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Imagine that you are inside a box made of iron. You are surrounded by iron and there is no gap between you and the iron. Assume you can breathe (but it doesn't matter). I said iron but it could be any resistant material that could resist an impact without any damage. The box is on top of a 50 meter tall building. What would happen to you if someone dropped the box from the top of the building? Assume the ground is made of concrete.

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The box stops abruptly when it hits the pavement (Let's say, for sake of argument, that it hits an outcrop of bedrock that happens to stick up right next to the building.)

The force between the box and the bedrock outcrop is enormous. ($F=ma$, and $a$ is enormous because neither the "box" nor the bedrock is willing to yield.)

Let's say you happen to hit feet first. Since the box experiences enormous acceleration, you experience enormous acceleration. The box applies enormous force to all of the parts of your body where "iron box" happens to be directly below your flesh. Let's take the soles of your feet as an example.

The flesh of your feet transmit the force to your leg bones, your leg bones transmit force to your hip bones (Now hear the word of the Lord!), your hips transmit it to your organs and your spine, etc.

Everything in your body feels a brief but enormous stress. The outline of your body doesn't change much because it's confined by that iron "box," but your insides? Probably get scrambled pretty badly.

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    $\begingroup$ This is what should happen to "Iron Man" in the MCU every time he hits the ground! My +1. $\endgroup$
    – Hans Olo
    Sep 3 at 10:19
  • $\begingroup$ basically, leg bones will try to get to the brain really quickly? $\endgroup$ Sep 3 at 15:18
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    $\begingroup$ @aaaaasaysreinstateMonica more likely, your brain will try to get to your leg bones really quickly. All your organs have inertia downwards and will try to keep going that way. Unless your leg bones bounce and try to meet your brain in the middle, that is. $\endgroup$
    – Drake P
    Sep 3 at 16:13
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    $\begingroup$ Note to the casual reader: the link to the gospel song is 100% relevant and 100% delightful. $\endgroup$
    – rob
    Sep 3 at 18:23
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    $\begingroup$ @spectras No--if you hit the ground directly you flatten out. The point of the box is that you don't flatten out. If your body were of truly uniform density you would survive unharmed--but your body most certainly isn't of uniform density and your lungs will really be a problem. $\endgroup$ Sep 4 at 16:08
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The force exerted on the occupant is the change in momentum divided by the time taken for the change.

The problem with being in the box is that the occupant is constrained in terms of shape and hence cannot increase the time of change by say, bending legs so that the majority of the body undergoes a longer time for the change in momentum and is thus subjected to smaller forces.

Just imagine jumping down a distance keeping “stiff” all the time. The force transmitted by your legs to change the momentum of your body would be very much larger than if you bent your legs as you hit the ground.
Those large forces exerted on you body would cause damage to all parts of your body.

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The box with you inside are both moving with the same acceleration, and therefore you would hit the ground with the same velocity $v=\sqrt{2gh}$ ($h$ is the height of the building).

Even if the iron box experiences zero deformation, you still rapidly decelerate while inside the box. Assuming the box does not experience any deformation, because of Newton's second law, you will experience a rapid deacceleration over a very small time. i.e., since $$F=\frac{\Delta p}{\Delta t}=\frac{m\Delta v}{\Delta t}$$ So if the building has a height of say $20\ m$, this means you're body must experience a changing speed $\Delta v=\sqrt{2g\times 20}\approx 20\ $m/s. Multiply your mass by this and divide it by a small impact time means a huge impulsive force is imparted to your bodies skeleton and organs.

It would seem that in terms of the probability of coming to great harm or death, the scenario is not much different to hitting the ground without being encased in the iron box.

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    $\begingroup$ The box will not come to rest instantly. It will bounce. $\endgroup$
    – PM 2Ring
    Sep 3 at 7:17
  • $\begingroup$ Yes, this is why I said "assuming". $\endgroup$
    – joseph h
    Sep 3 at 7:19
  • $\begingroup$ The comparison to the steel wall misses the point of the question: The steel wall is not shaped to your body, and so does not distribute the impact force equally over your entire forward surface. You have to expect much more damage from the steel wall than from the container in question. $\endgroup$ Sep 3 at 10:41
  • $\begingroup$ the question says there is no empty space in the box $\endgroup$
    – Jasen
    Sep 3 at 13:11
  • $\begingroup$ @Jasen That isn't relevant. You're still taking your entire body and accelerating it to a stop. Less surface damage maybe since the pressure is spread out but you're still going to die. You might as well say that instead of being inside an indestructible metal box you dropped yourself with perfect aim into a cavity in the ground shaped like your body. $\endgroup$
    – DKNguyen
    Sep 3 at 15:35
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As the indestructible box conforms perfectly to your body, which is safely able to deform at least a little under the influence of applied forces, the large acceleration itself is not much of a problem—the enormous force is evenly distributed over your surface exactly as if you were submerged deep in the ocean. Divers are able to survive pressures of many tens of atmospheres (equivalent to many tens of g’s of acceleration in the form fitting box).

