[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 rocket of mass [math]m[/math] is on its way from the earth to the moon along a straight line joining their centers. Two gravitational forces act simultaneously on the rocket and in opposite directions. One is the gravitational pull toward earth, equal in absolute value to [math]\frac{GM_1m}{{r_1}^2}[/math], where [math]G[/math] is the universal gravitational constant, [math]M_1[/math] the mass of the earth, and [math]r_1[/math] the distance between the rocket and the center of the earth. The other is the analogous gravitational attraction toward the moon, equal in absolute value to [math]\frac{GM_2m}{{r_2}^2}[/math], where [math]M_2[/math] is the mass of the moon and [math]r_2[/math] is the distance between the rocket and the center of the moon. Denote the radii of the earth of the earth and moon by [math]a[/math] and [math]b[/math], respectively, and let [math]d[/math] be the distance between their centers.

• Take the path of the rocket for the [math]x[/math]-axis with the centers of earth and moon at [math]0[/math] and [math]d[/math], respectively, and compute [math]F(x)[/math], the resultant force acting on the rocket at [math]x[/math].
• Set up the definite integral for the work done against the force [math]F[/math] as the rocket moves from the surface of the earth to the surface of the moon.