DrPhysics

The part you are missing is just how incredibly fast Mach 5.8 is. We know duralumin burns fast, but it still takes long enough that your body has time to perceive the effect. That means a push would take 0.1 to 1.0 seconds. Ignoring, reaction time (which averages around 0.1-0.2 seconds, but can be shorter if you know something is coming), our best-case scenario is that we can shorten our push into a 0.1-second burst. At Mach 1, a bullet will travel about 35 meters in that time. Mach 2 goes to about 70 meters. Mach 5.8? 200 m. At those distances, the bullet is just way too far away to give any significant push beyond those first few fractions of a second. Which is why you can get a Mach 1 up to Mach 2, but a Mach 2 bullet can only be sped up to about Mach 2.5 Reading through the books, you'll notice that Wax's push doesn't do much to the speed of bullets, but his pushes can add a lot to the stopping power. Hit a nearby target and add a full push as the bullet slows down inside the target: devastating.

Since the push gets weaker as the bullet gets farther away, at those speeds you can't push for very long before the bullet is too far away for the push to matter anymore. You max out at around Mach 2-3.

There just isn't enough momentum in the air to explain the deflection of ballistic objects. Also, the frothy effect can't be explained on a molecular level. To have wind change direction, the bubble would have to actively create (on the input side) and destroy (outgoing) matter. Actually, two things happen: They are energy-shifted so they maintain relative frequency across the border. Photons are added/removed so that the intensity of light is the same on both sides. Otherwise, the inside of a speed bubble would be very dim, and the inside a slow bubble would be blindingly bright. My current working theory: the edge of the bubble uses light weaving-like effects to essentially project the proper images like a television screen. That could explain the faint shimmer at the edge. Yeah, we see Wax get knocked out of a bubble in the Lost Metal. In the Alloy of law, chapter 18, an aluminum bullet actually goes through a bubble without popping it. Could be a plot hole, might need a retcon. Or, aluminum might not pop bubbles. Some other rules that I've noticed: They bend space-time so that gravity is what you expect inside the bubble (Gravity should be much weaker in speed bubbles and much stronger in slow ones so that dropped objects all accelerate at the same rate according to an outside observer). When an object crosses into a speed bubble, it considers the center of the bubble to be "at rest" (see this discussion about a cork and a train: https://wob.coppermind.net/events/10/#e6589) Objects aren't deflected if they leave a bubble as it drops/appears (See Wax's trick shot where he shoots another bullet at the end of Alloy of Law. He clearly fires, then the bubble drops). The center of the bubble stays at rest according to what the "user" sees as at rest (like the train). You could actually launch spaceships at incredible speed by getting them to move along a rail (to prevent deflection) into a speed bubble created by an etmetal cube, then somehow attaching the cube to the spaceship and convincing the cube that the spaceship is the rest frame. Our understandings of Special and General Relativity cannot describe how a bubble is made, so using them to describe what happens as you enter/exit is equally as useless.

If that is true, it would behave exactly as described in my post. The heating was a result of the massive compression air due to having much more air coming in than going out. That's what I simulated. And I have seen the WOB on the topic (I should have addressed them initially). I'm operating under the principle that it isn't Canon until it is in the books, time bubbles being air tight make for some potentially interesting story points, and people from Dragonsteel pop up occasionally in the forums.

I had this in my notes, but looks like I left it out of my post (I'll add it in an edit). I'd actually add that as an argument in favor of the bouncing molecules. The air inside sees itself as inside, so all of it wants to stay inside. True. This will probably be outdated once we get any information on what air does in a bubble (like during space travel, though keeping air inside has some really interesting applications here). This was just a simple "Here's what we know now, what would that mean if we take it to extremes".

TLDR; Letting air molecules pass through the bubble boundary the same way as bullets creates catastrophic conditions for those inside and outside a speed bubble. The molecules should bounce off the boundary instead. I'm a plasma physicist (really hot gasses) by training, which has made me think way too much about what would happen to air at the boundary of a speed bubble. One tenet of physics is that whatever happens on the microscopic (molecular) level should describe what happens on a macroscopic (regular) scale. So, I spent an afternoon building a particle simulation that shows what happens if air molecules behave the same way as bullets when they cross a time bubble boundary (i.e. they keep the same perceived speed, but are diflected in a random direction). The results were catastrophic. On the slow side of the border, pressure rapidly drops proportionately to the time slowdown (e.g. if time is 10 times slower, pressure is 10 times lower). That pressure drop spreads out at the speed of sound, freezing any nearby air and creating a vacuum. On the fast side, the molecules pile up, superheating the air and creating a shockwave that flies away at the speed of sound. Dropping a bubble would create a thunderous boom as the two sides slam back into each other trying to correct the pressure imbalance. Since we see none of this in the books, air molecules cannot interact with the boundary the same way a bullet does. What do the molecules have to do in order to show what happens in the book? Bounce off the boundary instead of going through. This makes the bubble airtight but stops the pressure drop, the freezing, the vacuum, and the loud boom, matching what we see in the books. (No one in the stories has stayed in a bubble long enough to come anywhere near the point where there is noticeably less oxygen). This matches what we know about the cognitive realm. Things are either all in or all out of a bubble. If the air sees itself as a complete fluid inside a bubble, its individual molecules won't want to leave the bubble because they all see themselves as part of the air inside. The only other option would be some odd interchange where the boundary counts molecules trying to cross and reflects some and bounces off others so that the count of molecules stays the same on both sides, but once you get to time differences of 100 or more (and the books show that differences of thousands are easily possible), that molecular exchange makes the bubble effectively airtight anyway.

