I guess the concept of a space elevator is pretty well known. The idea, first published by Konstantin Tsiolkovsky in 1895, and popularized (among others) by Arthur C. Clarke in The Fountains of Paradise, is to have a geosynchronous satellite with a very strong and light cable hanging down to the ground at some anchoring point on the equator circle, and a counterweight extending outward in space to keep the center of mass at the rigth geosynchronous altitude (this variant is due to Yuri N. Artsutanov). Whether it can be done for Earth remains doubtful, as we do not (yet?) know of materials that can sustain the strain (afaik), but the concept is interesting and has been reused in several science-fiction stories, for Earth and other planets.
In at least two of the novels I read that use the concept, the elevator cable breaks, or is broken, below the center of mass and the cable falls back to the ground (with or without the car). But these two novels do not seem to agree on how it falls.
In one novel, the cable falls in a heap on its ground anchoring place. In another, it falls on a big circle (the equator), making a "straight" line across the planet surface, though I do not recall whether it is forward (ahead of its anchor, with respect to the planet rotation) or backward (behind its anchor, with respect to the planet rotation). The planet may not be Earth.
My knowledge of mechanics no longer being was it may once have been, I am not sure I can analyze the problem correctly. I seriously doubt the cable would fall as a heap onto its anchor (or that the car would do that if it were to get loose from the cable, as suggested in one novel).
So my question is what are the machanical laws of the phenomenon. and where do the car and the cable fall and how.
They could fall ahead of the anchoring point, or behind the anchoring point (with respect to earth rotation). The cable could be taut on the equator, or zigzaging because it fell too fast with respect to earth rotation. It could start falling in one direction (forward or backward) and later reverse the other way for the remaining span. I just have no real idea of what might happen, and I wonder how it is to be analyzed.
I tried to get a first understanding by considering the car alone getting loose from the cable, and I am putting it in a first tentative partial answer, so that this question does not get too long. My conclusion for the car felt counter-intuitive at first, becoming obvious in retrospect. But what of the cable?