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Showing/Evidence/Feynman: what actually means "show that something is true"? We find out directly from Newton's law that the kinetic energy should change when we form its derivative by time, and then apply Newton's law.
dT/dt = d/dt(1/2mv²) = 1/2m2v (dv/dt) = mv (dv/dt)
From the 2nd Newtonian law, however, follows m(dv/dt) = F such that
dT/dt = Fv.
in the one-dimensional case, it is force times speed.
Vom Wesen physikalischer Gesetze München 1993
Vorlesungen über Physik I München 2001