Rocket engine
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A machine that burns fuel and shoots the hot gas out the back, so the push in the other direction moves the rocket.
Overview
There is no air in space, so a rocket carries not only fuel but also the oxygen (called the oxidizer) to burn it with. The engine mixes the two, burns them, and sends the gas out through a narrow nozzle. It works like a balloon you blow up and let go: it flies away from the side the air rushes out of.
Rockets mostly burn kerosene, liquid hydrogen or methane. Kerosene is easy to handle but leaves soot. Hydrogen is very efficient but must be kept extremely cold, so it is hard to work with. Methane leaves little soot, which makes it good for reusing an engine many times, so it is common in new reusable rockets.
An everyday example
An engine is the heart of a car, and it is the heart of a rocket too. It is the hardest part of a rocket to build, so whether you can recover an engine and fly it again changes the cost of a launch a great deal.
Why it matters
When a new rocket is running late, the engine is often the obstacle. That is why news that an engine passed its test firing is treated as a big milestone for a rocket program.