$($b$)$ Say the force applied by the space$-$ship is just gravitational attraction. If the asteroid has mass $\backslash ($ M$\_\{$a$\}\backslash )$ and the space$-$ship has mass $\backslash ($ M$\_\{$s$\}\backslash )$ and they are separated by a distance $\backslash ($ d $\backslash )$ then the gravitational force between them is $\backslash ($ G M$\_\{$a$\}$ M$\_\{$s$\}/$ d$^\{2\}\backslash )$ where $\backslash ($ G$=6\mathrm{.}67\backslash$times $10^\{-11\}\backslash$mathrm$\{$Nm$\}^\{2\}/\backslash$mathrm$\{$kg$\}^\{2\}\backslash )$ is the gravitational constant. $($Note to actually keep this force constant on the asteroid, the space$-$ship will have to keep the distance $\backslash ($ d $\backslash )$ constant against the pull of the asteroid's gravity, so it will take some jet propulsion to hold that distance fixed.$)$ To keep the ship safe from any other debris traveling with the asteroid $($e$.$g$.,$ knocked off by small collisions$),$ it's best to keep $\backslash ($ d$=10\backslash$mathrm$\{$~km$\}\backslash ).$

i$.$ What is the mass of the ship needed to have the asteroid miss the Earth by both distances in the first part of the problem?

ii$.$ If most of this mass is just rock loaded onto the ship from the Moon $($from where it's easier to launch$)$ then what would be the size of the ship in each case, assuming it's basically a cube $($i$.$e$.$ what is the length, width or height of the ship$)?$ Assume lunar rock density is $\backslash (3000\backslash$mathrm$\{$~kg$\}/\backslash$mathrm$\{$m$\}^\{3\}\backslash ).$ Are these plausible or implausible ship sizes $($e$.$g$.,$ is the ship the size of a planet $-$ which makes it implausible $-$ or the size of a building or an air$-$craft carrier $-$ which is more plausible$)?$