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What's a good way to explain dimensional analysis to a student?

Here's a simple question which this method would be useful:

Let's say a truck is moving with a speed of 18 m/s to a new speed of 13 m/s over a distance of 48 meters. How long did it take for the truck to break to its new speed, and what was the acceleration of the truck?

Edit

@TimGoodman and @Laar, Hmm. I was under the impression that you could solve such questions by analyzing their dimensional properties. For example, based on V1 and V0 in the example above I know the person decelerated 5m/s over a distance of 48 meters. If I want the time it took (first question) I can simply look at the units and see if I can eliminate all units except s (seconds). 48m/5m/s allows the meters to cancel out leaving 9.6 seconds.

Now I have a new peice of information and I can now solve the second question "What was the acceleration of the truck?" Since I'm solving for acceleration I know my final units will be m/s^2. I know the velocity of the deceleration was 5m/s and it took him 9.6s. Dividing 5m/s/9.6s = .52 m/s^2. My final units match and so by using dimensional analysis I've solved the problem.

Maybe I have a misunderstanding of dimensional analysis. Any suggestions?

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Dimensional analysis is not very useful in finding exact answers to questions (like yours). It's a way to find a relation between variables which could be of by a constant factor. For example, $s = g t^2$ and $s = 0.5 g t^2$ are both dimensional correct, but only the second describes the relation between travel distance $s$ and travel time $t$ of a free falling object. –  Laar Aug 16 '12 at 20:49
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This is just an elementary mechanics problem. To teach dimensional analysis I would suggest perhaps something where you have to convert between systems of units, or maybe a multiple choice question where students have to identify which of a set of unfamiliar equations is correct (where all but one answer can be ruled out due to inconsistent dimensions between the left and right hand sides). –  Tim Goodman Aug 16 '12 at 20:53

3 Answers 3

I think you really need to spend a lecture at the beginning of class talking about derived and basic units, and really reinforce this material every time you introduce something new. There seems to be something that students find deeply difficult about understanding what a $ \frac{\rm kg\cdot m}{\rm s^{2}}$ is, even when they get harder things.

And it seems that any misconceptions they have with dimensional analysis really come from not understanding how derived units work. Test them on this, and hold them to their unit conventions through the whole course, because they are pretty naturally inclined to not care in our age of magical numbers coming out of a calculator console. You have to force them to care, and then they will laugh at not caring six months ago.

That's at least my experience.

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Some possible hints: -To find formulas modulo a constant: ex the pendulum period. -To rapidily find a numerical range for an answer: ex the first US atomic bomb power has been estimated with dimensional analysis by the British physicist G.I. Taylor who was able to give a very accurate estimate of the strength from dimensional analysis by using available film of the expansion of the mushroom shape (link) - To reduce the parametric space of an experiment as it is often done in CFD. The idea is to use dimensional analysis to reduce the number of variables to be measured.

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Since I could not figure out how to write a comment on your post, I will post it as an answer, even if it's just a link:

You can find some inspiring dimensional analysis examples here:

https://particlephd.wordpress.com/2008/12/08/dimensional-analysis-for-animals/ (see also the comments)

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