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Age 16 to 18
Challenge Level

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?

Possible extension

Extension is naturally built into this question. Those who exhibit very clear thinking in mechanics might want to 'cross examine' the validity of the solutions of others.

Possible support

Start with the simple assumption of constant acceleration and discuss the weaknesses of such an assumption.