Tsiolkovsky Rocket Equation Calculator
Enter initial mass, final mass and effective exhaust velocity to compute Δv=v_e·ln(m₀/m_f), mass ratio and propellant mass. m₀=100000, m_f=10000, v_e=3000 → Δv≈6908 m/s (escape ~11200 needs bigger ratio).
Input Data
Results
At a glance:The Tsiolkovsky rocket equation (Tsiolkovsky 1903, Russian 'father of astronautics'): in the absence of gravity and air drag, the speed a rocket gains from burning its propellant is Δv=v_e·ln(m₀/m_f), where Δv is the velocity increment (m/s), v_e the effective exhaust velocity (i.e. specific impulse I_sp·g₀, m/s), m₀ the initial total mass (rocket + full propellant, kg) and m_f the final dry mass (after burnout, kg). The mass ratio m₀/m_f is how many times heavier the rocket is at liftoff than at burnout. Physical meaning: it shows that Δv depends only on the exhaust speed and the natural-log of the mass ratio — not on how the burn is timed. Because of the logarithm, Δv grows slowly: doubling v_e doubles Δv, but to multiply Δv by 2 you must square the mass ratio (e.g. ratio 5→25), meaning vastly more propellant. Classic example: m₀=100000 kg, m_f=10000 kg, v_e=3000 m/s → Δv=3000×ln(10)=3000×2.3026≈6908 m/s. To reach Earth escape (~11200 m/s) with v_e=3000 you need ratio e^(11200/3000)=e^3.73≈41.7 — almost impossible in one stage, which is why rockets stage. History: Tsiolkovsky derived it in 1903, 24 years before the first liquid rocket. Applications: (1) launch-vehicle design and stage sizing; (2) orbital transfer (Hohmann Δv budget); (3) multistage rockets; (4) ion propulsion (huge v_e); (5) return and landing budgets.
Formula
Rocket equation: Δv = v_e·ln(m₀/m_f)
Mass ratio: R = m₀/m_f
Propellant mass: m_p = m₀ − m_f
Propellant fraction: f = (m₀ − m_f)/m₀ = 1 − 1/R
v_e = I_sp·g₀ (specific impulse × g)
$$\Delta v = v_e \ln\frac{m_0}{m_f}, \quad R = \frac{m_0}{m_f}, \quad m_p = m_0 - m_f$$How to Use
- Enter initial mass m₀ (kg, with propellant).
- Enter final mass m_f (kg, dry).
- Enter exhaust velocity v_e (m/s; chemical 2500–4500).
- Optionally mass flow ṁ for burn time; the tool gives Δv, ratio, propellant mass/fraction.
Case Studies
Single stage to orbit and staging
Single stage: m₀=100000, m_f=10000, v_e=3000 → Δv≈6908 m/s, short of orbital ~9400 m/s.
Earth escape ~11200 m/s needs ratio e^3.73≈41.7 at v_e=3000 — impractical in one stage.
Staging sheds dead mass: each stage only needs a modest ratio, summing to the full Δv (Saturn V 3 stages).
Ion propulsion and probes
Ion thruster v_e up to 30000 m/s exhausted — ratio 20 gives Δv=30000×ln20≈90000 m/s.
Low thrust but sustained over months achieves huge Δv for deep-space probes (Dawn, Hayabusa).
Chemical rockets trade huge thrust for low v_e; ion engines trade time for efficiency.
Content review: Calculatorism Science Team. Results are for reference only; please refer to the relevant authorities for the official figures.