Rocket Equation Calculator

The rocket equation links three quantities together: the change in velocity of a rocket during combustion, the speed at which exhaust is ejected from the nozzle, and the ratio between the initial mass of the rocket to its final mass. It has been used in one form or another ever since it was first derived by Konstantin Tsiolkovsky in 1897, and is used to calculate things such as the fuel requirements for Apollo missions to the Moon, the orbital tweaks required by Starlink satellites, and the parameters needed for a Mars transfer orbit.

If you enter any three of the four values, the fourth is automatically calculated. This calculator also displays values that mission planners must check: mass ratio, propellant burned and its fraction of total mass, and specific impulse. A loss factor converts ideal values to actual delta-v for a mission, and two graphs show why staging and high specific impulse are so important.

What the equation means:

A rocket changes its velocity by ejecting mass in the opposite direction. A jet plane must suck oxygen from the atmosphere, while a rocket carries everything it needs for combustion, such as fuel and oxidizer. So after launch it becomes lighter gradually. The Tsiolkovsky rocket equation is:

where delta-v is the change in velocity, ve is the effective exhaust velocity, m0 is the mass with full tanks and mf is the dry mass after combustion. The problem is the natural logarithm. Engineers call this the tyranny of the rocket equation because doubling delta-v does not double the fuel requirement but squares the mass ratio. A spacecraft needs about 85 to 90% of its mass in fuel to get into orbit.

Modified formula

These four quantities are interrelated so that if three of them are known the fourth can be determined. The calculation tool will determine the missing values in the corresponding fields using one of the following formulas.

If the initial mass is known, you can calculate how big a spacecraft needs to be based on the required delta-v and available engines. Conversely, calculating the exhaust velocity allows you to determine what power an engine will need if the fuel budget is set.

Here's how to use this calculator:

Enter the ejection velocity and two mass values and leave delta-v blank to calculate it. Each of the speed and mass fields has its own option for switching units so you can mix metric and imperial or specify delta-v in km/s. To calculate a different quantity, simply leave that field blank and enter the delta-v. The main visualization shows the delta-v while the auxiliary card displays the mass ratio, propellant mass and fraction as well as specific impulse.

If the engine gives specific impulse (Isp) instead of exhaust velocity, multiply that by 9.80665 seconds to get exhaust velocity in meters per second. By enabling the loss layer you can account for effects of gravity and air resistance to determine actual delta-v of the mission.

Examples of calculations:

Let's take the top row as an example. A spacecraft starts with a mass of 24,000 kg, including propellant, and ends with a dry mass of 12,000 kg. The specific impulse is 320 seconds, which corresponds to an exhaust velocity of about 3,138 m/s. Half the mass is fuel. First we calculate the logarithm of the mass ratio.

We multiply by the ejection velocity.

So burning half the mass of the spacecraft will give you a speed increase of about 2.2 km/s. Note that ratio. Even if your fuel fraction is 50 percent, the mass ratio is only 2, and its logarithm is much less than 1. This explains why it's so hard to design deep space missions.

Why mass ratio is out of control

The logarithm makes the mass ratio compensation requirement grow ever larger. To double the available delta-v is usually not enough to simply double the amount of fuel without increasing the exhaust velocity. This is why staging and high specific impulse engines are central in rocket and spacecraft design. The "tyranny curve" on the analysis panel plots this directly. One can see how the mass ratio goes from 2 to 60 as the required delta-v increases, with no change of engine type.

Specific impulse and exhaust velocity

Specific impulse is the ratio of exhaust velocity to standard gravity, with the result given in seconds. For a motor with specific impulse Isp = 450 s, the exhaust velocity would be the product of 450 and 9.80665, which is approximately 4,414 m/s. Engineers prefer specific impulse because the value remains constant whether measured in kilograms or pounds, thus canceling out units. Typical values vary widely depending on propulsion system.

Ion engines have a much higher exhaust velocity than chemical rockets but very low thrust, at most a few hundred millinewtons. So they are only suitable for slow course corrections and not for launching from Earth.

The mass ratio for a given delta-v.

If you look at it from the other perspective, you get to the same result. Let's say we fix the engine and examine the mass ratio required for a given delta-v. If the exhaust velocity of hydrogen and oxygen is 4,400 m/s, then:

At a speed of 9400 m/s, about 7.5 kg of fuel must be carried for every kilogram of payload and structural mass. This mathematical relationship is the main reason that the external tank of Space Shuttle vehicles is much larger than the vehicle attached to its side.

Multi-stage rockets - A way for launch vehicles to get around some of the limitations of equations.

Because of the extremely high mass ratio penalties for single stage rockets, all real launch vehicles use multiple stages. The equations are applied to each stage individually and the delta-v's of each stage added together. By discarding empty stages, a rocket can reduce its weight unnecessarily during ascent as it sheds structures and engines that would otherwise have to be accelerated. Run the equations for each stage then sum the delta v's. Because the dry mass changes drastically with each stage jettisoned, a multistage rocket cannot be considered in one calculation.

Realistic takeoffs - losses due to gravity and air resistance

The Tsiolkovsky equation describes an idealized state. The rocket is in a vacuum with no gravity pulling it down or atmosphere impeding its flight. In a real launch the rocket has to fight against gravity throughout its flight, which results in losses of about 500 to 1,500 m/s. Additionally air drag and course corrections result in further losses of about 50 to 150 m/s. These losses have to be factored into the delta-v budget so that the engines must produce more than just what is needed for the actual mission. By activating the loss layer these losses can be taken into account to determine the actual delta-v required for the mission.

Assumptions and limitations

The rocket equation assumes an ideal and instantaneous combustion process and ignores gravity, air resistance, course changes, finite burn effects, and throttling. It also assumes that the rocket loses mass by expelling propellant so it cannot be applied to objects whose mass does not change during flight. The delta-v given is an idealized value; use the loss-of-mass layer for estimates and perform a full trajectory simulation if accurate values are needed.