Motion

Example: Unit conversions

“Units” are a pain and students can get themselves into a mess trying to change from one brand to another. Whether it’s SI units (meters, kg, seconds) into Imperial units (feet, pounds, seconds) or something more complicated, it can be confusing.

In QS&BB I don’t make a big deal out of it, plus one can use an infinity of tools to convert almost anything into anything. I usually just use Mr Google in the search bar. Or: Use this tool and type in “100 mph in km per hour” and see what you get.

Calculate it:

Feel free to use this when units are an issue or you want to calculate anything by writing out the formula.

The Question: What is 60 mph in meters per second (m/s)?

The Answer:

Conversion formulae look like equations, but they’re a little sneaky. Suppose I want to convert $10 to €…Today, $1 will get you 0.87€. How do we formalize this relationship?

It’s not correct to say that \[ 1 = 0.87\] is it. But there is a sort of equivalency in saying that \[ \$1 = 0.87\text{€}\] but it’s not a real “equals” in a mathematical way. So I’ll invent a new connector that I’ll call “equivalence,” \(\iff\) so now we can make an almost mathematical statement: \[ \$1 \iff 0.87\text{€}.\] Now let me define a new…“one”: \[\begin{equation} 1 \iff \frac{\$1}{0.87\text{€}} \iff \frac{0.87\text{€}}{\$1} \end{equation}\] and for the purposes of unit conversions, we can add factors of this “1” judiciously to make the conversion. So to convert 50€ to $, I’d set up this relationship, choosing the right hand representation of that “1” from Equivalence 1.

\[ \begin{align} x (\$) &\iff 50\text{€} \times 1 \nonumber \\ x (\$) &\iff 50\text{€} \times \frac{\$1}{0.87\text{€}} = \$\frac{50}{0.87} \nonumber \\ x (\$) & = \$57.47 \nonumber. \end{align} \] Notice that the € symbols “cancel” leaving the $ symbol. Once I start doing arithmetic, I converted my “equivalence” notation to “equal.”

That was simple, but what about our problem? Just how you would go about this depends on what you know, which might be a little or might be a lot. For example, since I was in high school I’ve remembered that 1 inch is 2.54 centimeters, that 1 mile is 5280 feet, and of course 1 hour is 60 minutes and 1 minute is 60 seconds and a meter is 100 cm. Without looking anything up, I can do the conversion with those things that I remember (you might remember smarter conversions). Let’s go: \[ \begin{align} x (m/s) &\iff 60 \; \frac{mile}{h} \nonumber \\ x (m/s) &\iff 60 \; \frac{mile}{h} \times 1 \times 1 \times 1 \times 1 \times 1 \times 1 \nonumber \\ x (m/s) &\iff 60 \; \frac{mile}{h} \times \frac{1\;h}{60 \;min}\times \frac{1\;min}{60 \;s}\times \frac{5280 \;ft}{1 \;mile}\times \frac{12 \;in}{1 \;mile}\times \frac{2.54 \;cm}{1 \;in}\times \frac{1 \;m}{100 \;cm} \nonumber \\ x (m/s) &= 60 \;\frac{mile}{h} \times 0.447 \; \frac{m \; h}{mile\; s} \nonumber \\ x (m/s) &= 26.8 \;m/s \nonumber \end{align} \] Sometimes unit conversions are linear… just a number or series of numbers multiplied. But sometimes they’re more complicated like converting Celsius to Fahrenheit where there is an offset for the freezing point of water of 32 F is 0 C, and boiling of 210 F is 100 C…and then there’s a linear relation between those two states of water. We’ll not go there. Just telling the whole story. 📖