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Tony's Intertwingled Memex • Reference Manual

Rocket Velocity Optimization via Rocket Equation (Aerospace Engineering)

Formulas & Definitions

An aerospace propulsion engineer is analyzing the acceleration profile of a vertical launch vehicle during atmospheric flight. The burnout velocity vv (in meters per second, m/s\text{m/s}) achieved by propellant consumption is governed by the Tsiolkovsky Rocket Equation:

v(m)=veln⁡(m0m)v(m) = v_e \ln\left(\frac{m_0}{m}\right)
where ve=2500 m/sv_e = 2500\text{ m/s} is the constant effective exhaust velocity, m0=50,000 kgm_0 = 50,000\text{ kg} is the initial fully fueled wet mass of the rocket, and mm is the instantaneous vehicle mass (in kg\text{kg}).

The rocket burns through its propellant fuel supply over time tt (in seconds) at a constant mass flow rate, meaning the instantaneous vehicle mass drops according to:

m(t)=50,000−250tm(t) = 50,000 - 250t

  1. Pre-process the velocity equation using logarithmic properties to separate ln⁡(m0/m)\ln(m_0/m) into a difference of logarithms before differentiating.
  2. Differentiate v(m)v(m) with respect to mass mm, and differentiate m(t)m(t) with respect to time tt.
  3. Apply the Chain Rule in Leibniz notation (dvdt=dvdm⋅dmdt\frac{dv}{dt} = \frac{dv}{dm} \cdot \frac{dm}{dt}) to establish an analytical expression for the vehicle's acceleration dvdt\frac{dv}{dt}.
  4. Evaluate the rocket's mass and instantaneous acceleration dvdt\frac{dv}{dt} at exactly t=60 secondst = 60\text{ seconds} into the burn sequence.