Rocket equation calculator: why is it so hard to reach orbit? • The Science 360

By The Science 360 · Updated October 2026 · 9 min read

Why is it so hard to reach orbit, and how much fuel does it take?

Reaching low Earth orbit needs a speed change of about 9.4 km/s, and the rocket equation says every extra km/s costs exponentially more fuel. Orbital speed itself is 7.8 km/s, and about 1.6 km/s more is lost to gravity and air drag on the way up. The equation is Δv = vₑ ln(m₀ ÷ m_f): the speed change equals the exhaust speed times the natural logarithm of the ratio of full mass to empty mass. With hydrogen and oxygen (4.4 km/s) a rocket needs a mass ratio of about 8.5, which means roughly 88% of it must be propellant. With kerosene and oxygen (3.0 km/s) it would need 23.0, which no single stage can build.

The simulation above uses a fuel share you can set, an engine, and a structure that is 8% of each stage's propellant plus structure, a good modern value. The one-stage and two-stage rockets carry the same propellant, structure and payload; the second one simply drops its empty first stage. Press "Fire the engines" to watch the burn (time-lapsed: a real ascent takes about 8 minutes) and compare the Δv each reaches with what orbit, the Moon and Mars need.

Key results from the simulation

Exponential cost: Δv = vₑ ln R. For kerosene (3.0 km/s) each extra 3.0 km/s multiplies the mass ratio by e = 2.72. Going from 6.4 to 9.4 km/s multiplies it by about 2.7.
One stage and kerosene cannot reach orbit: even with 92% of the rocket as fuel and no payload, 7.6 km/s is all it gets, 1.8 km/s short.
Hydrogen just makes it: a single hydrogen stage with 88.2% propellant reaches 9.4 km/s, leaving only 4.1% of the liftoff mass for payload.
Staging fixes it: dropping the empty tank lets kerosene reach orbit with 90.2% propellant and 2.0% payload, and lets hydrogen carry 8.2% instead of 4.1%.

Six rockets, one equation

Each section gives the numbers the simulation shows for that setting and has a button that loads it, including the engine. The slider is the propellant share of the launch mass. "Payload" is everything above the tanks, so it is what is left after propellant and structure. Orbit needs 9.4 km/s, a Moon landing about 15.9 and a Mars landing about 18.7, all counted from the ground.

Half the rocket is kerosene

Mass ratio 2, Δv barely 2 km/s

With half the launch mass as propellant, the mass ratio is 2.0 and Δv = 3.0 × ln 2.0 = 2.08 km/s. That is about a quarter of what orbit needs. The payload share is large, 45.7%, because there is so much room left.

This is why rockets look so unlike aircraft: a plane carries fuel for hours at a small share of its weight, but a rocket must spend most of its mass to gain a few km/s. Two stages give 2.13 km/s here, almost no gain, because staging only pays off when the empty tanks are a large part of what remains.

Mass ratio 2.0 Δv, 1 stage 2.08 km/s Δv, 2 stages 2.13 km/s Payload 45.7%

The most a kerosene stage can hold

7.6 km/s from one stage

Fill 92% of the rocket with propellant and the structure takes the other 8%, so there is no payload at all. The mass ratio is 12.5. Δv is 3.0 × ln 12.5 = 7.6 km/s, which is 1.8 km/s short of orbit.

This is the sharpest form of the rocket-equation problem. A single stage of kerosene and oxygen cannot reach orbit even in the best case, and the shortfall shows in the bar. Real kerosene rockets are always staged for this reason. The two-stage figure at this setting is 15.1 km/s, but with zero payload, so it is not a rocket anyone would fly.

Mass ratio 12.5 Δv, 1 stage 7.6 km/s Short of orbit 1.8 km/s Payload 0%

One hydrogen stage, just reaching orbit

88.2% propellant, 4.1% payload

Hydrogen and oxygen give 4.4 km/s of exhaust speed, so 9.4 km/s needs a mass ratio of 8.5, about 88.2% propellant. Structure at 8% of the stage leaves 4.1% for payload. The bar just crosses the orbit line.

