John Denker has recently said, "I find it unhelpful to focus too much
attention on KE."
I agree. On the other hand, it doesn't hurt to pay some attention to it
because it can differentiate between some "types of collision" even if
none of the "types of collision" are pure.
It is rather striking to launch an airtrack glider into another
stationary equal-mass glider and see the first glider suddenly stop
while the second takes off with the same velocity the first glider had
before the collision. Why does it do this? Conservation of momentum
does not require that the initial momentum all appear in the second
glider after the collision. There would be infinite pairings of p1f and
p2f that would add to p1i. Why don't we observe a range of
distributions of the final momentum between the two gliders?
Of course the answer is that conservation of momentum is not the only
thing that happened. What else happened? A lot of textbooks say that
kinetic energy was also conserved because no other pathway was available
for energy conservation. If, because of the construction of the
colliding objects, energy could not appear as internal energy in the
gliders, and likewise no other energy avenues existed, then after the
collision has occurred we had better see the same kinetic energy as we
saw before the collision. If this is so, then simultaneous solution of
p(final) = p(initial) and KE(final) = KE(initial) shows that the only
allowed result is for the first glider to stop and the second to go with
the same velocity the first one started with.
I don't see that the fact this is never 100% true is any reason not to
offer this explanation of why we only observe one outcome when
conservation of momentum alone would not have predicted the specific
outcome.
Regardless of whether the collision was mostly elastic, or the objects
stuck, or somewhere between, if we keep doing the collision over and
over, we keep getting the same result. What is forcing the result to be
the same when momentum conservation alone would allow lots of different
results? Of course the answer is that we are also conserving energy.
Notice that I just said "conserving energy" and did not say kinetic nor
any other named energy. However, if we leave it at that, it seems less
helpful than we could be. For some observed collisions, like the
"elastic" one described above, if you don't assume KE(final) is the same
as KE(initial) you can't derive the observed result. Even if the
observed result is not pure (the first glider only "almost" stopped),
you can measure that KE(final) was almost KE(initial) and you derived
that if KE(final) was exactly equal to KE(initial) then glider one
would have totally stopped.
And then we have those other cases such as when the airtrack gliders are
designed to stick. These collisions conserve energy also. So what is
different in the way the sticking gliders conserve energy compared to
the way the non-sticking gliders conserve energy?
It seems to me all we are trying to do with our usage of elastic and
inelastic is put some descriptions, or at least labels, on the various
ways energy is conserved. Was the energy after the collision mostly
associated with translational motion of the collided objects? If yes,
then that collision was mostly elastic. On the other hand, did a
significant portion of the energy end up as internal energy. If yes,
then that collision was not mostly elastic.
I think there is also utility in the words "perfectly inelastic" if we
would agree on the meaning. The meaning I see in the textbooks on my
shelf is that the maximum amount of energy switched from kinetic to
other forms. The reason there is a maximum is because conservation of
momentum might require some kinetic energy after the collision; i.e.
KE(final) cannot be zero because that would require a violation of
conservation or momentum. It seems to me this is a very reasonable
thing for students to realize and think about.
John said, "-- There is no comparable law of conservation of KE." I
agree.
John said, "For structureless particles, such as are used in
introductory discussions,
outside the interaction region *all* of the energy is kinetic, so there
can be no advantage to formulating the discussion in terms of KE instead
of simply E. And E has the advantage of being more fundamental."
I agree with this also, but this is not what introductory physics
students are doing in lab. In lab they are doing collisions on
airtracks and air tables. They also can observe or at least be
presented with problems that describe collisions between automobiles or
railcars in which the cars stick, or between cars with springy bumpers
where they bounce apart. It seems some discussion of KE versus internal
energy is appropriate in these cases.
Michael D. Edmiston, Ph.D.
Professor of Physics and Chemistry
Bluffton University
Bluffton, OH 45817
(419)-358-3270
edmiston@bluffton.edu