Well, the energy required to accelerate to any speed will be the same (1/2 mass*velocity^2). The difference between accelerating slowly or rapidly will be the relative efficiency of the engine (engine RPM/load).

I mentioned the GS as somewhat of a special case. A higher displacement engine will produce more power at, say ~2000 RPM (or whatever the ideal is) than a smaller one, so the most efficient acceleration rate will be faster for such a vehicle. The HCH-I has an engine sized as small as practically possible, to minimize pumping loss when cruising, so would reach its peak efficiency at a fairly low power output compared to the v6 engine in the Lexus.

We should be careful here when we talk about energy. The energy that you have described is the total amount of kinetic energy that that car needs to gain in order to get to that velocity. It does not represent the total amount of energy expended (by whatever means) to get that car to that velocity.

For example, let's take that car into space, free from gravity and friction. Let's equip ourselves with jet packs. If you push on the rear bumper with 60 lbs of force and I push on the front bumber with 50 lbs of force, the car, of course will accelerate forward with a net force of 10 lbs. Let's say the car moves 100 ft. Energy here is of course, force x distance. So, the car has gained 10 * 100 = 1000 ft-lbs of energy. However, we have expended 60 * 100 = 6000 ft-lbs and 50*100 = 5000 ft-lbs of energy, totaling 11,000 ft-lbs. So, between the two of us, we have spent a total of 11,000 ft-lbs in order to give the car 1000 ft-lbs of energy.

Even here, we're grossly under-reporting the amount of energy that we expended. The jet packs delivered that force, presumably from some chemical reaction. Only a fraction of the energy that the chemical reaction produced is translated into mechanical work. Most of the energy is in the form of wasted heat. Let's say we're idealistically generous and give that energy conversion 40% efficiency. That means the corrected total amount of energy expended is 11,000 / 0.4 = 27,500 ft-lbs of energy.

If we turn our attention to the car's internal combustion engine, we know that even under the ideal conditions, the efficiency for turning all those exploding gas fumes into mechanical work driving the pistons is around 35%. This is in contrast to electric motors, which typically achieve efficiencies of 90%.

So, when comparing ICE with the electric motor, side-by-side, the electric motor wins hands down in the minimizing the amount of energy that needs to be expended to get a vehcicle up to speed.

Energy aside, what's really important is how this all affects our fuel efficiency. If we're only looking at the time it takes to accelerate that car up to cruising velocity, of course the less ICE you use... the less gas you use... which translates into.. higher fuel efficiency.

Well, not so fast. What's the overall FE? For that, of course, we need to look beyond that acceleration to cruising speed and look at... the cruising. If we are able to acceleration from stop only on electric and get to a cruising speed that results in major battery drainage and there are insufficient opportunites to restore that energy via regenerative braking... well, then the ICE needs to spend its energy recharging that battery. So, in this scenario, your effective FE for that acceleration is not near-infinite, you need to pay for it later.

The question then becomes, what burns less gas: 1) ICE-assisted acceleration or 2) ICE used to recharge the batteries. I don't think the answer to this is trivial. At the end of the day, it's about the total miles covered for some amount of gas. When you're using the ICE during acceleration, the amount of gas you're burning to travel say cover a 1/4-mile of acceleration is certainly greater than the gas required to cover that same 1/4-mile of distance when at cruising speed. However, even in this case, this difference depends on the cruising speed. The higher the cruising speed, the more the wind resistance, the more gas that needs to be burned.

In the hybrid scenario, this is even more complicated because we're talking about the gas that's burned to re-charge the batteries which in turn (in principle) delivers the energy to cover that 1/4-mile during cruising. So, of course, that amount of gas is greater because there's always efficiency losses converting from one form of energy to another.

And, to top that off, the factor that really drives how all these tradeoffs are made is the terrain that the car is covering, flat, hilly, traffic conditions, etc.

I think what we can say without controversy, especially given the hard facts of the super hypermilers, is that the highest FE is obtained if you're accelerating with minimal to no ICE assist and you're able to recover that energy through regenerative braking. The ideal conditions for that of course, is a perfectly flat road.

