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In the idealized problem the rope/chain falls straight down, but if it is
moving horizontally just before leaving the table, then there must be a
horizontal force that opposes that motion just as the chain leaves the
table. That force must increase with time since the horizontal momentum of
each chain segment increases with time (or as we get near the end of the
chain) and the time period in which the chain segment must change its
motion from horizontal to vertical is decreasing. This is certainly a
problem that one does not want to look at too closely--at least not until
graduate school!