This much at least they did teach me in school.
Fall River would have been so named because, like so many mill towns and other industrial towns running on water power, it is located along the Atlantic Fall Line. “From Georgia to New England, the fall line along the Atlantic seaboard represents the boundary between the Piedmont and the coastal plain physiographic regions. It's called the Fall Line, for the number of waterfalls and rapids it features along this boundary.”
Atlantic Seaboard Fall Line
In other words it’s where the uplands end and all water everywhere has to fall over the edge onto the coastal plain. Tracing some Fall Line cities and towns:
End of today’s American geography lesson.