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Re: Gedankening



On Tue, 28 Oct 1997, LUDWIK KOWALSKI wrote:

... I am puzzled by simulation performed by John. It shows that up
to 10% difference (in times af fall) can be anticipated for the
horizontally fired bullet and the one dropped from rest. My gedankening
consists of saying that the horizontal motion (where the air resistance
is high) has no effect on the time of fall (where speeds are too small to
make the air resistance significant).

Does it mean that horizontal and vertical motions are no longer independent
of each other when air resistance is significant?

No. The significance of the air resistance is not important; the
form of the dependence on v is. When the air drag is not linearly
proportional to v, horizontal motion contributes to force in the
vertical direction as long as there is some vertical motion. For
example, if F = kv^2 then

F_vert = -k*sqrt(v_vert^2 + v_horiz^2)*v_vert.

In the case of the horizontally fired bullet, v_vert << v_horiz at
all times so, to a good approximation,

F_vert = -k*|v_horiz|*v_vert.

This is a far greater upward drag force than is experienced by the
bullet that is simply dropped.

Did you use Interactive Physics to perform simulations, John?

Yes.

I do not think that IP is smart enough to model the dependence (?)
of two components correctly.

Not true. Interactive Physics provides for two types of drag
force--linear and quadratic (and even if it didn't, you could
program one in.) I used the quadratic form, assumed a Mach 1
bullet, and adjusted the drag coefficient to produce a few Newtons
of drag force which, I think, is about right.

While I am at it, I'd like to apologize to Nick Guilbert who
apparently found my remarks and those of others to have been
nitpicking and/or to have missed the point of the original story.
I don't believe I missed that point at all, but I do still believe
that the teacher made a serious tactical error in suggesting that
the proposition was subject to "mathematical proof"--BOTH because
it isn't AND because this parent was clearly unlikely to be
receptive to such a "proof" anyway.

John
-----------------------------------------------------------------
A. John Mallinckrodt http://www.intranet.csupomona.edu/~ajm
Professor of Physics mailto:ajmallinckro@csupomona.edu
Physics Department voice:909-869-4054
Cal Poly Pomona fax:909-869-5090
Pomona, CA 91768-4031 office:Building 8, Room 223