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A person at rest 500 km above the Earth falls straight downwards. She has a snug magical force field around her that is totally rigid and completely protects her from outside heat. The force field does not change the shape of her body and DOES NOT protect her internal organs from rapid deceleration. Can anyone give me a rough estimate of the following:

  1. The time taken to fall the first 400 km to the top of the atmosphere.
  2. The time after that to reach the ground.
  3. The maximum deceleration she would experience.
  4. Whether she would survive the fall (ignoring the final impact).

I need this for a surprise book I am writing for my daughter, which I am hoping to keep generally within the bounds of believable science.

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    $\begingroup$ Terminal velocity will also depend on the drag coefficient, mass and size of the magical field.... unless it magically has no drag. But if it has no drag and there will be no deceleration. Can you stuff her in a pickle barrel instead? That would make the problem easier. /;O) $\endgroup$
    – Woody
    Nov 19, 2023 at 23:38
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    $\begingroup$ @Mark ... maybe a spherical pickle barrel ? $\endgroup$
    – Woody
    Nov 19, 2023 at 23:50
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    $\begingroup$ It's a cool question and I would like to answer, but now that you've said "the force field can take on any aerodynamic shape" it makes the problem a lot harder. I would have opted for a ~2 m diameter spherical shape so the solution is fairly straight-forward with a density scale height of 7 or 8 km up to the "knee" and about 14 or 15 km above. Now that it's a shape-shifting field, of course it would change to a space-plane-like reentry vehicle with a huge wingspan and therefore tiny loading (kg/m^2) and use aerodynamic lift to spend hours or days slowly descending. $\endgroup$
    – uhoh
    Nov 20, 2023 at 2:51
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    $\begingroup$ For what it's worth, you might get more appropriate answers on Worldbuilding. That community is good about not just checking the "truthiness" of fictional devices, but also at suggesting alternatives. Just make sure to use the "science based" tag (or whatever it is called). $\endgroup$
    – user52756
    Nov 21, 2023 at 17:02
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    $\begingroup$ @user71659, MOOSE was intended for orbital re-entry, not vertical re-entry. Very different load profiles. $\endgroup$
    – Mark
    Nov 22, 2023 at 3:31

1 Answer 1

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Your question is under-specified (you don't give the size or posture of your subject), so I'm assuming an average-sized woman falling in the classic face-down skydiver posture. I'm also modeling this using the "perfect gas model" of reentry, which is bordering on incorrect -- there's likely to be substantial compression heating at the point of peak deceleration.

Falling the first hundred kilometers takes about 153 seconds. Nothing really interesting happens here, or for the next few minutes. Look around and enjoy the view -- if you're falling near the poles, keep an eye out for the aurora.

After about 305 seconds, you reach the Karman Line, a hundred kilometers above the surface. At this point, you're moving about 2675 meters per second, and the atmosphere is thick enough that you're experiencing about a milli-g of drag.

Acceleration builds up quickly as the atmosphere thickens, and 16 seconds later, about 57 kilometers up, you hit your peak velocity of 2800 meters per second, with drag exactly balancing gravity. Acceleration is still building, though.

Twelve seconds later, at roughly 28 kilometers up, you hit peak deceleration, a crushing 25 times the force of gravity. Wikipedia's acceleration tolerance chart suggests that this is probably not survivable falling back-first (five seconds above 20 gs, where the chart gives a limit of one second). It's definitely not survivable in any other posture. If you didn't have the forcefield, your arms and legs would be dislocated and broken by the forces involved.

The forces fall off nearly as fast as they grew, and you'll be down to a tolerable 4 gs just thirteen seconds after peak, and under two gs just eight seconds later, having shed 2600 m/s in 32 seconds. Peak heat dissipation was roughly 25 megawatts, which Wikipedia says is similar to the reactor output of a Los Angeles-class nuclear submarine.

From here, you drift down, slowing as the atmosphere gets thicker. You hit the ground 560 seconds after you started falling, at a leisurely 50 meters per second. You probably won't even leave a significant crater.

If you're looking for survivability, your best bet is to spread the forcefield out into a circular plate at least two meters across. The fall to the Karman Line isn't changed much, but deceleration starts six seconds earlier, peak velocity is 2750 meters per second, and peak acceleration is only 21 gs. Time spent above 20 gs is roughly a second, right at the chart's limit.

The real change is the descent after slowing down: it really is a drift, lasting 860 seconds after peak deceleration, for a total fall time of 1190 seconds, landing at just 16 meters per second.

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    $\begingroup$ @CapIsland Head-first or feet-first is worse: you get deeper into the atmosphere before starting to slow down, so the forces are higher (but briefer), while your tolerance is lower. If you can spread the forcefield out into a sphere (or even better, a plate), deceleration starts earlier, and forces drop to (barely) survivable levels. $\endgroup$
    – Mark
    Nov 20, 2023 at 1:51
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    $\begingroup$ "just 16 m/s" is still 57 km/h so guaranteed injury and and over 40% risk of death based on pedestrian/car impacts at 50 km/h. $\endgroup$
    – Criggie
    Nov 20, 2023 at 5:11
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    $\begingroup$ @Criggie Surface impact was explicitly excluded in the question. $\endgroup$
    – gerrit
    Nov 20, 2023 at 7:53
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    $\begingroup$ If you can modify the shape in real-time, you can start with a disc for maximum drag in the less dense regions (though of course this would be unstable) and go progressively smaller in cross-section to keep the drag tolerable, before going wide again to decelerate $\endgroup$
    – Chris H
    Nov 20, 2023 at 14:48
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    $\begingroup$ @CapIsland You could try going to worldbuilding and ask there instead - I believe they do science-based answers... $\endgroup$
    – eirikdaude
    Nov 21, 2023 at 9:23

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