ohh I'm glad this is back
RivetGun Are you flying in Vacuo????
you can't take moles out of it the adiabatic gas laws don't allow that: Cp-Cv=nR, Y=Cp/Cp+nR
and I didn't say T dependent I said Rewrite as a function of T
ex: dw = -PdV and du = CvdT and dU = dw+dq adiabatically = du = dw
so I can say Cvdt=-Pdv; REWRITING P in terms of T
Cv dT/ T = -nR dV/V since Cv is a constant I can integrate then i have
<Tf/ (dT/T) / Ti> = -nR <Vf / (dV/V) /Vi > = lnTf/Ti = -nRln (Vf/Vi)
using a ln X =lnX^a
and gas laws,
I have Ti/Tf = (Vf/V1)^y
lastly, Pi Vi^y = Pf Vf^y.
applying this judiciously with the equation fo dynamic pressure you can derive the equations (corrected for adiabatic compressibility) provided by Old Smokey
with about four pages of algebra...I' m not doing it
p.s. please let me know if I made an error if anyone finds one
edited: to correct a error Cp DIFFERS from Cv by nR NOT Cp+Cv=nR
I was very wrong there!!!