Dimensional analysis flavor 2: Checking your answer

Let's talk about how you can use dimensional analysis to check your answers when you are finished with a problem. This is actually a more important use of dimensional analysis that the one described in Section 3.2, because it can help you keep from messing up -- if the units you get in answer to a question don't make sense, you've probably messed up somewhere.

Here's a simple equation for you:

 

In this equation, x is distance, v is speed, and t is time. What this equation says, of course, is that the distance you go is equal to the speed at which you are moving multiplied by the time for which you have been traveling that speed. This shouldn't sound too surprising, but how would I check to make sure that sounds right -- I mean, what if the equation should really be

 

How might you check that? You're probably familiar with speed measured in meters per second, which can also be written as ``m/s'' (see Section 3.2 if that's not familiar for some reason). As I said in Section 3.2, you can treat units as a fraction in dimensional analysis. So, let's use dimensional analysis on Equation 7 to prove that that equation is wrong. First, let's write it with the units one would expect for distance, speed, and time

and then let's use dimensional analysis to see if the units make sense.

So it looks like the units make no sense. Meters is not the same as meters-seconds, and who has ever heard of meters-seconds being used to measure distance? Nobody, because meters-seconds is a meaningless unit. So clearly, Equation 7 is wrong. Does Equation 6 work any better? Let's see. Again, we write out Equation 6 in terms of the units and use dimensional analysis to check them.

Yes, it works. Meters is the unit of distance, and since we get meters on both sides of the equation, Equation 6 works. So distance really is speed multiplied by time; but then, you knew that.

This is a simple example, but dimensional analysis works like this for any problem, provided you know what units you should get for a particular problem. If you get those units, you're not sure the answer is right, of course -- you may still have made a mistake in the math. But if you don't get those units, the chances are very high that either (a) you were wrong with what you should expect or (b) you got the answer really wrong. If you check to make sure that the expected units were what you thought they should be, then you are left with (b), and you know via dimensional analysis that you got the answer wrong.

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