I have learned a lot from the questions and answers that have
come up in this thread.
In particular: The theory says that in general it is not safe
to calculate the work in terms of the total force and the
average velocity. The fundamental laws say we "should" break
the system into a huge number of cells and apply the F dot dx
formula to each cell separately.
On the other hand, we all know in our bones that in a wide
range of familiar situations, it works OK to use the total
force and the average motion. Indeed the whole idea of
work would be almost useless if every time we wanted to use
it we had to pick the system apart into atomic-sized or
subatomic-sized cells.
So ... how do we make contact between fundamental theory and
common sense? Under _what conditions_ might we get away with
using the total force and average motion?
It is easy to identify two interesting cases:
a) If some system or subsystem is rigid and nonrotating, so
that it is moving with uniform velocity, then it doesn't
matter where you apply the force. It is OK to use total
force and average motion,
b) If the force is evenly distributed, so that each cell
feels the same acceleration (as in a uniform gravitational
field), then internal motions don't matter. The subsystem
doesn't even need to be rigid or nonrotating. Again you
can get away with using the total force and average motion.
*) There are undoubtedly other sufficient conditions.
This can be quantified in terms of the variations, δF and
δv. It is nice to know that common practice is actually
consistent with theory, in simple situations.
The results are exceedingly unsurprising ... but it has been
a gazillion years since the last time I thought about the
issue explicitly and consciously.
Also, I just noticed that this is tangentially related to the
physics of friction. Friction involves lots of tiny δF terms
correlated with tiny δv terms.