[Phys-L] Re: Momentum Agina

[John Denker <jsd@AV8N.COM>]

David Abineri wrote:

> If a stationary railroad car is struck by a moving railroad car of equal
> mass and coupled to it, why is that regardless of the material of which
> the cars are made and regardless of their aerodynamics and regardless of
> the density of the atmosphere in which they are moving and regardless of
> whether there is an atmosphere or not that half the kinetic energy is
> lost (converted to other forms)?
>
> To both my students and to me, this does not seem reasonable and yet, if
> momentum is conserved and the masses are equal this must happen!

I suspect that query is open to misinterpretation.  With apologies in
advance, if I were to take those words literally, the best response
would be for me to ask:
  -- Why does it seem unreasonable?
  -- What is the evidence that it is unreasonable?
  -- What would you "prefer" to happen instead?  Why would that be
   more reasonable?

But I suspect all that is just a distraction from the intended query.
But I'm really not sure what the real intention was.

Also, if the intention was to ask why the result is insensitive to
(say) the density of the air and the viscosity of the air, I thought
I already answered that.  In general, the result *does* depend on
those things.  For colliding railroad cars, the dependence is weak,
but in other settings (e.g. colliding bacteria) the dependence is
very strong: first order or indeed zeroth order.

Thirdly, if the intention was to ask why you get "one half" as opposed
to "one third" of the energy, there's a good answer for that.  It is
related to the "half" that appears in the law
    KE = .5 p v                  [1]
which is valid only in the nonrelativistic limit.  In contrast, for
fast-moving objects (including massless critters like photons) the
law is
   KE = p v                      [2]
and the collision dynamics is very unlike what you get with rail cars.

In the in-between regime (neither very slow nor very fast) the rule is
   E^2 - p^2 c^2 = m^2 c^4       [3]
and the kinetic energy piece thereof is
   KE = sqrt(m^2 c^4 + p^2 c^2) - m c^2
which you can easily show reduces to [1] or [2] in the appropriate limit.

So if you _really_ want to know where that "half" comes from, it comes
from the fact that E and p enter into equation [3] to _second_ order.
This is the same exponent "2" that appears in the Pythagorean theorem.
And if you want to know where that "2" came from, I surrender.  That's
just how Mother Nature rigged up the geometry of the universe.

Now for all I know, that still doesn't address the indended question.
Please clarify the question.
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