Ask Uncle Colin: Traffic Flow

Dear Uncle Colin,

I read that when cars are driving at 70mph on the motorway, they take up more space than when they travel more slowly (because you need to leave a longer safe gap between them). What’s the most efficient speed for motorway travel if you want to get as many cars past a given point as possible?

That’ll Really Annoy Fast Folk In Cars

Hi, TRAFFIC, and thanks for your message! What a lovely problem.

Stopping distances

Let’s suppose – seeing as we’re mathematicians, not transport planners – that cars ought to keep the minimum stopping distance as recommended in the Highway Code between them.

That’s actually a nice quadratic sequences problem in itself, especially if you use miles per hour and feet¹.

Speed (mph)

Stopping distance (ft)

20

40

30

75

40

120

50

175

60

240

70

315

If you fit a model² of the form $D_f = Av_{mph}^2 + Bv_{mph} + C$ to that, you find $D_f = \frac{1}{20}v_{mph}^2 + v_{mph}$. Splendid. But we really don’t like those units: $D_f$ is in feet and $v_{mph}$ is in miles per hour and just ugh³.

We measure distance in metres, and there are about $\frac{10}{3}$ feet in one of those, so $D_f = \frac{10}{3}D$.

We measure speed in metres per second, and (it turns out) 9 miles per hour is about the same as 4 m/s. So, $v_{mph} = \frac{4}{9} v$

Rewriting the formula: $\frac{10}{3}D = \frac{1}{20} \times \frac{16}{81}v^2 + \frac{4}{9}v$

Or, simplifying, $D = \frac{2v^2 + 90v}{675}$. Much nicer.

In terms of how much space a car takes up, we also need to add on its length – and let’s say the average car is 4m long.

So, a car travelling at $v$ m/s takes up $D_+ = \frac{2v^2 + 90v}{675} + 4$ metres - and I’ll pretend the car and its associated stopping distance simply a box that long.

How long does it take to pass?

Suppose I’m standing by the motorway⁴ and timing how long the box of the car takes to pass me. The front of the box needs to travel $D_+$ metres, it’s travelling at $v$ m/s, so it will take $T=\frac{D_+}{v}$ seconds to pass me.

That gives $T = \frac{2v + 90}{675} + \frac{4}{v}$ seconds.

Now for some calculus!

To maximise the number of cars going past a fixed point over any period of time, we need to minimise $T$ as a function of $v$. Let’s differentiate!

$\diff Tv = \frac{2}{675} - \frac{4}{v^2} = 0$ for an extremum⁵.

So, $v^2 = 1350$ and $v \approx 36.7$ m/s, which is 82.7 mph!

This is very naive analysis, of course, and just the first model that popped into my head – I don’t recommend driving at 82mph any more than I recommend standing by a motorway!

Hope that helps,

- Uncle Colin

* Many thanks to @realityminus3 for helpful comments and improving my work, as always.