Andrew Mason said:
Not my specialty.
Ok, I want start clean with your comments in mind, more than in mind, even corrections as it were, so that I can remain focused on the prize, unfettered by my errors of the past. And here, for me, the prize is a clear understanding what the 1/2 term really is, something I have been absolutely captivated by for a while now. While time differentials and such do a great job of explaining the why of the math, I find it very unsatisfying to use math equations to explain something that is very real and that we all live with the consequences of, all the time. I find that they serve me better in validating and predicting than they do at answering why.
Kinetic energy exists solely as a result of the perspective of one object to another and how the spatial relationship between those objects changes over time. Without an origin reference frame it cannot be defined and understood, and indeed even exist. Applying energy, or force, or creating positive or negative acceleration on an object, however it be termed, cannot be done without doing so to two opposing objects, such that momentum remains neutral. From either of the involved objects' perspectives, the kinetic energy of the opposing object will be equal to the energy applied to create separation (let's assume positive acceleration for two objects at rest respective to each other prior to the application of force, and I'm also accustomed to visualizing springs that experience no heat loss). Conservation of momentum will dictate the velocities of each such that total combined kinetic energy equals that which has been applied. If I said that kinetic energy is always balanced, then that is blatantly wrong. I did see a previous post in this thread use the term momentum field, which is a term that appeals to me, even if I don't recall its exact usage. The momentum field defines the split of kinetic energy which must remain equal in aggregate but need not be split equally.
Shifting the perspective of calculation from that of the objects involved in the forced separation to that of a third party observer, say one who is doing math problems on a piece of paper, will result in this observer likely only being interested in one of the two objects, which has an existing velocity of unexplained origin. The energy associated with that velocity is seen only as a time derivative of the objects velocity. In fact, the shift in perspective from having a balanced momentum field, to one which is not, creates a loss of one half of the energy involved since the energy applied was also used to "power" the opposing object headed in the reverse direction in the greater unseen portion of this picture.
This isn't my best effort so maybe I just need to keep trying until I make sense. But I am asserting that the energy applied to create velocity via acceleration does so in a way that creates balanced momentum in two directions. The absolute requirement by nature that this momentum field remain always balanced defines and creates the one half factor that exists when objects are looked at individually. If we only looked at kinetic energy as a part of total balanced momentum neutrality, then we don't need to decrement by one half at all, because energy is conserved (but not in its kinetic form, as we all know).