Super late to this one, but it is a problem I've looked at before. Here's my answers in case someone stumbles onto it. Yeah, with a bullet-shaped/sized object traveling at or above the speed of sound, skin drag will account for ~5% (at the absolute most) of the total drag. Most likely, it will be less than 1%. The problem with this math is that it assumes you deliver all the energy at once, so you can't translate the energy amount over to a bullet. There's a lot of evidence in the books that suggests allomancers are limited by both the force of the push and the amount of power delivered through the push. For a moving object, the power delivered is calculated by the force times the velocity (P=Fv), so the faster something goes, the bigger the force you need to keep adding energy to it. Also, as it gets faster it gets further away so the max pushing force also decreases. I put together a simple simulation once to ball-park a coin's max speed (can't find it to run it again) using Vin's duralumin jump from Luthadel to scale it. I found that we hit a max speed for coins somewhere around Mach 1 to 1.5. Starting a bullet at Mach 1 would let you get the speed up to about Mach 2. Starting at Mach 1.5 let us get up to about Mach 2.1 (The faster the object initially travels, the harder it is to add energy to it), so Mach 2 seems like a pretty reasonable limit.

Sorry I'm late to this thread, it's been a busy semester. I'm in this camp. I've tried running the numbers on a few things, and while general principles seem to hold, there just isn't enough detail and/or consistency to come up with any reasonable numbers. I don't know how much I can add to this conversation, but I will say that based on how we see lashings work, the "rods from God" thing could work, with one big caveat: we don't see anyone using any comparable amounts of stormlight in the stories, so there may be some unknown limits to lashings. Also, Investiture seems to be infinite in the sense that it collects itself again after being used so that you will never run out, but that doesn't mean that an infinite amount is available at any given time. Also, there are two rules of thumb you should use when discussing physics in the Cosmere (#1 I see referred to all the time and I see #2 frequently ignored): Cosmere physics matches our physics unless investiture is involved (I know there's a WOB on this, but I can't find it at the moment). Above all, physics relies on observations and describing how the world works. If something happens in the books, that's how the universe works, even if it contradicts what we'd expect based on our own physics. So, when I run into things like the energy released by Kaladin's flight post (barring bad physics - acceleration is zero at terminal velocity, among other errors) we have to compare the predictions to what actually happened. Releasing that much energy would leave at very least a superheated column of air behind Kaladin and create a thundering boom as he traveled. Since those things didn't happen, investiture can't work the way suggested in that post.

Yeah, many scenes don't work without some way for the allomancer to calibrate the strength of the push. I think I'm going to interpret the scenes where it talks about not controlling the strength to really mean that it's very difficult to go below some minimum strength, but then they can calibrate from that minimum up to flare (maximum). Your model is an interesting one. I like it. Some of these fine details we won't ever be able to solve without concrete data. Until then, one guess is just as good (scientifically speaking) as another.

I might be misunderstanding what you mean by distance and how it relates to alpha. Is it cognitive distance or a physical one? The math makes it look like a physical distance. If it represents a physical change in distance, a push would always move you away from an object, so slowing yourself to a stop while moving toward metal would be impossible. Only in the scene where Vin pushes herself as high as she can go then stops.

You didn't factor in how alpha depends on r. Unless you are assuming that alpha is constant. That would explain the sudden increase in force when the object being pushed hits something solid, but doesn't describe regular leaping through the air. I couldn't find an analytical solution, but I only fed it into Mathematica. If it has one, it isn't easy to find. Combining the forces to create this potential doesn't fix the first law problem, it just shifts the equilibrium point. To fix it, you'd need some sort of very strong damping, (proportional to the velocity) so that you have an overdamped system. But, that damping would also have to make sense with Wax speeding up bullets and Duralumin-enhanced jumping. Or we can use the copout that Vin pushing up on a coin to stop at the top has some cognitive realm-based damping because she was expecting to stop which isn't present in other examples of steelpushing.

How is that? See the Vin/door example in the very first post of this thread.

