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[Phys-L] Re: collision question (revisited)



On 02/25/05 09:43, John Barrer wrote:

Consider a 0.25 kg low-friction lab cart crashing into
a stationary 1 kg cart......
Is there
not momentum associated with these vibrating
particles?

There sure is.

While the time-average value of these
momenta is zero (due to the vector nature of p), the
instantaneous value(s) are not.

Yes, the instantaneous values are nonzero. (But
I'm not sure the vector nature of p is the key to
understanding the situation; electric field is
a vector and it can be shielded in ways that p
cannot.)

It would thus appear
that these non-zero instantaneous vibrating-particle
momenta could not have resulted from a transfer of CoM
momenta, but rather from the transfer of some of the
CoM KE (but NO CoM p) to internal and external
vibrations, the results of which are time-varying
values of both instantaneous non-CoM KE and p. Is this
mental model correct/reasonable?

I'm not 100% sure I follow that statement, but
if it means what I think it means, then yes, there
is an important (indeed beautiful) physics idea
here.

The cleanest version of the idea is as follows:
Imagine a box-car of mass M, with a trap door
on the back side. For simplicity, assume it
is initially at rest. We shoot it from behind
with a ball of mass m. The ball goes freely
in through the trap door and goes all the way
to the front wall of the box, where it bounces
elastically. It then travels back to the back
of the box, where it cannot get out the trap
door, so it bounces elastically, returns to
the front, et cetera ad infinitum.

This makes a great homework problem: Find the
*average* velocity of the M+m system.

For good students, you should not be explicit
about what you mean by "average". Make them
figure out what should be meant, and make
them tell you. This is good practice for
solving real-world ("Letter to Garcia")
problems. But for the dimmer bulbs, you'll
have to tell them what average you want.

Anyway, among other things, this gives a nice
unforgettable mental model of dissipation. If
you draw the world-line of the system in enough
detail, there is no dissipation. But if you
average over a moderately-long timescale, you
see only the average behavior, which is what
we call the dissipative case.

I emphasize that in many cases (maybe all cases
in principle) you can "undo" the dissipation if
you look closely enough, and know what to look
for. Spin-echo experiments are a spectacular
example of this.

Of course, this
would lead to another question: Why can CoM KE be
transferred to internal vibrations but CoM p cannot?

Well, yes and no. I'd say p _can_ be transferred,
within limits.

The point is that the transferred p cannot be
hidden -- not for long anyway -- not in a closed
system. You can hide some of the momentum some
of the time, but there's a strict upper bound
(tx) on how long you can hide it, and averaging
over any longer timescale will reveal the "hidden"
momentum, forcing it to contribute to the black-
box CoM motion.

For homework, find tx explicitly in terms of m,
M, and the size of the box.