Drive Slower, Save Money on Gas. Thanks, Physics!

it’s the season for summer road trips, but sky-high gas prices make that a costly proposition. Unfortunately, most of us are stuck with internal-combustion vehicles, so we have to pay up (in more ways than one—hello climate change). Want to save some bucks? It’s simple: Just slow down. Of course, that means it’ll take longer to get to your destination. A few years ago, I wrote a piece for WIRED on finding an optimal driving speed, assuming the extra drive time costs you money in lost wages (because you punch in later at work). Consider this the holiday edition, where time isn’t money. Instead, I’m going to focus on the physics and show why driving slower uses less gas. If you’ve ever been out on the interstate and realized you were low on fuel, and the sign says the next gas station is in 20 miles, you might have wondered if you should speed up to get there faster. That would be a losing move, and you’ll see why. Speed and Time Let’s start simple. Say you have to make a 30-mile drive from your home, and it’s a straight shot up the highway. The speed limit is 70 mph. How would the travel time change if you went faster or slower? First, let’s put this in physics terms. Imagine there’s a number line on your route, with the origin at your home. Your position (x) at any point is then given by your distance from home. Physicists call that “displacement”; you probably call it “distance traveled.” Your velocity (v), then, is the rate of change of position. It looks like this: This says velocity is the change in position (Δx) divided by the change in time (Δt). If you drove 100 miles in 2 hours, your average velocity was 50 mph. Now let’s apply this to our situation. We have a displacement of 30 miles with a velocity of 70 mph, so how long would this trip take? That’s easy, we just rearrange the equation above: Plugging in our values of Δx and v we get 30/70 = 0.428 hours, or 25.7 minutes. OK, what if you increase your speed to 75 mph? Using the same equation gives a trip time of 24 minutes. Yay, you saved 1.7 minutes! Just because I love graphs, here's a plot of minutes saved as a function of speed from 55 to 85 mph: Here you can see that reducing your speed to 65 mph on a 30-mile trip adds 2 minutes to your travel time. C’mon now, does that make a difference in your day? Speed and Fuel Consumption OK, I’m sure you know that you get better mileage on the highway than on city streets, so you might think fuel economy (miles per gallon) improves with speed. Not so. The advantage of highway driving is the fact that you’re traveling at a steady rate. In the city you’re always braking and accelerating, and it takes a lot of energy to get a car moving after a red light. In actuality, if we stick to highway driving, going faster always requires more gas to cover a given distance. Why is that? Well, let’s look at the forces acting on a moving car: There are three horizontal forces: (1) A static frictional force (Ff) between the back tires and the asphalt—that’s the “grip” that pushes the car forward; (2) a rolling friction force of the tires on the road (Froll); and (3) an air-resistance force (Fair), also called drag. The latter two push backward, resisting the car’s motion. Now, if the car is moving at a constant speed, we say it’s in equilibrium, which means the forward-pushing force must exactly equal the two backward-pushing forces. (Yep, Newton’s second law.) Let’s write this as an equation. Rolling friction doesn’t vary much with velocity, so we’ll treat it as a constant, C1. Air resistance, however, not only depends on velocity, it’s proportional to the square of velocity. We can write Fair = C2v2. Double your speed and the air drag quadruples. So for any constant speed: It's the frictional force on the left that propels the car, but of course it takes energy to rotate the wheels, and that comes from burning gasoline. At a higher velocity (v), the right side of the equation gets (much) larger, which means we need a greater force on the left side to maintain that velocity. And that means the engine must use gas at a faster rate, resulting in lower fuel efficiency. In a nutshell, your mileage drops at higher speeds because air resistance grows much faster than velocity. So if you’re in that situation where you’ve got 20 nervous miles till the next gas station, you want to slow down, even though it takes longer, because you’ll use less gas to cover the same distance. The slower you go, the farther you’ll go—at least down to about 50 mph, where air drag is no longer much of a factor. How much does your mileage vary with speed? Every car is different, but the US Department of Energy estimates that for every 5 mph increase in speed over 50 mph, fuel efficiency declines by 7 percent. In equation form: Here e0 and v0 are the efficiency and velocity at some benchmark, and this tells you the efficiency for any other velocity. The 0.93 is that 7 percent decrease. Let's do a quick example. Maybe your car gets 30 mpg (e0) at a speed of 70 mph (v0). Then at 75 mph it would get 27.9 mpg, and at 65 mph it would get 32.3 mpg. See how that works? Time and Money Now let's pull this all together. If you drive faster, you save time. But you use more fuel, so it costs more. What are the terms of the trade-off? Let’s go back to our 30-mile trip, and I’ll assume gas costs $4 a gallon. If you drive at 70 mph with an efficiency of 30 mpg, you use 1 gallon of gas, which costs $4.00. Bump the speed up to 75 and you use 1.08 gallons, for an extra cost of 32 cents. No big whoop, right? But remember, you’re only saving 1.7 minutes, or 0.028 hours. If we divide the added cost by the time savings (0.32/0.028), that’s a penalty of about $11.15 per hour saved. Finally, let’s say you have a 500-mile round trip drive planned for the July 4th weekend. If you set cruise control to 70 mph, that’ll be 7.14 hours of driving. Cut that to 60 mph (10 mph less) and you’re looking at an extra 30 minutes behind the wheel each way. But your fuel efficiency will increase from 30 to 35 mpg. That means you’d use 2.5 gallons less to drive the same distance, for a cost savings of $10. Or another way to think about it: Driving in the slow lane is equivalent to paying $3.40 a gallon for gas instead of $4.00 a gallon. You’d probably go out of your way for that kind of price difference, right? Oh, and you’d reduce your CO2 emissions by more than 50 pounds for the trip. So my advice? Ease up on the pedal, ease up on the planet, and enjoy the scenery.