18 July 2008

Lower speed limits, part two

One thing people complain about in regards to slower speed limits, which I wrote about earlier today, is that when speed limits are lower it takes longer to get places. This is, of course, true. But on the other hand you use less fuel.

From Wikipedia on fuel economy in automobiles: "The power to overcome air resistance increases roughly with the cube of the speed, and thus the energy required per unit distance is roughly proportional to the square of speed." Furthermore, this is the dominant factor for large velocity.

So let's say your fuel usage, measured in fuel used per unit of distance (say, gallons per mile), at velocity v, is kv2. (k is some constant that depends on the car. A typical value of k, for a car using 0.05 gallons per mile at 60 mph, is 0.000014.) Let's say you value your time at a rate c -- measured in, say, dollars per hour, and the price of fuel is p.

Then for a journey of length d, you'll spend dpkv2 in fuel, and cd/v in time. Your total cost is
f(v) = d\left(pkv^2 + {c\over v}\right)
and differentiating and setting f'(v) = 0, the optimal speed is (c/2pk)1/3. The cost of the journey at this speed is
f \left( \left( c/2pk\right)^{1/3} \right) = {3d \over 2} (c^2 pk)^{1/3}

So according to this model, if you value your time more you should go faster; not surprisingly your value of time c and the price of fuel p show up only as c/p -- effectively, your value of time measured in terms of fuel.

Also, the optimal speed doesn't go down that slowly as p increases -- it only goes as p-1/3. But a doubling in gas prices still leads to a 20 percent reduction in optimal speed -- perhaps roughly in line with what people are suggesting. Taking c = 10, p = 4.05, k = 0.000014 gives an optimal speed of 45 miles per hour, although given the crudeness of this model (I've assumed that all the fuel is used to fight air resistance) I'd take that with a grain of salt, and I won't even touch the fact that different people place different values on their time and get different fuel economy. We can't just let everyone drive at their optimal speed.

Besides, part of the whole point of this is that if we use less fuel, demand for fuel will drop significantly below supply and oil prices will go down. So to forecast the effects of a lower speed limit I'd have to factor in that gasoline could get cheaper -- and let's face it, I can't predict the workings of the oil market.


Anonymous said...

Question: roughly how does the optimal speed vary with fuel efficiency? I get significantly more than 20mpg, for instance, since I almost never drive in a city.

And anyhow, since the highway is about the only place you can hold a steady speed of (more or less) your own choosing for any length of time, we should be working this out for a typical highway efficiency, no?

Anonymous said...

unapologetic: Your first question can be addressed by finding your value for k. I believe Isabel got that "typical" value by inverting 20 mpg and then dividing by 60 mph squared. You can easily put in your known efficiency at any speed to find your value of k. Highway speeds, however, will give you the most accurate value, since you won't be stopping and starting as much. That is, you'll be exerting, as nearly as is practical, all of your fuel fighting wind resistance.

For the second question, the term kv² already scales the efficiency to varying speeds, if we can trust the "roughly proportional to the square of speed" comment to be a good approximation.

Anonymous said...

Well that's what I'm asking: How did Isabel come up with a k-value? As far as it's been explained here it's just a lookup "typical value". Great for engineers, but pretty weak for mathematicians.

Michael Lugo said...

How did I come up with k? I made it up. I'm playing engineer here.

Anonymous said...

I thought as much... heretic.

Anonymous said...

I can see all this, and I think many people realize the time value of various speeds on trips of considerable length. (Short trips and all trips involving much in town driving are irrelevant for this discussion.)
My daughter and her family live 920 miles from where I live. This is a two day auto trip for me at current speed limits. Reducing highway speed limits by 15 mph would increase the driving time from about 15 hours to about 18.5 hours. When stops for food and gas are added, this amounts to a three day trip, including an extra motel stay and meals, about $100 extra. So the 15 gallons of gas saved has to be worth about $6.66 per gallon to make the economics work out. Of course, when I add in what I think a day of my remaining life is worth, I get $75 per gallon.

Anonymous said...

When people drive slower, there are more cars on the road (because it takes longer to go from point A to point B). Lower speed limits generate increased traffic. Increased traffic means gas consuming delays (unless people drive less to avoid the traffic). Increased traffic also takes a heavy mental toll in many urban areas.

Also it is really hard to argue that lower speed limits save lives. For a classic treatment from statistical researchers, see the article by Campbell and Ross on the Connecticut Crackdown on Speeding. They use lower speed limits and traffic fatalities as an exemplar of the challenges of making causal inferences about public policy.

Anonymous said...

Hey, Isabel... or anyone. Can anyone tell me if there are any progams than you can use to interface between the TI-89 and Mathematic &or maple. That is work with a set of equations on your computer and then easily push them down the TI-89 and vice versa.

Or do you even use a calculator, or just calculate everything in your mind like that swiss mathematian.. Beuler, i think it was?

-Stuipd Kid

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