Walking pushes the ground. A boat pushes water, an aircraft pushes air. Space offers none of these.
The only way to move is to throw part of yourself backward. The momentum that goes back with it exactly balances the momentum you gain going forward.
The total stays at . That is conservation of momentum, and it is the whole reason a rocket moves.
So how much speed can be gained is settled by what you can throw, and how fast.
There is an awkwardness in this. Each throw also makes the rocket lighter.
Lighter, it is pushed further by the same throw. So each later throw does more than the one before. Adding all of that up produces a logarithm.
, with the exhaust speed, the starting mass and what is left. Tsiolkovsky wrote it in .
Only the ratio of the masses appears. A large rocket and a small one with the same ratio gain the same velocity change.
Gaining only as a logarithm means, turned around, that the mass ratio required grows as the exponential of the velocity change.
To gain the exhaust speed itself, two thirds of the mass must be propellant. For twice it, six sevenths. For three times, nineteen twentieths.
A good chemical rocket exhausts at about km per second, and low orbit around the Earth costs , so the ratio exceeds two doublings. That is why a launch vehicle is almost entirely propellant.
Raising the exhaust speed eases everything. An ion engine exhausts at km per second, but pushes so weakly that it must run for months.
Once a tank is empty, carrying it further is waste. Drop it and the remaining propellant pushes something lighter.
That is staging. Each separation lowers , so the same propellant yields a larger velocity change.
Reaching low orbit on a single stage is technically possible but leaves almost no payload. Everything that actually flies has two stages or three.
Tsiolkovsky set out the idea of dropping stages in the same paper.
The vocabulary of this chapter is now settled. Every destination has a velocity change attached, and that is its price.
From the surface of the Earth to low orbit, km per second. From there to escape, . On to the orbit of Mars, . Add them and you have what the journey costs in propellant.
Because propellant enters exponentially, saving even km per second changes the launch mass by tens of percent. That is where the value of a cleverer trajectory lies.
The following articles look at those trajectories, beginning with the speed required never to come back.