Fight scene steel mechanics

[The_Lopen279]

Hey guys. first post here. I was recently rereading mistborn again and I was on the fight scene between Vin and Shan. It goes like this,

"Shan slashed, and Vin felt a flare of pain in her cheek as she dodged.
She dropped the staff in shock, reaching up to her face and feeling blood. She stumbled back, seeing the smile on Shan's face.
And then Vin remembered the vial. The one she still carried-the one Kelsier had given her.
Atium.
She didn't bother to grab it from the place she had tucked it at her waist.
She burned steel, Pushing it out into the air in front of her. Then she immediately burned iron and yanked on the bead of atium. The vial shattered, the bead heading back toward Vin. She caught it in her mouth, swallowing the lump and forcing it down."

I have 3 problems with this. I being the nerd I am, drew a diagram to scale by hand, then modeled it in CAD (autodesk fusion) the cad is better and undoubtedly more accurate so I'll only show it. ↓

in this diagram I took some liberties with measurements but 63" is average height of a female, and most humans are 8" spine to belly button, especially because she is described as slight, and the 45" mark is her center of mass (or self) best I could figure based on average proportions, and the 36" mark is her hips (where her metal is) by similar reasoning. The diagonal dotted line is the line made by connecting the 2 points, which is the vector the metal would be going on. the curved line (spline) is my best guess at attempting to fit the metal path into what is described. The reason it varies slightly from the vector is due to gravity. The circle is a very rough estimate at her head dimensions and the chord of that circle being her mouth. 

with the description out of the way we can now address the issues. Firstly the initial push, she pushes it directly away from her and "into the air in front of her." How does this work. we can only push directly away from the center of self. You might say that the center of self  is lower than that because its modified by perception heavily, but to do as described it would ideally be below the object being affected, which is in her waist(band). This seems incredibly unlikely, especially because it's normal for the rest of the fight. 

Next is the pull sequence. After pushing the vial "into the air in front of her" she then, "immediately burned iron and yanked on the bead of atium. The vial shattered, the bead heading back toward Vin. swallowing the lump and forcing it down." Here more problems arise. From the diagram drawn it would be entering her mouth at a a very high speed and a very steep angle. The speed comes from the shattering of the vial. admittedly, this detail was likely just a writing tactic, so she isnt swallowing glass, but I fear he's created a far greater problem. I just did some calculations and the nugget is likely moving at 90+ mph. this is bad. catastrophic even. I made this under the assumption that the vial was tempered glass (they jump all over w/o them breaking) and the nugget was about 0.5-1 cm in diameter. 

So we now have a bead about the size of a small caliber bullet heading at breakneck speeds over a short distance directly to her hard palate. After some research I'm actually unsure if it would break it. If it did, that would go straight to her brain which feels like a problem. thoughts on this? I know that I'm lacking in the physics area and anatomy area. any insights from yall? 

[Treamayne]

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25 minutes ago, The_Lopen279 said:

I know that I'm lacking in the physics area and anatomy area. any insights from yall? 

Keep in mind that Intent is important, and center-of-self is different from center-of-gravity (and that is not a hard limit, just the way they understand that in era 1; which already has many examples of not being an explicit limitation). We know experienced allomancers can adjust how they use their pushes and pulls (e. g. Kelsier pushing and pulling on opposite ends of the same bar to produce spin). Era 2 spoilers:

Spoiler

Wax is not pushing from his "center" on the bullets - or he could not aim with his eye and push the bullet without wildly changing the trajectory

Tempered glass seems like an odd assumption here (they do break sometimes, and they are padded so without an outside force acting upon them they should not randomly break from a small bead immersed in a liquid). A 1 cm bead is probably way to large to calculate with as well - most vials are just a few flakes in a suspension - 1-2 mm is far more likely here.  You also have to account for the vial's velocity away from the body as well as the reverse pull on the bead for the combined force exerted to break the glass so a much higher mass weight moving away, being struck by a very small object on a cruved plane with all of the force of the iron pull in a very small area.

 

Does this scene play to "rule of cool" a bit? Sure. But it is also Brandon's second published work. . . 

 

@DrPhysics has a great post on Steelpush Physics

WoB:

Spoiler

Questioner

I have a theory. Because the center of gravity for a female is naturally lower, but when Vin burns iron or steel, the blue lines come from her chest, does that come from her center of self, rather than the center of gravity?

Brandon Sanderson

Yeah. That's probably a more accurate way to put it. 

Questioner

Would it be possible for that to change, then? 

Brandon Sanderson

Yes, that is possible. I should say it like that, because it's not going to actually be... Because center of gravity, where you would actually put it, is not where I'm having those lines come from. You came in costume. You can just make that canon now and we will put that on all of the lists that that is what it is.

Starsight Release Party (Nov. 26, 2019)

Hope that helps

[Returned he/him]

You can be as generous or stingy as you want in assessing the description of most action sequences. I'm no physicist, and probably less skilled in that area than you given that you went to the trouble of diagramming it, so the below are only possibilities and definitely not firm judgements. I'm not suggesting that any of the below are necessarily the case, but if the scenario as described is bothering you maybe some of them can soothe that, at least a bit:

• I would be shocked if the bead were larger than 0.5 cm, but as I looked for evidence to support my shock I came up with very little so I can't say this confidently; the beads are generally described as tiny, which is obviously imprecise, but I personally wouldn't think of a 1 cm bead that way. If we're trying to enjoy the scene in light of the issues raised in the OP I think it makes sense to err on the side of estimating the bead to be smaller. Also, we don't know much about the density of atium, which is relevant to its dangerous momentum as well as its loss of velocity to air resistance
• A person's waist isn't the same as their hips (the waist is higher up the body), so the vial is already higher up from what your measurement suggests if the vial is indeed at her waist
• She could have been crouching (seems very Mistborn-y), or leaning back (she did stumble away from Shan), or moving in some other way, any of which could change the center of mass' angle relative to the metal vial, including at different moments of this (very rapid) sequence of events. If she bobs and weaves a bit then there might be any number of adjustments to the atium bead's vectors
• The vial will still be in the air while she's doing the pushing and pulling (even if also headed downward). The whole sequence is rapid enough that the vial doesn't need to hit the ground before the bead is ripped out. For that matter, Vin is very practiced at pushing and pulling by this point in the book and so it seems fair to think she's timed and modulated this push and pull to accomplish what she wants (atium in the mouth)
• When she pulls on the atium bead, the vial will still be moving away from her due to her initial push, so the bead will hit the side of the bottle nearest Vin with the force of her pull as well as the opposite-vector force of the glass moving in the opposite direction (which you may have already factored into your calculation, but if not the velocity of the bead doesn't need to be as high as might otherwise be suggested). If she pushed and pulled with equal strength this would double the impact force and not require the bead to move towards Vin at such a fast speed. This might explain why the pull caused the vial to shatter while the push did not
• I see little reason to assume they use tempered glass for their metal vials, though of course it's possible. If that's the sticking point, then Occam's Razor strongly suggests that it's not tempered
• Vin is burning pewter, if I remember correctly, which would give her extra physical resilience against the bead and any glass that came along with it. Even if it hurt her it wouldn't necessarily kill her as described, or even meaningfully injure her. Pewter lets Shan survive being stabbed in the heart and Vin survive being disemboweled by an axe-- it's a hugely important factor
• The angle of the bead's travel towards Vin is unclear because she is presumably moving (particularly suggested by the description of her "[catching] the bead in her mouth"), so it might not be as fatally directed as you have suggested and probably not directed at her mouth or head at all; the angle should be shallower. Additionally, if she's moved (say, slightly to the side) then more tweaks with iron or steel to get the bead into her mouth could bleed away velocity in the dangerous vectors as the direction of the bead's motion is changed
• Vin could easily (instinctively or due to her training and practice) have pushed on the bead again when it was near or in her mouth, to mitigate its velocity somewhat, not unlike how she modulates her pushes and pulls to control her descent and objects' angles of motion in other circumstances
• The bead's shape is irregular, certainly not bullet-like, and it's not rotating like a bullet shot from a rifled barrel, so it will lose velocity more quickly and be more responsive to other deviations in its direction than any modern bullet. I wouldn't expect those to matter over a distance of a few feet, but if we're grasping at straws we might as well grasp at all of them that we can!
• The bead is almost certainly not moving faster than coins shot by Allomancers burning steel, and while those hurt they don't tear through Mistborn like bullets, so a tiny bead of atium probably wouldn't either

Edited February 26 by Returned

[DrPhysics he/him]

21 hours ago, The_Lopen279 said:

we can only push directly away from the center of self.

That is true at long ranges, but there are several instances where it appears that allomancers are pushing/pulling with only part of their body. I mentioned a few in the post that @Treamayne linked, but I'll not list them here because I don't know what level of spoilers we are at. This will not be the only time you see that contradiction.

As far as breaking the glass goes, the best I could find was this paper that shot BB's at sheets of glass, and observed the breakage pattern. They found that if you had an impact energy of around 3-4 J would give glass failure at a level where the BB went through (for reference, a .22 rifle would have a 40 J impact energy). They used double-strength window glass, which would probably be comparable with a well made glass vial. We couldn't produce tempered glass for anything except curiosities (like Prince Rupert's Drop) until the late 1800's, so they wouldn't have had the tech available in the Last Empire either.

Since neither metal nor glass has much give, impact energy will be a better predictor for shattering than anything using PSI (Only the front most part of the bead would touch the glass before shattering it, so using the entire cross-section of a sphere would be a bad predictor).

So, 3-4 J does mean a velocity of around 130 m/s for the BBs, which sounds insanely fast, but it doesn't have enough energy to do much to you. A 1 cm diameter steel ball bearing would only have to travel at 45 m/s to shatter the glass, which matches your 90+ mph calculation, but it would still only have 3-4 J of impact energy, which is equivalent to a baseball going about 16 mph.

But, we have two other things to consider: (1) these speeds would be the speed of the bead relative to the glass, which would be slower than the speed of the bead towards Vin and (2) that assumes that the bead doesn't lose any energy or speed on impacting the glass, which is impossible. Doing the exact calculation for speed involves many more assumptions (thickness of the glass, speed of the vial away from Vin, cross-sectional area of the bullet, which matters if you are going through, but not for whether or not the glass breaks, ect.) than I feel comfortable assigning an exact number to, but some back of the envolope calculations suggest that the bead should lose somewhere between 25%-75% of its enery, which works out to be 45%-85% of its speed.

 

[The_Lopen279]

thanks for such quick responses! I'll open by saying that I've read everything. all novels and novellas and series. (and i will go make an intro post). I will think about all this and really appreciate all the contributions. 

[Returned he/him]

Welcome to the Shard, @The_Lopen279! As you may have noticed, forum members go wild for this kind of thing and your question is interesting and very much appreciated as well. I had fun thinking through it, even if I can't match @DrPhysics.