[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]

An object is dropped from an airplane which is flying in a straight line over level ground at a constant speed of [math]800[/math] feet per second and at an altitude of [math]10,000[/math] feet. The horizontal coordinate of the velocity of the object is constant and equal in magnitude to the speed of the plane. The vertical coordinate of velocity is initially zero. However, the vertical component of acceleration (due to gravity) is [math]-32[/math] feet per second per second. (These data are realistic only if we neglect air resistance, the curvature of the earth, etc.)

• Define a parametrization [math]P(t)=(x(t),y(t))[/math] which gives the position of the particle at time [math]t[/math]. Assume that the object was dropped when [math]t=0[/math] and that [math]P(0) = (0,0)[/math]. Compute the velocity and acceleration vectors [math]\vec v(t)[/math] and [math]\vec a(t)[/math].
• How long does it take the object to fall to the ground?
• Identify and draw the curve in which the object falls.
• Express the distance traveled along the curve as a definite integral.