Uniform Acceleration Motion: Equations with Explanations

The equations that quantitatively describes uniform acceleration motion are explained.

Let
a be the acceleration
u be the initial velocity at time t1
v be the final velocity at time t2
t = t 2 - t1
x is the displacement between t1 and t2
x0 is the initial position

The relationship between all the above quantities are given by the following equations:
definition of acceleration: a = (v - u) / t

v = a t + u 1 (deduced from above definition)
x = (1/2) a t 2 + u t + x0 2
x = (1/2)(u + v) t + x0 3
v 2 = u 2 + 2 a (x - x0) 4

If x
0 = 0 (start from origin) , the above equations simplifies to
v = a t + u 1
x = (1/2) a t 2 + u t 2
x = (1/2)(u + v) t 3
v 2 = u 2 + 2 a x 4

If x
0 = 0 (start from origin) and u =0 (starting from rest) the above equations simplifies further to
v = at 1
x = (1/2) a t 2 2
x = (1/2) v t 3
v2 = 2 a x 4