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The fluff:

My protagonist finds herself in an empty patch with everyone running away. As she looks up she sees a space ship come through the clouds at an angle, coming straight at her. Its on fire, pieces are falling off and still able to brake its decent. The protagonist starts running a perpendicular prometheus run (perpendicular to the direction the ship is falling in for those who dont get that reference) as she tries to avoid the ship landing on her. The ships nose lands behind her and she makes it to the edge before the thicker middle section passes her.

Statistics:

The space ship is 3km long and 200m high&wide with an average weight per volume of an aircraft carrier (if necessary, pick one).

The question:

What is the maximum speed that the space ship can hit the ground with, without killing the protagonist?

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    $\begingroup$ What exactly would kill the protagonist? The shock-wave of the impact (like from a meteoroid)? In that case the mass should be in the question directly. Or do you mean the space craft directly hitting her? The critical speed would, assuming a simple triangular shape, simply be 'distance between ship's nose to where it's width reaches 200 meters' divided by 'time protagonist needs to run 100 meters'. If the ship moves faster than that speed the protagonist won't make it. $\endgroup$
    – user91641
    Commented Nov 12, 2022 at 20:34
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    $\begingroup$ What the heck is "perpendicular prometheus run"? Is it a sick stateboard trick? $\endgroup$
    – Daron
    Commented Nov 12, 2022 at 22:16
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    $\begingroup$ @Daron a reference to the movie Prometheus, where the protagonists will run away in a straight line away from a danger that can be dodged by stepping sideways. I edited the question. $\endgroup$
    – Demigan
    Commented Nov 12, 2022 at 22:25
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    $\begingroup$ Guys, I really don't think the basic question is as unclear as everybody's making it out to be. "what would kill the protagonist?" Anything in the scenario described that would likely kill the protagonist. "The edge of what?" The edge of the area in which she has a reasonable chance of not dying. Demigan is obviously asking the question because they do not know all of the possible causes of death, etc, and are asking for help determining what all the factors are and what each one would mean for survivability at a given impact speed and (runnable) distance from the point of impact... $\endgroup$
    – Foogod
    Commented Nov 12, 2022 at 23:46
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    $\begingroup$ @AlexP "Oh here it comes" and then "Oh there it goes." $\endgroup$
    – Daron
    Commented Nov 13, 2022 at 13:39

4 Answers 4

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Very, very slowly!

(About 60-100mph)

The basic numbers:

Modeling the ship as a cylinder 3000m long and 100m in radius, we have a:

Volume of 9.4*10^7 cubic meters

At the average density of an aircraft carrier (just under the density of water, so let's round to 1000kg/m^3) we have:

Mass of 9.14*10^9 kg

Clouds are between 2000 and 5000 meters overhead, so we'll round to 3000m to make things simple.

Distance traveled by the ship: 3000 meters

Potential speeds:

Orbital Impact Velocity

Even minimal orbital impact velocity is a non-starter. At ~11.2km/s, the ship will hit the ground a quarter of a second after she spots it. I won't even bother doing the math for that.

Your ship is actively braking, so it may be going considerably slower. Let's try some smaller numbers.

Terminal velocity

According to this terminal velocity calculator, the terminal velocity of an object like this is about 3km/s.

Since kinetic energy is 0.5mv^2, we have 4.05*10^16J coming in hot.

Okay, so that would give her one second to run, at which point she will be smashed flat by the ship, which will hit the ground with the force of six megatons of TNT.

Airliner cruising speed

At 900km/hr or 250 m/s, roughly the speed of a modern airliner, our protagonist has a full 12 seconds to run. If she's a star D1 college athlete specializing in the 100m dash, she can clear just over 100m (the radius of the ship) in that amount of time. So she still gets squished flat. But we've gotten that impact down to... 60kilotons of TNT! Progress.

Highway speed

Let's say the we've gotten down to the sedate pace of a car driving down the highway, 70mph or ~31.3m/s. This gives our heroine plenty of time to run, 96 seconds! A star college athlete running at their 800m sprint speed would cover about 576 meters in that time.

According to BLAST OVERPRESSURE AND SURVIVABILITY CALCULATIONS FOR VARIOUS SIZES OF EXPLOSIVE CHARGES, enter image description here

if the impact is equivalent to < 10^7 lbs, TNT, she has a 50% chance to survive the blast wave! Let's check our kinetic energy... 4.47*10^12J! Just around a kiloton of TNT. She's got a solid chance to survive the blast wave. Shrapnel is more difficult to calculate (and your ship is going to generate a lot of shrapnel) but unlike overpressure, with shrapnel, you can get absurdly lucky.

So I'd say you've got a chance!

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    $\begingroup$ None of the kinetic energy goes into plastic deformations? $\endgroup$
    – fectin
    Commented Nov 13, 2022 at 20:10
  • $\begingroup$ Frankly at 60-100MPH at collision the ship's captain could call this "hard landing" instead of a "crash", thus people on that ship could well survive in numbers. That would add a twist to the story... $\endgroup$
    – Vesper
    Commented Dec 1, 2022 at 15:06
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Too many variables ...

