Rocket Equation Calculator
Enter exhaust velocity, initial and final mass to get the velocity change and mass ratio. v_e=4500, m₀=100, m₁=10 → Δv≈10,362 m/s.
Input Data
Results
At a glance:The rocket equation (Konstantin Tsiolkovsky, 1903, Russian spaceflight pioneer): under no external forces (gravity and drag neglected), the velocity change gained by ejecting propellant is Δv=v_e·ln(m₀/m₁), where v_e is the exhaust velocity relative to the rocket, m₀ the initial mass and m₁ the final mass. Derivation: conservation of momentum. In time dt the rocket ejects mass dm (dm<0 means the rocket loses mass) at relative speed v_e, and gains momentum dmv=v_e·(−dm); integrating gives Δv=∫v_e·(−dm/m)=v_e·ln(m₀/m₁). Physical meaning: (1) exponential growth — doubling Δv needs e²≈7.4× the mass ratio, so multi-stage chemical rockets are inevitable; (2) exhaust velocity sets efficiency — higher v_e gives more Δv per unit fuel, which is why ion propulsion (v_e~30 km/s) beats chemical (4.5 km/s); (3) the payload fraction is pessimistic: m₁/m₀=e^(−Δv/v_e). With Δv=10 km/s, v_e=4.5 km/s → m₁/m₀=e^(−2.22)≈0.108. History: derived by Tsiolkovsky in 1903, alongside Goddard (USA) and Oberth (Germany) as the three founders of astronautics. Applications: (1) orbital mission design; (2) multi-stage mass allocation; (3) interplanetary Δv budgeting; (4) ion and nuclear propulsion design; (5) fuel estimation.
Formula
Rocket equation: Δv = v_e·ln(m₀/m₁)
Momentum derivation: dm·v + m·dv = 0
Mass ratio: R = m₀/m₁
Payload fraction: m₁/m₀ = e^(−Δv/v_e)
Staging: Δv_total = Σ v_ei·ln(m₀i/m₁i)
$$\Delta v = v_e \ln\left(\frac{m_0}{m_1}\right), \quad R = \frac{m_0}{m_1}, \quad \frac{m_1}{m_0} = e^{-\Delta v / v_e}$$How to Use
- Enter exhaust velocity v_e (m/s; chemical ~4500).
- Enter initial mass m₀ (kg, with fuel) and final mass m₁ (kg, after burn).
- The tool gives Δv, mass ratio and propellant mass.
Case Studies
Falcon 9 and Falcon Heavy
Falcon 9 first stage v_e~340 s×9.81=3335 m/s, m₀=549e3 kg, m₁~28e3 kg → Δv≈9.0 km/s (LEO needs 9.4 km/s incl. gravity/drag losses).
SpaceX lands the first stage, cutting cost ~30%; multi-staging is the direct answer to the rocket equation.
Falcon Heavy: three boosters, 63.8 t to LEO, 16 t to Mars; used for commercial crewed lunar flybys.
Ion propulsion — Deep Space 1
NASA Deep Space 1 (1998) used a xenon ion thruster v_e=30 km/s, mass ratio 1.21 → Δv=5.6 km/s on only 74 kg of fuel.
A chemical rocket for the same Δv would need mass ratio 1.36 (80% more fuel). Ion thrust is tiny (92 mN) but efficient over long burns.
Ion propulsion is the future of deep space — Dawn, Hayabusa and SMART-1 all used it.
Content review: Calculatorism Science Team. Results are for reference only; please refer to the relevant authorities for the official figures.