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In elementary physics courses one is taught that energy is a measure of an object's ability to do work (this in itself seems a little flaky as how does one then define exactly what "work" is, other than stating that when a force acting on an system causes a displacement of that system, then the force is said to have done work on the system, or resorting to a tautological statement that describes it in terms of energy?!). Then in more advanced courses this description usually becomes more abstract in as such that energy is a conserved quantity attributed to a physical system, or an emergent quantity as a natural consequence of time translation invariance of a physical system (via Noether's theorem).
My question is, though, what exactly is energy? Is it possible to give a qualitative description of what it is a measure of?
I have read through the answers given in What's the real fundamental definition of energy? and although they are somewhat useful, I feel that they don't really answer my question in terms of whether we can give a qualitative statement of what energy is, by this I mean, for example, velocity is a measure of the rate of change in position of an object per unit of time, is there a similar statement that can be ascribed to energy (or momentum for that matter)?