Re: [Phys-l] momentum and energy (post edit)

[John Denker <jsd@av8n.com>]

On 11/13/2006 12:31 PM, Spencer, Rob wrote:
> the fact that the recoil must be slightly less does not imply that the collision is inelastic

Agreed.  See also below.

Earlier:

> b)  The wall (earth) does recoil with a non-zero momentum...the velocity being ridiculously
> small...the bigger the mass ratio of wall to ball, the smaller the velocity ratio of wall to
> ball...but implying that there really is an associated nonzero K for the wall (earth)

I assume K represents the spring constant;  if this assumption is
wrong, please clarify the question.

I don't think spring constant is relevant to the spirit of the original
question.  We can simplify the discussion by drawing a "black box" around
the scattering event and asking only about the long-before and long-after
states.

1) In the CM frame, for an elastic collision, the golf ball picks up
a delta_momentum of 2p and the earth picks up a delta_momentum of
-2p.  Each keeps 100% of its incident energy:
          before            after
   Ball:  4p^2/2m           same
   Earth: 4p^2/2M           same

That's exact in the elastic limit.

2) In an inertial frame /initially/ comoving with the earth, to a
verrry good approximation the ball picks up the same delta_momentum
i.e. 2p and the earth picks up the same delta_momentum i.e. -2p.

          before                     after
   Ball:  4p^2/2m * (1+eps)          4p^2/2m * (1-eps)
   Earth: 0                          4p^2/2M

where eps is a small correction term, of order m/M.

We see that in this inertial but non-CM frame, the ball loses a
tiny fraction of its energy, even though the collision is elastic.
Here we are using "elastic" to mean no dissipation and no deformation
of the objects.

The energy lost by the ball is negligible compared to its overall
energy, and the energy gained by the earth is tiny in absolute
terms, but arguably it is large in /relative/ terms compared to
the initial KE, since the initial KE was zero.

3) If you analyze this in a "lab frame" rigidly attached to the
earth, it's a horrible mess, because the non-inertial behavior
of the frame is significant to leading order as soon as you
start asking about the delta_KE of the earth in this situation.

I'm not saying the analysis couldn't be done in a non-inertial
frame, but I'm not enough of a masochist to bother with it.