If I'm on an elevator that breaksloose and plummets down theshaft, can I avoid harm by jumpingat the last second?

First of all, elevators never plummet down their shafts. For the past century, elevators have had a backup break that automatically engages when an elevator starts to fall. If all the cables snapped (highly unlikely), the elevator would only fall a few feet before the safety breaks would activate. The safety breaks are mechanical so that they work even if power is out or no one is around. Secondly, even if a terrorist managed to saw off the huge metal breaks and cut all the cables, the plummeting elevator would land on a cushion of air at the bottom. The elevator shaft traps air much like a giant air bag, which would soften the blow. But the essence of the original question is still interesting.

Jumping at the last second before a plummeting elevator hits the ground would slightly reduce your speed, but not enough to make much of a difference. Public domain image, source: Library of Congress.Imagine that you are standing on a high, heavy, outdoor platform (with holes in it so that air resistance is negligible). Suppose the platform's supports break so that you and the platform fall together. Could you minimize the damage of hitting the ground by jumping off the platform at the last second? If we neglect air resistance, the answer is a weak yes. But because you are both in free fall, you would not really be “jumping” off the platform, but would be pushing it away. By the law of conservation of momentum, pushing the platform downwards will give it some of your momentum and slow you down. (Think of a speeding ice skater pushing another stationary ice skater. The speeding skater slows down.) The platform has to be heavy in order to be able to give it much of your momentum. Pushing against a penny won't do much unless you shoot the penny off at high speed using a gun. But pushing away a heavy platform will give it more momentum.