What an interesting discussion - I appreciate Slim's theoretical approach as (although I am supposed to be an engineer, albeit of the electronic kind) I hadn't thought it through from these first principles.
The criticism that there are some variables not taken into account (e.g. that the acceleration is not constant due to the way say fixed pitch props perform and due to increase of parasitic draq etc.) are all fair, but don't detract too much from the value of the theoretical illustration for me.
I was never taught, nor have I seen in any publication, a good rule of thumb for speed halfway down the runway. The only thing I have read is that one should pace out a halfway point at home airfield and then on subsequent take-off runs, note the airspeed speed at halfway point in order to build up a practical knowledge of the normal numbers, so that one's brain might notice when this isn't achieved. But I have not made a habit of doing this (perhaps I should!)
So the value of this discussion for me is provocation of thought!
Thanks guys
Andy