I just realized another hole in my model when answering a question on another thread. While metal seems to follow F=ma, there are several instances where powerful objects (like an allomatic grenade) or people behave as if their mass is much larger than it actually is. Well, back to the drawing board.

It is unusual, but not the only place where this oddity pops up. Sometimes the push/push back follows Newton's third law, (I push on something, it pushes back just as hard), and sometimes it doesn't. Another similar scene happens in Hero of Ages where Elend and a steel inquisitor both push on the same object and the inquisitor moves much more than Elend, (the inquisitor thought that they'd both be thrown back the same amount, like what would happen with Newton's third). The book purposely makes a point of highlighting the oddity, so my best guess is there is some sort of "push mass" that depends on the allomancer's strength or perception of their strength that makes them respond less strongly to pushes than they should. The same effect must be what is going on with the grenade.

Well, that makes it much harder for the physics to make sense, possibly to the level that we can't consider a push a force. I'll have to chew on this one for a while. Controlling power instead of directly controlling the force of the push would still explain the discontinuity, just not quite how I described it in the example.

Either that or he does it in really short jerks so that by the time he is moving he isn't pulling anymore.

It stores mass (otherwise you wouldn't change speed mid-air). The short version of momentum is that without outside forces, the value of mass times velocity doesn't change. So, if you decrease your mass (without pushing on the mass that you are losing), you will speed up. If iron feruchemy changed how hard gravity was pulling on you, instead of speeding up in mid-air, you'd keep going at the same speed, but you wouldn't fall as quickly. There are a few people who argue about whether or not iron feruchemy conserves momentum if you change reference frames (e.g. Wax is in the train, what does someone outside the train see), but all of those problems go away if you imagine that the mass that you store (or tap) changes its speed so that it is at rest with respect to whatever the allomancer considers at rest (e.g. in the train example, any stored (or tapped) mass is moving at the speed of the train).

The short answer: magic makes time go faster/slower (and yes, I know that answer is facetious). They break so many physical laws, that I can't make any good predictions or explain them any other way. Beyond redshift, you'd also have to worry about things like super fast air molecules crossing the boundary and heating the rest of the room (Wayne would just cook everthing). They also increase (or reduce for slow bubbles) the pull of gravity. The way gravitational forces work, if someone dropped something in a bubble, someone outside the bubble should meaure it falling at the same rate we'd expect in the regular room, and anyone in the bubble would see it fall much too slowly (and they'd be apparently much lighter), and we dont see that happening. We do have ways to slow down/speed up time, but there is no way to create something like a speed bubble using those principles (bending spacetime to that degree would shred everything that passes through the edge of the bubble and the energy released would kill everyone in the room, even after compensating for redshift). Therefore, since there is no real world physics that can even approximate speed bubbles, we're stuck with "magic makes time go faster/slower".

When constructing a physical model, you need a justification for each piece. So, we could talk about relative velocity and show that the bigger the relative velocity, the harder it is to push the thing, but that doesn't give us a reason. Saying that an allomancer is limited by the power they can produce (the rate they can convert investiture into energy), however, tells me that I'd expect the force to be proportional to 1/v, rather than 1/v^2 or exp(-v), all of which get weaker when you have a larger velocity. Those kinds of justifications are especially important if you don't have solid data to put a fit to. If we had a graph of Vin's speed over time and we saw that the fit was actually 1/v^2, then we'd have to figure out why that is, but until then, it's better to find physical, direct cause reasons for any models that you build. As another example, since the pushing applies to metals, we'd have to guess that it is at least somewhat related to electromagnetism, so I'd guess that we follow Coulomb-like forces and we'd expect the push to be proportional to 1/D^2. So what I was really trying to say is that any model I'd use would have the force proportional to 1/V and 1/D^2, unless I had a really good reason not to. Part of the problem is that pushes/pulls aren't as consistent over Era 1 as they are through Era 2. I think Sanderson was still developing his model, so we see lots of oddities in Era 1 that don't pop up as much in Era 2. For example, this scene from TFE: That wouldn't happen if we could describe pushes and pulls as a force exerted on the object that points directly at the allomancer. Instead, we'd see them curve and arc like we do when Vin travels with horseshoes in the next book. Era 1 was pretty bad with Newton's first and third law problems (like the example above), but in era 2, they are much more consistently applied. Does that mean the first law applies or doesn't? What about third law? At some point we need a little hand waving and justification, and it looks like we don't agree on which pieces to handwave away (and there's no reason that we should have to). For me, assuming those rules I posted above and that Allomancers have more unconscious control than they think that they do is how I'm able to justify most of the oddities that I run into. (Except for that Kelsier example. I can't think of any way to justify that other than Sanderson didn't understand inertia when he wrote it.)

