Two of the recent models discussed illustrate limitations of the modeling
process, the limits of applicability.
In the discussion of "derivatives", the model assumes that the release is
discontinuous, but is then used to argue for the continuity of the
acceleration
As others have pointed out, if you are going to look at the transition from
rest to "falling" then you have to model the release as a continuous process
with a variable supporting force.
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I take exception to the accuracy of the last paragraph. I think the
discussion did not conclude that you *have* to model the release as a
continuous process. I think people were questioning whether modeling as a
discontinuous process was precisely 100% accurate or not. I don't even
think people questioned the discontinuous model as being all that bad
(others may disagree here). My guess is that one can model quite well,
under most typical circumstances, the motion of the object by a
discontinuous model. I admit you could probably cook up some set of
circumstances where the discontinuous model would break down; but that is a
far cry from saying that you *have* to model it as a discontinuous process.