Galileo's thought experiment

In class, we have looked at the problem to compute position from velocity or to compute position from acceleration. This involved taking anti derivatives. From the constant acceleration x''(t) = -10, we got
x(t) = -5 t2 + x'(0) t  + x(0) 
where x'(0) is the initial velocity, a constant and where x(0) is the initial position (height) which is also a constant. If we drop a ball for example from a height of 4 meters with zero vertical velocity (as in the movie), the motion is
x(t) = 4 -5  t2

Source: NOVA: galileo's battle for the heavens.