Daniel B calculated the energy of the impact. But where, exactly, does that energy go? Some of it will deform the ship. Some of it will deform the ground. Some of it will accelerate fragments of the ground or the ship to high velocity. Much of it will ultimately become heat, to be radiated over time and over the length of the ship.

I've stood a meter from a train passing at reasonably high speed. Not airliner speed, of course, but it was still impressive. Yet the train wasn't imparting the kinetic energy on me. I've also stood a meter from a braking train. In that case, the train was imparting heat and a slight deformation on the wheel and tracks. Again, observers were not harmed.

I think the two key questions are how the spacecraft crumples, and following from that if the character is hit by any parts. Too many variables in that.

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  • $\begingroup$ I suspect that if that train had slammed into a concrete wall just after passing your nose, you would have experienced its imparted energy very differently, despite it being orders of magnitude smaller in mass and in velocity than the ship would likely be at. $\endgroup$
    – Daniel B
    Commented Nov 13, 2022 at 23:56
  • $\begingroup$ @DanielB, likely from the fragments mentioned in my first paragraph, or from the deformation. $\endgroup$
    – o.m.
    Commented Nov 14, 2022 at 5:34
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At or slower than terminal velocity.

People have survived structures of similar size to your one falling on them when buildings collapsed, such as at 9/11.

What followed immediately was an enormous rumble, nuclear and otherworldly that overtook us. The ground rippled around us, as if a volcanic eruption had exploded from beneath the earth.

In a split second, the light I was optimistically following out of the building was disappearing. In its place, as I stood frozen in that one moment permanently carved into my consciousness, a brown colossus just feet away, advanced toward us and upon us, a wall, stories high, moving with such momentum across the Plaza like a runaway bullet train full of the now infamous toxic stew that was number 2 World Trade Center. It was collapsing, carrying tons of concrete, asbestos, glass, its dead, those on the Plaza and we were next. The plate glass window that separated us from it was swallowed up in the path of the moving wall – gone in a second.

With large fragments flying around and the ground ripping from the collision if it's moving much faster you're unlikely to make it- from this description, it's already pretty bad when it hits the ground at terminal velocity, which is about 50m/s. You should probably lower the speed a bit below this, because the above description doesn't sound like something you can run away from. It's hard to run during an earthquake.

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  • $\begingroup$ Fixed that issue $\endgroup$
    – Nepene Nep
    Commented Nov 13, 2022 at 0:06
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    $\begingroup$ "Terminal velocity" depends on the nature of the object falling. The terminal velocities of a feather, mouse, human, steel crowbar and pointy-shaped 3km long aircraft carrier are very, very different. You seem to be going with the terminal velocity of a human, but why would that be the terminal velocity of a huge pointy metal thing (albeit with lots of internal voids). $\endgroup$ Commented Nov 13, 2022 at 7:00
  • $\begingroup$ The world trade center fell at 55m/s, so I would assume the ship has a similar speed. $\endgroup$
    – Nepene Nep
    Commented Nov 13, 2022 at 10:24
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    $\begingroup$ The WTC may have been travelling at 55 m/s when it hit the ground, that does not mean it had reached terminal velocity. A human doesn't reach terminal velocity until they have fallen around 450 m, the top of the WTC was only about 100 m higher. $\endgroup$ Commented Nov 13, 2022 at 12:13
  • $\begingroup$ This post is about making a rough estimate. IRL, we don't have any real way of estimating the chance of survival if a skyscraper sized ship hits near a person as such a thing hasn't ever happened in the past because we can't build flying ships that big. As such, about 50m/s is a reasonable estimate for the terminal velocity. $\endgroup$
    – Nepene Nep
    Commented Nov 13, 2022 at 19:13
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How big is a rock?

Even at glacial speeds, the ship could fragment and throw deadly shrapnel.

At orbital speeds, impacting near you could be like a bomb going off nearby. But the actual effect is highly subject to terrain and the specific impact conditions, and normal people can be shockingly resilient besides.

At super-orbital speeds, it's probably even more subject to the specific conditions. While that's likely to be a Bad Time for anyone on the same planet, it wouldn't be that surprising if, say, the speed was high enough that ship simply penetrated the ground like a bullet, and the ground shaped the blast up. Don't count on it though.


Probably though, you're looking at a ship doing a long crash roll though, not an impact. That means you're basically only looking at reaction times and running far enough away to not get hit.

So, assume your protagonist is super unlucky, and this spaceship is coming straight at him. You can spot nav aids on a carrier at something like a mile out, so assume that's when you can get a good view of the ship's heading. It probably takes until a half-mile out to see that it is coming actually at you, so that's your distance. You need to sprint approximately 100 meters to avoid the ship, which for most generally-fit people is about 13 seconds (track times are faster, but tracks are more favorable in a few ways). 1/2 mile is about 800 meters; 800/13 is about 60 meters/second is about 130 miles per hour.

So I guess a rock is about 130mph big.

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