The rocket equation and delta-v

In 1903 a deaf schoolteacher in Kaluga, Konstantin Tsiolkovsky, wrote down the equation that still sizes every rocket. It explains why a Falcon 9 is about 89% propellant, why the payload is only a few per cent of the weight at liftoff, and why rockets drop parts of themselves on the way up.

Every rocket is a race between the speed you want and the mass you must carry to get it.

Look

A rocket moves by throwing mass backwards. Throw it faster, or throw more of it, and you go faster. There is a catch: the fuel you have not burned yet must also be carried and accelerated. That makes the gain logarithmic: to add the same amount of speed again, you must multiply the fuel, not add to it.

Delta-v (Δv, “change in velocity”) is the currency of spaceflight. Each manoeuvre has a price in km/s:

• Earth’s surface to low orbit: about 9.4 km/s, including losses.
• Low orbit to geostationary orbit: about 3.9 km/s.
• Low orbit to leaving Earth for Mars: about 3.6 km/s.

A rocket’s budget is how much Δv its engines, fuel and mass can deliver.

Understand

Δv = I_sp · g₀ · ln(m₀ / m_f)

• I_sp: specific impulse, a measure of engine efficiency in seconds (exhaust speed divided by g₀).
• g₀ = 9.80665 m/s², standard gravity.
• m₀: mass at the start of the burn; m_f: mass at the end.

Example with the second stage of the simulated rocket (Isp 348 s): full, it weighs about 112 tonnes including a 15-tonne payload; empty of propellant, about 19 tonnes. Δv = 348 × 9.81 × ln(112 / 19) ≈ 6.0 km/s. The first stage adds the rest.

A single stage with a mass ratio of 10 and Isp 350 s gives only 7.9 km/s, not enough for orbit once losses are counted, and a structure that is only 10% of the total is already very light. That is why nearly every orbital rocket uses two or more stages.

Master

Staging multiplies mass ratios. For a two-stage vehicle,

Δv_total = I₁ g₀ ln(m₀₁/m_f₁) + I₂ g₀ ln(m₀₂/m_f₂)

where the second stage’s starting mass excludes everything the first stage threw away. Optimising the split between stages (for equal Isp, roughly equal Δv per stage) is a classic problem: the payload fraction falls exponentially with the required Δv, which is why lowering the Δv to orbit by a few hundred m/s (launching east, near the equator) matters so much.

The equation also explains the logic of reusable first stages: they fly back using fuel that would otherwise have been payload capacity, trading a few per cent of payload for not throwing away the most expensive part of the rocket.