The concept behind a space elevator would be the many massive payloads can be hauled into orbital space with very little energy use as compared to the present use of large rocket boosters. It would also not be necessary to subject human occupants to 9 Gs of acceleration to get to space.
When the space elevator is severed from the space station, the station falls in the film as if the elevator is supporting it. In real life, the system of elevator - station - counterweight should have its center of mass on the geosynchronous orbit, and if the connection is severed the new center of mass should fling the station out into space.
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