If I'm on an elevator that breaks loose and plummets down the shaft, can I avoid harm by jumping at the last second?
Category: Physics Published: December 5, 2012
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.
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.
The first question to approach is: when is the best time to push off the platform? If you go through the kinematic equations (neglecting air resistance), you can indeed show that your ground impact velocity is minimized by jumping close to the last second. If you jump exactly at the last moment, the platform will not have time to receive your momentum, and hit the ground. So it is best to jump close to the last moment. Simple mathematics show that near impact your impact velocity is just your free-fall-acquired final velocity minus your jump velocity. Your jump velocity is how much upward velocity you gain by pushing away the platform. Assuming the maximum human jump velocity to be 7 mph, if your platform fell from 33 feet (about three floors), you would reduce your impact velocity from 31 mph to 25 mph by pushing the platform down with all your might near the last moment possible. If you fell from 165 feet (about 15 floors), pushing at the right moment would reduce your impact velocity from 72 mph to 65 mph. The bottom line is that “jumping” at the last moment would help, but not much. Of course, the actual damage depend not only on your impact velocity, but also on the solidity of the ground. Cement will do more damage than haystacks. These calculations neglect air resistance, which would also help slow you down. Air resistance would be another reason to stick close to the platform until late in the fall. In practical situations where the ground is soft and the platform is hard, it may be more important to jump clear of the platform early on to avoid its impact shrapnel than to try to give it your momentum.