Two identical arrows, one with speed

*v*and one with speed 2*v*, 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 2*v*, 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.
## No comments:

## Post a Comment