Rocket Propulsion
Throw mass backwards fast enough and you go forwards, with nothing to push against.
Lesson at a glanceShow the key points
- A rocket ejects exhaust at speed u relative to itself at a mass rate r = −dM/dt.
- Momentum conservation gives the thrust ru and acceleration dv/dt = ru/(M₀ − rt), which grows as mass falls.
- Starting from rest (no gravity): v = u ln[M₀/(M₀ − rt)] = u ln(M₀/M).
- Doubling the fuel fraction does not double the final speed: the dependence is logarithmic.
In space there is nothing to push against. A rocket pushes against its own exhaust: momentum conservation in its purest form.
Setting up
Fuel burns and hot gas is expelled backwards through a nozzle. Let the rocket plus remaining fuel have mass at t = 0; gas leaves at a steady rate , at speed relative to the rocket. At time t the mass is and the speed is v. Ignore gravity and air.
A momentum balance
In a short time Δt, a mass of gas leaves. Relative to the ground it moves at , while the rest of the rocket speeds up to . No external force acts, so
Divide by Δt and let it shrink:
Integrate from rest at t = 0:
A rocket’s exhaust speed is 2.5 km/s and 75% of its launch mass is fuel. (a) What speed does it reach after burning all its fuel, ignoring gravity? (b) If it burns 50 kg/s and starts at 10 000 kg, what is its initial acceleration?
- 1Thrust = ru; acceleration ru/M grows as fuel is spent.
- 2From rest, without gravity: .
- 3Final speed is logarithmic in the mass ratio; staging helps.
Check yourself
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