C/C++ :: Trapezoidal Motion Profile Using Discrete Method?

I'm trying to program an arduino to generate a Trapezoidal Motion Profile to control a DC motor with a quadrature encoder.

Essentially, the user will input the desired Target Position, Max Velocity and Acceleration (decel = -accel) and the code will calculate the target position versus time which will then be compared with the actual position. The result will then be subject to a PID calculation

My initial assumption was that I could use basic Newtonian physics to determine position (i.e. PT = P0 + V0T + 1/2AT2, VT = V0 + AT). However, after reading through documentation for pre-existing motion controllers, I discovered that the prevalent method was to use a discrete time method, which is as follows:

VK = VK-1 + A (A = Acceleration)
PK = PK-1 + VK-1 + A/2

I'm having a hard time understanding quite how this equation would generate the target position versus time. In the case of Velocity, it seems to just add the acceleration to the current velocity. But what about everything in between?