IMPULSE
THE PHYSICS OF SPACEFLIGHT, FROM FIRST PRINCIPLES TO FUTURE ENGINES

A propeller pushes against air. A car's tyres push against the road. A rocket has nothing to push against once it leaves the atmosphere — and yet it accelerates faster in vacuum than it does at sea level. The answer is Newton's third law, applied to the simplest possible system: a machine and the mass it throws away.

Conservation of momentum doesn't care what's outside the machine. Throw propellant out the back at high speed, and the vehicle gains velocity in the opposite direction — in a vacuum, in an atmosphere, in orbit, anywhere. A rocket is, physically, nothing more than a very fast, very controlled way of throwing mass.

Integrating the momentum balance from Chapter 1 gives the single most important equation in spaceflight, derived by Konstantin Tsiolkovsky in 1903 — decades before anyone had flown anything. It says the velocity a rocket can gain depends only on how fast it throws propellant, and what fraction of its own mass that propellant is.

The brutal part is the logarithm. Doubling your mass ratio doesn't double your Δv — it only adds a fixed amount. To go faster, you either need a much higher exhaust velocity, or a mass ratio that grows exponentially, mostly as propellant carried just to accelerate more propellant.

Thrust is just the rate you throw mass, multiplied by how fast you throw it. A fire hose and a rocket engine obey the identical relationship — mass flow times exhaust velocity — the only real difference is how fast the exhaust actually leaves.

Specific impulse, I_sp, restates exhaust velocity in seconds — how long one kilogram of propellant can produce one kilogram-force of thrust. It's the fuel-economy number of rocketry: a low-Isp engine is a muscle car, huge thrust for a short time; a high-Isp engine sips propellant through a very long, gentle burn.

Low Earth orbit only requires around 7.7 km/s of horizontal velocity — plug Earth's parameters into the circular-orbit formula below and that's what falls out. Real rockets need closer to 9.4 km/s of Δv to get there, because climbing out of the atmosphere costs velocity you don't get to keep: gravity losses from flying mostly upward before you're moving fast, and drag losses from pushing through air.

Staging exists because of Chapter 2's logarithm. A single-stage rocket has to carry the dead weight of its own empty fuel tanks all the way to orbit. Drop that structure the moment it's empty, light a fresh, lighter rocket on top, and its mass ratio starts over — with a huge head start in velocity already banked.

Newton's own thought experiment still holds: fire a cannonball fast enough horizontally, and by the time gravity has pulled it down, the curved Earth has fallen away beneath it by the same amount. It never lands — it's still falling, forever missing the ground. Kepler had already found the resulting paths are ellipses, a law ahead of Newton explaining why.

One formula, vis-viva, gives the speed anywhere on any orbit — circular or wildly elliptical — using nothing but the current distance and the orbit's overall size. It's the equation behind every transfer-orbit calculation ever made, including the ladder of costs in the next chapter.

Every chemical engine, from the V-2 in 1944 to the engines flying today, is limited by the same thing: how much energy is released when a fuel and oxidiser react, and how much of that energy converts into exhaust speed. There is a ceiling near 4.5 km/s of exhaust velocity for anything that burns — set by chemistry itself, not by engineering.

The propellant pair closest to that ceiling is hydrogen and oxygen, burned in engines like the Space Shuttle's SSME. Everything flown since — Merlin, Raptor, RD-180 — has mostly optimised cost, reusability, and thrust-to-weight instead, because the exhaust-velocity ceiling itself hasn't moved since the 1960s.

Ion and Hall-effect thrusters don't burn anything — they use electric fields to accelerate charged propellant, usually xenon, to exhaust velocities chemistry can't touch, ten times higher than the best hydrolox engine. The price is thrust: an ion engine pushes with roughly the force of a sheet of paper resting in your hand.

That trade only makes sense away from a planet's gravity well, where there's no rush — NASA's Dawn spacecraft spent years slowly spiralling outward on ion thrust alone, gaining more total Δv over its mission than most chemical rockets ever will, one whisper of thrust at a time.

Nuclear thermal propulsion swaps the combustion chamber for a fission reactor: pump liquid hydrogen through the reactor core, let it flash to superheated gas, and expel it through a conventional nozzle. No chemical reaction is doing the work, so there's no chemical ceiling — NASA and DARPA's DRACO programme targets roughly double the exhaust velocity of the best hydrolox engines.

Nuclear electric propulsion goes further: a reactor generates electricity instead of heat, feeding ion or Hall thrusters with far more continuous power than any solar array could manage past the orbit of Mars — the only realistic way to run high-power electric propulsion in the outer solar system, where sunlight is too weak to matter.

Reaching even the nearest star within a human lifetime needs Δv budgets and exhaust velocities nothing on this page's chart can supply. Fusion engines, like the conceptual Direct Fusion Drive, aim to push thrust and exhaust velocity up together rather than trading one for the other — Isp in the tens of thousands of seconds, without the dead-weight reactor shielding nuclear thermal propulsion still carries.

The most radical concepts skip propellant altogether. Breakthrough Starshot proposes pushing a gram-scale sail to roughly a fifth of the speed of light with a ground-based laser array — a rocket equation with an exhaust velocity of light itself, and a mass ratio of one, because nothing has to be carried at all.