From: burton@cfm.brown.edu (Joshua W. Burton)
Newsgroups: sci.physics,sci.math
Subject: Re: Elevator mayhem
Date: 15 Sep 1992 23:44:20 GMT
Organization: Brown University Center for Fluid Mechanics
NNTP-Posting-Host: qcd2.het.brown.edu

In article <1992Sep15.211616.27615@CSD-NewsHost.Stanford.EDU> pratt@Sunburn.Stanford.EDU (Vaughan R. Pratt) writes:
>
>There's a very simple argument showing that jumping can't help.  If you
>could absorb all your kinetic energy by jumping at the right instant,
>you could with the same jump from a motionless elevator jump back up to
>where you started falling from (in the absence of braking and other
>resistance), putting you in a league with that other famous
>jumper-over-tall-buildings.  And if you could do this you would have to
>be well calibrated in this Olympic event, since if you jumped exactly
>twice as hard as was necessary your head would have the identical
>impact at the roof of the elevator that we've been trying to avoid on
>the floor.
>
>-- 
>======================================================| God found the positive
>Vaughan Pratt   pratt@cs.Stanford.EDU   415-494-2545  | integers, zero was
>======================================================| there when He arrived.

Let's keep this analysis semi-quantitative, in the spirit of my earlier
posting.  Our hero is trying to survive a fall of 100 meters at 0.8 g, give
or take a bit, and the spring at the bottom of the shaft is going to stop
the elevator in about one meter, yielding an average deceleration of 80 g's
over a period of 50 to 100 msec.  The strategies so far proposed are:

1.) Lie flat on your back and take it as it comes;

2.) Try to roll out like a paratrooper;

3.) Leap at just the right moment.

As I mentioned earlier, 80 g is a very survivable load on your skull for the
first millisecond or so (before it reaches maximum deformation and attempts
to rebound), but is near the outer limit for prolonged deceleration like
this.  The actual deceleration profile experienced is probably going to be
critically important---if we assume a perfect Hookean spring, the load will
peak at pi/2 times the time average, unless the spring bottoms out---in
which case we get a nasty acceleration spike.  In this case, or if there is
no spring, I think we can all agree that we default to solution (4), which
involves putting our head between our knees and...or, as another poster
eloquently put it:

>you might as well
>just press all the buttons and laugh, 'cuz yer hamburger. 

Now, what about solution (3), which I glibly dismissed but which seems to
appeal to a lot of respondents?  Certainly, for the reasons Mr. Pratt puts
forward, this alone isn't going to solve the problem---we have 40 m/s of
downward velocity to shed, and 3 m/s is a very respectable upward jump even
when one isn't in a falling elevator in near-freefall.  (It corresponds to
raising your center of gravity a half meter from a standing start, which is
worlds different from rolling or striding over a half-meter bar!).  Still,
that's 7% or so off our speed wrt the ground, or more than 13% off our
kinetic energy.  Is it worth it?

I still say no.  Look at it in our falling (near-inertial!) frame.  The floor
is about to start accelerating up into us, hard.  If we are on the way up
already when this happens---in the falling frame, remember!---we get some
benefit, but if we have reached the top of our jump and are on the way down
we get a comparable DISadvantage.  So to be safe we jump a mere tenth of a
second before impact, as the floor indicator flashes 3.  (Actually, we _try_
to jump between 8 and 7, allowing for a 300-msec reaction time.)  At L, the
elevator begins decelerating, slowly at first but with a ferocious jerk of
several g per millisecond.  Still falling at 40 m/sec in the rest frame, we
hit the floor a few tens of milliseconds late...*just as the deceleration is
reaching its murderous peak*.  In other words, we have only succeeded in
substituting our jump (several g from -0.1 s to 0 s) for the `easy' part
(zero to fifty g, from 0 s to +0.02 s), and we get to enjoy the `hard' part
in whatever awkward position we are caught in in mid-jump.  No, thanks.

In deciding between (1) and (2), a key insight is that the internal springs
in our body---after all, `breaking your fall' in this context means using
parts of your body you can spare to cushion your gourd---are *not* Hookean
over the full range of accelerations involved.  Our intuition is tuned to
the 5-g range, where a paratrooper or a gymnast can count on tendons to
behave elastically in tension, and muscles to do likewise in compression.
At ten or twenty times this acceleration, loose knees---and probably spines
as well---will crumple without much noticing the soft tissues in their way.
Thus if we are going to give up the fatalism and clear advantages of (1)---
first among these being that lying flat keeps your neck bone connected to
your head bone---we should stop thinking of legs as springs, and start 
thinking of them as `crush zones', like the front of a Volvo.

Maybe the thing to do is to *lock* your knees, hoping that Euler buckling
of first the tibias and then the femurs will smooth out the peak acceleration
to the skull.  Of course, this advantage is negated and more if you fall
over sideways and crack your head on the ground before you've lost most of
your downward velocity.  You could stand against one wall, with your legs
braced rigidly a few inches out.  The main problem I see is that the
shock-absorption below the pelvis has to be not merely better than just
the spring below the elevator, but actually good enough to keep your spine
(at least the part above C4 that you need to breathe with) more or less in
one piece.  I don't like the odds.

One more really desperate expedient occurs to me.  Whatever deceleration the
floor of the elevator experiences, the *roof* will stop at least as smoothly,
and more so if there is any significant crumpling of the elevator body.  You
are nearly in freefall on the way down, and should have little trouble
reaching up and grabbing a convenient fixture on the ceiling.  If it's strong
enough to break only near peak deceleration, and if you are attached to it
(belt?  wedged fingers?) tightly enough that you don't drop back to the
floor, you might be able to get some more load averaging that way.  Further,
since you are now straining expendible body parts (like your arms, or your
clothes if you are quick with knots) in tension instead of compression, the
limiting injuries become very different.  I have a perfectly horrifying
anecdote about how human tendons fail in tension, which I am certainly not
going to jeopardize my Usenet access by recounting here.

Finally, there is always the reflection that there are worse things than
dying quickly and cleanly.  A good friend of mine is a flight surgeon on an
American aircraft carrier, and a marvelous source of grisly tales with
interesting and unusual physics content.  (Remind me to tell you the one
about the candy machine someday....)  He tells me that the standard pilots'
manual describing ejection procedures gives fairly explicit directions for
getting into a head-down position in case of double parachute failure, since
a really frightening number of victims who fail to take this last precaution
live for minutes or even hours after impact.  I think that option (1) combines
the best of both worlds by guaranteeing a relatively clean escape---if any.

Cheers,
Joshua W. Burton          <really should get to work on a .signature file!>








































Newsgroups: sci.physics