# Dangerous driver?

A particular speed camera is located a short distance down the road from a particular set of traffic lights, at which there is always a queue at rush hour. At around this time a driver was caught out by the camera and challenged the ticket in court, claiming that he started moving from rest at the lights and that it would be impossible to reach the speed shown on the ticket over the short
distance between the lights and the speed camera.

Prosecutor: "Although I accept that you left the traffic lights at rest, you were snapped by the camera doing $133\mathrm{kmh}^{-1}$, which is found at a distance of $338$ metres from the traffic lights. I claim that this is ample distance to reach the speed shown on the ticket."

Defendant: "But the specifications in the manual of my cheap car show that the maximum acceleration is $0$ to $96\mathrm{kmh}^{-1}$ in $10.5\mathrm{s}$. I could never have accelerated to such a high speed in such a short distance!"

Analyse this case carefully. Could the penalty reasonably be rejected on mathematical grounds?

Don't forget units.

Don't forget to consider carefully the validity of any modelling assumptions, such as that of constant acceleration.

We had two very interesting solutions, which were beautifully presented Word documents -- click below to read them

Michael from Ecclesbourne

Michael believes that the penalty could not reasonably be rejected as the situation stands -- more information will be required about the specific car in question because, quite rightly, acceleration will not be constant in a real situation.

Henry, from Elizabeth College

Henry carefully converted all of the units and performed a calculation based on constant acceleration and various equations of mechanics. Based on these assumptions, he concludes that the car could be going as fast as 41m/s. This is greater that the speed that the camera recorded and that the case should not be dismissed on mathematical grounds.

Steve notes

In reality I wondered if constant power produced by the car might be a more solid starting point for a calculation. In principle this could be inferred from the solid data point of acceleration from 0 to 96 km/h in 10.5 seconds. A big unknown would be the retarding effect of wind resistance. Another big unknown would be the road configuration. Is it curved, straight, flat or up/downhill?

Patrick sent us his solution, which considers how air resistance might affect the problem.

He used the formula for air resistance, $F_{air} =\frac 12 A C_d D v^2$, where $A$ is the cross-sectional area of the car, $C_d$ is a constant saying how resistive air is and $D$ is the density of the air.

He also used the chain rule to write acceleration as:

$\frac{dv}{dt} = \frac{dx}{dt} \frac{dv}{dx} = v \frac{dv}{dx}$

### Why do this problem?

This problem gives an interesting situation in which simple equations of mechanics can be used in a non-trivial way. The problem arose from a real-life query which would work well as either a homework task or a discussion point at the start or end of a mechanics module.### Possible Approach

It might be fun to take the solutions into court. Have students create their best solutions in groups. When they are created, switch solutions and spend 10 minutes thinking through the strengths and weaknesses of these. Have the creators of the solutions cross-examined by a prosecutor whose job it is to try to pick holes in the argument. As teachers, you can stand as judge. This will certainly encourage clear mathematical communication!