Here are a few interesting things I learned from this discussion.
(1) I originally tried to solve this from a conservation of energy
viewpoint. I suppose this technically does "solve the problem" because
it indeed results in the answer that the final velocity does not depend
on mu, and that was the question asked. But I'll admit that the energy
approach I used only gets an approximate answer for V_f. The problem is
that I assumed the dissipation was purely mu*N*delta(x). [See point (3)
below.]
(2) I should have realized conservation of momentum must be used. Just
as in linear collisions, conservation of momentum is the "stronger tool"
because it has one solution whereas there are multiple avenues for
energy conservation.
(3) Speaking of multiple avenues for energy, I find it interesting that
thermal energy because of mu*N*delta(x) cannot be the only dissipation
in this problem. If we assume that slipping dissipation is the only
avenue of energy loss from the system, we end up with a final velocity
that is slightly higher than the final velocity calculated by
conservation of momentum. Thus, this situation not only requires some
amount of dissipation (as others have nicely pointed out) it requires
more dissipation than simple slipping.
This is relevant to the posts Brian Whatcott has made. Others are
responding to him that even if we could have zero slipping that we would
need some form of energy dissipation. I am going a further step and
saying that I think we also need another avenue of energy dissipation
even if we do have slipping at the tire/road interface.
Michael D. Edmiston, Ph.D.
Professor of Chemistry and Physics
Bluffton College
Bluffton, OH 45817
(419)-358-3270
edmiston@bluffton.edu
This posting is the position of the writer, not that of SUNY-BSC, NAU or the AAPT.