$($a$)$ Provide an expression for dynamic pressure, and explain why it is important and how

it is reduced during launches.

$($b$)$ Draw a diagram to show that the transverse and retarding forces $(F_L$ and $F_r$ respectively$)$

on a vehicle flying with an angle of attack $$ through an atmosphere are given by

$F_l=$Lcos$+$Dsin$$

$F_r=-$Lsin$+$Dcos$,$

and provide expressions for $L$ and $D,$ the lift and drag forces, in terms of the dynamic

pressure.

$($c$)$ The height of a rocket launched in the presence of gravity, but ignoring atmospheric

drag, can be written as

$S=v_e\frac{M_0}{(m^{})}[1-\frac{1}{R}(lo{g}_{e}R+1)]-\frac{1}{2}g{t}^2,$

where $M_0$ is the initial mass of the vehicle, $m^{}$ is the mass flow rate, $v_e$ is the exhaust

velocity, $g$ is the acceleration due to gravity, and $R$ is the mass ratio at time $t.$

A solid fuelled launch vehicle has a cross$-$sectional diameter of $3$ metres, and a mass $M_0$

at launch of $350{10}^{3}kg.$ It ascends vertically, burning propellant at a rate $m^{}=2400$

$kg{s}^{-1},$ with an exhaust velocity of $v_e=2600m{s}^{-1}.$ Including the effects of gravity,

but ignoring drag, calculate the value of the following parameters $40$ seconds into the

flight assuming that the atmospheric density is given by $=1\mathrm{.}22e^{-\frac{S}{8\mathrm{.}0}}kg{m}^{-3}($where

$S$ is in units of $km)$:

i$.$ the altitude

ii$.$ the velocity, accounting for gravity losses

iii. the dynamic pressure

$($d$)$ Assuming a drag coefficient of unity, calculate the drag force acting on the rocket due

to dynamic pressure and compare it to the thrust of the vehicle.

$($e$)$ Explain, qualitatively, the concept of a gravily turn in the launch phase of a mission.