$($a$)$ The vertical range $s$ of a launch vehicle in the presence of gravity can be written as

$s=v_e\frac{M_0}{(m^{})}[1-\frac{M}{M_0}(ln\frac{M_0}{M}+1)]-\frac{1}{2}g{t}^2,$

where $m^{}$ is the mass flow rate $($assumed to be constant during the burn$)$ and $g$ is the

acceleration due to gravity at Earth's surface $($assumed to be constant$).$

The Terrier$-$Black Brant IX is a sounding rocket used by NASA for sub$-$orbital missions.

It is a two$-$stage rocket in which the Terrier booster is the first $($lower$)$ stage, and the

Black Brant is the second $($upper$)$ stage.

Assuming a vertical launch from sea$-$level and neglecting the effects of drag, use the

data in Table $1$ below to calculate:

i$.$ The mass flow rate, specific impulse and exhaust velocity for the two stages.

ii$.$ The velocity at Terrier burnout.

iii. The altitude at Terrier burnout.

iv$.$ The altitude at the start of the Black Brant burn.

v$.$ The total $V$ achieved by the complete rocket, accounting for the delay in staging.

vi$.$ The apogee altitude of the rocket.