Re: elastic collisions

[Hugh Haskell <hhaskell@MINDSPRING.COM>]

At 21:02 -0400 9/30/02, Jeff Weitz wrote:
> Discussing momentum and collisions before energy does raise a questio=
> n, though.  What do we do about elastic collisions, which are usually=
>  defined as collisions that conserve total kinetic energy, or so I've=
>  always thought.

I don't see anything wrong with the definition, but I think there is
a way to avoid it when doing collisions before energy.

In the center of mass reference frame, the total momentum of the
incoming objects is always zero. In a perfectly inelastic collision,
the two objects hit and stick together. Don't worry about possible
rotation of the merged objects at this point, just note that the
linear velocity of each object is now zero. It doesn't take much to
argue that a totally elastic collision is one in which the two
objects bounce off with the same velocity as they came in with.
Conceptually, that squares with the idea of a perfect ball bouncing
as high as it started from when hitting the floor. Not only does this
avoid the consideration of energy, but it also allows a very
straightforward approach to real collisions--ones where the final
velocities in the C0fM frame are less than the initial velocities. It
is not difficult to show that the fractional change in outgoing
velocity over the incoming one must be the same for both objects, and
this fractional change is just the coefficient of restitution.

This enables you to discuss all sorts of collisions (except the ones
involving rotation) without reference to energy. And the mathematics
is much simpler.

Then when you get to energy, revisit collisions in the new context.
You can now do the rotational collisions, and show how energy
conservation connects with elastic collisions. This has two benefits.
First, it allows you to revisit a subject, for more reinforcement,
and second, it provides some real motivation for the energy section,
by using it to explain an earlier idea that may not have been
well-understood and to do additional problems that couldn't be done
before (the rotational stuff).

I like the idea of being able to come back to subjects more than
once. It helps with the digestion.

Hugh
--

Hugh Haskell
<mailto:haskell@ncssm.edu>
<mailto:hhaskell@mindspring.com>

(919) 467-7610

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