Deriving equation 3 and 4 from the v-t graph

The standard velocity-time graph.

Equation 3 is written as d = V1Δt + ½aΔt².

On a v-t graph, when we are looking for distance, we find the area of the trapezoid shape.

We can do that by dividing the trapezoid into a square and triangle.

The formula to find the area of the triangle is ½(V2-V1)t. Equation 1 states that at= V2-V1. Sub equation 1 into the formula. ½at²

The formula to find the area of the rectangle is V1*t, also known as V1Δt.

Combine these 2 formulas to create equation 3, d = V1Δt + ½aΔt².

Equation 4 is written as d = V2Δt - ½aΔt².

On the v-t graph, we can derive equation 4 by making the graph look like this big rectangle.

First, find the area of the rectangle as a whole. V2*t, or V2Δt.

Then find the area of the triangle within the rectangle. ½(V2-V1)t. We know from equation 1 at = V2-V1.
½aΔt².

Combine the two formulas to create V2Δt - ½aΔt².