The reason you would not survive the fall from 50 m is not the peak magnitude of the acceleration, but its extremely rapid rate of change: the magnitude of the jerk. The air in your lungs and other areas of your body is very compressible. As a diver gradually moves deeper, this air is allowed to gently compress, but when you hit the concrete in your iron box, it squashes violently. Your body would be damaged in the same manner as it would by the nearby blast of a bomb. The sudden collapse of lungs, sinuses, etc. would rip surrounding tissues apart.

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    $\begingroup$ "the large acceleration itself is not much of a problem—the enormous force is evenly distributed over your surface exactly as if you were submerged deep in the ocean" This is untrue; there's no load pushing down on the payload at the moment of impact, for example. The fact that the stress state isn't equitriaxial (as hydrostatic stress is) is the entire point. $\endgroup$ Sep 3 at 17:54
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    $\begingroup$ @Chemomechanics you are mistaken. Your body basically is water when it comes to how it behaves in this situation (fully supported on all sides). It is essentially hydrostatic, and I believe OP’s intuition that this is the case is what prompted the question, which is why I upvoted it. Most people’s intuition, like yours, is that there’s something anisotropic about the stress that develops in this case. It’s not true. $\endgroup$
    – Ben51
    Sep 3 at 20:49
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    $\begingroup$ @Ben51 Surely the immediate consequence of hitting the ground would be a fairly intense shockwave which propagates from the point (or surface) of impact. I'm not sure how you're imagining this will be in any way isotropic. $\endgroup$
    – J. Murray
    Sep 4 at 3:57
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    $\begingroup$ +1 for realizing that when the contour of your body is indeformable you are actually more akin a liquid falling in a rigid container than an structure. The effect of impact will be much less than falling without the box, although compressible part (air in lungs) and the differences in density of body parts may be enough to still kill you. $\endgroup$
    – Pere
    Sep 4 at 23:20
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    $\begingroup$ And for a simple comparison: you can easily mix a half fill bottle of liquid or jelly by shaking it, but shaking a completely full bottle (without any air gap) of liquid or jelly nearly doesn't work. When your body is perfectly enclosed in the iron box, it's more like a completely full box. $\endgroup$
    – Pere
    Sep 4 at 23:23
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I guess you are assuming that the iron does not get deformed by the impact and it decelerates and becomes at rest in a very short timescale. If that is so, you can imagine the problem as being equivalent to having the iron below you at rest in the floor (with your shape), and you falling into it (and forget the iron above you, which does not have any effect). You would obviously crush.

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The above answers can be clarified a bit.

Technically, the force isn't a problem at all. Spread out over enough time or matter, or area, or other mitigating stuff, the numerical force is manageable. The problem is the accelerations involved.

No collision, however intense,is instantaneous. They all take time. If you watch in slow motion they evolve, peak and diminish. A collision could take millions of years (continents colliding), or tiny fractions of a second (protons colliding,or as close as they get to collision anyway), or anything in between. The substances collide and slightly (or greatly) give way change state or whatever, and that has the result that the collision is spread out over time.

Say a collision starts with some object moving at velocity A, and ends with it moving at velocity B. In a traffic accident A might be 50mph forward, B might be zero. Where the materials cause a collision to spread out over time, that change of velocity takes longer. Acceleration is (crudely) change of velocity divided by time taken. If the same change of velocity takes longer, the acceleration is less. If the collision is over much quicker, its more.

Now let's look at what happens to an object which is forced to undergo rapid deceleration. Suppose the object is a human lower leg, facing the same direction as the traffic collision. Suppose the collision takes "t" seconds (crudely, again its the principle that matters).

The whole of the leg is travelling forward. The foot (say) abruptly stops, because of the very intense deceleration. If the leg is to stay intact, the ankle, shin and knee must also stop that fast. To do that in a very short time, there must be a lot of force applied via the stopped foot, ankle, and lower leg bones.

The foot stops (say). The forces built up in the bones and the rest of the leg begins to slow. But the leg can't transmit the needed forces. Bones and connecting tissue begin to distort then buckle,sheer, fracture, and break as the forces being transmitted start to exceed the legs capacity to handle force along its bones and joints. The forces still grow and the leg shatters, and fragments, totally overwhelmed by the forces involved.

