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I recognize that the result can be obtained using
pseudowork in fewer steps ...
but only because the problem was chosen to be
an exceptionally good match to the pseudowork
technique.
1. The static frictional force does "pseudowork" on the car.
(where "pseudowork" = external force dotted into the displacement
of the center of mass of system)
2. The pseudowork is equal to the change in the bulk translational
kinetic energy of the car.
(This is because static friction is the ONLY force that does
pseudowork on the car--neglecting any drag forces--and because net
pseudowork is ALWAYS equal to the change in the bulk translational
kinetic energy of a system.)
3. The static friction does no "real work" on the car
(where "real work" = external force dotted into the displacement
of the point of contact.
4. Thus, the total energy of the car does not change.
(Because NO "real work" is done on the car and the net real work
is ALWAYS equal to the change in *total* energy of a system as
long as there is no heat transfer to or from the system. This is,
however, a bit of a stretch in this case and, beyond neglecting
the drag forces, also requires us to pretend that the oxygen used
in the combustion processes as well as the exhaust products remain
part of the system. One could imagine a car that carries its own
oxygen tanks and diverts its exhaust into a big bag or something
like that! This statement is also contingent on the assumption
that we take into account "thermal energy" and that there are no
thermal losses from the radiator or any other source. The
pseudowork analysis is FAR simpler because it is subject to far
fewer caveats of this nature.)
Carpenters deal with wood.
Dairy farmers deal with milk.
Physicists deal with energy and momentum.