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        riddles >> hard >> Minimize time. (Some physics)
        
(Message started by: Earendil on Apr 18^th, 2005, 2:10pm)

Title: Minimize time. (Some physics)
Post by Earendil on Apr 18^th, 2005, 2:10pm

Definition: You have 2 points, A and B, on a plane. There is a gravitational field of equal intensity and direction, paralel to the Y axis, acting on every point of the plane. A has more potential energy then B

Question: What is the curve which conects A to B and minimizes the time an object on A takes to slide to B (the only force acting on it is gravity).

(There is a proposital ambiguity on defining what a 'curve' is)

Title: Re: Minimize time. (Some physics)
Post by towr on Apr 18^th, 2005, 2:32pm

So.. Basicly you want to throw a waterballoon from ontop roof A onto the head of passerby B below, along the arc which takes the least time.
Under the usual assumptions (flat earth, etc)

[hide]Just throw at B with an infinite speed ;)
But for all intends and purposes it's a parabola[/hide]

Do you need any proof? (Assuming I interpreted the problem correctly)

Title: Re: Minimize time. (Some physics)
Post by markr on Apr 18^th, 2005, 9:09pm

I think you're looking for the brachistochrone:
http://mathworld.wolfram.com/BrachistochroneProblem.html

Title: Re: Minimize time. (Some physics)
Post by towr on Apr 19^th, 2005, 12:27am

Then you have a centripetal force working on it aside from gravity, don't you?

Title: Re: Minimize time. (Some physics)
Post by Deedlit on Apr 19^th, 2005, 3:15am

Well, there's also the normal force from the surface it's sliding on. I think saying that there's also a centripetal force is misleading, since centripetal force is just whatever part of the net force that is making the object go around in a circle.

Title: Re: Minimize time. (Some physics)
Post by towr on Apr 19^th, 2005, 4:37am

on 04/19/05 at 03:15:44, Deedlit wrote:

Yes, in this case it's just the normal force of the surface (or wire, or whatever).
But there has to be some force that keeps it from following a parabola, which is what it would do if there was only gravity.

Title: Re: Minimize time. (Some physics)
Post by Earendil on Apr 19^th, 2005, 1:28pm

Ops... indeed, there are many forces acting upon it. I guess what was meant to be said is that at time 0 the object is standing still and after that it only gains more cinetic energy from the potential energy. The curve is something like a 'slideboard' (I don't know if that is the correct word in english).

[hide]Indeed, this is the Brachistochrone Problem. But there are some intuitive and very interesting solutions to it.[/hide]

Title: Re: Minimize time. (Some physics)
Post by atulsinghk on Apr 20^th, 2005, 12:55am

hi
I am working to develop a sprinkler by having a perforated pipe where small diameter (1.5 mm) holes are located at different trajectory angles and at different locations with respect to its centre of axis of rotation. Pipe is pivoted in the middle and holes are in two opposite direction to facilitate the rotation by formng a couple when water is allowed to pass.
I am struggling to develop its hydraulics a). static condition and b). while in rotation.
As per irrigation objective we need to ensure uniform water application and we want that it requires minimum of energy to facilitate rotation for desired wetting front.
atul

Title: Re: Minimize time. (Some physics)
Post by SWF on Apr 20^th, 2005, 5:33pm

A related problem may be found in this forum under, Abel's Wire Problem (http://www.ocf.berkeley.edu/~wwu/cgi-bin/yabb/YaBB.cgi?board=riddles_hard;action=display;num=1061633861). In that one, the shape of wire that results in time being a desired function of starting height is asked. If time is independent of starting height, the answer turns out to be the same as for this problem.