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My original premise is that associated with a change in direction is a change in speed

It follows from Newton's third law that an object will resist changes in its direction of motion; so the sun pulls on the earth, then the Earth resists this pull with a force, which is an acceleration, and the Earth and sun are continually in "tug of war". This is also Newton's first law; an...

what do you mean when you say w

So this is the same as what I said?

Thanks... That is another way to derive it.

Please explain further ...I am not familiar with the mechanics of calculus notation.

why does kinetic energy depend on half the square? if we add up all the velocities does it not produce half a square?

I agree. Are you a teacher? If you are then maybe you can tell me what my physics teacher was talking about, when he was writing 1/2 on the parabola, and crossing each one off before proceeding to writing the next 1/2. I mentioned this above.

and how was the factor of "2" treated by men of science when thinking in terms of twice the KE or twice the work?

Did your physics teacher ever define work or KE as MV^2 as opposed to 1/2MV^2?

I've been trying to figure it out for the longest time.

I think it's arbitrary too.

this was before the midterm

But there was a time, looking back when he seemed to be (just from a fragment of what he said, as this is all I remember) talking about MV^2, or twice the work. The fragment was exactly this: there was picture of a parabola on the board in lab or pre-lab in which he was marking off points and...

My teacher always taught us that KE or Work = 1/2MV^2, not MV^2