Optimum fuel usage: Driving a 'sine wave' shaped road...

This is rather random, so apologises in advance:

I occasionally wonder about this when driving up and down hills – is there a mathematically optimum method of driving to minimise fuel usage and maximise distance travelled, for a specific amount of fuel ‘X’?

Consider a car driving along a sine wave (constant, repetitive hills and troughs) :

If the car had a fixed amount of petrol to drive along the sine wave, what would be the best times to burn the fuel? Would you be best to:

(a) Drive with as constant as possible a speed, maintaining even fuel usage throughout the journey
(b) Use petrol to accelerate rapidly on the downhill portion to gain enough speed to ‘free-wheel’ up the next hill to just clear the brow, or
© ‘Free-wheel’ down the hill and only use petrol when the car threatens to come to a stop on the upward slope.

Is there a relatively easy way to calculate this? Does anyone have a intuitive feeling for which would be the most effective method?


Ignoring friction, wind resistance and engine gearing, I think it wouldn’t make any difference at all (as long as the vehicle never actually started rolling backwards, in which case you’d have to expend energy to regain ground)

I can’t help you with the math, but as for practical experience with this I can relate our findings from a recent spin through the Colorado mountains.

I pay very close attention to my gas mileage. I know exactly what it gets in the flats. For three days we went through mountain passes with lots of climbs and descents. I would (traffic permitting) kick it in neutral and coast from the peaks to the valleys and then put it back in gear at the point when I needed to maintain my speed and momentum to climb the next hill. I think this would correspond most closely to your © choice. I saw an improvement of about 25% over what I get on a level surface.

This came as quite a surprise as I thought the extra effort to climb might cancel the benefits of the coast but we validated this over the course of several fillups.

Coasting in neutral might be counterproductive, Most modern cars have a deceleration fuel cutoff feature that lets the wheels turn the engine with no fuel input when the foot is off the gas.

If you put it in neutral, though, the car has to burn fuel to maintain idle. So you’re using fuel in neutral during a time when you wouldn’t be using any in gear.

But since staying in gear slows you down more due to engine braking, maybe it all balances out in the end.

Ignoring friction and air resistance, you wouldn’t need to use any gas at all, provided you started on a hill.

Let’s suppose friction is pretty much constant (may be a bad assumption). Air resistance increases with speed, so I guess driving at a constant speed would be optimal.

I suspect you paid for your gas savings with your next brake job. With the engine in neutral you couldn’t engine brake in a lower gear and probably had to ride your brakes too hard.

Burning out brakes on a downhill run can be less than fun.

When necessary I did kick it back in gear to slow down as well as out of OD. I used to drive trucks commercially so these efforts were performed judiciously as traffic, curves and slope permitted but they did seem to work to an appreciable degree.

Assuming that you never need to brake, your optimum milage is going to be from driving at nearly constant speed, which means you would step on the gas only when climbing a hill, and give it a varying amount of gas at different points on the hill. I say “nearly constant”, because you’re not going to be able to avoid changing speed going down the hill, without using your brakes. And brakes will cost you energy. The ideal would be a regenerative braking system such as you find on hybrid cars. If the regenerative breaking were perfect, then you could get extremely high milage by travelling always at a constant, low speed, using the gas and brakes as needed to maintain that speed. In reality, even regenerative brakes still waste some energy, so your optimum speed is probably going to be close to the speed you would have at the bottom from coasting down the hill.

Air resistance goes as square of speed, so energy lost to air resistance per unit distance is greater at higher speed. So energy lost to air resistance per unit distance is greater at higher speed. Which means the lower the speed, the higher the fuel efficiency.

On the other hand, using a brake to kill speed is counter-productive. I’d say the most efficient method is to coast down the hill with the engine off (or neutral), then run engine just enough to reach the t op of the next hill. This minimizes the average speed, which minimizes energy loss due to air resistance.

That’s what I meant to say.

It would depend a great deal on the gearing of the vehicle.