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The bottom line is that it does not fundamentally matter how you
(legitimately) divide up dw and dQ, as far as the answer to any real
physics/engineering question is concerned - as you illustrated in the
various interpretations of your inelastic collision example. Rutherford
would say that these arguments are about stamp collecting (taxonomy), not
physics. I think this flexibility should be part of our teaching, just as
a history teacher should admit of another "ratio divisionis" (other than
his) of civil war causes, to his students.