Driving Faster Takes Longer

I often drive between Boston and New Haven. While on the road, I find myself pondering a simple question: If my only goal is to arrive as fast as possible, how fast should I drive?

Ignoring things like ethics (or fuel efficiency), the solution would seem to be simple. Drive as fast as possible. But there’s a catch: If I crash and die, then the next 50 years (which I had planned on spending alive) are spent dead. It’s only fair to count this as a penalty towards the length of the trip.

So, the expected trip length isn’t just ${\frac{\text{distance}}{\text{speed}}}$ . It’s actually

${\frac{\text{distance}}{\text{speed}} + \Pr[\text{Death}] \cdot 50 \text{ years}.}$

How much time do I spend dead on my 140 mile trip? Roughly speaking, one fatality occurs for every 100,000,000 miles driven. This ignores a lot of things (e.g., the fact that different speeds lead to different fatality rates, which we’ll come back to later), but it’s good enough to give us a sense of what’s going on. My expected time spent dead on the trip is about

${\frac{140 \text{ miles}}{100,000,000 \text{ miles per death}} \cdot 50 \text{ years per death} = 37 \text{ minutes}}.$

That’s quite a lot considering that, when I don’t die, my entire trip length comes out to about 2 hours and 30 minutes!

But what happens to my probability of death if I drive faster or slower? A somewhat dated analysis from the University of Alabama gives a bit of insight. To a first approximation, if you crash at a given speed, your probability of dying doubles each time you add 10 mph to that speed. This is to say that ${\Pr[\text{die }\mid\text{ crash}]}$ doubles each time you increase your speed by 10 mph. Of course, what we actually care about is $\Pr[\text{crash and die}]$ , which by Bayes’ rule is $\Pr[\text{die }\mid\text{ crash}] \cdot \Pr[\text{crash}]$ . It seems likely that both $\Pr[\text{die }\mid\text{ crash}]$ and $\Pr[\text{crash}]$ increase as you drive faster, but since I only have data on how the first quantity changes, I’ll ignore changes to the second.

Now let’s calculate the expected length of the trip (including dead time) at different speeds. As a baseline, let’s suppose that our previous analysis (i.e., our estimate of 37 minutes dead) holds when we travel at the speed limit of 65 mph. Now suppose we travel at 65 + x miles per hour for some x. Our actual time on the road is ${\frac{140 \text{ miles}}{65 + x \text{ miles per hour}}}$ , but our expected time spent dead is ${37 \cdot 2^{x / 10}}$ minutes. So our expected trip length, including time spent dead, is:

${(60\text{ minutes / hour}) \cdot \frac{140 \text{ miles}}{65 + x \text{ miles per hour}} + 37 \cdot 2^{x / 10}} \text{ minutes}$

${= \frac{8400}{65 + x} + 37 \cdot 2^{x/10}}$ minutes.

Graphing the length of the trip (in minutes) as a function of x gives:

So the fastest speed to drive is… about 62 miles per hour.¹

Of course, this model bakes in your remaining life expectancy at 50 years. An odd feature of the model is that, the older you get, the quicker your trips become, at least in expectation, and the more you should speed. Here’s the same graph if you model the cost of dying as 25 years:

Now your optimal speed is closer to 70 mph, still on the slower end of what people do on I-95. If you want to rationalize the 85 mph speeds that the fastest drivers travel at, you would need to reduce your estimated cost of dying to a morbidly short 5 years.

Now, if you really do want to optimize how fast you get places, it’s worth noting that this model has a cheat code: different cars have very different fatality rates. (Although, unfortunately, they are typically reported per registration year rather than per mile). The trick to getting somewhere fast isn’t to speed, it’s to get a Volvo XC90.

Okay, to be fair, there are a bunch of factors we ignored. Maybe most deaths are caused by really irresponsible drivers that are speeding by a lot, so little-old-us traveling at 65 mph is actually super safe. Maybe our guess that 65 mph was the speed at which our original 37-minutes-dead-analysis should hold was totally wrong. Maybe highway miles are much more dangerous than non-highway miles, and we shouldn’t be on the highway at all. This is really not my area of expertise, so take everything I say with a giant hand full of salt. ↩︎