Dangerous driver?
Was it possible that this dangerous driving penalty was issued in
error?
Age
16 to 18
Challenge level
Problem
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?
Getting Started
What equations of mechanics do you know which relate to
acceleration?
Don't forget units.
Don't forget to consider carefully the validity of any modelling assumptions, such as that of constant acceleration.
Don't forget units.
Don't forget to consider carefully the validity of any modelling assumptions, such as that of constant acceleration.
Student Solutions
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:
$\frac{dv}{dt} = \frac{dx}{dt} \frac{dv}{dx} = v \frac{dv}{dx}$
Teachers' Resources
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
This problem draws together the following points
1) Modelling assumptions in mechanics
2) The implications of motion under constant power
3) The implications of motion under constant
acceleration
4) Conversion of units
As such, it would be a good activity to draw strands of work
together at the end of a mechanics module.
Discuss the problem together. How would students answer this
question if called in as an expert witness by a genuine court? What
factors would need to come into the analysis? They should be
encouraged to question and challenge assumptions rather than
blindly to perform a constant acceleration analysis. Don't forget
to stress that this car is a REAL car. Students could refer to
their own experiences of being in a car and how that performs when
accelerating flat-out.
It is worth pointing out that this problem does not easily
yield a clear conclusion (in the opinion of its author), which
makes it interesting.
It is very important that students realise that good
mathematical modelling is all about making statements such as IF (the following is true or the
following approximations are made) THEN (these results follow with
certainty). Good modelling is also about realising where
information which might be relevant to the problem is lacking (such
as the road layout, in this case).
Starting with simple assumptions (such as uniform, constant
acceleration) is fine, so long as they are clearly and explicitly
stated. The solution can then be refined by challenging these
assumptions in a clear and structured way.
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!
Key questions
What mechanics will definitely be of relevance to this
problem?
What mechanics might be of relevance to this problem?
What are the weaknesses of the assumption of constant
acceleration?
Would you be able to defend your
arguments against well-constructed, logical criticism?