PROBLEM I

A lunar excursion module $($LEM$)$ was used in the Apollo moon$-$Janding missions to save fuel by making it unnecessary to launch the entire Apollo spacccraft from the moon's surface on its return trip to earth. Check the eflectiveness of this approach by computing the encrgy per pound required for a spacecraft $($as weighed on the earth$)$ to escape the moon's gravitational field if the spacecraft starts from $($a$)$ the moon's surface, $($b$)$ a circutar orbit $50$ mi above the moon's surface. Neglect the cffect of the earth's gravitational field. $($The radus of the moon is $1081$ mi and its mass is $0\mathrm{.}0123$ times the mass of the carth.$)$

HOBLEW $2$

A space shutle is in an elliptic orbit of eccentricity $0\mathrm{.}0356$ and a minimum altitude of $300$ km above the surface of the carth. Knowing that the radius of the earth is $6370$ km $,$ determine the periodic time for the orbit.

PROBLEM $3.$

Observations show that a celestial body traveling at $\backslash (1\mathrm{.}2\backslash$times $10^\{6\}\backslash$mathrm$\{$mi$\}/\backslash$mathrm$\{$h$\}\backslash )$ appears to be describing about Point $\backslash ($ B $\backslash )$ a a circle of radius cqual to $60$ light years. Point $\backslash ($ B $\backslash )$ is suspected of being a very dense concentration of mass called a black hole. Determine the satio $\backslash ($ M$\_\{$B$\}/$ M$\_\{$S$\}\backslash )$ of the mass at $\backslash ($ B $\backslash )$ to the mass of the sun. $($The mass of the sun is $330,000$ times the mass of the earth, and a light year is the distance traveled by light in one year at a velocity

of $\backslash (186,300\backslash$mathrm$\{$mi$\}/\backslash$mathrm$\{$s$\}\backslash ).)$