Rocket Equation (Δv) Calculator
Work out how much a rocket can change its speed. Solve the Tsiolkovsky rocket equation for delta-v, propellant, mass ratio or specific impulse, check thrust-to-weight, and add up to four stages against the budget for orbit.
How to Use
- Pick what to solve for: Delta-v, the Propellant needed, the Mass ratio, the Isp, or several Stages.
- Enter the starting (fuelled) mass and the final (empty) mass in kg, tonnes or pounds; the difference is the propellant burned.
- Enter the engine’s specific impulse in seconds, or its exhaust velocity in m/s or km/s.
- Optionally add the thrust to get the thrust-to-weight ratio at ignition and burnout and the burn time.
- For a multi-stage rocket, set the payload and, for each stage, its full mass, empty mass and Isp. Stage 1 fires first.
- Compare the delta-v bar with the shaded budget for reaching low Earth orbit, and read Show Work for every step.
Worked Example
A first stage. A 500 t rocket burns 400 t of propellant, ending at 100 t, with an Isp of 300 s. The exhaust velocity is v_e = 300 × 9.80665 = 2,942 m/s, and Δv = 2,942 × ln(500 ÷ 100) = 2,942 × 1.6094 = 4,735 m/s. With 7,000 kN of thrust it lifts off at a thrust-to-weight of 1.43 and burns for 168.1 s.
Propellant for a job. An upper stage that weighs 20 t empty must add 3 km/s with a 450 s engine (v_e = 4,413 m/s). The mass ratio is e^(3,000 ÷ 4,413) = 1.974, so it starts at 39.47 t and needs 19.47 t of propellant.
The common mistake: using log₁₀ instead of the natural log. For the first stage, 2,942 × log₁₀ 5 gives 2,056 m/s, less than half the real 4,735 m/s. The rocket equation needs ln (log base e); on most calculators that is the ln key, not log. Forgetting g₀ and using 300 s as if it were 300 m/s is just as bad: 300 × ln 5 = 482.8 m/s.
Show Work
Formulas
From Tsiolkovsky to Orbit
The equation is named after Konstantin Tsiolkovsky, a Russian schoolteacher who published it in 1903 in Exploration of Outer Space by Means of Reaction Devices and proposed liquid hydrogen and oxygen as propellants; in 1929 he followed it with the case for multi-stage and drew from it the case for multi-stage rockets and liquid hydrogen and oxygen as propellants.ldquo;rocket trains and drew from it the case for multi-stage rockets and liquid hydrogen and oxygen as propellants.rdquo;. The same relation had appeared earlier: the British mathematician William Moore worked it out for war rockets in 1813, but it had no space application then.
Robert Goddard derived it independently in his 1919 paper A Method of Reaching Extreme Altitudes and flew the first liquid-fuelled rocket on 16 March 1926; Hermann Oberth did the same in Die Rakete zu den Planetenräumen (1923). Every launch since has been planned around the logarithm in this equation, which is why rockets are mostly propellant and why they drop empty stages on the way up.
The orbital figures used here are computed from Earth’s gravitational parameter GM = 3.986004418 × 10¹⁴ m³/s² and equatorial radius 6,378.137 km; the ground-to-orbit budget of 9.3–10 km/s is an approximate range, because gravity and drag losses depend on the rocket and its path.
About This Tool
This calculator solves the ideal rocket equation for any of its unknowns: the delta-v a stage can give, the propellant a job needs, the mass ratio, or the specific impulse an engine must have. Isp can be entered in seconds or as an exhaust velocity, masses in kilograms, tonnes or pounds. Add the thrust and it also gives the thrust-to-weight ratio at ignition and burnout and the burn time. The stages mode stacks up to four stages on a payload, works out each stage’s delta-v with everything above it as dead weight, and compares the total with the budget for reaching low Earth orbit and with the same masses flown as a single stage.
The equation is ideal: it leaves out gravity and drag losses during the burn, which is why the orbit budget is larger than the orbital speed. Everything runs in your browser; nothing you enter is sent anywhere.
Related tools: Gravity Calculator, Special Relativity Calculator, and Inverse-Square Law Calculator.
Frequently Asked Questions
What is the Tsiolkovsky rocket equation?
Δv = v_e × ln(m₀ ÷ m_f): the change in speed equals the exhaust velocity times the natural log of the starting mass over the final mass. A 500 t stage that burns down to 100 t with an Isp of 300 s (v_e = 2,942 m/s) gains 2,942 × ln 5 = 4,735 m/s.
What is specific impulse?
Isp is the exhaust velocity divided by standard gravity, g₀ = 9.80665 m/s², so it is measured in seconds. 300 s means an exhaust velocity of 2,942 m/s; 450 s means 4,413 m/s. As a rough guide, kerosene–oxygen engines reach about 300–350 s in vacuum and hydrogen–oxygen engines about 450 s.
How much delta-v does it take to reach orbit?
The circular orbital speed 200 km up is √(GM ÷ r) = 7.784 km/s, but climbing through gravity and air costs more, so launches from the ground budget roughly 9.3–10 km/s (an approximate figure that depends on the rocket and trajectory). From that low orbit, escaping Earth takes about 3.224 km/s more.
Why do rockets use stages?
Because the equation punishes carrying empty tanks. One stage at 350 s needs a mass ratio of 15.47, so 93.5% of it must be propellant, to reach 9.4 km/s. The two-stage example here reaches 10.02 km/s; the same masses as a single stage would only manage 7.18 km/s.
Why does adding more fuel help less and less?
Delta-v grows with the logarithm of the mass ratio, and the extra fuel also has to be carried. Each doubling of the mass ratio adds the same v_e × ln 2: at 300 s that is 2,039 m/s whether you go from a ratio of 2 to 4 or from 8 to 16.
How do I use the Rocket Equation (Δv) Calculator?
Just type your numbers. The answer shows up right away — there is no button to press. Change anything and it updates by itself.
Does it cost anything or need an account?
No. The tool is completely free, there is no account to create, and it keeps working offline after the page first loads.
Is anything I type uploaded?
No. The tool works entirely on your device, so the values you enter never leave your browser.
Common Use Cases
Launch vehicle sketches
A 510 t two-stage rocket (400 t first stage at 300 s, 100 t second stage at 350 s) puts 10 t of payload past the ~9.4 km/s orbit budget, with 10.02 km/s.
Upper-stage sizing
A 20 t dry stage needing 3 km/s from a 450 s engine must carry 19.47 t of propellant, a mass ratio of 1.974.
Electric propulsion
An ion thruster at 3,000 s gives a 1,000 kg probe 6,565 m/s while burning only 200 kg of propellant.
Liftoff checks
7,000 kN of thrust on a 500 t rocket gives a thrust-to-weight of 1.43 at ignition and a burn time of 168.1 s for 400 t of propellant.
Engine requirements
To get 2 km/s from a craft going from 10 t to 6 t, the engine needs an Isp of at least 399.2 s.
Last updated: