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

Units

If you called the doctor and asked how long you had to wait to get your results back and they said "5", you would probably be very confused. Five minutes? Five hours? Five days? Five weeks? Most quantities in physics (and life) come with units, which are as important as the value itself. Units represent things like length, speed, time, or mass, and they give an indication of how large the quantity is (a minute versus a year, for instance).

When working with numbers in physics, it is important to state your answer in the proper units, to avoid confusion. It is also a useful way to check our work, to make sure we are combining quantitites in the right way.

Wideexample

Suppose I have to travel 40 kilometers at a speed of $60\u{mi/hr}$, and I want to know how long it will take. I vaguely remember that I need to either multiply or divide these, but I don't remember what the rule is. (True story.) So I try each of the three possible combinations, keeping track of units.

$$ \def\ccl#1{\color{red}\cancel{#1}} \begin{align} 40\u{km} \times 60\u{km\over hr} &= 2400\u{km^2\over hr}\\ 40\u{km} \div 60\u{km\over hr} &= 40\u{\ccl{km}} \times \frac1{60}\u{hr\over \ccl{km}} = 0.67\u{hr} \color{green}{\checkmark}\\ 60\u{km/hr} \div 40\u{km} &= 60\u{\ccl{km}\over hr} \times \frac1{40\u{\ccl{km}}} = 1.5\u{/hr}\\ \end{align}$$ Notice how the units work a lot like variables do, and you can cancel them in the same way. Only the second one gives an answer in time units (hours, specificalloy) so that must be the answer we're looking for.

Optional

If we know an equation using a particular quantity, we can usually predict what units it will have. For example, Einstein's famous equation \(E=mc^2\) is a relationship between energy $E$ and mass $m$, but what is $c$? First we solve the equation for $c$: $$c=\sqrt{E\over m}$$ Next, we note that energy has units of $[E]=\u{kg{m^2\over s^2}}$ (we sometimes use square brackets to mean "the units of") and mass has units of $[m]=\u{kg}$, and so $$[c]=\sqrt{kg {m^2\over s^2}\over kg}=\sqrt{m^2\over s^2} = \u{m\over s}$$ or meters per second, which is a unit of speed. Thus $c$ must be some sort of speed, and because there's only one speed that is universally important everywhere (the speed of light), $c$ is the speed of light.