There’s not a lot of math to show until we start talking specifics of the elevator and know the materials we are working with. In general, space is pretty empty.

Are you taking about the orbiting station? Because that won’t crash into Earth, it will fly away (slowly) because it is orbiting faster than it should to have a stable orbit in order to balance out the elevator cable. A few thrusters can help it stay in a stable orbit if the cable breaks.

The cable itself isn’t a huge threat. It could potentially kill some people if it landed on them, but the scenario of it whipping into the Earth at supersonic speeds isn’t going to happen due to wind resistance and basic safety precautions.