[Atomic-S (OP)]

Which uses less fuel:  to drive up a mountain on a road whose slope is moderate enough to allow the use of high gear, or to drive up the same mountain via a steeper but shorter road that requires the use of a lower gear?

[syhprum]

There are several factors to consider, the overall thermal efficiency will generally in terms of MPG will be lower when driving in a lower gear but you must bear in mind how the lengths of the different routes compare.
How long is a piece of string ? 

[wolfekeeper]

Probably the moderate one, if you do squirt and coast and don't go too fast.

[ProjectSailor]

Typically your engine will be more efficient at lower revs, so the moderate in higher gear would seem better. So most fuel efficient..

BUT a practical experiment may show that you use less fuel to get from A-B the steep way than the moderate way due to less time and distance.. so yes.. without facts and figures.. how long is this piece of string compared to that piece of string.

[puppypower]

The maximum torque for most engines with a red line of about 7000 rpm, is about 3500-4000 rpm. Maximum horsepower is higher and closer to 5000-6000 RPM. Which ever scenario is able to use the maximum torque RPM will have the easiest time up the hill; least resistance. The torque will drop off quickly in both directions from the peak (more or less RMP) and will waste gas.

If you use to high a gear you may not be able to reach the maximum torque RPM and will waste gas. If you use too low of a gear, you might have the best RPM, but you might waste horsepower; waste gas. The idea is the top of the torque curve, while making use of all the horsepower to just climb the hill.

[Pecos_Bill]

Assuming that you have a rear wheel drive vehicle, if you back up the incline you will get better traction when the wheels pull you up the hill than if you go up in whatever gear pushing the car forward in drive.

That way will use the less gas.

[chris]

Assuming no losses, because the change in gravitational potential is the same in both circumstances (i.e. you are gaining the same amount of height), then the energy required to lift you and the car from the bottom to the top of the mountain would be the same.

However, cars and engines are not lossless (and we'll leave Volkswagen out of it too); that is, they are inefficient for a variety of reasons, so there will be an optimum rate at which to climb the hill and this will be determined by these factors.

These include engine speed and vehicle speed; the faster you drive, the greater the impact of air resistance or drag. This scales at roughly the square of the velocity, so if you drive twice as fast you actually feel four times the air resistance; that is, you need to burn 4 times as much fuel to travel the same distance.

Engines tend to become lossier (more inefficient) the faster they turn, because they shake about and produce more noise, so higher revs will tend to waste more fuel.

Also, the rubber tyres are lossy, because they deform as they revolve. This flexes the rubber, leading to heating. Between ten and thirty per cent of the energy in moving a vechicle is this form of rolling resistance.

Therefore, when considering this interesting problem you have set, it is not simply a case of how fast to run the engine but also a case of how fast you drive the car and for how long.

If you are in a convertible, whether the roof is up or down will also have an impact...