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This is possible via the conservation of momentum. A rocket ejects matter from its back and this matter carries momentum. To conserve momentum the rocket must gain an equal amount of momentum in the opposite direction. Since momentum is related to velocity this causes the rocket to move. This is basically an application of Newton's third law.

You might have been confused with the situation in air. For example an airplane moves through air by pushing air past it (using its propeller). This accelerates the air, and thus increases its momentum. The airplane must therefore gain momentum in the opposite direction, and accelerates itself as well.

The only difference between the airplane and the rocket is that the rocket carries the stuff it accelerates (and ejects) while the airplane can use the air that is already present all around it.

Note that while this is the basic explanation, the situation with the rocket is made somewhat more complicated by the fact that as the rocket loses fuel it also loses momentum because momentum is velocity times mass and the lost fuel lowers the mass of the rocket. You might want to look up Tsiolkovsky's rocket equation if you want more information.

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