In a recent diatribe against slower driving speeds, the Daily Mail recently announced The Slow Death of driving”[i]. And the prime minister Rishi Sunak has just announced[ii] plans to restrict the ability of councils to impose “anti-motorist policies” such as 20mph speed limits.

So is driving more slowly really "anti-motorist"?

Well that, as they say, is “a complicated question”.

Let us consider a car driving along a single traffic lane (perhaps
a lane on a motorway or one side of a single carriageway road) between two points
A and B that are 1km apart:

Obviously, the faster the car drives, the shorter the
journey time will be:

Graph 1:
Journey times

Though it is interesting to note that this is not a simple linear
relationship and faster speeds provide diminishing returns. In round numbers
(my underlying calculations are all in metres per second) at 30mph, the journey
will take 70 seconds; at 20mph, this will increase to 110 seconds. Going from
60mph to 70 mph only cuts the journey time by 5 seconds.

Now let us consider a more realistic scenario where the lane
in question is saturated with cars:

Each car will have to drive so that there is a safe gap
between it and the car in front. In the calculations below I have used the
stopping distance to define this gap – allowing for the worst-case scenario
where the car in front comes to a sudden and complete and unexpected halt
(perhaps crashing into the back of a stationary lorry in fog) – but the same
principles would still hold if I used figures that were more reflective of the
less-safe gaps that average drivers actually maintain.

Graph 2:
Stopping distances

The length of the safe gap increases in a non-linear and
quite dramatic fashion. It doubles from about 12m at 20mph to about 24m at
30mph.

By calculating the time each car takes, at the relevant
speed, to drive the distances represented by these gaps (plus its own car
length), we can calculate the throughput of cars (e.g. the number of cars
passing point B in one minute) along the lane of traffic for different traffic
speeds when the lane is fully saturated with traffic:

Graph 3: Throughput

At 20mph there is a throughput of about 34 cars per minute.
At 30 mph, this drops to 30 cars per minute and, apparently paradoxically
(though not really paradoxically if you understand what is going on), the
faster the speed at which the cars go between A and B thereafter, the fewer cars
will get from A to B in a given time.

Of course, for the individual driver in the situation
presented thus far, Graph 1 above still holds. The faster he or she is allowed
to drive, the quicker his or her journey from A to B will be. Why should that
driver care about throughput?

The problem is that there are often more cars trying to make
the journey (from A to B in our example) than the throughput for a given speed
can cope with. If the cars are going at 30mph and there are hundreds of cars
(rather than 30 cars) trying to make the journey between A and B during the
same minute, a queue will form and most of the cars will come to a complete
stop. Waiting in stationary traffic can dramatically increase journey times - especially if you are at the back of the queuing traffic.

At times when we became stuck in an endless queue of
stationary traffic, my late father would ask rhetorically (and, obviously, jokingly)
“Why doesn’t whoever is at the front simply drive off?”. Let us imagine just
such a situation:

The car at the front of the queue drives off and accelerates
to whatever the speed limit is on this road, and the next in line does the same
as soon as it is safe to do so.

The car at the head of the queue obviously has zero waiting
time. The tenth car in the queue has to wait for 16 seconds before setting off if
the moving cars are going at 20mph; 18 seconds if the moving cars are going at 30mph.
At a speed limit of 70mph, the waiting time goes up to 28 seconds:

Graph 4: Waiting times for 10th queuing
car for different speed limits

The waiting times for the 100^{th }car in the queue are,
respectively, 177 seconds, 197 seconds, and 301 seconds at 20mph, 30mph, and 70mph:

Graph 5: Waiting times for 100th queuing car for different speed limits

If the queue gets up to a 1000 cars (that is about 5km long
and I have certainly been in queues as long as that) waiting times for the last
car in the queue goes up to 30 minutes, 33 minutes, and 51 minutes at 20mph,
30mph, and 70mph,

## Conclusions

The main argument in favour of 20mph speed limits in
built-up areas is road safety. The calculations presented above are, therefore,
only of partial relevance to this particular debate. Moreover, most of those
calculations are based on what happens when the roads are fully saturated with
traffic. This is encountered often enough, but will vary in different locations
and at different times of day.

Nonetheless, the point has been established that, given the
simplifying assumptions detailed below in *Methodology*, lower speed
limits will, in certain circumstances, keep traffic moving and get more cars along
the road in a given time than higher driving speeds would. These kinds of
considerations are the basis of variable speed reductions on smart motorways,
which keep traffic flowing in situations where it would grind to a complete
halt if everyone tried to drive at 70mph.

Far from being a way to
persecute motorists, the enforcement of slower driving often *improves *life for car
drivers and their passengers, while killing fewer pedestrians, cyclists, and
animals in the process.

**Methodology**

Graph 1 simply
uses the equation for time, distance, and velocity[iii] [t=s/v] to calculate the Y-axis values.

Graph 2 adds thinking
distance to braking distance to provide stopping distance. Thinking distance is
assumed to be 0.7 seconds in line with the Highway Code estimate[iv].
Braking distance is calculated using the equation for distance, final velocity
[which is 0 after braking], initial velocity, and acceleration [s=(v^{2}-u^{2})/2a]. Acceleration is
calculated using the formula for acceleration, force and mass [a=F/m] and
assumes a braking force of 7 Newtons per kg of car – an assumption that
provides braking distance broadly in line with those given in the Highway Code[v].

Graph 3
divides 60 seconds by the time to travel the stopping distance at the relevant
speed plus the time to travel a car length at the relevant speed. The time to
travel the stopping distance and the time to travel a car length are calculated
using t=s/v and an average car-length is assumed to be 4m in line with the
Highway Code assumption[v].

Graphs 4 and
5 divide the queue position minus 1 by the throughput per second.

All the calculations
are based on average cars, and no attempt has been made to factor in larger vehicles
like busses and lorries. There has also been no attempt to take account of
overtaking or changing lanes – as might happen on a motorway or dual
carriageway.

[iii] Velocity is a vector so, strictly speaking, we are talking about the magnitude
of the velocity or “speed” here.

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