[Phys-L] Re: Momentum Again

[John Denker <jsd@AV8N.COM>]

Rick Tarara wrote:
> I still think most people are missing the reason for the question about
> exactly 1/2 of the KE being transferred in a perfectly inelastic collision
> between same mass objects, one at rest.
>
> The question, as I read it, is HOW is exactly 1/2 transferred?  Restating
> the math of momentum conservation or stating that no real-life situation
> actually behaves this way, is not, if I interpret the question properly, an
> answer.  Here is what I imagine a student thinking:
>
> "A ball is moving along at speed v, has momentum mv, and KE .5mv^2.  Now
> there is this collision, and after the collision, the math says that only
> half of the energy is KE.  But where did the rest of the energy go?  Heat,
> sound, deformation--OK, but HOW does exactly 1/2 of the original energy find
> its way into these other channels, none of which are very well defined nor
> necessarily exactly the same from collision to collision."
>
> Again, I offer that the problem here is the reification of
> energy--considering it to be 'contained' in the first ball and somehow
> 'flowing out' into these other channels during the collision.  In that
> model, it is natural to ask 'how does exactly 1/2 remain in the kinetic
> channel and half moves into the others?'

I don't see it that way at all.

To my ears, the question of "why" the system loses half of its KE is
closely analogous to asking why the hypotenuse of a 45/45/90 triangle
is ~41% longer than the other legs.  Why?  Because if it were any bigger
or any smaller it wouldn't fit the gap between the other legs, i.e.
it wouldn't satisfy the stated conditions of the problem.

To apply this analogy:  If the KE(after) were any larger or any smaller
than half of the KE(before), it wouldn't satisfy the known laws of
physics and the stated conditions of the problem ... namely conservation
of momentum plus the stated requirement that the objects *stick*.  If
they didn't lose energy, they wouldn't stick.

Indeed this is one of the factors that made my thesis experiment possible:
when a hydrogen atom collides with a hydrogen atom, it does not stick,
for the simple reason that it has no way to conserve both energy and
momentum.  Sure, there is a huge attractive potential, but that is
nowhere near sufficient to cause things to stick.  A comet is strongly
attracted to the sun, but after falling into the potential well it
shoots back out again ... as it must, because there is not much
dissipation.

Rail cars collide and stick because they *do* have dissipation.
   http://www.google.com/search?q=draft-gears

By way of contrast, consider what would happen in the following example,
which is fairly typical of what happens when there is not enough
dissipation.

We have a 100kg cannonball intially moving with a speed of 100m/s.
We also have a 100kg boxcar initially at rest.  The boxcar has a
special trapdoor, such that the cannonball freely enters the
boxcar but can never leave; it just bounces back and forth, fore
and aft.  (For simplicity assume a one-dimensional geometry.)

So the picture looks something like this:

           ________________________________
          |                                |
          |                                |
          |          O -->                 |
          |                                |
          |                                |
          |________________________________|

and then after the next bounce

              ________________________________
             |                                |
             |                                |
             |                    <-- O       |
             |                                |
             |                                |
             |________________________________|


Assignment:  draw the world-line of the boxcar.  The slope of this line
indicates the speed of the boxcar as a function of time.

Also:  Calculate the long-term average speed of the boxcar.  Compare it
with the min and max speeds.  Does this have anything to do with the
dreaded "half" in the statement of the original question?

 From this we learn that it is simply not true that when objects collide,
they "magically" (or otherwise) lose half their KE.  It is perfectly
possible to have other types of collisions.

The key word in the problem is the requirement that they collide *and stick*.
Things don't stick unless there is dissipation!

=========

Keeping track of the energy (and momentum) via the conservation laws -- which
are in a very precise sense *flow* laws -- makes this problem easier, not harder.
Energy flows.  Momentum flows.
   http://www.av8n.com/physics/conservative-flow.htm

Also perhaps
   http://www.av8n.com/physics/euler-flow.htm
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