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Re: [Phys-l] Final velocity of bullets



another nit:

Coriolis:

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

bc

p.s. includes the esoteric Magnus effect.

David Bowman wrote:

Regarding Ken's mea culpa.


Very nice! Thank you, David, for the lucid explanation.
I stand corrected.
I was certainly guilty of an error (ignorance), and to my mind the
worse error of being dogmatic in my ignorance. Well, well, I had
not seen this before, although I've used 3 different books in
teaching analytic mechanics and have referred to others many times,
I simply had missed it. As explained in my post, I was under the
impression that I had tried everything possible in attempting to
separate variables in this situation (not all shown here). But I
should not have been so quick to assume that!


Don't beat yourself up so much. It's perfectly understandable that
you didn't previously see the solution method for separating the
variables for the ballistics problem with a quadratic-in-velocity air
resistance. The result does not seem to be very widely known. In
fact, I have never seen it before myself, either, in any book I've
ever looked at. I happened to try to work it out about a year ago &
just accidentally happened to hit upon a technique that worked to
separate the problem. I'm fairly certain that that the result just
*must* be known, and be even familiar, in the appropriate circles.
Perhaps it is in some obscure 18th or 19th century army book
explaining the theory of aiming artillery, or some such thing. It's
just that I've never seen it anywhere, and nobody I showed the result
to has yet claimed to have ever seen it somewhere else. I've been
too lazy to do a thorough literature search to try hard to find it.
I just really can't believe that the result is new, or even less than
150 years old.


The derivation of Kepler's Laws from Newton's Law of Universal
Gravitation, for instance, showcases some similar tricks and
manipulations to separate variables, and I give a 4-page handout on
that to my students, showing various false starts, some of the
logic behind the things we try.

It's an art form, not merely following predetermined processes.


True. Finding exact solutions to certain standard nonlinear
dynamical problems or to some famous statistical mechanical models
does seem to be much more of an art than a science, or at least much
more than a set of fixed algorithmic methods for obtaining solutions.
The process can be quite exhilarating if it is successful. But even
if it is not there often are side benefits in the form of little side
discoveries that happen to pop up here & there along the way. For
instance, when working on this ballistics problem with air resistance
I happened to discover (what I think, at any rate is) a cool little
identity that generates a bunch of other similar identities. That
identity is:

arctanh(sin(A)) == arcsinh(tan(A))

This is just one of a whole slew of cool identities that exist
between hyperbolic functions and inverse trig functions, and between
trig functions and inverse hyperbolic functions.

Another memorable side discovery for me was when I was investigating
a duality between the high temperature and the low temperature
behaviors of the free energy/partition function for a single quantum
particle in a 1-dimensional box. In that case I found a remarkable
function that was analytic and even in its argument, but which
generated a whole series of identities concerning the moments of a
discrete Gaussian probability distribution. One of them, I recall,
is that the particular discrete Gaussian probability distibution
that has the form:

p(n) = N*exp(-[pi]*n^2)

with the discrete random variable n taking on all the integer
values from -[infinity] to +[infinity] (and where N is some
appropriate normalization constant) has a variance that is
*exactly* 1/(4*[pi]). What makes the result especially remarkable
is that the exact value of the normalization constant N is not
available in terms of commonly known numbers, but the variance is
still has a nice exactly known value (in terms of [pi]) anyway.
The more one plays around with trying to come up with exact solutions
to various problems the more of these amazing side discoveries that
tend to accumulate.


Thanks, David, for adding to my appreciation of this particular
aspect of that beauty today. :-)

Ken


You're welcome. So I'm glad I decided to post that long complicated
explanation. I was afraid that it would bore all the readership of
the list.

BTW, I'm interested in any other cool side discoveries that various
list members may have happened to stumble upon, but which have not
seen much light of day in public because the discoverer thinks they
are not worth looking into the literature to see if they are actually
new/original results or not. So if others out there have some cool
little results to report, I'm listening.

David Bowman
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