I already have. There is no logical reason to take the mean velocity (accept to add the extra velocity value that results is v^2).
Do you know why the math works? Look where it started. Kinetic Energy is derrived from the work-energy theorim Ef=Ei + W[ext] which itself is based off of the conservation of energy which itself stems from the idea of conservation of momentum.
At this first step, there are several debunking options. The foremost being that in a newtonian environment (such as deep space), there would be no external force (or non-conservative force). This, of course, leaves us with only Ef=Ei, which is an assumption based on observation of a non-newtonian environment in the first place. It comes at no surprise then that idea of conservation of momentum [sum]mv would be objected by engineers who believed "that conservation of momentum alone was not adequate for practical calculation and who made use of Leibniz's principle."
http://en.wikipedia.org/wiki/Conservation_of_energy
At any rate, assuming the above is still true (for the sake of argument), then another problem occurs here:
vf-vi is a single term delta v. Even mathamatically, it is not logically applicable to seperate this term. delta v means a 'Change in Velocity'. When you seperate it into two seperate terms, you destroy that meaning.
The next problem occurs here following the assumption that energy is conserved (before successfully derriving it) by taking the mean and then destroying the relationship of velocity (delta d/t by multiplying the t value through the (1/2)sum of v) and then utilizing the destroyed relationship from above and providing the extra d/t=v varibles to later give the needed v^2:
*Note, the REAL reason they use the 'mean' velocity is to get rid of the delta relationship. Velocity = delta (d/t) and using the 'mean' is how they 'fiddled' their way out of having to deal with that relationship.
Later, it eliminates the extra additive velocity (-vi) value (from where the extra d/t value was originally added) by assuming the object started at rest (which is rarely the case except when arbitrary evaluating from the same rest frame).
[*Note, it was never "Given" that the object started at rest. It was "assumed" for the sake of making the relationship between Kinetic Energy and Conservation of Energy PE=-KE. In actuallity, the reason the vi value goes away (when you dive into the reasons why energy is observed to be conserved) is because impact (or other such interaction where energy is exchanged or evaluated) occurs at vf and vi is irrelevant and since the initial velocity is irrelevant so is the distance as the graph on the first page indicates and why there is absolutely no logical reason to take the mean velocity (d/t)).
Then they just plug and play the values back in:
And POOF, there's that magical v^2. Aren't mathamatical fiddles fun?
The underlying point is that the whole process for deriving Kinetic Energy is wrong. They whole derivation rely's on the assumption that energy is conserved before they even know what the deffinition (mathamatically) of energy is. So from the begining of the derivation, the resulting KE equation is going to be restricted to that assumption. This just reinforces the idea that they had an idea of where it should start and where it should end and they just muddled the stuff in the middle.
Again, Mr. Novak covers all of this and much more on his site in great detail. He also correctly derrives Kinetic Energy from Force.