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Re: [Phys-l] time to bottom of ramp



Regarding John M.'s disagreement with Jack U'.s calculation:

On Oct 27, 2006, at 2:29 PM, Jack Uretsky wrote:

For a frictionless quarter-circle arc of radius h, the time
to travel the arc is (barring arithmetic errors) about 2.6 times
to slide down an inclined ramp between the same 2 points. The
answer can be expressed as a beta-function.

Nah. Doesn't make sense. Here's my stab: I get that the curved
ramp takes 7.4% less time. The ratio of curved ramp time to
straight ramp time is

B(1/4,1/2) / [4 sqrt(2)] = 0.927 ...

John Mallinckrodt

I agree with John's number. BTW, the particular beta function
can be written in terms of a complete elliptic integral of the
first kind K(k) with a modulus k = 1/sqrt(2). And that also can be
written in terms of a Gamma function of 1/4. If we let r == the
ratio of the quarter circle time to the straight ramp time then I
get

r = K(1/sqrt(2))/2 = 0.9270280 ...

If we compare this with the corresponding ratio for a curved ramp
in the shape of a brachistochrone I get

r_min = 0.91284109470 ...

for the brachistochrone case. Thus, the quarter circle time is
much closer to the brachistochrone time than the straight ramp is.

David Bowman