Tsiolkovsky Ideal Rocket Equation
The fundamental equation of astronautics: computes the maximum velocity change of a spacecraft as propellant is expelled at effective exhaust velocity.
\[ \Delta v = v_e \cdot \ln\left( \frac{m_0}{m_f} \right) = I_{sp} \cdot g_0 \cdot \ln\left( \frac{m_0}{m_f} \right) \]
Plain Text Formula:
Delta_v = v_e * ln(m0 / mf) = Isp * g0 * ln(m0 / mf)
📐 Variable Definitions & Parameters
| Symbol | Variable Name | Standard Unit | Physical Definition |
|---|---|---|---|
| \[ \Delta v \] | Velocity Increment | m/s | Net velocity change achievable by the rocket stage |
| \[ v_e \] | Effective Exhaust Velocity | m/s | Speed of escaping combustion gases: Isp * g0 |
| \[ I_{sp} \] | Specific Impulse | seconds (s) | Thrust efficiency per unit propellant mass flow |
| \[ m_0 \] | Initial Wet Mass | kg | Total vehicle mass including propellant at ignition |
| \[ m_f \] | Final Dry Mass | kg | Vehicle mass remaining after fuel burnout |
💡 Worked Engineering Example
Problem Statement:
A rocket stage has Isp = 350 s, wet mass m0 = 100,000 kg, and dry mass mf = 10,000 kg (mass ratio = 10). Find Delta-v.
Step-by-Step Solution:
- v_e = 350 * 9.80665 = 3,432.3 m/s
- ln(m0 / mf) = ln(10) = 2.3026
- Delta_v = 3,432.3 * 2.3026 = 7,903 m/s = 7.90 km/s
🎯 Result: The stage produces 7.90 km/s of Delta-v.