Deriving Equation 4 from the Graph

Equation #4 is: d=V2Δt-½aΔt²

In order to derive the equation from the graph, we must find the area between the slope and the x axis. Unlike the steps for Equation 3, this time we must first find the area of the large rectangle and then subtract the smaller triangle from it.


The equation for the rectangle is ab, where a = base, and b= height. From the graph, we can determine the base and height of the rectangle. The length is Δt and the height is V2.

So, area of rectangle=V2(Δt)

To find the area of the triangle, we know that the equation is ab/2, where a is the base and b is the height. For the small triangle that we are going to use, it appears that its height is V2-V1, and its length is Δt.

So, area of triangle=½(V2-V1)Δt
                           
From Equation 1, we find that aΔt=V2-V1!

Substituting that into the equation gives us the following:

area of triangle=(½aΔt)Δt
                      = ½aΔt²

Finally, by subtracting the area of the triangle from the area of the rectangle, this gives us equation 4: 

d=V2Δt-½aΔt²