$(35$ points$)$ A supersonic wind tunnel is to be constructed as shown in the figure below with air at atmospheric pressure passing through a converging$-$diverging nozzle into a constant$-$area test section and then into a large vacuum tank. The test run is started with a pressure of $0$ psia in the tank. How long can uniform flow conditions be maintained in the test section $($that is$,$ how long will it be before the tank pressure rises to a value such that a shock will appear in the test section$)?$ Assume the test section to be circular, $4$ in $.$ in diameter, with a design Mach number of $2\mathrm{.}4.$ The tank volume is $\backslash (100\backslash$mathrm$\{$ft$\}^\{3\}\backslash ).$ Atmospheric conditions are $14\mathrm{.}7$ psia and $\backslash (520^\{\backslash$circ$\}\backslash$mathrm$\{$R$\}\backslash ).$ Assume the air to be brought to rest adiabatically in the tank.

$(35$ Points$)$ Repeat problem $2$ but assume that there is a diffuser of area ratio $2$ to $1$ between the test section and the tank as shown in figure below.