The anchor would need to be beyond geostationary orbit to keep the center of mass geostationary, so a broken tether would result in the anchor departing "outward". The reason to use an anchor is to avoid creating a tether that's twice as long as it needs to be.

Depends where the tether breaks. If it's somewhere along the middle, then the Earth-side section would fall to Earth. KSR's Mars trilogy examines the impacts (pun intended) of this.

Even an optimal space elevator needs to support a sizable portion of its own tether weight with the tether itself.

For a solid non-magnetic tether to be at all realistic, the tether material would likely be so light relative to it's length/volume, it'll never be at all dangerous regardless of how high you drop it from - its terminal velocity would be tiny.

I can drop some yarn, fishing line, whatever, from whichever height I want and it will never be dangerous to anything on the ground. Same principle.

I dont know about magnets, but I suppose the same applies here: If your tether isn't light, its own weight will add a stupid amount of stresses that would likely deform any load-bearing metal. Probably the magnets themselves in this case.

But I thought the whole point of an elevator is you elevate things. Which means the line falling wouldn't be the biggest problem- that would be the payload falling to earth surely?

The falling line would be a problem still - using fishing line as an example ignores scaling. The seed cable for example in some designs is 20 tons by itself: https://en.wikipedia.org/wiki/Space_elevator_construction

So possible 100s to thousands of tons falling for 5+ days…

Spread out over some drop area that's still just a nuisance. Even falling on one area chances of loss of life are minimal, since due to the gradient of gravity accelerating the top end of the wire the least. It should experience way too much drag to have some sort of whip effect on the ground rather than entering a stable configuration before arrival. You'll see it more-or-less neatly arriving in the right order. The first few grams of material landing on your house is maybe a good warning to get out of the way before the remaining 20 tons are done arriving in a few days.

If you have some thin wire design you could also consider just spooling it up as it drops, either at the base or with portable infrastructure you'll have plenty of time to deploy. If you do that you're just dealing with grams of material/second that you can deal with piecewise. Do this faster than terminal velocity and it'll land exactly where you want it to.

That part is not exactly an engineering challenge if you consider what other other stuff humanity likes to get up to with kilometers of much heavier cables and chains.

There shouldn't be much of an issue with designing the the payload as an entry vehicle. Give it the ability to destructively remove itself from the tether, then parachute down. Or just build your tether next to a body of water and have the payload steer into that.

That is if your payload wasn't so high up, it's now on its way to an orbit around earth. I think physically the place of departure should be the periapsis of its orbit, so even with orbital decay through atmospheric drag you'll have long enough to figure where to steer it.

If you use an asteroid as an anchor, then you are doing it wrong. You made your space elevator half as long as it could be!

A longer elevator can fling you into the outer solar system. Just climb to the correct height, wait until the right phase angle, and let go.

But if you break the tether halfway down the bottom half falls at 16,000 mph right? And then it’s burning and then it cracks like a whip. I don’t know about extinction, but not a fun time for anyone.

Aren’t they talking about a mess up on the asteroid trajectory? There is no way a space elevator cable snapping wipes out all life on earth, although I’m sure any populated areas it hits would be quite devastated.

A falling elevator would not wipe anyone out. They would place a protective balloon containing approximately 5 quadrillion tons of gas beneath it. The elevator would be paper thin and a couple of meters wide in the middle where it needs to be strongest. The parts that don’t burn will flutter and slowly fall to the ground.

Not 16,000 mph. A very lightweight tether (thing tie down straps) will drift down and land softly.

might be a good idea to have it (explosively?) disconnect it's segments if it's found to be broken. You'll spread the impact but avoid something like a whip crack.