The Rocket in Empty Space
A small rocket floats at rest beside a satellite in deep space, where there is nothing to push against. It burns its fuel as a series of separate puffs of hot gas, each thrown backward at speed u relative to the rocket. Four sliders set the exhaust speed, the mass of fuel, the size of each puff and the empty mass of the rocket. Press play to watch the puffs leave one by one until the fuel is gone, and read the two momentum bars: the rocket's and the gas's grow together, equal and opposite, and their sum never leaves zero. Drag the timeline to any moment of the burn. Distances travelled are drawn 600 times shorter so the rocket stays in view; every number is true.
इस सिमुलेशन का उपयोग कैसे करें
- Exhaust speed u — 200 to 3000 m/s, starting at 2000 m/s: how fast each puff leaves, relative to the rocket.
- Fuel mass — 5 to 40 kg, starting at 20 kg.
- Puff size Δm — 0.5 to 5 kg, starting at 1.0 kg; when the fuel does not divide evenly, a last, smaller puff burns what is left.
- Empty rocket mass — 40 to 150 kg, starting at 80 kg. The rocket is drawn larger for a larger total mass, and a change keeps the timeline where it is.
- Play/pause, back to the start, a timeline you can drag to any moment, and ½×, ¼×, ⅛× slow motion.
- Drag anywhere on the scene to look around.
क्या देखें
- The total momentum reads 0 at every moment: at rest, after each puff and while the rocket coasts.
- At the start values the rocket ends at 443.8 m/s, carrying +35 504 kg m/s; the 20 kg of gas carries −35 504 kg m/s.
- Every puff flies backward far faster than the rocket ever moves forward, from 1980 m/s for the first to 1556 m/s for the last: 20 kg of fast gas balances 80 kg of slow rocket.
- Halve the exhaust speed to 1000 m/s and the final speed halves to 221.9 m/s. With 40 kg of fuel the rocket reaches 806.8 m/s; a 150 kg empty rocket reaches only 249.5 m/s.
- Later puffs add more speed than early ones, 20.0 m/s from the first and 24.7 m/s from the last; forty 0.5 kg puffs give 445.0 m/s, four 5 kg puffs only 434.0 m/s.
इसके पीछे की भौतिकी
Take the rocket and all of its exhaust as one system. With no gravity and no air, nothing outside acts on it, so its total momentum stays at the 0 it started with: Δp rocket = −Δp gas. Each puff of mass Δm leaves at velocity v′ − u, where v′ is the rocket's velocity just after the puff separates. Conservation for that one puff, m v = (m − Δm) v′ + Δm (v′ − u), gives v′ = v + Δm·u/m, so each puff adds a little more speed than the one before, because the rocket it pushes is lighter. With 20 kg of fuel in twenty 1 kg puffs on an 80 kg rocket, v = u(1/100 + 1/99 + … + 1/81) = 0.2219u: 443.8 m/s for u = 2000 m/s, with +35 504 kg m/s in the rocket and −35 504 kg m/s in the gas. Smaller puffs come closer to a smooth burn, u ln(100/80) = 446.3 m/s. The rocket pushes the gas and the gas pushes the rocket; lifting off from the ground also needs the thrust to be greater than the weight.