Rocket Equation Calculator
Use the Tsiolkovsky rocket equation to find the Δv of a rocket stage from its specific impulse and wet and dry masses, or the propellant needed to reach a target Δv.
The equation that rules rocketry
Konstantin Tsiolkovsky published it in 1903: Δv = vₑ × ln(m₀ ÷ m_f). The change in velocity a rocket can achieve depends on only two things, the speed of its exhaust vₑ and the ratio of its full mass m₀ to its empty mass m_f. Exhaust speed is usually quoted as specific impulse Isp in seconds, and vₑ = Isp × g₀, with g₀ = 9.80665 m/s² by definition. The calculator takes the masses in tonnes, but any unit works because only their ratio matters.
Take a stage with an Isp of 450 seconds, typical of a hydrogen and oxygen engine in vacuum, a wet mass of 100 tonnes and a dry mass of 25 tonnes. The exhaust velocity is 4,413.0 m/s, the mass ratio is 4 and the propellant fraction is 75%. Δv = 4,413.0 × ln 4 = 6,118 m/s. That is the ideal figure in empty space, before gravity and drag losses.
The tyranny of the logarithm
Because Δv grows only with the logarithm of the mass ratio, each extra kilometre per second costs exponentially more propellant. To reach a Δv equal to one exhaust velocity the rocket must be 63% propellant; for two, 86%; for three, 95%. Run the equation backwards for the same 25 tonnes of dry mass and a target of 6,000 m/s: with Isp 450 s you need 72.37 tonnes of propellant, but with a kerosene engine at Isp 311 s you need 153.78 tonnes, more than twice as much, for the same Δv. Reaching low Earth orbit takes about 9.4 km/s including losses, which with chemical engines would demand propellant fractions above 90% in a single stage. Staging solves this by throwing away empty tanks and heavy engines on the way up, so the later stages accelerate less dead mass.
Common mistakes
The dry mass must include everything that is not burned: structure, engines, payload and any upper stages riding on top. Leaving out the payload overstates Δv badly. Using the sea-level Isp of an engine for an upper stage, or the vacuum Isp for a first stage, shifts results by several percent; most engines lose 8 to 15% of their Isp at sea level. Mixing up the natural logarithm with log base 10 gives a figure 2.3 times too small. And the equation gives Δv, not speed: a stage that starts at 2 km/s and supplies 6 km/s ends at about 8 km/s only if all of it is applied along the direction of motion.
How to use it
- Choose whether to solve for Δv from masses or for the propellant needed to reach a target Δv.
- Enter the specific impulse in seconds for the conditions the stage flies in, vacuum or sea level.
- For Δv, enter the wet mass and the dry mass in tonnes, including payload in both. The defaults, 450 s, 100 t and 25 t, give 6,118 m/s.
- For propellant, enter the dry mass and the target Δv in metres per second and read the propellant mass and the resulting wet mass.
- For a multistage rocket, run each stage separately, counting all upper stages and the payload as part of the lower stage’s dry and wet mass, and add the Δv values.
Questions from the flight desk
What is the Tsiolkovsky rocket equation?
It gives the ideal velocity change of a rocket: Δv = Isp × g₀ × ln(m₀ ÷ m_f), where Isp is the specific impulse in seconds, g₀ is 9.80665 m/s², m₀ is the full mass and m_f the mass after the propellant is burned. It assumes no gravity or drag losses and an engine with constant exhaust speed.
What is specific impulse?
A measure of engine efficiency: the thrust produced per unit weight of propellant flow, expressed in seconds. Multiplying it by 9.80665 m/s² gives the effective exhaust velocity. Solid motors manage around 250 to 290 s, kerosene engines about 300 to 340 s in vacuum, hydrogen engines about 450 s and ion thrusters several thousand seconds at very low thrust.
Why do rockets use multiple stages?
Because Δv depends on the logarithm of the mass ratio, a single stage carrying its empty tanks all the way to orbit would need an impractically high propellant fraction. Dropping empty stages means later engines accelerate much less mass. Running the equation stage by stage and adding the results shows how much staging gains.
How much delta-v is needed to reach orbit?
About 9.3 to 9.5 km/s for low Earth orbit. The orbit itself needs about 7.8 km/s; gravity losses during the climb, air drag and steering add roughly 1.5 to 2 km/s. Launching eastward near the equator gives up to 0.465 km/s free from the Earth’s rotation. The rocket equation result must cover the full budget.
Disclaimer: Orbital mechanics results are ideal two-body figures with instantaneous burns, computed from the values you enter and the constants stated on each page. They are not mission analysis, flight dynamics or launch safety advice. Real orbits are perturbed by Earth’s oblateness, the Moon, the Sun and atmospheric drag, and real burns carry gravity and steering losses. Never use these numbers to operate a spacecraft or to fly a rocket; follow the safety code of your rocketry association and the advice of qualified engineers.