To floor it or not to floor it

I'm trying to understand the effects of rate of acceleration on minimizing fuel consumption over a given distance. We've all read and heard that "jackrabbit starts" waste fuel. How can this be shown? Well, fuel contains energy in hydrogen and carbon chemical bonds. We oxidize fuel to release this energy. The energy does two things in our vehicle - it adds kinetic energy, thus changing the potential energy of the chemical bonds to the kinetic energy of the moving vehicle, and it does the work of moving the car against the sum of the forces acting to resist motion. These include engine and driveline friction, pumping fluids, aerodynamic drag, tire rolling resistance, etc. For the purists, it also can increase our potential energy by utilizing chemical potential energy to raise our position in the Earth's gravitational field (i.e., take us up hills). But since, on average, our vehicle stays at one elevation (that is, at the elevation of wherever it lives) we can ignore this.

It's easy to show that the more slowly we accelerate, the farther we go in the process of adding a given amount of kinetic energy to the car (that is, getting up to a given speed since kinetic energy is one half the product of the mass of the vehicle and its contents times the square of the velocity or speed). Thus, if we accelerate from 0 to 60 in twice the time, that is, accelerate at half the rate, we will go twice as far to add the same amount of kinetic energy, which comes from burning the fuel. So, the same amount of fuel will take us twice as far while getting us up to the same speed. It will take longer but be more fuel efficient. As I said, this is quite easy to show mathematically.

So it seems like a no-brainer. But... It's also true that there is a speed at which an automobile gets the best fuel economy. Because it burns fuel while idling (my Grand Cherokee burns 0.38 gal/hour as nearly as I can determine), the car gets 0 m.p.g. standing still. As we start moving and achieve higher speeds the fuel economy, that is, the m.p.g., increases. Above some speed, different for each vehicle, the aerodynamic drag, which increases approximately with the square (at least according to my calculations - others say cube) of speed begins to take its toll and the efficiency decreases. So, the vehicle is more efficient (gets better gas mileage) the faster you go, up to some speed at which efficiency begins to decrease. I'm still working on the mathematics of this but it seems intuitively reasonable and agrees with the nuggets I've found on the web (see here for example) and some "back of the envelope" calculations.

So let's say, for conversation's sake, that for my Grand Cherokee, the most efficient speed is 50 m.p.h. I think this isn't too far off, based on my experiments and a website I found (but since lost). And let's say I'm going to take a 20 mile trip. To make it easy, the trip is on a level road, no stops. I could, in principle, accelerate so slowly that I don't get to 50 m.p.h. before reaching my destination. Or, I could floor it and reach 50 m.p.h. as quickly as possible and drive the greatest possible portion of my trip at the most efficient speed. So flooring it gives me the maximum number of miles at the most efficient speed. On the other hand I could, in principle, accelerate so slowly that I only get to, say, 5 m.p.h. before reaching my destination and drive the entire trip at a very inefficient speed. This seems to imply that quick acceleration is most efficient. So which is best?

I'm still working on it, so more to follow.....