by Matthew Cobb
Readers are invited to explain the first one, which involves calculus of some kind, I believe.
When the shortest path is not the shortest path pic.twitter.com/GcytACNe1G
— Matt Lieberman (@social_brains) March 21, 2017
And this might amuse you
Aliens is a reasonable idea… pic.twitter.com/zZJB42LSQ8
— David Bressan (@David_Bressan) March 21, 2017
My guess on the first one is that after the ball goes over the “jumps” it’s not slowed down by friction for a time. Either that or hidden magnets. 🙂
First one: The ball averages a greater speed in the wavy path. The farther the ball drops, the more kinetic energy it has (greater velocity), and the shape of the path is such that the increased length of the track through each dip is more than compensated for by the increased velocity when the ball is low in the dip.
Pretty much this. The faster you can convert height to speed, the higher your average velocity.
The flat path accelerates slowly over its entire length so the first portion of the trip is very slow.
The curvy immediately reaches its maximum speed at the first drop. The waviness beyond that point is just obfuscation.
See “curve of fastest descent” for math.
I think there’s more to it than merely being the brachistochrone stretched out – see how the ball separation increases suddenly right after the first one turns around?
Ah yes – because gravity only works in one direction.
… that is I think the illusion or trick being played on an intuitive level. is that the up is the same as down, when of course it doesn’t work that way.
… brachistochrone curve Ryan ^^^ – good call.
This is the right answer. The wavy path isn’t even the fastest one possible, that is the Brachistochrone Curve:
https://en.wikipedia.org/wiki/Brachistochrone_curve
Just eyeballing it, it looks to me like the two paths follow the same curve in their first and last segments. Moreover, the two intermediate peaks in the wavy path seem to be the same height as the bulk of the straight path. So neglecting friction, the ball on the wavy path will be traveling the same speed as the ball on the straight path at the start, the end, and the two peaks in the middle.
At all other points in the wavy path, that ball will be traveling faster, regardless of whether it’s accelerating downhill or decelerating uphill. So it gets to the end first.
how much would the inclines on the ends of the straight path have to change to match the curvy path?
That is, to make the average age velocities equal?
Not age. Auto-incorrect fault…
Yes, exactly. More height (potential) energy available therefore more speed.
cr
Increased velocity due to gravity. I have no idea.
I’m not sure why the video is a puzzle to intuition… it would be an even shorter path if there were no slope at all and were completely flat! There of course, the ball on that path wouldn’t move at all.
A steep drop at the very start gives the ball a velocity boost right away. Jacob’s post illustrates this concept clearly: Zero slope at the start means zero velocity which means infinite time.
Check out ‘brachistochrone’ in Wikipedia for a detailed description of the dynamics.
This one is intuitively easier to see
Nice clear demonstration, thanks.
Very good. I learned something new, but shouldn’t it be the Brachistochrone Problem?
I thought it was?
Yes, but it seems to me a little bit feeble that they hadn’t bothered to modify their apparatus by adding the optimum ramp to it. How hard could it be?
cr
This isn’t the same thing. The original video has the balls returning to the starting line.
Well in the original video they didn’t return quite to the start line – that would require zero friction and zero energy loss.
The ‘brachistochrone’ video relates, I guess, to the fact that the track with dips in is faster than the ‘straight’ track. It’s a demonstration of the principle of potential (height) energy being converted into kinetic (velocity) energy.
cr
It seems to me that the addition of gravity (the hilly course) in this race is a cheat. An even more extreme version could have the ball dropping down a steeper, longer track even faster.
My first thought was “brachistochrone” too – it’s a very well-known problem and story.
But I smell some slight whiff of dead rat too. It’s hard to be sure from the GIF, but isn’t the launch point of the nearer ball higher than the further.
There are also suspended portions of the “straight” track – which would have bent under the weight of the ball, converting kinetic energy into work of deformation. Some of that energy would have come back, but not all of it. Or am I just being suspicious?
OK, OK – guilty as charged. Yes, I had a train set, and yes, I learned that they mean it when they say you’ve got to support your elevated track at 5-6 cm intervals, closer on curves.
Verily, one can learn through play.
You’re just being suspicious. I think the two ramps are ‘level’ so far as the ends go. The greater speed of the ‘wavy’ ball is due to each depression accelerating the ball (and it slows to its original speed at the top of the next hump, but by then it’s gained on the ‘flat’ ball. And repeat…
cr
Many underground Metro railway lines use that same principle. The line deliberately slopes down away from stations to help accelerate the trains and the up to the next station.
