Q2.13 Choosing a speed limit (T)
Assume that drivers on a (single lane) road behave according to the following rules:
- They go as fast as they legally can (ie at the speed limit).
- 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 km⋅hr-1. The speed limit is raised to 75 km⋅hr-1; how long does it now take them to leave the city?
Hint
Reveal
Solution
Reveal

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)):
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
![\[ L =Nd = \frac{Nu^2}{2a} \]](mastermathpng-0.png)
![\[ t= L/u = \frac{Nu}{2a} \]](mastermathpng-1.png)