Problem $1$: A stationary, normal shock stands in the diverging section of a converging$-$diverging nozzle. The Mach number immediately upstream of the shock is $3\mathrm{.}0.$ The nozzle area at the shock is $A_1=500m{m}^2.$ A reservoir supplies the nozzle with air at $1100$ kPa at $400$ K $.$ Calculate the following:

a$.$ Mach number, stagnation pressure $(kPa)$ and static pressure $(kPa)$ after the shock

b$.$ percentage loss in stagnation pressure due to the shock

c$.$ nozzle throat area $(m{m}^2)$

d$.$ entropy change across the shock $(k\frac{J}{k}gK)$

e$.$ taking the nozzle exit area to be $A_3=600m{m}^2,$ calculate the exit Mach number, $M_3$

f$.$ considering the result calculated in $($d$),$ discuss whether the actual exit Mach number is higher, lower, or the same

Sketch the process on a $Ts$ diagram that includes all static and stagnation states.