$1.$ A rocket uses propellants with a specific impulse equal to $270$ seconds. It has an empty weight $($i$.$e$.,$ without propellants$)$ of $200$ lbf $($measured at sea level$).$ The desire is to accelerate it from rest to a velocity of $8,000$ feet per second. Assume that drag forces are negligible and gravity acts normal to the flight direction.

$($a$)$ What is the required mass of propellant?

$($b$)$ If the rocket had an initial horizontal velocity of $1000$ feet per second $($e$.$g$.,$ with a launch from an aircraft$),$ how much propellant mass is required to achieve the $8,000$ feet per second speed?

$($c$)$ If the specific impulse were increased to the new value of $290$ seconds, what would be the answer to part $\backslash ((\backslash$mathrm$\{$a$\})\backslash )?$