In talking about displacement vs time graphs in my high school Physics
class it was clear to everyone that displacement must be a
differentiable function in order to represent a real motion. No object
can suddenly (instantaneously) go from one velocity (slope) to another.
Can the same be said for the velocity vs time graph? It seems that it
is possible for an object to go from one acceleration (slope) to another
instantaneously. When releasing an object to free fall the acceleration
goes from 0 to 9.8 at the instant it is released does it not?