The whole collision thing is beginning to make sense of the original
question.
OK, a skater pushes off the wall. There are a number of processes here.
Within the arm, internal forces distort the muscles. These are internal
forces and cannot change the momentum, but can store energy. This energy
comes from oxidation of carbon or whatever, but is stored as elastic
potential.
As the skater pushes off the wall, notice how long the hand is in contact
with the wall - it is a lot longer than the simple deformation of the wall
or the hand could allow for. While this contact force is applied, there is
the impulse (change in momentum) FROM THE WALL, and the work (change in KE)
FROM WITHIN THE ARM. That is the release of that elastic potential that
the body just stored in tightening the muscles. It is applied over a large
distance because the arm straightens.
There is also some elastic potential stored in the wall / hand contact
deformation, but it is negligible because of the distances involved (which
is really small because the characteristic force is actually rather small,
use Hookes Law).
Don't believe it? Try pushing off by flicking the wall with a finger tip.
The force is similar (it hurts - thats a pretty good indication of how far
your finger deformed and thus how big the force was) but it is applied over
a very short time and very short distance. Work and impulse are both
small - you go nowhere fast. However, the wall/finger deformation are the
same.
In a collision, some of the energy is stored in the deformation of ball &
wall/ ball & floor/ head & hammer/ oh wait, thats not a good idea. Since
the energy ~ kx^2, the most pliant stores the most energy, but that really
doesn't matter. AS they seperate, that potential energy is converted back
to kinetic. How efficient that conversion is depends on the materials
involved (particularly the most pliant as this 'absorbs' the most energy -
and 'decides' whether it go to potential, or heat)
But how can that happen when the contact point doesn't move? Well that all
comes down to frames of reference. The distance moved is the distance of
the deformation. That ball sees the force move - right into it. It is,
after all, the ball that is being worked on; we should be considering that
distance. Same for the putty wads colliding. They both have negative work
done, the work is not equal and opposite - it has to be the same sign. The
forces are opposite, the directions are opposite, how can their products be
opposite?
Back to the car. Certainly the friction provides the impulse. The engine
provides the work in the same way that the arm did for the skater. Without
the force between the engine and the tires AND the friction force, the car
cannot go. It would lack either work or impulse - hence either momentum or
kinetic energy. The weight is also required (for friction and for
balancing torques). The question was, indeed, bad. I guess the driver /
pedal force is really the thing that starts the whole thing going - but
PLEASE lets not go there:)
Finally, it is certainly possible to do lots of work with minimal momentum
and lots to the momentum without much to the energy - it all depends on the
mass involved.
It is also possible to change the momentum without the kinetic energy
(uniform circular motion) and to change the kinetic energy of internal
components without changing the bulk momentum.
However, when you are talking about applied forces and objects accelerating
(particularly in a straight line), it is usually not wise to throw one away
and say that it is not relevant. It is still there. If we really want our
students to UNDERSTAND what is happening & not just plug&chug, we have to
face these issues. Both are relevant, both are vital, both are the real
reason for humans to describe a force. Philosophically, a force is just
the medium that allows a change in the fundamental quantities of momentum
and energy. It is where the momentum comes from and where the energy comes
from that are the really big questions.