This can be seen particularly well on Ligne 1 of the Paris Metro, which now has driverless trains so you can sit up front and see the line ahead – the ‘hump’ up to the next station is quite pronounced.
cr
I might point out the shortest path is still the shortest (length) path. It just may not be the fastest. Any mall walker with a Cinnabons knows this.
Still hard for me to get my head around this. As a cyclist, “rollers” are great fun to ride, but in my experience with closed-loop time trials, the flattest courses always allow the fastest average speed. The same holds for distance running — very fast marathons are always on very flat venues, like the Berlin course, not hilly ones like Boston.
Of course in both disciplines, but especially cycling, aerodynamic drag is a dominant force, and drag increases more than linearly with airspeed. For the ball bearing experiment, I assume that both friction and drag are low, so that gravity is the dominant force.
I’d guess that your ‘rollers’ most commonly consist of a series of humps (i.e. little hills) with the start-finish line on the flat at a lower level? So each hump means that you first reduce your speed (up the hump) then regain it down the other side, hence a net reduction in average speed over that bit. Whereas the demonstration above is a series of hollows so the speed first increases to a max at the bottom then drops back up the other side, hence a reduction in the time taken for the section.
With longer hills, where momentum plays very little part (e.g. distance running), the loss of speed uphill is more significant than the gain downhill (quite aside from any considerations that there’s probably a limiting speed to how fast one can run regardless of ‘assistance’ by a downhill slope).
But suppose we assume 15mph on the flat, 12mph uphill, 18mph downhill. Over 2 miles, on the flat takes 8 minutes; 1 mile up plus 1 mile down takes 8.3 minutes, (quite aside from any extra distance due to the slope). Time lost by slowing down requires a greater speeding-up afterwards to regain the average, and this is quite aside from the drain in energy required to climb the hill.
cr
This is all very well illustrated in the original Youtube video what I found here:
http://www.youtube.com/watch?v=_GJujClGYJQ
cr
I think another large confounding factor comparing runners and steel balls is the mode of movement of each. The balls are not themselves dynamic. They are dead weight being acted upon by gravity and the surface of the path they are following.
Humans running have to exert energy themselves to move and the forces of gravity and the surface of the path they are following are secondary. A human running down hill does not fall freely, they have to expend energy and actually resist the force of gravity. Running down hill can be dangerous. The chance of injury is increased running downhill too.
Riding a bike seems like it should be much closer to the balls, or perhaps even at an advantage. A cyclist can add velocity going up and down hill pretty efficiently but an inert ball can’t. But which is more efficient friction and aero wise? The ball or the human on a bike?
Yes, I agree that runners and steel balls (or bikes!) are different cases.
The fact that any short-term loss of average speed has to be compensated by a (numerically greater) gain, though, is just arithmetical and independent of the moving entity.
I do agree that downhill (other than very gentle inclines) does runners no favours and uses more energy than the flat.
Friction – I wouldn’t care to say. Steel ball on *hard* surface (like steel track) is very hard to beat, though I suppose a racing bike with those skinny rock-hard tyres on a dead-smooth concrete or asphalt surface might come close.
Aerodynamically – I’d take the ball. More compact mass, I think.
But the ability of the rider to add energy at appropriate moments is probably an advantage. The huge advantage of a bike over a runner downhill is obviously that its top speed is not limited by ‘gearing’ (i.e. a runner only has one ‘gear’), only by air resistance.
cr
Shortest length != shortest time.
They forgot the Mega Evolution on Charizard.
Ah, ninja’d! I was about to mention that myself. 🙂
Video is cut off before both balls roll to a stop therefore we don’t know which ball was faster.
It was cut off a fraction of a second before the ‘wavy’ ball stops, way ahead of the ‘straight’ ball. If you think the straight ball somehow accelerated to pass the other ball in the final second I’ve got a zillion dollars says you’re wrong. 😉
Now if someone can find the original video on Youboob…
cr
… and here it is, in the first few seconds of this video.
http://www.youtube.com/watch?v=_GJujClGYJQ
The ‘take’ cuts off a second or two after the Twitter GIF posted by Matthew, but it’s quite clear that the ‘wavy’ ball has stopped and started to reverse direction before the ‘straight’ ball reaches it.
cr
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I remember “geodesics” in general relativity, loosely – that comes to mind somehow.
After some more thought, I’d like to go on record as saying, about the ball bearings :
This is NOT the brachistochrone problem.
It is related to the brachistochrone problem as much as the tautochrone problem – that is, there’s more to it. But I think what Ryan (?) said – the average velocity – helps understand this a great deal.
I don’t know what exactly there is yet, but still I’m hooked by this delightful puzzle, thanks for all the discussion.
G*d prefers wavy paths, of course.