Time Motion - Increasing Velocity

1) The two Top equalities of Equations govern the movements in this system:The Bottom Left Equation; a(i)-were looking for Integers when, a = 1,4,16,25, etc. 2) I updated this to show the two vectors velocity and acceleration. 3) Notice the different orientations the two vectors can obtain in relationship to each other.
This is a Math Table Based on all My work up to this Point using my Original Hypothesis.