You are planing a mission to Mars. You want to send a $3-$ton spacecraft there $(3$ tons wet mass, it is the initial mass

of the spacecraft$).$ As all the engineers working for you are calling in sick, you will have to design the mission

yourself.

To assess the amount of fuel required to leave the Earth, you first need to compute the different $v$ :

$1-$ What is the excess velocity required $v_d^{}($in $k\frac{m}{s})$ when leaving the Earth to reach Mars orbit, assuming a Hohmann

transfer trajectory?

$2\mathrm{.}94$

$2-$ What $v$ should be provided to your spacecraft to leave from a $500km$ low$-$Earth orbit $($in $k\frac{m}{s})?$

$128\mathrm{.}36$

$3-$ Your spacecraft has an engine that uses a fuel providing a specific impulse of $350$ s$.$ How much propellant will be

required to acheive the $v$ computed above? $($give your answer in $kg)$

$4534\mathrm{.}37$

Instead of doing the above $v$ and spending a significant part of your mass on propellant, you decide to use the

space elevator that a colleague of yours just designed. It allows to bring payloads up to an altitude of $55\text{'}000$km$.$

$4-$ At what altitude should be the center of mass of the elevator system $($as always, assume that center of mass $=$ center of

gravity$)?$