cycloid

As it turns out, this shape provides the perfect combination of acceleration by gravity and distance to the target. The steep slope at the top of the ramp allows the object to pick up speed, while keeping the distance moderate. There is a necessary trade off between the two variables… the shortest distance (the straight line) also has the least amount of acceleration. Alternatively, the fastest acceleration (the extreme curve) also has the longest distance. The brachistochrone curve is the baby bear – it’s juuuuust right.

BUT WAIT – THERE’S MORE!

This curve has a super amazing bonus feature – it’s also a tautochrone curve, meaning “same time.” An object released at any point on the curve will take exactly the same amount of time to reach the end… no matter if it starts at the very tippy top, in the middle, or near the bottom.

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(forgiving slight variation by human error and friction)

It’s almost as if they’re waiting for each other! But really it’s just physics at work.

I find this relationship between the radius of a circle and the properties of motion to be SO ELEGANT! The simplicity of their demonstration is perfect, and I’m happy to say that I learned something new today.

🙂

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