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 vo at time t = 0, and a speed v at time t = T. We know that v = vo + 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, vave is:
Now, if v = vo + at, then at = v -vo, and:
Which is what we set out to prove.