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01 December 2014

Teaching semi-quantitative reasoning: first, ask students to derive a useful equation.

Two identical arrows, one with speed v and one with speed 2v, are fired into a bale of hay.  Assume that the hay exerts the same friction force on each arrow.  Use the work-energy theorem to determine how many times farther into the hay the faster arrow penetrates.

Typical students know how to apply the work-energy theorem if the problem is stated in numbers.  In fact, if you told these students to answer this question by calculating the distances penetrated by a 10 m/s arrow and then by a 20 m/s arrow, they'd get the answer right.

But if those students try to solve in variables only, without making a couple of calculations with made-up numbers, they get lost.  They don't know where to put the factor of 2... they solve for v rather than for the distance penetrated... they get lost doing random algebra.  (Don't believe me?  Try assigning this problem.)  Nevertheless, I need to teach even my not-so-mathematically-fluent students how to answer this type of question with algebra rather than numbers.  

The trick, I think, is to rephrase the question.  Consider this version:

Two identical arrows, one with twice the speed of the other, are fired into a bale of hay.  Assume that the hay exerts the same friction force on each arrow.

(a)       Use the work-energy theorem to determine an expression for the distance into the hay that an arrow of speed v will penetrate.

(b)       How many times farther into the hay will the faster arrow penetrate?  Justify your answer.

When I explicitly require an algebraic solution for the relevant variable -- the distance penetrated -- in terms of the variable v rather than 2v, the question becomes straightforward.  Students see that the speed v appears in the numerator, and squared; so, doubling v quadruples the penetration distance.

The difficult part of the problem was figuring out to solve for distance in terms of v.  So I've told them to do that first.  As the year goes on, I will gradually take off the training wheels, and ask the question straight-up, like at the top of this post.  However, I want to start establishing good habits of answering problems involving semi-quantitative reasoning, so I'll guide students to deriving a useful equation first.  

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