Equation 3 Relation to the V/T Graph

Equation 3 is: d=V1Δt+½aΔt²

and...

there are two standard points in the graph (t1, v1) and (t2, v2).

To determine the displacement (d), we are required to calculate the area between the slope and the x-axis, and there are two sections for this. The first one (lower one) is a rectangle and the second one is a triangle.

Coincidently, there are two parts for equation 3 as well. v1Δt and ½aΔt².

The area of a rectangle is defined by the formula Base x Height, where the height is v1 and the base is t2-t1.
A = v1(t2-t1)
A = v1(Δt)

The area of a triangle is defined by the formula Base x Height / 2. As we learned earlier, base is represented by Δt, and the height in this case is v2-v1. Also, from equation 1, v2-v1=aΔt.
A = ½Δt(v2-v1)
A = ½ΔtaΔt
A = ½aΔt²

Thus, d=vΔt+½aΔt²