Deriving v_ave = (v_o + v)/2 (AP)

A Graphical Approach:

An Analytical Approach:

This innocent-looking equation requires some mathematical knowledge that you might not get until your second semester of calculus, since the average value of a continuous function is a calculus concept, but here goes:

Suppose that an object has a speed v_o at time t = 0, and a speed v at time t = T. We know that v = v_o + at if the object's acceleration is constant. The average value of the function y = f(x) between x = a and x = b is defined as , so the average velocity, v_ave is:

Now, if v = v_o + at, then at = v -v_o, and:

Which is what we set out to prove.