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In rocketry, staging is the use of multiple independent rockets to reduce the total amount of mass that needs to be accelerated. As the rockets, known as stages, run out of fuel, they are discarded.
With this system the final mass of the rocket is lower than it would otherwise be, as empty fuel tanks and no longer used engines are thrown away. On the downside, staging requires you to loft engines which are not being used until later, as well as making the entire rocket more complex and harder to build. Nevertheless the savings are so great that every rocket that launches payloads into orbit uses staging.
Many rockets use linear staging, in which a number of rockets are stacked on top of each other and fire one after the other. An example of such a rocket is the Saturn V. In order to increase the efficiency of the staging, the "upper stages" were fueled by hydrogen, meaning there was much less mass to lift than had they used kerosene.
Alternatively, stages are overlapping, i.e. the "next" stage fires before the previous one is disconnected, or even right at the start.
Examples include:
Several attempts have been made to build completely parallel stages, in which an economy of scale could be achieved by using a large number of identical stages strapped together into a bundle. The most complete was the OTRAG project, which failed for political reasons.
In more recent times the usefulness of the technique has come into question. As the costs of space launches appear to be almost entirely the operational costs of the people involved (as opposed to fuel or other costs), reducing these costs seems like the best way to lower the costs. Since staging is expensive in terms of manpower, a new movement has concentrated on SSTO designs that have no stages.