Tsiolkovsky Equation Calculator Formula
Understand the math behind the tsiolkovsky equation calculator. Each variable explained with a worked example.
Formulas Used
Required Mass Ratio (m0/mf)
mass_ratio = exp(delta_v / exhaust_velocity)Fuel Mass Fraction
fuel_fraction = 1 - 1 / exp(delta_v / exhaust_velocity)Fuel Fraction (%)
fuel_fraction_pct = (1 - 1 / exp(delta_v / exhaust_velocity)) * 100Variables
| Variable | Description | Default |
|---|---|---|
delta_v | Required Delta-V(m/s) | 9400 |
exhaust_velocity | Exhaust Velocity(m/s) | 3000 |
How It Works
Tsiolkovsky Equation: Mass Ratio Form
Given a target delta-v and an engine's exhaust velocity, this tells you how massive the rocket must be at launch relative to its dry mass.
Formula
m0/mf = exp(dv / ve)
The fuel fraction is 1 - 1/(m0/mf). For orbital velocity (9.4 km/s) with a 3 km/s exhaust, over 95% of the rocket must be propellant.
Worked Example
Reach LEO (dv = 9400 m/s) with ve = 3000 m/s.
- 01m0/mf = exp(dv / ve)
- 02m0/mf = exp(9400 / 3000)
- 03m0/mf = exp(3.133) = 22.94
- 04Fuel fraction = 1 - 1/22.94 = 0.9564 = 95.64%
Frequently Asked Questions
What limits the mass ratio in practice?
Structural mass. Tanks, engines, and the payload set a floor for the dry mass. Typical first stages achieve mass ratios of about 8 to 12.
How does exhaust velocity affect the mass ratio?
Higher exhaust velocity dramatically reduces the required mass ratio. Doubling ve halves the exponent, which can reduce the mass ratio by orders of magnitude.
What is a typical exhaust velocity?
Chemical rockets: 2500 to 4500 m/s. Ion engines: 30 000 to 70 000 m/s, but with very low thrust.
Ready to run the numbers?
Open Tsiolkovsky Equation Calculator