I've introduced myself in other places, but I'm a physics professor who is also a big fan of Sanderson's works. There are a lot of in-depth physics discussions on various topics all over this forum. This isn't that. I wanted to start a thread for those who have simple questions about Cosmere (or other Sanderson works) physics, but without diving off the deep end. So, if you have a question but don't want to debate the proper way to model a steelpush (or if those threads just feel way too intimidating), ask it here and I'll do my best to answer. If we run into a topic that could use a more in-depth discussion, we can spin it off into a new thread and I'll edit this original post to include a brief summary and a link.

I set up a simple Euler's method code in Python. There isn't an analytical solution that can include both power delivered and drag force. Newton's third law and statics. If I push with 140 N, 140 N pushes back on me. An average (70 kg, 1.7m tall, chest height at 1.3m, 0.5 m stance) person, without bracing, will fall over if pushed by a force larger than 140 N at chest height. Most of the time that we see coinshots pushing coins, they are pushing multiple coins and aren't bracing. 140 N gives the net force they could exert without bracing against something behind them. Kelsier says that is it hard, (Exact quote, while training Vin early on: “Varying the strength with which you Push or Pull is difficult, but possible. It’s better to just fall a bit, then Push to slow yourself. Let go and fall some more, then Push again. If you get the rhythm right, you’ll reach the ground just fine.”), but we have to remember that he is not a physicist, and "strength" does not necessarily mean the same thing as force. Also, when he does say it, he's talking to a brand new Mistborn just starting with her powers. My best guess is that conscious control is hard, but we modulate our strength all the time in unconscious ways. Therefore, any model that tries to describe pushing as an allomancer simply setting some "force of push" is bound to fail. That would be like trying to model our muscles based on tension in our biceps. We don't think about how hard we are flexing our bicep to move our arm or lift a thing, we're thinking about what kind of path our arm takes as we lift (or fail to lift) the thing that we are trying to move. We can find some limits (human bicep will tear if it exceeds a certain tension), but those limits don't let us model every single motion. I misunderstood your notation. It makes sense now. Sorry. I agree, but I think it makes more physical sense to model the force based on Power (which is proportional to Velocity (P=FV)) and Distance. Power would reflect how quickly the allomancer can convert Investiture into kinetic energy. The faster they can make that conversion, the more powerful they are. Whereas, there is no good physical model that would suggest why velocity alone would matter.

I've never made that argument. Only that that was a scene where we could approximate a force. Then it must be much less than 1 MW. Why isn't that possible? We can use the colt single action army revolver for a stand in. Those would fire a .45 caliber, 16 gram bullet at about 300 m/s. At that speed, drag alone is about 1 kw. 10 kW of a push would have the bullet moving at 350 m/s at 10 meters from the gun, 400 m/s at 25 meters. That's a big increase without much power. (And the push averages to about 30 N) Pushing a coin at that constant power does get it up to 500 m/s, but that involves very large initial forces that would knock an allomancer down. An average person (70 kg, 1.7m) in a good stance could take a 140 N force without being knocked over. If you run the numbers limiting a maximum force to 140 N and a maximum power to 10 kW, you get coins that hit fast but still subsonic speeds (260 m/s) at 10 meters, which fits what we see in Era 1. Also, thanks for finding that WOB. Now I know what my next deep dive topic will be: what flying with lashings would really feel like.

I'd have to find the exact WOB, but he says that in the Cosmere matter, energy, and investiture are all equivalent, but doesn't want to put an exact E=mc^2 equation together because it would limit the storytelling.

Air resistance is proportional to speed, and while you can have big forces at high speeds, as you slow down they get weaker and will take a very long time to slow you to a "stop" (technically, air drag can never truly stop you). Also, your model seems to be relying on a "force of motion" (i.e. when force is zero, you stop). That isn't the case. We also treat gravity as if it pulls on the center of mass, but in reality, it pulls on every single piece of us. The behavior of steelpushes suggests that they follow similar rules (see rule #3 in the original post). I went looking for examples, and I couldn't find where it's talked about. Do you remember where this happens? They could for a very short time, but wouldn't be able to maintain the acceleration (see rule #2, the push is limited by both a maximum strength and power delivered).

Couldn't you just apply the inefficiency constant to the process of storing/removing mass from the spiritual realm? So maybe priming the pump was a bad analogy. Instead, imagine it like an electrical relay. We use a little bit of investiture to open access, but keeping the access open uses up the stored investiture. Really, the point I was trying to make was that if you literally stored all of the mass in the metalmind as investiture, you would see a large increase in the mass of the metalmind. If it follows an E=mc^2 like relationship as Brandon has said, then the metalmind's weight would increase just as much as the mass you stored, and storing mass would have no meaning. You'd still be just as heavy, it's just that some mass would be in the metal, and the rest would be in your body. Something else must be going on, and as far as what that something else is, your guess is as good as mine.