Article 13525 of rec.arts.startrek:
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From: thiagara@silver.ucs.indiana.edu (raja thiagarajan)
Newsgroups: rec.arts.startrek
Subject: Re: The Most Toys and warp speed
Summary: c w^3.337 works fine if you discard warp 9.9
Message-ID: <45271@iuvax.cs.indiana.edu>
Date: 19 May 90 02:43:20 GMT
References: <1456.2651061B@atlsbo.UUCP>
Sender: root@iuvax.cs.indiana.edu
Organization: Indiana University, Bloomington IN.
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In article <1456.2651061B@atlsbo.UUCP> jmp@atlsbo.UUCP (Marshall Presnell) writes:
>The TNG Production office uses the following chart for warp speed calculation.
>             Warp Factor      # of times the Speed of Light
>             -----------      -----------------------------
>                  1                   1
>                  2                  10
>                  3                  39 <<-- Pretty close (up to mil spec)
>                  4                 102
>                  5                 214
>                  6                 392
>                  7                 656
>                  8                1024
>                  9                1516
>                  9.6              1909
>                  9.9              3503
> As you can see, it dovetails with your calculations well. - Now, if one of
> you mathematics weenies can find a formula for this curve, all of us software
> weenies would be happy
>! <grin>

Glad to oblige, though I'm an ex-astrophysics weenie rather than a
math weenie. It's good to have some *real numbers* to play with. So,
anyway, here's

			RAJA'S GUIDE TO WARP SPEEDS
		(dedicated to my math-weenie friend Julie)

I assume that warp speeds take the form
	s = k * c * w ^ p
where
	s is the "speed" (eg, 39 c)
	k is some constant, to be determined
	c is the speed of light (299792458 m/s)
	w is the warp factor (eg, 3)
	p is the to-be-determined exponent to which w is raised before
		being multiplied by k c

To emphasize: We are looking for values of k and p that will fit the
table that Marshall posted.

Okay. I determined k and p by taking the logarithms of s and w and
doing a linear fit. If you plot ln (w) versus ln (s), a curve of the
form k w ^ p will have a slope of p and a y-intercept of ln (k). Using
MathCAD on my PC compatible (PCs forever!) I got

	k = 0.936
	p = 3.406

(If you think I'm being sloppy with my sig-digs now, wait a few
paragraphs.) Now, if we use these values and assume that 
s = 0.936 * c * w ^ 3.406, we get the following results

warp	speed (* c)	 error 
----	-----------	-------
 1	    0.936	 -6.4%
 2	    9.922	 -0.8%
 3	   39.478	 12.2%
 4	  105.170	  3.1%
 5	  224.889	  5.1%
 6	  418.465	  6.8%
 7	  707.424	  7.8%
 8	 1114.809	  8.9%
 9	 1665.044	  9.8%
 9.6	 2074.396	  8.7%
 9.9	 2303.612	-34.2%

That last one is a terrible fit, by my standards. (Probably even by mil
spec :-) But on the other hand, does it make sense that warp 9.6 is
1909c and warp 9.9 is 3503c? If you plot these things, the curve is
nice and smooth, until you reach the value at 9.9. Then there's a
sharp spike in the curve. Could this be a sign that warps above a
certain threshold don't use the same power law? Could it be a sign
that there is a regime that violates the natural laws of warps just as
warp speed violates the natural laws of Einsteinian Special
Relativity? Or the worst possibility: Could there be a typo in the chart?!??

What do we get if we ignore the value for warp 9.9? Well, then a good linear
fit for the remaining ten points gives k = 0.996, p = 3.337. Going a little
further: As I recall, the sum of squares of the differences between the
predicted and observed values is a reasonable criterion for judging goodness
of fit (hence "least squares."). If you calculate this sum of squres of the
error terms, you find that k = 1.000 gives a slightly better fit (the total
of the squares is 420 rather than 513). Again, all of this is predicated on
the assumption that we can ignore the value for warp 9.9; if we don't (ie, if
we use k = 0.936 and p = 3.406) we get a total error of about 1,500,000. I
think the difference in error between 1.5 million and 420 might emphasize the
fact that the warp 9.9 figure really doesn't belong with the rest ....

Anyway, here are the results I got from ignoring the last value and doing the
fit with k = 1.000 and p = 3.337

warp	speed (* c)	 error 
----	-----------	-------
 1	    1.000	  0.0%
 2	   10.103	  1.0%
 3	   39.087	  0.2%
 4	  102.074	  0.1%
 5	  214.921	  0.4%
 6	  394.900	  0.7%
 7	  660.497	  0.7%
 8	 1031.276	  0.7%
 9	 1527.768	  0.8%
 9.6	 1894.880	 -0.7%

So *I'm* fairly convinced that speed = c w ^ 3.337. I'd be interested in seeing
results that other people got, either using my technique or a completely
different one. Comments and criticisms about my methodology are welcome, though
this is far enough off the main topic that e-mail is probably the best way
to handle further discussion.

And can *anybody* rationalize the value for warp 9.9?

Raja


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