For a solid propellant rocket motor with a sea$-$level thrust of $207,000lbf,$ determine: $($a$)$ the $($constant$)$ propellant mass flow rate $m^{}$ and the specific impulse $I_s$ at sea level, $($b$)$ the altitude for optimum nozzle expansion as well as the thrust and specific impulse at this

$44$ DEFINITIONS AND FUNDAMENTALS

optimum condition and $($c$)$ at vacuum conditions. The initial total mass of the rocket motor is $50,000lbm$ 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 $(A_t)$ is $164\mathrm{.}2$ in$.{?}^2$ and its area ratio $(\frac{A_2}{A_t})$ is $10.$ The chamber pressure $(p_1)$ is $780$ psia and the pressure ratio $(\frac{p_1}{p_2})$ across the nozzle may be taken as $90\mathrm{.}0.$ Neglect any start$/$stop transients and use the information in Appendix $2.$

Answers: $($a$)m^{}=873lb\frac{m}{sec,237sec}.,($b$)F=216,900lbf,I_s=248\mathrm{.}5sec.,($c$)F=231,000lbf,I_s=295sec.$