Variable Mass and the Rocket Equation
A rocket accelerates by throwing mass backwards, so the second law has to be applied to a system whose mass is changing. Integrating gives a speed that depends on exhaust speed and the logarithm of the mass ratio.
What a learner can do afterwards
- Sets up the momentum balance for a short interval in which a small mass is ejected
- Derives the rocket equation and identifies exhaust speed and mass ratio in it
- Explains why staging beats carrying one large tank
1 · Read
A rocket carries no road, no water, and no air to push against. It moves by throwing mass backward: every second the engine shoots hot gas out the back, and momentum conservation shoves the craft forward. Freeze a short interval dt: a small mass dm leaves at the exhaust speed relative to the rocket. Balance momentum of rocket plus gas over that slice, shrink dt toward zero, and you hold a differential equation for speed.
Integrate from ignition to burnout and you get the rocket equation: delta v equals exhaust speed times the natural log of start mass over end mass. Only the ratio matters, never the absolute tons. The log is unforgiving: doubling the ratio barely moves the speed, so going a bit faster wants a lot more fuel.
The log rules mission design. Rockets fly as mostly fuel with a sliver of payload, since the mass ratio grows exponentially with the speed you want. Dropping empty tanks mid flight, called staging, stops you hauling dead weight around. Each fresh stage restarts the ratio in your favour.
Draw the system boundary around the rocket plus the gas it is about to eject. The inside push and shove of the engine then cancels out, leaving only outside forces like gravity on the force side. Get the boundary right and the engine mess becomes one clean equation.
Eject mass backward, integrate the momentum balance, and speed grows with the log of the mass ratio.
2 · Watch
Take it off screen
Where it sits
This opens up
Nothing builds on it yet.
Where this leads
Jobs that lean on this skill. Follow one to see everything it is built on.
8 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.