\(\def \u#1{\,\mathrm{#1}}\) \(\def \abs#1{\left|#1\right|}\) \(\def \ast{*}\) \(\def \deg{^{\circ}}\) \(\def \ten#1{\times 10^{#1}}\) \(\def \redcancel#1{{\color{red}\cancel{#1}}}\) \(\def \BLUE#1{{\color{blue} #1}}\) \(\def \RED#1{{\color{red} #1}}\) \(\def \PURPLE#1{{\color{purple} #1}}\) \(\def \th#1,#2{#1,\!#2}\) \(\def \lshift#1#2{\underset{\Leftarrow\atop{#2}}#1}}\) \(\def \rshift#1#2{\underset{\Rightarrow\atop{#2}}#1}}\) \(\def \dotspot{{\color{lightgray}{\circ}}}\)
Appendix A: Miscellany
9.

Proportionality

Equations are useful things when you want to solve for one of the variables in them, but they also tell you something about the relationship between these variables.

Consider the equation $a={bc^2\over d}$.

$$v=\sqrt{k\over m}d$$
Thus, equations tell us how different quantities depend on one another. For instance, the equation shown here is the speed of a ball that is launched from a spring: it depends on the stiffness $k$ of the spring, the mass $m$ of the ball, and the distance $d$ the spring is pulled back. We see that we can make the ball go faster by