Let us consider an object moving in straight line with a constant acceleration. For such a situation the first equation of motion gives the final velocity (v) after time t given that the object is having an initial velocity of u and constant acceleration of a. Now for deriving it, let us consider the following velocity time graph:

Initial velocity of the body, u = OA ... (1)

Final velocity of the body, v = BC ... (2)

From the graph BC = BD + DC

Therefore, v = BD + DC ... (3)

Again DC = OA

So, v = BD + OA

Now from equation (1), OA = u

So, v = BD + u ... (4)

We should find the value of BD now

Now, from the velocity time graph, if we calculate the change in velocity to change in time, then we can find the acceleration, a. This is nothing but the slope of the graph.

Thus,

Acceleration, a = Slope of line AB

a =

But AC = OC = t

Hence we get

a =

BD = at

Now putting this in equation (4) we get

v = u + at