Problem $6$: Air enters a circular duct $0\mathrm{.}5$ m in diameter with a $($supersonic$)$ Mach number of $3\mathrm{.}0.$ The friction factor is $0\mathrm{.}01.$ Calculate the following:

$($a$)$ the length of a duct required to reduce the Mach number to $2\mathrm{.}0$ ;

$($b$)$ the percentage change in temperature $T,$ pressure $P,$ and density $$;

$($c$)$ the entropy increase of the air $(\frac{J}{k}gK)$;

Assuming the same length of duct as computed in $($a$),$ but with a subsonic initial Mach number of $M=0\mathrm{.}50$ and $M_2=0\mathrm{.}84,$ calculate the percentage change in temperature, pressure, and density, and the entropy increase for the air.

Comment on the differences between the subsonic and supersonic cases in terms of changes in thermodynamic state properties.