What does the Tsiolkovsky rocket equation say?

Δv = v_e ln(m₀/m_f): the speed change a rocket can achieve depends only on its exhaust velocity v_e (Isp × g₀) and its mass ratio m₀/m_f, not on how big the thrust is or how fast it burns. Because of the logarithm, going from a mass ratio of 5 to 10 adds only about 43% more Δv. To get much more you need a higher Isp or staging.

Why does reaching low Earth orbit take about 9.4 km/s when orbital speed is only 7.8 km/s?

The extra 1.5–2 km/s is lost along the way. While the rocket climbs, part of the thrust only holds it up against gravity (gravity loss, usually the largest). Pushing through the dense lower atmosphere costs drag loss, and thrust that isn't aligned with the velocity costs steering loss. Launching east gains up to about 465 m/s from Earth's rotation at the equator. The simulator reports the total loss for each flight.

Why use stages instead of a single stage to orbit?

Reaching 9.4 km/s on kerosene and oxygen (v_e ≈ 3 km/s) in one stage needs a mass ratio of e^(9.4/3) ≈ 23. That means 96% of the liftoff mass is propellant, leaving 4% for tanks, engines and payload, which is extremely hard to build. Staging drops each empty stage so the rest of the rocket stops carrying dead weight, and the total Δv is the sum of the stage Δvs.

What is Max-Q and why does it matter?

Dynamic pressure q = ½ρv² measures how hard the airflow pushes on the vehicle. Speed rises after liftoff while the air thins, so their product peaks at roughly 10–15 km altitude. That peak is Max-Q, typically around 30 kPa. It is the moment of highest aerodynamic load, and many rockets throttle down around it. The orange dot in the simulator marks where it happens.

How accurate is the simulation?

It is a teaching model: 2D, a spherical Earth, an exponential atmosphere with an 8.5 km scale height, a fixed drag coefficient, and constant Isp and thrust per stage with no throttling. The preset rockets use approximate public figures. It correctly shows how gravity turns, staging, Max-Q and orbit insertion work, but it is not a mission-design tool.

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🚀Rocket Equation Calculator - Tsiolkovsky, Multi-Stage, Launch Simulator

Enter the specific impulse (or exhaust velocity) and the initial and final masses to get a rocket's delta-v, or solve backwards for the Isp or mass you need. The multi-stage tab comes with approximate Falcon 9, Saturn V and Long March 5 figures and gives Δv, thrust-to-weight and burn time for each stage, checked against the budgets for low Earth orbit, GTO, the Moon and Mars. The launch simulator integrates thrust, gravity and exponential-atmosphere drag with RK4 over a to-scale round Earth, plots altitude, speed, dynamic pressure and g-load, and tells you whether you made orbit.

Free · No sign-up · Runs in your browserLast updated: 2026-10-29

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How to use Rocket Equation & Launch Simulator

1On the Rocket equation tab, pick the quantity to solve for and fill in the other three, or choose an engine preset for the Isp
2On the Multi-stage tab, choose a rocket preset or enter propellant, dry mass, Isp and thrust for each stage
3Read each stage's Δv, TWR and burn time, and see whether the total reaches LEO, GTO, the Moon or Mars
4Switch to Launch sim, set the pitch-over kick angle and target altitude (or let the tool find a kick angle), then play the launch

FAQ

What does the Tsiolkovsky rocket equation say?: Δv = v_e ln(m₀/m_f): the speed change a rocket can achieve depends only on its exhaust velocity v_e (Isp × g₀) and its mass ratio m₀/m_f, not on how big the thrust is or how fast it burns. Because of the logarithm, going from a mass ratio of 5 to 10 adds only about 43% more Δv. To get much more you need a higher Isp or staging.
Why does reaching low Earth orbit take about 9.4 km/s when orbital speed is only 7.8 km/s?: The extra 1.5–2 km/s is lost along the way. While the rocket climbs, part of the thrust only holds it up against gravity (gravity loss, usually the largest). Pushing through the dense lower atmosphere costs drag loss, and thrust that isn't aligned with the velocity costs steering loss. Launching east gains up to about 465 m/s from Earth's rotation at the equator. The simulator reports the total loss for each flight.
Why use stages instead of a single stage to orbit?: Reaching 9.4 km/s on kerosene and oxygen (v_e ≈ 3 km/s) in one stage needs a mass ratio of e^(9.4/3) ≈ 23. That means 96% of the liftoff mass is propellant, leaving 4% for tanks, engines and payload, which is extremely hard to build. Staging drops each empty stage so the rest of the rocket stops carrying dead weight, and the total Δv is the sum of the stage Δvs.
What is Max-Q and why does it matter?: Dynamic pressure q = ½ρv² measures how hard the airflow pushes on the vehicle. Speed rises after liftoff while the air thins, so their product peaks at roughly 10–15 km altitude. That peak is Max-Q, typically around 30 kPa. It is the moment of highest aerodynamic load, and many rockets throttle down around it. The orange dot in the simulator marks where it happens.
How accurate is the simulation?: It is a teaching model: 2D, a spherical Earth, an exponential atmosphere with an 8.5 km scale height, a fixed drag coefficient, and constant Isp and thrust per stage with no throttling. The preset rockets use approximate public figures. It correctly shows how gravity turns, staging, Max-Q and orbit insertion work, but it is not a mission-design tool.

Features

Solve the Tsiolkovsky equation for any of its four variables, with ten common engine presets
Up to four serial stages: per-stage Δv, TWR, burn time and liftoff mass
Δv budget bar for LEO, GTO, GEO, trans-lunar injection, Mars transfer and Earth escape
2D round-Earth launch with thrust, gravity, exponential-atmosphere drag and Earth's rotation, integrated with RK4
Altitude, speed, dynamic pressure (with Max-Q marked) and g-load plots, plus periapsis, apoapsis and leftover Δv
Gravity turn with optional closed-loop upper-stage guidance and an automatic kick-angle search

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