B$3.$ In a space mission, a single stage rocket containing a space shuttle is launched into space from Earth's surface. During the launch, a chunk of mass $m$ is ejected at a speed of $v_{ex}$ relative to the rocket, causing the rest of the rocket to gain a velocity increment $v.$

$($a$)$ Derive the differential equation for the rocket flight by ignoring gravity and air resistance. Show by integration that the total velocity increase from mass $M_0$ to $M_l$ is $v=v_{ex}ln(\frac{M_0}{M_l}).$

$(4$ marks$)$

$($b$)$ After the launch, the space shuttle enters a circular geostationary orbit around Earth. The orbital seed of the space shuttle is $3074\frac{m}{s}.$ Determine the orbital radius from the Earth's centre.

$(4$ marks$)$

$($c$)$ The total initial weight of the rocket is $28,000kg($rocket plus satellite plus fuel$),$ and the fuel exhaust speed is $v_{ex}=4k\frac{m}{s}.$ Ignoring the effect of gravity and air resistance, determine the amount of fuel needed to place the space shuttle into the geostationary orbit.

$(4$ marks$)$

$($d$)$ In reality, gravity and air resistance cannot be neglected. Describe what engineering design strategies are used to minimise these effects in a real space shuttle launch.

$(4$ marks$)$

$($e$)$ The space shuttle is equipped with a propulsion device that enables it to gain speed. What is the minimum gain in orbital speed needed in order for the space shuttle to escape from Earth's gravitational pull?

$(4$ marks$)$