\(\def \u#1{\,\mathrm{#1}}\) \(\def \abs#1{\left|#1\right|}\) \(\def \ast{*}\) \(\def \deg{^{\circ}}\) \(\def \tau{\uptau}\) \(\def \ten#1{\times 10^{#1}}\) \(\def \redcancel#1{{\color{red}\cancel{#1}}}\) \(\def \BLUE#1{{\color{blue} #1}}\) \(\def \RED#1{{\color{red} #1}}\) \(\def \PURPLE#1{{\color{purple} #1}}\) \(\def \th#1,#2{#1,\!#2}\) \(\def \lshift#1#2{\underset{\Leftarrow\atop{#2}}#1}}\) \(\def \rshift#1#2{\underset{\Rightarrow\atop{#2}}#1}}\) \(\def \dotspot{{\color{lightgray}{\circ}}}\) \(\def \ccw{\circlearrowleft}\) \(\def \cw{\circlearrowright}\)
Appendix A: Miscellany
7.

Unit Conversion

If we are given information in units other than SI units (e.g. hours, miles, pounds, etc), then we will usually need to convert them into SI units before we can proceed. To do this, first we need a conversion rule between the two units, like this: $$1\u{hr}=3600\u{s}$$ Since this is an equation, we can divide one side by the other, and get: $$\frac{1\u{hr}}{3600\u{s}} = 1 \qquad \hbox{ or } \qquad \frac{3600\u{s}}{1\u{hr}} = 1$$ Since these two fractions are both equal to one, and multiplying by one doesn't change that number, we can multiply any quantity by one of these fractions without changing the value of that quantity. For example, $$\begin{align} 1000\u{s} &=1000\u{s} \times 1\\ &= 1000\u{\redcancel{s}} \times {1\u{hr}\over3600\u{\redcancel{s}}}\\ &=0.28\u{hr}\\ \end{align}$$ And so 1000 seconds is the same as 0.28 hours.

It is useful to think about units the same as any other variable: if the unit appears in the numerator and the denominator, then we can cancel them both.

Example

Suppose we want to convert $1\u{m/s}$ into miles per hour. To do this, we need to replace the meters with miles, and the seconds with hours. The seconds are in the denominator, and so we need a "seconds" in the numerator as well to cancel it out. We can get this by putting the $3600\u{s}$ in the numerator, and $1\u{hr}$ in the denominator, like so: $$1\u{\color{blue}{m}\over \redcancel{s}}\times {3600\u{\redcancel{s}}\over 1\u{\color{blue}hr}} = 3600\u{\color{blue}{m\over hr}}$$ The conversion factor for miles and meters is $1\u{mi}=1609\u{m}$. Since we want to cancel the meters in the numerator here, we need another "meters" in the denominator: $$3600\u{\redcancel{m}\over \color{blue}hr}\times \frac{1\u{\color{blue}mi}}{1609\u{\redcancel{m}}}=2.24\u{\color{blue}{mi\over hr}}$$

Now that we have this useful relationship $1\u{m/s}=2.24\u{mi/hr}$, we can use it to do other conversions. For instance, a car on a highway moving at $65\u{mi\over hr}$ has a speed of $$65\u{\redcancel{mi/hr}} \times {1\u{\color{blue}{m/s}}\over 2.24\u{\redcancel{mi/hr}}} = 29\u{\color{blue}{m/s}}$$