Q2.13 Choosing a speed limit (T)

Assume that drivers on a (single lane) road behave according to the following rules:

  1. They go as fast as they legally can (ie at the speed limit).
  2. They maintain a separation d from the car in front equal to the stopping distance (the distance they need to stop, given a fixed breaking acceleration a).

It takes 2 hrs for 10000 cars to leave a city, travelling according to these rules, when the speed limit is 50 kmhr-1. The speed limit is raised to 75 kmhr-1; how long does it now take them to leave the city?

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Hint

It may be helpful to think of the cars in a long line... and ask yourself how long it is.
 

Solution

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Solution

First let us think about the stopping distance d. This is the distance required to come to rest from an initial speed u, say (remember to use symbols to begin with, and leave the numbers till last: Guideline 0.6) given an acceleration of magnitude a. The distance follows from the constant acceleration equation (Key Point 1.4 (c)):

0=u2-2ad
being careful with signs (the breaking acceleration acts in the opposite direction to the initial speed). Thus d=u2/2a.

Now imagine our N=10000 cars strung out in a long line, each one separated from its neighbours by that distance d. The total length of this line is

\[ L =Nd = \frac{Nu^2}{2a} \]
ignoring the length of the cars themselves (and, if you are picky, the difference between N and N-1.) At the moment the first car leaves the city boundary the last car has a distance L to travel, and does so at speed u. So the time taken for the whole lot to clear the city is
\[ t= L/u = \frac{Nu}{2a} \]
Increasing the speed limit from 50 to 75 kmhr-1 will thus increase the time required from 2 to 3 hours. The reason for this unexpected result (that the time required increases in proportion to the speed limit) is that the distance between the cars increases as the square of the speed. If you stand at the city boundary, when the speed limit is increased you will see each car moving past you faster but the number of cars passing you per unit time will go down. Although the ‘model’ is implausible in some respects (perhaps you can think of one...apart from the assumption that the speed limit is observed!) it does capture a real effect.