Blimey. Here we go again. I'll be honest and say I didnt have the patience to read all those posts in detail, but it just seems to me that everyone is coming to a roughly equivalent understanding, but are still arguing about minutae that got posted ages back.
I also think its difficult to have a meaningful debate because the definitions haven't been sorted. Also, I think that for myself as well as others, existing pieces of knowledge are preventing us from properly assimilating other bits of knowledge, and also we are making statements that apply to a particular context without necessarily qualifying that context.
When I step back from cars and engines and gearings for a moment, and just think about raw physics, it all seems blindingly obvious.
If over period of time t, an average power of p is expended, the amount of kinetic energy added to the object will be p*t.
So any change in the average power over the period of time will result in a proportional change in the amount of kinetic energy added.
So lets ignore transmissions or drivetrain losses etc. and say that our CVT produces a dead flat power curve (line?) at max power, so the average power for the time t = max power.
In any manual transmission, the power curve will rise and fall depending on gear changes, but will always be below or equal to max power, so average power over time t < maxpower.
So that means that after time t, less kinetic energy has been added in the second example, which means that any time our engine is below maximum power, we are adding less kinetic energy.
Now i'm starting to state the obvious now, but if we have initial velocity u, our final velocity v will always be higher when more kinetic energy has been added. And if our final velocity is higher, we have 'accelerated better'.
So, for a given engine, with complete freedom to gear appropriately, maximum final velocity will be acheived when the engine spends as much time as possible as close as possible to maximum power.
Just for completeness, I going to make a few statements that have come up in the debate, together with the context needed to make them correct.
1. For a given fixed gear ratio, the highest instantaneous acceleration acheivable in that gear will occur at the moment when the engine is producing maximum torque.
2. For a given road speed, the highest instantaneous acceleration acheivable at that speed will occur if the gearing is such that the engine is producing maximum power.
3. Over a given period of time, ignoring any gearing limitations, the greatest overall increase in velocity will occur when the area under the power/time graph is maximised.
And thats it from me.