I think this might be a case of a little knowledge is worse than no
knowledge.
When a disk or hoop rolls without slipping, the velocity of the center of
mass is numerically the same as the tangential velocity relative to the
center of mass, and that's the velocity I'm using in the following
explanation. The translational kinetic energy is (1/2)mv^2 and the
rotational kinetic energy of a disk is (1/4)mv^2. Thus, a translating (but
non-rotating) tire has twice the energy of a rotating (but non-translating)
tire (if the tangential velocity of the rotating tire is the same as the
velocity of the translating tire). This has probably lead someone to
conclude that if a non-rotating tire falls off a trailer and lands in a way
that it rolls, it will have to roll twice as fast to conserve energy. This
is wrong on two counts. First, the energy doesn't go completely into
rotation, the tire still has translational energy. Second, the velocity is
squared in the equations so the velocity does not scale the same as the 1/2
and 1/4.
We have (1/2)mv(i)^2 = (1/2)mv(f)^2 + (1/4)mv(f)^2 where v(i) indicates the
initial velocity of the tire sitting on the trailer, and v(f) indicates the
final velocity of the tire as it rolls along the highway. This of course
assumes all the energy stays kinetic.
Simplifying this we have v(f) about equal to 0.8 v(i). If a tire were
sitting on the bed of a trailer, and it fell off and started rolling down
the highway, its velocity would be about 80% of the velocity of the trailer.
If we model the tire as a hoop instead of a disk, then it would go slower
yet [71% v(i)] because the rotational energy would be (1/2)mv(f)^2 rather
than (1/4). It depends on the wheel construction whether the tire is
modeled better by a disk or hoop. It would probably be somewhere between.
Of course all this ignores friction.
So my guess is someone got this backwards and also missed the effect of the
squared velocity.
Michael D. Edmiston, Ph.D. Phone/voice-mail: 419-358-3270
Professor of Chemistry & Physics FAX: 419-358-3323
Chairman, Science Department E-Mail edmiston@bluffton.edu
Bluffton College
280 West College Avenue
Bluffton, OH 45817