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Re: [Phys-L] quickest route



On 7/27/2013 12:51 AM, John Denker wrote:
On 07/26/2013 07:31 PM, Anthony Lapinski wrote:
... wins since ball has highest acceleration at start.
Consider the right triangle shown here, with vertices
A,B,C facing sides a,b,c respectively.


B
|\
| \
a | \ c
| \
|________\ A
C b


The base b is horizontal and the altitude a is vertical,
aligned with the local gravitational field.

The ball starts at rest at point B. The ball then moves
under the influence of gravity along some path to A. The
only forces are gravity and the forces of constraint. The
question is this: Is it faster for the ball to approximately
follow path BCA, with a reasonably sharp 90 degree turn near
point C, or is it faster to follow the direct path BA?

For simplicity, assume the "sharp" turn is rounded enough
to keep the forces finite, but sharp enough to have no
significant effect on the timing.

This question is well within the scope of the introductory
algebra-based physics course.

And yes, I am quite aware that there is something ill-posed
in the question as I have stated it. Students should learn
to deal with this sort of thing. Some hints can be found at:
http://www.av8n.com/physics/ill-posed.htm#sec-how-to


A sure sign that a particular puzzle is a Physicist's delight, is the idea
that a ball can be constrained by a triangle: the next giveaway is
mention of the idea of infinite forces. Reducing these abstractions
to something approximating the real world, leaves us with two
immediate lines of reference:
1) Galileo's experimental methodology, which 'diluted' gravity using a
ball rolled down a grooved lath.
2) The historical mathematical effort leading first to the isochronous curve,
thence to the brachistochronous cycloid.

I start by asking, "What provides the horizontal displacement of the ball?"
The motivation is evidently the points of contact at the foot of the vertical surface:
this will evidently be provided by one or two contact patches, depending on details
only available in the real-world. Moreover, I am faced with considering the energy
loss associated with the idea of the Coefficient of Restitution. Then, saluting Galileo,
i need to consider the angular momentum which is likely to increase on the
horizontal leg.
But I see I have already said too much...

Brian Whatcott Altus OK