Problem $1$

In Lecture $3,$ we derived the equation $s_2-s_1={\displaystyle \underset{1}{\overset{2}{}}}{c}_p(T)\frac{dT}{T}-R{\displaystyle \underset{1}{\overset{2}{}}}\frac{dp}{p}($E$1).$ Using $($E$1)$ as a starting

point, provide a detailed derivation $($show all steps$)$ of the equation $\frac{T_2}{T_1}=(\frac{p_2}{p_1}{)}^{\frac{-1}{}}($E$2).$

Then use $($E$2)$ along with the equation for stagnation enthalpy, $h_0=h+\frac{u^2}{2},$ to provide a detailed

derivation of the isentropic flow equations $\frac{T_0}{T}=1+\frac{-1}{2}{M}^2$ and $\frac{p_0}{p}=(1+\frac{-1}{2}{M}^2{)}^{\frac{}{-1}}.$ Show all steps

and state any necessary simplifying assumptions for each step.