However, my experience at least with the HiHy (of surprisingly similar weight as the GS 450h) is that accelerating without the ICE is relatively rare unless you intend to get to a cruising speed of about 30-35 miles per hour, and you have a nice flat road to accelerate on. The sweet spot, so far that I've found is to accelerate with light to moderate ICE-assist. If I try to "force" electric-only, I find that I end up draining my batteries too much and the ICE ends up charging the batteries before I have time (and opportunity that I'm willing to tolerate!) to recharge them through regenerative braking. Qualitatively, I don't think I have a net fuel efficiency win when my ICE is overly re-charging my batteries because of the cruising speeds that I typically drive at ~65-70 miles per hour.

This is exactly why I said already. The kinetic energy is the same irrespective of time, the difference is specific engine/drivetrain efficiency, based on RPMs and throttle position. Generally the consensus on discussions about engines is that the best thermal efficiency will occur by minimizing RPMs (low RPMs, least mechanical friction), and running as wide of a throttle position as possible (reduced manifold vacuum/pumping loss) without running the engine with an enriched fuel mixture, commonly about ~1/2 throttle shifting near the engine's torque peak (CVT should shift this way automatically). This is why pulse-and-glide delivers better MPG for a given (relatively slow) average speed than steady cruising, by operating the engine at this peak in cycles, and storing the excess energy as momentum.

The key however with the hybrid is that all the elecrtical energy came from the engine at one point -- either from stored up kinetic energy (braking) or more commonly, direct charging from the engine running at say 33%. Assuming the engine->battery->motor process is about 66% efficient, that means using the elecrtic motor beyond what is recovered from regen will be more efficient in situations where the engine would be running at 22% or less efficiency, such as near idle (low speed creeping) or wide-open-throttle.

In situations of moderte demand, such as an acceleration, it would be best to run the engine at its most efficient output, along with just as much elecrtic assist as could be recovered by braking later.

Determining this all experimentally in terms of driving/FE, as you went on to say in the rest of your post is going to be extremely tedious. What I will say is that in my Honda, in which the engine always run, is that it's best to accelerate at throttle position at or slightly above where elecrtic assist first kicks in. The system logically would not "assist" the engine if it would be more efficient to apply more load, so the point where elecrtic assist first kicks in (on the Honda) should mark the beginning of the engine's most efficient operating range.

This tends to produce a relatively slow acceleration, as my car has a very small engine. However, in the case of the 450h, with an engine producing about three times as much horsepower, at the same RPM/throttle position, the larger engine should produce much more power, and thus a relatively fast acceleration at its most efficient output.

This is a good point, and is very relevant in a certain situation: climbing long grades at highway speeds. The electric/gasoline ratio may be too high, and deplete the battery before reaching the top, forcing the engine to handle both the full climbing load and extra charging load. This is why the hybrids should offer user input to be able to separately control the amount of elecrtic and gasoline output. I know the MIMA system experimented with on the Insight has led to about 10-20% improvements in FE due to this.

This also makes sense, however, in most of these scenarios, the best thing to do would be to allow the engine to operate in its sweet spot, and to vary the amount of elecrtic assist independently, using relatively more assist when the final speed is lower, or for climbing short inclines, and using less when climbing long inclines or before driving at a higher speed. This is because adding additional load when the engine is already under heavy load (high-speed cruising) is less efficient than charging off the engine when load is light. Without the ability to anticipate the road ahead though, a computer cannot do these thing.

I believe your conclusions about power/efficiency is incorrect. For the larger V6 and V8 ICEs, the sweet spot for maximum power efficiency is typically obtained at the higher speeds (50+ miles per hour). Your Honda is probably a V4, and the power band should peak at lower RPMs. At lower speeds, the larger V6/V8 ICE engines are traditionally poorer in both acceleration and fuel efficiency. That's what's great about hybrids. The electric power compensates where the ICEs are weak, at the lower speeds to provide a big win for overall power and acceleration.

I also found this PDF off of the Lexus website:
http://www.lexus.com/assets/models/p...ance_guide.pdf

On page 8, they list the Top 10 tips for improving fuel economy for the GS 450h:

Their #2 tip is: Accelerate slowly.

Interesting enough, the #1 tip is: Plan ahead to combine short trips in order to minimize cold starts.