The Physics of Propulsion

Dynamics of Variable Mass Systems & Orbital Injection

1. Conservation of Momentum for Variable Mass

The fundamental derivation of rocket motion begins with Newton's Second Law applied to a system where mass is being exhausted at a rate \(\dot{m}\). Considering the system momentum \(P\) at time \(t\) and \(t + dt\):

\[P(t) = mv\] \[P(t+dt) = (m - dm)(v + dv) + dm(v - v_e)\]

Neglecting the second-order term \(dm \cdot dv\) and setting \(dP/dt = F_{ext}\), we arrive at the equation of motion:

\[m \frac{dv}{dt} = -v_e \frac{dm}{dt} + F_{ext}\]

2. The Tsiolkovsky Equation (Ideal Case)

In the absence of external forces (\(F_{ext} = 0\)), integrating from initial state \((m_0, v_0)\) to final state \((m_f, v_f)\) yields:

\[\Delta v = I_{sp} g_0 \ln \left( \frac{m_0}{m_f} \right)\]

This logarithmic relationship dictates the "tyranny of the rocket equation," where the propellant mass required increases exponentially with the required \(\Delta v\).

3. Flight Dynamics & Delta-V Losses

For a rocket ascending through an atmosphere, the instantaneous acceleration is governed by:

\[\frac{dv}{dt} = \frac{T \cos(\alpha)}{m} - \frac{D}{m} - g \sin(\theta)\]

Integrating this over the burn time \(t_b\) reveals the total required \(\Delta v\) budget:

  • Gravity Loss: \(\Delta v_g = \int_{0}^{t_b} g \sin(\theta) dt\) — Minimized by pitching over quickly (Gravity Turn).
  • Aerodynamic Drag: \(\Delta v_d = \int_{0}^{t_b} \frac{D(t)}{m(t)} dt\) — Significant in the dense lower atmosphere.
  • Steering Loss: \(\Delta v_s = \int_{0}^{t_b} \frac{T}{m} (1 - \cos\alpha) dt\) — Resulting from thrust misalignment with the velocity vector.

4. Multi-Stage Optimization

To bypass the limitations of a single stage, serial staging is used. The total \(\Delta v\) is the sum of the changes in velocity for each stage \(i\):

\[\Delta v_{total} = \sum_{i=1}^{n} v_{e,i} \ln \left( \frac{m_{0,i}}{m_{f,i}} \right)\]

Structural efficiency is measured by the structural coefficient \(\epsilon = \frac{m_s}{m_s + m_p}\). Optimization involves distributing the mass ratios such that the payload delivery is maximized for a given total \(\Delta v\).

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