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outrun
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### Can you motivate this speed difference??

Cuchulainn
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### Re: Can you motivate this speed difference??

Yes. I think this is the sort of toy John Bernoulli made for his sons.

The question is? shortest travel time?
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outrun
Topic Author
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Joined: April 29th, 2016, 1:40 pm

### Re: Can you motivate this speed difference??

Yes, the question is: why is the wavy one faster then the flat one?

outrun
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### Re: Can you motivate this speed difference??

And this one: why doesn't the bottom fall?

Cuchulainn
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### Re: Can you motivate this speed difference??

Yes, the question is: why is the wavy one faster then the flat one?

The wavy one is not always faster everywhere; but the total transit time is less. If you know y' then you can compute the integral by finite difference etc.

If want to get from start to finish ASAP, then minimising the integral will be a cycloid. Solve by Euler's equation.
Last edited by Cuchulainn on July 23rd, 2017, 4:45 pm, edited 2 times in total.
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http://www.datasim.nl

Approach your problem from the right end and begin with the answers. Then one day, perhaps you will find the final question..
R. van Gulik

Cuchulainn
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### Re: Can you motivate this speed difference??

And this one: why doesn't the bottom fall?

A family that plays together, stays together.
My guess is it does not have enough kinetic energy. You can put the motion in Lagrange's equation and solve.
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http://www.datasim.nl

Approach your problem from the right end and begin with the answers. Then one day, perhaps you will find the final question..
R. van Gulik

outrun
Topic Author
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Joined: April 29th, 2016, 1:40 pm

### Re: Can you motivate this speed difference??

I've read more "intuitive" explanations instead of the exact equations you've mentioned! I'll wait a bit with disclosing the explanations I found, ok?

Cuchulainn
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### Re: Can you motivate this speed difference??

Maybe for that annual Dutch quiz 2017?

Mrs. Cuchulainn got the first one right (without any maths!)
http://www.datasimfinancial.com
http://www.datasim.nl

Approach your problem from the right end and begin with the answers. Then one day, perhaps you will find the final question..
R. van Gulik

Cuchulainn
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### Re: Can you motivate this speed difference??

What is the shape of a bull's hide?

"The Kingdom you see is Carthage, the Tyrians, the town of Agenor;
But the country around is Libya, no folk to meet in war.
Dido, who left the city of Tyre to escape her brother,
Rules here--a long and labyrinthine tale of wrong
Is hers, but I will touch on its salient points in order....Dido, in great disquiet, organised her friends for escape.
They met together, all those who harshly hated the tyrant
Or keenly feared him: they seized some ships which chanced to be ready...
They came to this spot, where to-day you can behold the mighty
Battlements and the rising citadel of New Carthage,
And purchased a site, which was named 'Bull's Hide' after the bargain
By which they should get as much land as they could enclose with a bull's hide."
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Approach your problem from the right end and begin with the answers. Then one day, perhaps you will find the final question..
R. van Gulik

Traden4Alpha
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Joined: September 20th, 2002, 8:30 pm

### Re: Can you motivate this speed difference??

Initially, the bottom of the spring is in equilibrium between the downward pull of gravity and upward pull of the spring (and the distribution of gravity. When the top is released, the top is double-accelerated by BOTH gravity and the spring force but the bottom remains in equilibrium with gravity pulling down and the residual per-unit-length spring force pulling up.

Traden4Alpha
Posts: 23951
Joined: September 20th, 2002, 8:30 pm

### Re: Can you motivate this speed difference??

The ball track one may be caused by the m*g*h potential energy differences in the two tracks in which the wavy track has deeper valleys that induce the ball to "proportionally"* higher velocities and yet the distances are not proportionally longer.

* "proportionally" in an O(mgh) = O(mv^2) sense and reflecting the relative effects of linear and rotational inertia of the rolling ball.

Alan
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### Re: Can you motivate this speed difference??

Clearer with a half pipe, I think.
If this half pipe was very shallow, it's perhaps obvious the time to reach the bottom would be longer. By symmetry, the time to reach all the way across would be twice that time.

Cuchulainn
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### Re: Can you motivate this speed difference??

http://www.datasimfinancial.com
http://www.datasim.nl

Approach your problem from the right end and begin with the answers. Then one day, perhaps you will find the final question..
R. van Gulik

Alan
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### Re: Can you motivate this speed difference??

BTW, the cycloid is also the geodesic for the Heston model asymptotics: Gulisashvili &
Laurence

Cuchulainn
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### Re: Can you motivate this speed difference??

BTW, the cycloid is also the geodesic for the Heston model asymptotics: Gulisashvili &
Laurence
I browsed through the article couple times but was unable to see the goal, neither in the Abstract nor the Conclusion. It's a bit high-falutin' maths and excellent maths.
Does the paper relate to a  minimisation of a functional?
http://www.datasimfinancial.com
http://www.datasim.nl

Approach your problem from the right end and begin with the answers. Then one day, perhaps you will find the final question..
R. van Gulik
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