I already looked at one consequence of increasing the speed limit from 70 mph to 85 mph. There is obviously another question to answer: what will this do to your car's fuel consumption?
Cars are complicated things. You put gasoline in, and the engine essentially converts this to mechanical energy needed to keep the car moving (because of air resistance and friction). How much energy does it take to keep a car going? This isn't quite so easy to answer. Internal combustion engines are not all too efficient, there is friction in the transmission and in the tires and everything.
The question is about increasing your speed. If you do this, what will change? One thing that will obviously change in the air resistance on your car. So, here is what I am going to do. I am going to estimate the extra amount of fuel you would need to use for a trip on the highway going 70 mph vs. 85 mph. for the same car. Maybe your engine will be less (or more) efficient at a higher rpm, but I am going to ignore that.
Here are my starting assumptions:
- The air resistance force is proportional to the velocity squared.
- The frictional force doesn't depend on speed.
For the air resistance, I will use the following model:
In that model:
- A is the cross sectional area of the car.
- ρ is the density of the air.
- C is the drag coefficient that depends on the shape of the car.
For this calculation, let me assume two identical cars drive a distance s. The first at a speed v1 and the other at a speed v2. How much energy would be "lost" due to the air drag?
To look at the energy "lost" (where lost is in quotes because you can't really lose energy), I can use the work energy principle. If I take just the car as my system then:
Here there is work done by both the frictional force (whatever that is) and the air pushing on the car. I am neglecting the energy needed to increase the speed of the car (just assume it is a long enough trip the energy needed to speed up is small compared to the rest). If the car is traveling at a constant speed, then:
This is the change in "fuel energy" for the car going at speed v1. How much more energy would car 2 (at speed v2) use?
Faster cars use more energy. I guess that is obvious. How about something useful? What would it do to your fuel efficiency (gas mileage)? Let me go back to car 1. To travel the distance s, it used an energy Δ E1. What are the units for energy? In the above expression, the energy is in units of Joules. However, the more energy you use, the more gasoline you need. The amount of energy is proportional to the amount of gas.
If I call m the gas mileage for a certain speed then:
Where the constant K contains both unit conversions and an efficiency of the engine. I know the change in energy for the car from above. Let me introduce a new constant B to represent all the stuff in front of the velocity squared term in the air resistance.
Now let me make an assumption. What if the frictional force is much less than the air resistance force? Maybe this isn't such a bad approximation - especially at highway speeds. This means that I can write:
What next? Well, let's take an average car. Suppose this is a pretty nice car and gets 30 mpg at 70 mph. Using these values, I get:
Now what about the second car? If it has a speed of 85 mph, what would its fuel efficiency be?
Oh wait, if the speed limit were 85 mph people would really drive 90 mph. This would make a fuel efficiency of 30 mpg at 70 mph go to 18 mpg. Notice that in the manner I calculated the efficiency, it doesn't matter what units you use for the speed as long as both speed have the same units.
One more calculation. What if you have a huge SUV that gets 18 mpg at 70 mph? Boom. You would only get 12 mpg at 85 mpg. That pretty much sucks. But What if you drive your SUV at 65 mph? 20 mpg. Not too bad. This is the reason that I usually drive around 65 mph on the interstate. Yes, I know you are going to claim this is super dangerous because someone will crash into me. Well, maybe driving too fast is super dangerous and I am just being reasonable.
Ok, I can't leave this without a graph. Here is the fuel efficiency for two different cars as a function of speed.
Final note. Yes. I know this isn't the best model for fuel efficiency. Like I said there are lots of factors involved. However, it gives a fairly accurate representation of what happens as you drive faster.
