On the CART racing circuit (the series of races for Indy cars), it is not uncommon to see the cars reach average speeds of 230 miles per hour. What if I asked you to tell me how many meters an Indy car travels in one second, at an average speed of 230 miles per hour? Here's how I would do it. I know that there are 1609 meters in a mile, and I know that there are 60 seconds in a minute and 60 minutes in an hour. I would write the problem down like this:
where I have used the abbreviations for miles (mi), hours (hr), meters (m), and seconds (s) and the fact that miles per hour can be written ``mi/hr''. Now, the next step is to convert miles per hour into meters per hour. To do that, I have to multiply by the number of meters in a mile, like this:
Now, I have meters per hour. What's left? I have to convert meters per hour to meters per second, which means I have to convert seconds to hours. But I don't know the number of seconds in an hour, so first, I have to convert from hours to minutes and then from minutes to seconds, like this:
So now I know that there are 3600 seconds in an hour. So the final step is to convert meters per hour to meters per second. To do that, I just multiply by 1/3600, like this:
So, the average speed of an Indy car is 102.8 meters per second, or 102.8 m/s. That's about 112 yards in a second -- longer than a football field!
The key to dimensional analysis when you are trying to convert units is that you think of the units like fractions. Instead of numbers like 1/4, though, you have things like ``m/s'' or ``km/yr'' or ``g/ ''. And when you convert, you end up multiplying the units just as if you were multiplying fractions. That's really the key thing. Let's use our Indy car example, and do all the conversion in one step, like below:
Here, the ``mi'', ``hr'', and ``min'' all drop out, because there is one of each in the top of the units at one point and in the bottom of the units at another point; in other words, the ``numerator'' of one unit is the same as the ``denominator'' of another, and we cancel those parts of each unit. That's the key thing -- we cancel tops with bottoms just like when you multiply fractions!
Here's another quick example, just to show you that this works with non-metric units just as well as with metric ones. How many 8-ounce glasses of milk can you get from a standard 1-gallon milk bottle? Well, I would go straight to a table and get the conversion from gallons to ounces, but let's say you can't. Here's another way:
So you can get 16 8-ounce glasses of milk from a standard 1-gallon milk bottle.
So while it's not thrilling, by any means, dimensional analysis can help you convert from one unit to another, even when your conversion tables don't have both units related to each other.