[math] \newcommand{\ex}[1]{\item } \newcommand{\sx}{\item} \newcommand{\x}{\sx} \newcommand{\sxlab}[1]{} \newcommand{\xlab}{\sxlab} \newcommand{\prov}[1] {\quad #1} \newcommand{\provx}[1] {\quad \mbox{#1}} \newcommand{\intext}[1]{\quad \mbox{#1} \quad} \newcommand{\R}{\mathrm{\bf R}} \newcommand{\Q}{\mathrm{\bf Q}} \newcommand{\Z}{\mathrm{\bf Z}} \newcommand{\C}{\mathrm{\bf C}} \newcommand{\dt}{\textbf} \newcommand{\goesto}{\rightarrow} \newcommand{\ddxof}[1]{\frac{d #1}{d x}} \newcommand{\ddx}{\frac{d}{dx}} \newcommand{\ddt}{\frac{d}{dt}} \newcommand{\dydx}{\ddxof y} \newcommand{\nxder}[3]{\frac{d^{#1}{#2}}{d{#3}^{#1}}} \newcommand{\deriv}[2]{\frac{d^{#1}{#2}}{dx^{#1}}} \newcommand{\dist}{\mathrm{distance}} \newcommand{\arccot}{\mathrm{arccot\:}} \newcommand{\arccsc}{\mathrm{arccsc\:}} \newcommand{\arcsec}{\mathrm{arcsec\:}} \newcommand{\arctanh}{\mathrm{arctanh\:}} \newcommand{\arcsinh}{\mathrm{arcsinh\:}} \newcommand{\arccosh}{\mathrm{arccosh\:}} \newcommand{\sech}{\mathrm{sech\:}} \newcommand{\csch}{\mathrm{csch\:}} \newcommand{\conj}[1]{\overline{#1}} \newcommand{\mathds}{\mathbb} [/math]

The velocity [math]v[/math] of a freely falling body depends on the distance [math]s[/math] that it has fallen according to the equation [math]v = \sqrt{2gs}[/math], where [math]g[/math] is the constant gravitational acceleration.

• lab{1.3.7a} Using an [math]s[/math]-axis and a [math]v[/math]-axis, plot the dependent variable [math]v[/math] as a function of the independent variable [math]s[/math].
• lab{1.3.7b} If [math]s[/math] depends on the time [math]t[/math] according to the equation [math]s=\frac12gt^2[/math], how does [math]v[/math] depend on [math]t[/math]?

Note that the variable [math]v[/math] in \ref{ex1.3.7a}, which depends on [math]s[/math], is not the same function as the variable [math]v[/math] in \ref{ex1.3.7b}, which depends on [math]t[/math]. Without knowing which is referred to, the meaning of the value of [math]v[/math] at 2 is ambiguous.