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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.