The text Physics : Calculus by Eugene Hecht uses the locution "work done
against" in a number of contexts. This includes work done against
friction, against gravity, and against inertia. Work done against
friction is "defined" as the work done by a force pushing a block through
a displacement along a horizontal surface at constant speed. The surface
exerts a kinetic frictional force opposing the motion. Work done against
gravity is "defined" as the work done by a force in lifting an object at
constant speed (or at least with no net change in kinetic energy). The
metaphor changes for the work done against inertia, which is "defined" as
the work done by the force F acting on a block which moves in a straight
line along a horizontal frictionless surface with increasing speed.
When discussing work done by a force has been (and is now) my preference
to avoid the "work done against" locution and to only refer to the work
done BY a force, rather than to refer to both the work done BY a force and
the work done AGAINST a force, as Hecht does. I discussed this with him
in Denver last month, and I'm not sure I fully understood his reasons for
using "work done against". I think his purpose was to give the student a
sensible "definition" for work rather than to define it solely as an
outcome of integrating F = ma (the work-energy theorem). (In this context
sensible means observable through the senses, as in sensible heat.)
We are currently using this text, and in a few weeks we will be into the
chapter on work and energy. I would appreciate any input as to how to
treat "work done against".