And we go cruising to entertain ourselves

Philip Hammond, Transport Secretary, will announce the government's intention to bring in a new speed limit of 80 mph the Conservative Conference next week. Apparently, this is considered to be a win-win initiative in some circles, with the Liberals in government negotiating to bring the speed limit down to 20 mph in many urban areas.

Apparently, Ministers and experts argue that raising the speed limit could boost the economy. They say that a modest rise would mean quicker journeys for employees, reps and delivery vans.

Let's have a look at some numbers, here.

First, the basic Physics:

Let's imagine a basic journey of 100 miles, covered at constant speed. This 100 mile journey would take 1 hour 25 minutes to complete at 70 mph (allowing no time for breaking or acceleration). At 80 mph, this same journey would take 1 hour and 15 minutes, so you have, at maximum, a saving of ten minutes. This ten minutes is the upper limit - it takes longer to accelerate to and from 80 mph than it does to 70 mph, and in real life, the journey would most probably be broken by several speeding ups and slowing downs.

The saving in time is, really, negligible; there are few places in the UK where a vehicle can travel for 100 miles with no interruptions, if any, and even fewer for journeys longer than that. So we are looking at a saving in time of around 14%, maximum.

Now, the fuel consumption:

For a car to acquire a given speed, it has to be provided kinetic energy from the fuel its burning. The kinetic energy increases with the square of the speed - if you double the car's speed, it needs four times as much energy. Thus the increase in energy required to reach a speed of 80 over a speed of 70 is (80/70)^2 or (64/49) or 1.31 - this is an increase in required energy of 31%.

As the car travels at a constant speed, it has to keep pumping energy into the kinetic energy 'pool' it posses because of frictional forces. The frictional forces keep slowing the car down, or turning the kinetic energy into heat and sound and other less useful (from the viewpoint of the driver) energies, and the engine has to keep topping up the kinetic energy that car has to have to travel at that speed  (In GCSE we'd say that the work done by the engine equals the work done by the frictional forces).  It's a bit like a stream supplying a pond with water at the same rate as the water leaves the pond. 

The main frictional force to consider when thinking about the forces slowing the car down is air resistance; air resistance increases roughly with the cube of the speed of the car. 

The increase in energy lost to air resistance and therefore the increase in the energy that must be supplied by the car engine is therefore around (80/70)^3 or (512/343) or 1.49. This looks like an increase of around 49%. In practice, it's likely to be less than that. As the car is travelling more quickly, the air resistance forces are acting for a shorter time, so the increase in energy lost to air resistance isn't quite as high. 

However we're already getting a bigger increase in fuel costs than reduction in time-spent costs, percentage wise, if we look at the energy needed both to get the car to the higher speed and the energy needed to keep it at that higher speed.


And, to be fair, different vehicles have different aerodynamic profiles. If a car is aerodynmic and therefore needs less energy to keep it going at a high speed, then the increase of 49% may be 49% of a small amount and of no significance. However, the sorts of vehicles which are used commercially tend not to be that aerodynamic - I've not seen a Ferrari deliver toilet rolls to Tesco recently, and although your average sales rep might like to drive a Ferrari, a company boss who is concerned about economics and financial efficiency probably isn't going to supply him with one.

There are more details on fuel economy here, and the Wiki article also provides a handy graphic.

Now, the economics.

The Wiki article graphic is quite old; although the Physics won't have changed, the aerodynamics of cars that are commonly used today may be more efficient.  I have found some more up to date numbers from this article.

Interestingly, the most fuel efficient cars seem to have the highest drop in fuel efficiency once you get past the 'sweet spot'.  A 5 mpg drop for an increase in speed of 10 mph from 65 mph to 75 mph seem reasonable from the data.

I'm going to use the Toyota Camry numbers (a sort of standard rep car) and assume that there will be a similar drop in fuel efficiency from 70 mph to 80 mph as there is from 65 mph to 75 mph. While this is only an assumption, it is a reasonable (and possibly conservative) assumption, as the gradients of graphs of fuel efficiency tend to be constant above 65 mph (i.e. they will show a straight line directed downwards) - and then there's the cubed law as explained above.

So, the Toyota Camry will have a fuel consumption of around 30 miles per gallon when driven at a constant speed of 70 mph. This means that it will use (100 / 30) or 3.3 gallons of fuel to drive that 100 miles at a constant speed of 70 mph.

At 80 mph, it has a fuel consumption of around 25 miles per gallon. This means that it will use (100 / 25) or 4 gallons of fuel to drive the 100 miles.

The latest prices in petrol I've been able to find are detailed here. There are roughly 4.5 liters in a gallon, so, at £1.31 per liter, we get these costs for driving this 100 miles at each speed:

At 70 mph, it will cost the driver:   3.3 gallons * 3.8 liters/gallon * £1.31/liter = £16.43
At 80 mph, it will cost the driver:   4 gallons * 3.8 liters/gallon * £1.31 = £19.91**

That's an increase in fuel costs of £3.48 for a saving of, maybe 10 minutes.
In general terms, that's an increase in costs of 21% for a saving in time of 14%

Remember, although this is a thumb-suck estimate, all these numbers are conservative. The increase in fuel costs will be much larger for, say, a lorry delivering commercial goods. Fuel-efficient cars will have a much bigger percentage drop in mpg over this range, so, although fuel efficient cars are cheaper to run in the first place, the increase in fuel costs for those cars driving at 80 mph is probably about the same in your actual pounds cash.

Conclusion:

The idea that a higher speed limit will boost the economy by providing quicker journey times is risible, and totally ignores the effect of the higher fuel costs contingent upon driving at these increased speeds.

Oh, and there's the fact that we'll actually be burning more of our dwindling supply of oil, as well as releasing commensurately larger amounts of CO2 into the atmosphere, but these considerations don't seem to have the same importance to our 'greenest government ever'.



This is all basic GCSE physics and maths, by the way, and part of the National Curriculum that Gove seems to think is too easy.

Article edited on 5.10.11 to reflect the fact that the data given by the Consumer Report website is in American gallons.