Re: [Phys-l] Interactive Physics Simulations

[John Denker <jsd@av8n.com>]

On 10/12/2006 04:02 PM, John Mallinckrodt wrote:

> Better yet, consider the (not quite physically realizable) case of a  
> particle attached to a massless vertical stick and another particle  
> initially moving horizontally that strikes and sticks to the end of  
> the stick.  Is this a totally inelastic collision?  Note that, in the  
> CM frame, the final kinetic energy is equal to the initial kinetic  
> energy.

Bingo!  What a delightful gedankenexperiment!  It shows the richness
of the physics.

As a generalization, let's say the second particle hits the stick
at a distance b from the first particle, i.e. the impact parameter
is b.  The final state where the second particle is attached to the
stick can be considered the maximally inelastic outcome, i.e. the
maximally disspative outcome ... but we note that this is a
*conditional* maximum, conditioned on the value of b and the other
"givens" of the problem.  That's because other values of b will
lead to other amounts of dissipation.  The extreme case b=0 will
least to the *unconditionally* maximal dissipation.

I assume that the particles have no internal structure, and that
any internal modes in the stick are highly damped, i.e. highly
dissipative.

> Now some (myself included) would claim that rotational kinetic energy  
> is "internal" energy.  Thus, in the CM frame, the collision  
> completely "internalized" the initial bulk translational kinetic  
> energy of the two colliding objects and, in that sense, was indeed  
> totally inelastic.  (Note also that the separation speed is zero in  
> keeping with the coefficient of restitution based definition of  
> "totally inelastic.")

I'm not sure I buy that.  The following seemingly-similar reasoning
leads directly to trouble:

Let the first particle be attached to one end of a massless non-
dissipative spring (rather than a massless stick).  Let the collision
take place in one dimension, such that the second particle attaches
to the other end of the spring.  As a result, the compound object
moves away with some herky-jerky motion.  AFAICT, the energy in
the spring is entirely "internal", as is the KE of the relative
motion of the two particles in the CM frame.  So by the reasoning
quoted above, we ought to call this a "totally inelastic" collision.

Now suppose that the second particle detaches from the spring at
just the right moment, at the end of any full cycle. It flies off
with 100% of its original energy.  So the interaction wasn't
inelastic at all.

===

Therefore it seems to me that the key physics that defines what we
mean by "inelastic" has to do with /dissipating/ (i.e. thermalizing)
a certain part of the energy.  Merely "internalizing" the energy
isn't sufficient IMHO.

The example with the stick (instead of the spring) is tricky because
for some reason people assume that it is "easy" for the second
particle to attach to the stick ... whereas it must in fact be
tremendously complicated, if you look closely enough, because there
is necessarily some dissipation going on.