Refer to the competition introduction and overview for more information on where the above buttons will take you.

- As long as there's no obstruction (another car or a red light) in front
of a car, it will travel at the speed limit of
*c*kilometres per hour. - If a car gets within a distance of
*L*metres from an obstruction (where, to avoid having to think about the car length as a separate factor, we measure distances from the back of the car), it will reduce its speed proportionally to that distance. - If a car gets within a distance of
*l*metres from an obstruction, it will stop altogether.

Is this model realistic? Not really; it does not take into account drivers' reaction times, limits to how fast a car can brake or accelerate, and other factors. But, although it's not completely realistic, the inadequacies of the model don't make a whole lot of difference to the basic nature of the answers you get. Working with a more realistic model requires much, much harder mathematical calculations, and the final answers aren't all that much different! So, we'll stick with this simple model.

Let's be a little more precise about how this model works.
Let *x* denote the distance in metres between two cars (measured
from the back of one to the back of another), or between the back of
one car and a red light up ahead.

This is indicated below:

The speed *v* of the car is:

In the second case,

c,if x>=Lbetween 0 and c,if l<x<L0, if x<=l.

*v*=0 when*x*=*l**v*=*c*when*x*=*L*;

In summary, then, we're assuming that the speed of a car is given by
the following function of the distance *x* to the closest obstacle
in front of it:

where

v=c,if x>=Lv=m x+b,if l<x<Lv= 0,if x<=l.

This page last updated: April 12, 1997

Original Web Site Creator / Mathematical Content Developer: Philip Spencer

Current Network Coordinator and Contact Person: Joel Chan - mathnet@math.toronto.edu