Unfortunately by the time the leg has shattered, joints higher up are having the same happen. Any joint that can't decelerate in the given time, and sustain the transmitted forces needed to do so, breaks or shatters. The upper bones are less damaged because they have longer time, so the force/acceleration to slow them is lower. In effect the lower body sacrifices itself to save the upper. If there was an airbag, it sacrifices itself the same way, to save both. By extending the time over which parts of the body must decelerate.

Meantime the inner body structures must slow down as well. As they're mushy, they can't slow down any way except pressing against lower body parts. The forces rip loose muscles, blood vessels and entire organs and pulverise them.

And thats roughly what happens.

The only difference with your scenario is, cars have design stuff to force collisions to take longer time by the time a human is affected. Crumple zones, the engine compartment, more crumble zones, typical bent knees on pedals.....

Your scenario takes those away so the time is shorter. Apart from that, same broad stuff as above.

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Concrete is extremely strong against compression, but not so much under tension or shearing.

Depending on the thickness of the concrete and the mass of the box, this might turn into an exercise in shearing stress, such that the box punches out a box-shaped section of the pavement and depresses it into the subsoil.

Let's make a few assumptions and see where this goes...

I assume the concrete is rated for driving "normal" trucks on, so it would be about 20 cm thick.

I assume the box is a cube, and that you want to stand up inside it.

Since the box is thick enough to "resist an impact without any damage", I'm going to make its volume 50% steel, 49% air, and 1% human; inside 2×2×2m, outside 2.52×2.52×2.52m, walls 26cm thick).

Steel has a density of 7.85kg/L, meaning that the total weight of the box is almost 63 tonnes. Yes, the box is 0.14% human by weight. In fact, let's include two humans, to round the weight up to 63 tonnes exactly.

Now for the first problem. While setting up the exercise, we parked the cube on the carpark while we got the crane into position. And it's already cracked the concrete.

Calculating the impact velocity is trivial, since air resistance can be ignored: √(2×g×h) = √(2×9.8×50) = 31.3 m/s = 112.7 km/h = 70 mph.

The impact yields just shy of 2 megajoules, or about the same as two sticks of dynamite, or one litre of instantly vapourized water.

Couple that with the standing weight, and the concrete doesn't stand a chance; we're going through.

The only question is "how far", since the rate of deceleration will dictate the level of injury sustained by our passengers.

(Incidentally, your best chance of survival is "stand on top of the other passenger", but let's not go there.)

The atmosphere will make so little impression on the outside of the box that your next problem will be weightlessness making it difficult to keep yourself positioned for landing. So let's strap you in for good measure.

Our hollow box will be around 1.5× as dense as the concrete, and anywhere up to 3× as dense as the substrate.

A quick back-of-the-napkin calculation suggests that the box will penetrate between 50% and 100% of its own depth into almost any "normal" substrate.

The quick way to calculate the stopping force is simply the drop height divided by the stopping distance. Stopping in 1 metre means about 50g, stopping in 2 metres means about 25g.

Landing on your feet effectively adds another 0.7 metres to the stopping distance (of your torso), as your legs get pulverized and take up most of the shock, so this is touch-and-go survivable. In any other orientation survival is unlikely.

Assuming you have a door to get out of the box, and it hasn't been jammed by the impact, and you make it to hospital in time, then you have a fighting chance.

Note: do not attempt this using an airport control tower, as the concrete of the tarmac is 3× as thick, designed to sustain the weight of an A380 (circa 400 tonnes), so the stopping distance for your box will be ... small. The concrete will suffer stress fractures for many metres around, but it won't move far. And your "damage resistant box" will be, erm, damaged.

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  • $\begingroup$ I strongly doubt that to sustain the weight of A380 you need two metres of concrete, even on a runway. $\endgroup$ Sep 4 at 20:27
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    $\begingroup$ @MartinBaláž I double-checked this and you're correct; I'll update my answer accordingly. $\endgroup$ Sep 10 at 0:19
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I think it would be better to change the question a little bit.

Instead of a steel box that exactly fits you, make it a steel box that is full of salt water, salt water that has the same density as you.

Then it looks like a whole lot of the force gets converted to pressure that doesn't move you any appreciable distance, but is close to being the same pressure on all sides.

Will that change the answer by very much? I don't think so, but it might make it clearer what the question is that needs to be answered. How much of a sudden sharp pressure on all sides does it take to hurt you? In addition to the sudden deceleration.

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