That is how thin the margin is. Make the structure a little heavier or the engine slightly weaker and a single stage no longer reaches orbit, which is why single-stage-to-orbit rockets have stayed on the drawing board. The two-stage version at the same setting has 11.6 km/s, enough to spare.

Propellant 88.2% Mass ratio 8.5 Δv, 1 stage 9.4 km/s Payload 4.1%

Kerosene with a dropped stage

two stages: 90.2% propellant, 2.0% payload

Keep the same engine but let the rocket drop its empty first stage. With 90.2% propellant, split about 88% to the first stage and the rest to the second, the two-stage rocket reaches 9.4 km/s. The single stage in the left column reaches only 7.0 km/s.

The trick is that the rocket stops carrying the empty first stage and its engines. Total propellant, structure and payload are the same in both rockets, but after the first burn there is far less mass to push, so the same fuel adds more speed. This is the layout of real orbital rockets, and a payload of 2.0% is close to what they deliver. The Falcon 9 lifts roughly 4% of its liftoff mass to low orbit when the booster is not recovered.

Propellant 90.2% Δv, 1 stage 7.0 km/s Δv, 2 stages 9.4 km/s Payload 2.0%

Hydrogen all the way to the Moon

two stages: 91.3% propellant, 0.8% payload

Landing on the Moon needs about 15.9 km/s counted from the ground. Even with hydrogen, two stages that carry 91.3% propellant reach 16.1 km/s and leave 0.8% of the launch mass as payload, while one stage manages 10.7 km/s.

The real Apollo missions solved this with three stages, and a landing craft that burned propellant again at the Moon. Every kilogram landed on the Moon costs hundreds of kilograms on the launch pad. The same arithmetic is why Mars, at about 18.7 km/s, needs refuelling in orbit or a very large rocket. For the flight itself, see the Mars launch window simulator.

Moon needs 15.9 km/s Propellant 91.3% Δv, 2 stages 16.1 km/s Payload 0.8%

An ion thruster: huge Δv, no liftoff

20.8 km/s from half fuel

An ion thruster throws its propellant out at about 30 km/s, seven to ten times faster than a chemical rocket. With half the launch mass as propellant it reaches 20.8 km/s, more than the Moon needs. Reaching orbit takes only a mass ratio of 1.37, about 27% propellant, leaving 71% for payload.

It cannot be used on the ground. An ion thruster pushes with a force of around a tenth of a newton to a few newtons, far less than the weight of the spacecraft, so it can never lift off. It is used in space, where a gentle push applied for months adds up. The Dawn probe got about 11 km/s this way. See the future of space exploration.

Exhaust speed 30 km/s Mass ratio 2.0 Δv, 1 stage 20.8 km/s Orbit needs 1.37:1

What each engine needs to reach low Earth orbit

Engine	Exhaust speed	Mass ratio needed	Propellant, 1 stage	Payload, 1 stage	Propellant, 2 stages	Payload, 2 stages
Kerosene	3 km/s	23.0 : 1	impossible	none	90.2%	2.0%
Hydrogen	4.4 km/s	8.5 : 1	88.2%	4.1%	84.4%	8.2%
Ion	30 km/s	1.4 : 1	26.9%	70.8%	26.7%	71.0%

Values come from the simulation: 9.4 km/s from the ground, structure at 8% of each stage's propellant plus structure, and an optimal split of propellant between two stages. "Impossible" means that even a rocket with no payload falls short. Ion thrusters are shown for the arithmetic only: they cannot lift off. See Top 20 interesting facts about our solar system for the scale of the planets we are trying to reach.

How the simulation works

Tsiolkovsky derived the rocket equation in 1903. A rocket in empty space that ejects propellant at speed vₑ changes its own speed by Δv = vₑ ln(m₀ ÷ m_f), where m₀ is the mass at the start and m_f the mass after the propellant is spent. The slider sets the propellant share ζ of the launch mass, so the mass ratio is 1 ÷ (1 − ζ). At ζ = 85% the ratio is 6.7 and kerosene gives 5.7 km/s, hydrogen 8.3 km/s.

