For a solid propellant rocket motor with a sea$$level thrust of $207,000$ lbf$,$ determine: $($a$)$ the $($constant$)$ propellant mass flow rate and the specific impulse Is at sea level, $($b$)$ the altitude for optimum nozzle expansion as well as the thrust and specific impulse at this optimum condition and $($c$)$ at vacuum conditions. The initial total mass of the rocket motor is $50,000$ lbm and its propellant mass fraction is $0\mathrm{.}90.$ The residual propellant $($called slivers, combustion stops when the chamber pressure falls below a deflagration limit$)$ amounts to $3\%$ of the burnt. The burn time is $50$ seconds; the nozzle throat area $($At$)$ is $164\mathrm{.}2$ in$\mathrm{.}2$ and its area ratio $($A$2/$At$)$ is $10.$ The chamber pressure $($p$1)$ is $780$ psia and the pressure ratio $($p$1/$p$2)$ across the nozzle may be taken as $90\mathrm{.}0.$ Neglect any start$/$stop transients and use the information in Appendix $2.$ Answers: $($a$)=873$ lbm$/$sec$,237$ sec$.,($b$).,($c$).$