Rocket Propellant Mass Flow Rate (Aerospace Engineering)

During a static hold-down test of a high-altitude solid rocket motor prototype, real-time load cell telemetry records the total mass of the motor assembly $M(t)$ in kilograms ($t \in [0, 8]$ seconds). As the propellant grain combustion progresses, the overall mass decreases according to the quadratic time profile:

The instantaneous time rate of mass depletion, $\frac{dM}{dt}$, defines the propellant mass flow rate ($\dot{m}$) expelled through the supersonic nozzle.

1. Using the difference quotient limit definition, compute the general derivative formula for $\frac{dM}{dt}$.
2. Calculate the instantaneous mass flow rate of the motor at $t = 5\text{ seconds}$ into the burn.
3. If the effective exhaust velocity of the expelled gas is $c = 2500\text{ m/s}$, compute the instantaneous rocket thrust $F_{\text{thrust}} = c \cdot \left|\frac{dM}{dt}\right|$ generated at $t = 5\text{ seconds}$.