Mathematical problem

[Bullit]

ICE made me more aware and more interested in Maths. One of way i go is to read Math History. People, how theories born etc. Since i don't like theory for theory sake.

The question:

If a ball is to roll down a ramp which connects two points, what must be the shape of the ramp’s curve be, such that the descent time is a minimum?

So what is the answer? I know but i was empirically wrong albeit not by much.



It also would be interesting to test if 3D applications rigid body dynamics systems respect the physical result.

[rray]

No idea - I believe this will either have a very simple solution or a very complex one ;)
A great site where you're nearly guaranteed to get an answer for these kinds of questions is: http://math.stackexchange.com
-rr

[angus_davidson]

ICE made me more aware and more interested in Maths. One of way i go is to read Math History. People, how theories born etc. Since i don't like theory for theory sake.

The question:

If a ball is to roll down a ramp which connects two points, what must be the shape of the ramp’s curve be, such that the descent time is a minimum?

So what is the answer? I know but i was empirically wrong albeit not by much.



It also would be interesting to test if 3D applications rigid body dynamics systems respect the physical result.

Just off hand I would have thought the fastest decent would involve getting as close to 90 degree straight down as you can, so that your still technically a ramp and not a drop ;) Note this is different from a brachistochrone curve as you specified decent (ie movement in one axis)and not the total move from point a to b.

And thats enough thinking for a sunday ;)

[Bullit]

Note this is different from a brachistochrone curve as you specified decent (ie movement in one axis)and not the total move from point a to b.

brachistochrone curve problem it is.

My answer was an inverted circle so i wasn't far from the cycloid that is the solution:

http://visualizingmath.tumblr.com/post/ ... -about-one



Cycloid
http://en.wikipedia.org/wiki/Cycloid

Also a cycloid inverted curve is the curve where any place you put a sphere they will arrive all at same time if started at same time.

http://en.wikipedia.org/wiki/Tautochrone_curve

[csaez]

What an interesting reading! thanks for sharing

[ChrB]

I presume these calculations take place in an ideal space, e.g. there are no friction or wind resistance. Taking these factors into account, I would presume (sorry no time for calculations right now) that the straight curve would likely be the fastest path?

[angus_davidson]

I presume these calculations take place in an ideal space, e.g. there are no friction or wind resistance. Taking these factors into account, I would presume (sorry no time for calculations right now) that the straight curve would likely be the fastest path?

As far as I am aware there is friction, other wise the ball would not be considered rolling, but rather sliding.

[Mathaeus]

I presume these calculations take place in an ideal space, e.g. there are no friction or wind resistance. Taking these factors into account, I would presume (sorry no time for calculations right now) that the straight curve would likely be the fastest path?

I think it became much more accelerated along first half of cycloid, in another half, all needed is a curve type that doesn't decelerate it too much. By the way, I have to admit I've found a spoiler, just by pasting the question into Google search, so don't even know what's my personal presumption, unfortunately.

[Bullit]

Without friction says in the text. Yes showing spheres might lead to the idea there is friction.

Edit: a bit more history http://en.wikipedia.org/wiki/Christiaan_Huygens

Christiaan Huygens who invented the pendulum clock and discovered the solution to tautochrone problem was a cycloid not like Galileo thought a circle http://en.wikipedia.org/wiki/Horologium_Oscillatorium

So the time problem, with pendulum led to this discoveries

Also:
http://en.wikipedia.org/wiki/Anchor_escapement