[Coda]

No, the equation only cares about the initial velocity. The acceleration term provides the information necessary to handle the change over time.

Actually FINDING the velocity at a given time is a completely different thing. In calculus, you would determine this by using the first derivative (which is why there's the 1/2 factor), but at this level, you just have a different equation to work with.

The equation for the position of an accelerating object generalizes to this form:
s = (1/2)a*t^2 + (v_0)t + s_0
where:
s is the position at time t
a is the acceleration
v_0 is the initial velocity
s_0 is the initial position

You can find how fast the object is going at any time using this equation:
v = a*t + v_0

The fun part of having all of these equations available to you is that they're all tools that can build up more solutions. Knowing this equation, you could figure out what the acceleration is for an object if you know how fast it's going at two different points in time (because then you would have v, t, and v_0 defined and solve for a) and then you could plug that back into the position equation to figure out where the object is at any point in time, past or future.