B$3.$ Ina space mission, a single stage rocket containing a space shuttle is launched into space from Earth$$s surface. During the launch, a chunk of mass Am is ejected at a speed of vex relative to the rocket, causing the rest of the rocket to gain a velocity increment Av$.$ iitj$4($$t i$$i $($a$)$ Se $4$ Derive the differential equation for the rocket flight by ignoring gravity and air resistance. Show by integration that the total velocity increase from mass Mo to M$/$ is Av $=$ Vex In $($Mo$/$M$1).($b$)(4$ marks$)$ After the launch, the space shuttle enters a circular geostationary orbit around Earth. The orbital seed of the space shuttle is $3074$ m$/$s$.$ Determine the orbital radius from the Earth$$s centre. $(4$ marks$)($Cc$)($d$)($e$)$ The total initial weight of the rocket is $28,000$ kg $($rocket plus satellite plus fuel$),$ and the fuel exhaust speed is vex $=4$ km$/$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$)$ 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$)$ 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? $-$ END OF PAPER$(4$ marks$)$