[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]

A particle moves in the plane during the time interval from [math]t=0[/math] to [math]t=2[/math] seconds. Its position at any time during this interval is given by the parametrization

[[math]] P(t) = (t,t^2-t) , [[/math]]

where it will be assumed that the unit of distance in the plane is [math]1[/math] foot.

• lab{10.5.2a} Identify and draw the curve which the particle traces out during its interval of motion.
• Compute the velocity vector [math]\vec v(t)[/math]. Find the position, velocity, and speed at [math]t=0[/math], [math]t=1[/math], and [math]t=2[/math]. Show these positions and draw the velocity vectors in the figure in part \ref{ex10.5.2a}.
• Compute the acceleration [math]\vec a(t)[/math]. Find the times and corresponding positions (if any) when the acceleration and velocity vectors are perpendicular to each other.
• lab{10.5.2d} Write a definite integral equal to the distance (in feet) which the particle moves during the interval from [math]t=0[/math] to [math]t=2[/math] seconds.
• Evaluate the integral in \ref{ex10.5.2d}.