Everything that is not propellant is dry mass. In the model the structure (tanks, engines, plumbing) is 8% of a stage's propellant plus structure, so a stage holding propellant mass P has structure 0.087 P. The payload is what remains: 1 − ζ ÷ 0.92. Real stages range from about 4% to 10%, so this is a representative middle value. The 9.4 km/s that orbit needs is 7.8 km/s of orbital speed plus 1.6 km/s of gravity and drag losses, which depend on the trajectory and are taken as fixed. Orbital speed alone means 30.4 MJ for every kilogram, about the energy in 0.7 kg of kerosene, which is part of why rockets are so large. For the scale of energies like this see this much energy can blow up the entire Earth.

The two-stage rocket carries the same total propellant, structure and payload. The first stage burns P₁ and is dropped with its structure, then the second burns the rest: Δv = vₑ [ln(1 ÷ (1 − P₁)) + ln(m₂ ÷ (m₂ − P₂))], where m₂ = 1 − P₁ ÷ 0.92. The simulation picks the split P₁ that maximises Δv. The burn on screen is time-lapsed to about 6 seconds. During it the bars follow Δv at each moment as propellant drains, and the first stage falls away at the moment its tank is empty.

What the model leaves out

Gravity and drag losses are fixed. The 1.6 km/s losses depend on how quickly the rocket gets off the pad and on its shape. A rocket with weak thrust loses more, so a real design must also have a thrust-to-weight ratio above 1 at liftoff, which the model does not check.
Constant exhaust speed. Engines are more efficient in vacuum than at sea level. Real exhaust speeds vary by 10 to 15% between the two, and the model uses a single typical value.
Fixed structure fraction. Real structure share changes with size, fuel density and design. Hydrogen is light but needs big tanks, so a hydrogen stage is bulkier than a kerosene stage of the same mass.
Only two stages. Real rockets use two or three, and the Saturn V used three. The model shows the idea but not the diminishing return from the third.
No recovery or reuse. Reusable boosters keep part of the structure and fuel for landing, which lowers the payload but cuts the cost.

Common misconceptions

"Orbit is about height." Getting 200 km up takes only a small part of the effort, about 2 MJ per kilogram. Staying there needs a sideways speed of 7.8 km/s, which takes about 30 MJ per kilogram. Most of the rocket is spent going sideways, not up.

"A bigger rocket gets proportionally more." Doubling the fuel does not double the speed. Because of the logarithm, the second half of the fuel adds much less Δv than the first, so a bigger rocket has to be vastly bigger, or staged, to go a little faster.

Frequently asked questions

What is the rocket equation?

It is Δv = vₑ ln(m₀ ÷ m_f): the speed a rocket gains equals its exhaust speed times the natural logarithm of its full-to-empty mass ratio. It was derived by Konstantin Tsiolkovsky in 1903 and holds for any rocket, however it is built.

Why is it so hard to reach orbit?

Orbit needs a sideways speed of 7.8 km/s, plus about 1.6 km/s lost to gravity and air drag, 9.4 km/s in all. Chemical exhaust speeds are only 3 to 4.4 km/s, so the mass ratio must be 8.5 to 23.0, and nearly all of the launch mass is propellant.

How much fuel does a rocket need to reach orbit?

A single stage with hydrogen needs about 88% of its liftoff mass as propellant, and a kerosene stage cannot do it at all. Real two-stage rockets are roughly 90% propellant at liftoff. For example, a two-stage kerosene rocket needs about 90% in this model.

Why do rockets have stages?

Because the empty tanks and engines of a finished stage are dead weight. Dropping them means the remaining fuel pushes a smaller mass. In the simulation, the same propellant, structure and payload give 9.4 km/s with two stages instead of 7.0 km/s with one, at 90.2% propellant and kerosene.

Why can't ion engines launch rockets?

Their exhaust is very fast (about 30 km/s) but the thrust is tiny, far less than the weight of the spacecraft, so they cannot lift off. They are used in space, for example on the Dawn probe, where a small push applied for months adds up. See the future of space exploration.

How much delta-v does it take to get to the Moon or Mars?

Counted from the ground, about 15.9 km/s for a Moon landing and about 18.7 km/s for a landing on Mars with rockets alone, against 9.4 km/s for low Earth orbit. These are approximate and depend on the route; aerobraking lowers the Mars figure. See the Mars launch window simulator for the transfer part.

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