A liquid propellant rocket utilizes liquid oxygen $($LOX$)$ and RP$-1($kerosene$)$ as its propel$-$

lants. This propellant combination produces a characteristic velocity of $5800$ ft$/$s$.$ The chamber pressure is $1000$ psi, the throat diameter is $9$ inches, and the nozzle expansion ratio is $30.$ If the ratio of specific heats is $1\mathrm{.}2$ and the engine is operating at sea level, answer the following questions:

$($a$)$ Determine the mass flow rate, thrust, and specific impulse assuming no separation within the nozzle.

$($b$)$ Determine the mass flow rate, thrust, and specific impulse assuming that the flow separates.

$($c$)$ What expansion ratio would we need if we wanted the flow to be perfectly expanded for this chamber pressure $(1000$ psi$)?$

$($d$)$ What is the thrust and Isp for the condition described in $($c$)?$