🚀 Tsiolkovsky's Rocket Equation

Written and maintained by the PhDino author · Last reviewed 21 September 2026 · Checked against 3 independent reference calculations · how PhDino checks its numbers

Why a rocket needs to be mostly propellant, and how efficiently it converts that propellant into velocity change.

A rocket in space has no road, water, or air to push against — the only way it can accelerate is by throwing part of its own mass (propellant) out the back at high speed, gaining velocity in the opposite direction as a consequence of momentum conservation. The faster the exhaust leaves and the more of the rocket's mass gets thrown out this way, the more total velocity change (delta-v) the rocket can achieve.

Specific impulse (Isp) is the standard way of describing how efficiently an engine converts propellant into thrust — expressed, by a somewhat historical convention, in seconds regardless of unit system. A higher Isp means more delta-v extracted from the same amount of propellant mass, which is why more efficient (but often more complex) engines like ion thrusters trade lower thrust for dramatically higher Isp than chemical rockets.

The equation's logarithmic relationship between mass ratio and delta-v is the reason rockets look the way they do — mostly propellant tank, with the actual payload a small fraction of the total. Doubling the delta-v you need doesn't just double the propellant required, it requires the mass ratio itself to be squared, which is why every extra bit of delta-v gets disproportionately expensive as a mission's requirements grow.

Key formula

Δv = Isp × g₀ × ln(m₀ ÷ m_f)

Variables

Isp

specific impulse — a measure of engine efficiency, in seconds

g₀

standard gravity, a fixed reference constant (9.80665 m/s²) used here regardless of where the rocket actually launches from

m₀

initial ("wet") mass, including all propellant

m_f

final ("dry") mass, after the propellant is expended

How to use the Rocket Equation calculator

Use this to link three things in a rocket: how much change in velocity (delta-v) it can achieve, how much of its mass is propellant, and how efficient its engine is (specific impulse). Fill in three of the four values, leave the one you want at 0, and it solves for it.

The equation is the reason spacecraft are mostly fuel. Delta-v grows only with the logarithm of the mass ratio, so each extra bit of velocity needs a disproportionate amount of propellant, and small differences in engine efficiency matter a great deal.

Specific Impulse, Isp (0 = solve for it) (sec)

The engine's specific impulse in seconds, a measure of propellant efficiency. Solid motors are around 250 to 280 s, kerosene and oxygen 300 to 350 s, hydrogen and oxygen about 450 s in a vacuum, and electric thrusters several thousand seconds.

Initial (Wet) Mass (0 = solve for it) (kg)

The total mass before the burn, including propellant and payload, in kilograms.

Final (Dry) Mass (0 = solve for it) (kg)

The mass after the burn, with the propellant consumed: the dry mass plus payload.

Delta-V (0 = solve for it) (km/s)

The change in velocity the burn must provide, in kilometres per second. Mission planners work out the budget for each maneuver.

Worked example: propellant for an orbit-raising burn

A 1,200 kg satellite carries a thruster with a specific impulse of 320 s and must change its velocity by 1.5 km/s to reach its final orbit. How much of the launch mass must be propellant? Leave the final mass at 0 so it is solved for.

Worked by hand:

1. Effective exhaust velocity. v_e = Isp × g₀ = 320 × 9.80665 = 3,138 m/s.
2. Rearrange the equation. Δv = v_e × ln(m₀ ÷ m_f), so m_f = m₀ ÷ e^(Δv ÷ v_e) = 1,200 ÷ e^(1.5 × 1,000 ÷ 3,138).
3. Solve. The exponent is 1.5 × 1,000 ÷ 3,138 = 0.478, so the mass ratio is 1.613 and the final mass is 744 kg.
4. Propellant. 1,200 − 744 = 456 kg of propellant, which is 38% of the starting mass.

The satellite must be 38% propellant to deliver 1.5 km/s, leaving 744 kg of dry mass. Engine efficiency changes the picture: the same 1,200 kg to 744 kg burn with a specific impulse of 450 s would deliver 2.11 km/s instead.

Reading the result: the tyranny of the exponent

Because delta-v is proportional to the logarithm of the mass ratio, the propellant needed grows exponentially with delta-v. A mission that needs twice the delta-v does not need twice the fuel; it needs the mass ratio squared. This is why reaching orbit takes a launch vehicle that is 90 percent propellant, and why staging (dropping empty tanks) helps.

• The delta-v budget for a launch to low Earth orbit is about 9.4 km/s once gravity and drag losses are included, well above the 7.8 km/s orbital speed itself.
• Higher specific impulse is the most direct fix: it multiplies the delta-v for the same mass ratio. That is why electric propulsion, with tiny thrust but enormous efficiency, dominates long deep-space missions.
• Separate stages each obey the equation, and their delta-v adds, which is the whole point of staging.
• The equation assumes the exhaust velocity is constant and ignores gravity and drag during the burn; for a launch those losses must be added to the budget separately.

Notes & limitations

• This describes a single stage burning continuously — a multi-stage rocket (nearly all orbital rockets) applies this equation separately to each stage, since dropping spent stages partway through flight is exactly what makes multi-staging so effective: the rocket stops hauling along tankage it no longer needs.
• g₀ here is a fixed reference constant tying Isp's conventional "seconds" unit back to a real velocity, not the actual local gravity the rocket experiences — this is a historical convention of how Isp is defined, not a physical statement about where the rocket is.

Common mistakes

• Forgetting that the final mass includes the payload and the structure, not just the empty tank. Only the propellant counts as the mass that is burned.
• Mixing kilometres per second with metres per second. The calculator uses kilometres per second for delta-v.
• Confusing specific impulse in seconds with exhaust velocity in metres per second; they differ by the factor g₀ = 9.80665.
• Using vacuum specific impulse for a burn at sea level, where the engine is less efficient.
• Ignoring gravity and drag losses on a launch, which add up to a large fraction of the total delta-v.

Frequently asked questions

Papers worth reading

A method of reaching extreme altitudes Goddard, R. H. (1919), Smithsonian Miscellaneous Collections. Goddard’s early quantitative analysis of how efficiently a rocket turns propellant into altitude, and how much of its mass has to be propellant.