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That is certainly made explicit the first time relativity is taught,
at least by me. I can't recall ever noticing a misconception of the
sort you suggest among my first year students. They seem to be able
to cope with F = ma satisfactorily. The only problem I encounter is
that many of them who know how to do the problems don't seem to
believe that F = ma works in the real world. That is a much deeper
problem.