Ball bearing ramp challenge: which reaches the bottom first?

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Originally published at: Ball bearing ramp challenge: which reaches the bottom first? | Boing Boing

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This works because each segment of the curvy path is much closer to being a brachistochrone (“curve of fastest descent”) than the linear part. The little hill between each curve is surmounted very quickly so it doesn’t rob the marble of all that much speed.

Wikipedia goes into the math of why the brachistochrone is a cycloid.

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I liked how in the middle of the video a Title Card came up after he listed the options and it just said “YOU DECIDE!”. I thought that was going to be the end of the video.

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I don’t like how he so casually uses a pencil as the starter. Reminds me too much of using a screwdriver to keep the demon core from reaching criticality.

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Related: Missing the giant slide we had at Joyland Amusement Park in Wichita, Kansas when I was a kid. You’d gain acceleration on the slopes so much that small kids like myself would fly off the humps like a Hollywood car chase set in San Francisco.

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Thanks, Rob. Personally, I find the subject of mathematicians’ balls endlessly fascinating.

Here’s another video discussing cycloids in general, and the brachistochrone in particular, with Adam Savage and Michael from Vsauce making and demonstrating one.

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Well I was going to be less technical and note that the ‘up’ part of the slope was not very ‘up’ at all (looks not much above level) and the ‘down’ part was longer and steeper, so obviously it was going to be faster.
But we were not shown a proper side view of the shape of the dips/curves.
I’d like to see it with dips/curves that are more even.

But some mathematicians do not have balls, just curves - equally fascinating.

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The question i’ve occasionally wondered about is: what’s the curve of slowest descent? (yet not infinitely so) Is linear that already? Or is there some sort of tilted upside down brachistochrone? uhrm… (Greek roots 101)… makristochrone ?

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Reminds me of this (long, sad, infuriating) story.

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If you insist it stay at or below the straight line, I expect (but haven’t proven) it’s the straight line. If you allow it to go above, then it’s easy to list a family of curves whose times are each finite but get longer than any given bound, i.e., there’s no slowest one.

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Gravity always wins.

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Then you might be interested in the story of the constipated mathematician … he worked it out with a pencil.

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And as a thought experiment, this shows why the “If they start and stop at the same level, ,they should get there at the same time,” thinking is so flawed. It is easy to imagine a curve that starts out very gradual so that the ball is still only just starting to move as the ball on the straight ramp gets to the bottom.

Of course my original thinking was that the straight line would be slightly quicker than the wavy one because the longer path meant that more of the potential energy would put into spinning the ball bearing rather than increasing it’s speed. Shows how well my logic works.

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  1. Is there a hard requirement that it every reaches it’s destination? If no, then slope upwards and keep ball at starting position
  2. Are sharp corners allowed? If yes, then a very slight slope until the end, (as allenk suggests) with a drop to the end point.
  3. Are we confined to the a confined space? if no, then a slope which goes around the world.
  4. Can we add foreign substances? If yes, place a small amount of pitch in front of the ball.
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How do those requirements apply to the discovery of the brachistochrone?

  1. it must get to the bottom eventually (hence “not infinitely so”)
  2. sharp corners are allowed (“Then it’s a fractal!” [shrug])
  3. there is a confined space it has to fit in the same 2d profile as the linear ramp (as the brachistochrone does)
  4. no foreign substances, no rocket motors, springs, sources of friction, energy sources, no hidden trained cockroaches… just gravity.

It’s just a thought experiment. Given the same constraints that lead to the brachistochrone, (and the calculus therein) is there a solution which maximizes the descent time, entirely under gravity? Currently, i’d hold that it is the linear ramp. That the slowest is right there in the initial statement of the puzzle.

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Relax, just a Hollywood car chase in San Fran. Not like Evel Knevel.

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Great video.

There are many cycloids that go through the beginning and end point. They can dip lower than the end point. Would a rolling thing take longer if the cycloid dipped below the end point and then came back up?

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The way I heard it: he worked it out with a pencil and paper.

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