A new tool

A velocity-time graph is a good way to visualize the relationships between velocity, acceleration, time and distance. Since velocity is an intermediate concept between acceleration and distance, let the vertical axis represent velocity and the horizontal axis represent time.

Now, if the velocity is constant -- which is to say that acceleration is zero -- it is easy to see that distance in this mathematical space is represented by area. That is the trivial case. We wouldn't need velocity-time space to visualize that, but things get more interesting when acceleration is something other than zero. If the acceleration is constant, it is represented as a straight line in our drawing.

The easiest case to visualize is when the initial velocity is zero, when the object starts at rest.  Then the distance traveled is just one-half the distance above. The triangle is just half rectangle.

As before, the distance traveled is equal to the velocity times time, but the velocity is an average velocity Vave. The average between zero and the final velocity Vf is just half the final velocity. You can see from the geometry of the diagram that this works because the object spends half the time traveling less than the average velocity Vave , shown as a green line in the diagram above, and half the time traveling faster. The distance traveled in the second half the time exactly makes up for the distance not traveled in the first half. The part of the grey triangle above the green line is congruent to the clear triangle below the green line.

We can generalize the above argument to constant acceleration situations where the initial velocity is not zero. Let's say that the velocity starts at some initial value, Vi and accelerates to a final value, Vf, at a constant rate, a

Again, the average velocity is half way between the initial and final velocities. That is, the average velocity is the average of the velocities as long as the acceleration is constant. In fact, if we define the average velocity as distance over time, we can say

Or, writing this in terms of the sum of velocities,

Now acceleration, defined as the change of velocity over time, can be written in terms of the difference of velocities.

Here we have two equations in five "unknowns," and we can easily combine them to eliminate one of the variables. Take initial velocity Vi , for example. Add the two equations and Vi is gone. It is more usual, however, to eliminate the final velocity Vf  . Just subtract the two above equation and solve for d, and there it is. You will get your old
friend d= Vi t + (1/2) a t^2 . Try it and see!

The real payoff from this approach, however, comes from eliminating time t in the above two equations. Just multiply.

Recalling that the sum times the difference of two variables is the difference of the squares, we get

This is your new tool.