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Rocket Propellant Mass Flow Rate (Aerospace Engineering) Solution

Formulas & Definitions

Problem Overview

We are given a rocket motor mass profile M(t)=800−12t2 kgM(t) = 800 - 12t^2\text{ kg}. We calculate dMdt\frac{dM}{dt} using the limit definition and determine the instantaneous mass flow rate and thrust at t=5 st = 5\text{ s}.

Step-by-Step Mathematical Solution

Step 1: Set up the Difference Quotient Limit

dMdt=lim⁡h→0M(t+h)−M(t)h=lim⁡h→0[800−12(t+h)2]−(800−12t2)h\frac{dM}{dt} = \lim_{h \to 0} \frac{M(t+h) - M(t)}{h} = \lim_{h \to 0} \frac{[800 - 12(t+h)^2] - (800 - 12t^2)}{h}

Step 2: Expand the Binomial (t+h)2=t2+2th+h2(t+h)^2 = t^2 + 2th + h^2

dMdt=lim⁡h→0800−12(t2+2th+h2)−800+12t2h\frac{dM}{dt} = \lim_{h \to 0} \frac{800 - 12(t^2 + 2th + h^2) - 800 + 12t^2}{h}
dMdt=lim⁡h→0800−12t2−24th−12h2−800+12t2h\frac{dM}{dt} = \lim_{h \to 0} \frac{800 - 12t^2 - 24th - 12h^2 - 800 + 12t^2}{h}

Step 3: Cancel Non-hh Terms and Factor out hh

dMdt=lim⁡h→0−24th−12h2h=lim⁡h→0h(−24t−12h)h\frac{dM}{dt} = \lim_{h \to 0} \frac{-24th - 12h^2}{h} = \lim_{h \to 0} \frac{h(-24t - 12h)}{h}
dMdt=lim⁡h→0(−24t−12h)\frac{dM}{dt} = \lim_{h \to 0} (-24t - 12h)

Step 4: Direct Substitution (h=0h = 0)

dMdt=−24t−12(0)=−24t kg/s\frac{dM}{dt} = -24t - 12(0) = -24t\text{ kg/s}

Step 5: Evaluate at t=5 secondst = 5\text{ seconds}

M′(5)=−24(5)=−120 kg/sM'(5) = -24(5) = -120\text{ kg/s}
The mass flow rate is m˙=∣dMdt∣=120 kg/s\dot{m} = \left|\frac{dM}{dt}\right| = 120\text{ kg/s} (the negative sign indicates system mass loss).

Step 6: Compute Rocket Thrust

Fthrust=c⋅m˙=2500 m/s×120 kg/s=300,000 N=300 kNF_{\text{thrust}} = c \cdot \dot{m} = 2500\text{ m/s} \times 120\text{ kg/s} = 300{,}000\text{ N} = 300\text{ kN}

Engineering Conclusion

At t=5 secondst = 5\text{ seconds}, propellant mass is exiting the nozzle throat at 120 kg/s120\text{ kg/s}, generating a reaction thrust of 300 kN300\text{ kN} (67,440 lbf67{,}440\text{ lbf}). Tracking instantaneous dMdt\frac{dM}{dt} through calculus allows aerospace propulsion engineers to predict vehicle acceleration curves and verify structural stage separation timing.