Problem $1(30$ points$)$

In our discussion of the ideal rocket, we defined the thrust coefficient $C_F$ by

$C_F=\sqrt[\phantom{2}]{\frac{2{}^2}{-1}(\frac{2}{+1}{)}^{\frac{+1}{-1}}[1-(\frac{p_e}{p_0}{)}^{\frac{-1}{}}]}+\frac{p_e-p_a}{p_0}\frac{A_e}{A^{*}}$

Consider the case where $=1\mathrm{.}2$ and $\frac{p_a}{p_0}=0\mathrm{.}012.$ If these values are fixed, then $C_F$

becomes a function of $\frac{A_e}{A^{*}}$ only, as the nozzle exit pressure ratio is related to the exit Mach

number $M_e,$ which is in turn related to $\frac{A_e}{A^{*}}$ through the area$-$Mach number relation.

In this problem, you will look at the effect of nozzle exit ratio on the thrust coefficient for

fixed $$ and $\frac{p_a}{p_0}$ in an ideal rocket. The final result will be a plot somewhat similar to the

one on page $14$ of the Lecture $7$ notes $($which will be discussed further in Lecture $8).$ The

nozzle exit ratio in the plot should vary from $1-300.$ I$\text{'}$m breaking this into unscored parts

to help walk you through the steps.

$($a$)$ First, you should write your solver for the area$-$Mach number relation that you intend

to use for the project. The solver should find $M_e$ as a function of $\frac{A_e}{A^{*}}$ for an array of

area ratios. For our ideal rocket, we are only interested in supersonic values for $M_e.$ You

can check that your code is working properly by plotting your results and comparing

to the plot from the lecture notes or one of the course reference textbooks, though this

is not required.

$($b$)$ Next, determine $\frac{p_e}{p_0}$ as a function of $M_e$ using isentropic relations.

$($c$)$ Finally, plot $C_F$ vs$.\frac{A_e}{A^{*}}$ for $\frac{A_e}{A^{*}}$ ranging from $1-300.$ On the same figure, inde$-$

pendently plot the first term in the square root and the "pressure thrust" second term.

Comment on the results and where the assumptions of an ideal rocket may break down

in the plot.