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"There is every reason to believe [F and ma] are
simultaneous, and no reason to believe otherwise. If anybody has
theoretical or experimental evidence to the contrary, please let us
know."
In the model we expect the unbalanced applied net force (John
Clement's push) and the acceleration of the point particle or the
rigid body to be simultaneous.
However in the real world, where elastic forces come into play as
existence of the unbalanced force is "communicated" to all parts of
the non-rigid body, simultaneity is not obvious.
Instantaneous is also limited by special relativity.