DoomslugTD

I love these! I have been looking for some more really good fanart and I found it. The watercolor style really fits stormlight!

As the title has suggested I was wondering what level of "death" must someone attain before they manifest conciously in the cognitive realm and then pass on. Heart has stopped (I doubt its as easy as that), Brain dead (I also doubt because I would think someone being supported while being brain dead in a coma for example would still be revivable by regeneration or something similar, though this doubt is just vibes), Brain cell death (Feels arbitrary, though difficulty / timeline wise it makes more sense) or full cell death (That can take more than a day, so I doubt.) Maybe after one of those conditions are met a timer is started that is either a constant for everyone, maybe determined by probability, maybe by how invested or connected they are and after the timer is up they move on. These are just me spitballing however, so if someone has other ideas let me know. Additionally I am aware that people with more investiture last longer in the cognitive realm, but do we have any sense for how much investiture relates to how long, even vaguely? Additionally I believe that regrowth or other similar healing can still save someone once they are in this state of being conciously in the cognitive realm, though I may be mistaken so I wanted confirmation on that. Basically I want to gain more understanding of when the cosmere considers someone "dead" and what healing can do for someone at any of these levels of "dead"ness. Thanks!

The model for the motion does not, but the book itself suggests that it is that way which are where the assumptions are coming from. We know that the steelpush always gives the same force at the same distance and its a fact that the force of gravity is independent of path (when we assume its a constant) from there we can then determine what the force of the steelpush is by asking what forces must be present on an allomancer for them to follow a certain motion. I chose this motion to be htanh(at)^2 because it is one of many that fits the motion as described in the books, in this simplified example we assume there are only two forces acting on this allomancer, the push and gravity. We know what gravity is so there is only one unkown, the force of the push, we solve for it and put it in terms of the distance from the metal in this motion because we know the steelpush should only depend on distance and not velocity or time. We now have a force in terms of the distance of this very particular motion htanh(at)^2 but from the first assumption we made (that was not made from the motion but instead made from the material of the book which the motion was derived from) we can say that the force of the steelpush should not depend on what path the allomancer is following and should instead only depend on their distance regardless of what path it took to get there and what path it will continue to follow. We can now confidently say that the same force equation we came up for that took in the distance when distance(t) = htanh(at)^2 as an input can take in any distance(t). Essentially we know that Fₚ(x(t)) = Fₚ(distance(t)) where distance(t) is an arbitary function, not from any part of the equation of x(t) but from the underlying laws that come from the work this entire discussion is based off of. We know this because we know that the force of a push is the same at a given distance regarldess of the overall motion because the book has told us this not because of some underlying physical principle.

Exactly this. x(t) can potentially describe a push while under gravitational pull, with a max of h. Anything you derive using your x(t) will have that same limitation. Even if you find a general and particular solution for your differential equation, it will still be bound by h because it was bounded by h when you created it. It only blows up at infinity if you don't let h go to infinity as well. I guess I must just have a fundemental misunderstanding of what is going on. But if you could point out where exactly my argument breaks down when switching between using the constrained x(t) to a non constrained distance I would love it, I have iterated it in multiple different ways in both my original post and other replies including my last one. I just know that in this model the force of gravity is constant regardless of the motion of the particle, and that the force of the steelpush is always the same at a certain distance regardless of the motion of the particle as well, I used this to justify why it should not matter if we replaced x(t) in the force equation with d(t) which is also represents distance but is not constrained to the original motion of the particle. Essentially the force of the push should not depend on the overall motion of the particle so we can generalize the force equation for a steelpush we find in terms of x(t) to a force equation for a steelpush in terms of a general distance d(t) that does not depend on path. Your example seems like it might have extra complexities that this simple idealized system does not have and might not be able to generalized because of the very specific requirments I layed out in order to get from x(t) to d(t) so I dont know if it proves what I am doing inccorect, though it does cast some doubt. Thanks!

Why does limiting x do have a maximum of h limit d? We have real world examples of systems that dont tend to infinity in the presence of gravity but would if they were not. Like if you a negativly charged plate fixed to the ground and a postivly charged ball above it they charges would push away to a certain distance before gravity became stronger then the colobmn force and it settled at a final height, but if you removed gravity from this equation the ball would accelerate from the plate but never stop therefore tending towards infinity. It is a very similar system here where we have a pushing force that when combined with gravity (x(t)) results in a motion that maxes out at h, but when not in the presence of gravity there is no pulling force that could slow the ball down therefore it should tend to infinity (d(t)) I explained this in more detail in my previous post if you missed that. There is another force at play in x(t) that is not present in d(t) so as far as I know it does not make sense to say that because x(t) is limited to h d(t) should also be. I dont understand why I cannot build a model based on how the motion looks in the presence of gravity, determine a force function that takes distance as an input that is the combination of steelpushing and gravity, subtract off gravity and get a new equation of motion that would behave differently as a fundemental part of the forces on it have been changed. There is a part of the original post which explains how I switched the force equation from depending on x(t) (the distance specifically in the equation of motion I started with) to a more general d(t) that is not constrained to x(t). I would suggest re-reading that portion of the original post if that is where the misunderstanding is routed from. But essentially it stems from the fact that the part of the force equation that represents the steelpush should not depend on whether the system is in gravity or not, so I can convert x(t) from the distance specifically when the motion is x(t) to a more general for where I represent distance as d(t). As an example the strength of the steelpush should be the same at 1 meter regardless of weather the distance is following the motion of x(t) or any other arbitrary motion. A more thourough explenation, in case there is still confusion. Fₚ(x(t)) = F(x(t)) + mg because by definition F(x(t)) is the combination of the force of the steelpush Fₚ(x(t)) and the force of gravity -mg. We also have the fact that Fₚ(x(t)) = Fₚ(d(t)) because the force of the push at any given distance does not depend on overall motion of the particle, only on the distance at that given point in time, so we are able to replace x(t) (the motion constrained to htanh(at)^2) with d(t) (which is the path when the particle is not under the influnece of gravity) or any other path of our choosing. The same logic can be applied to get F(x(t)) = F(d(t)) because the only additional force is a constant one which should also not change based on the path of the particle therefore the total force still only depends on the distance from the metal regardless of the overall path of the particle. This leaves us with Fₚ(d(t)) = F(d(t)) + mg which is not constrained to htanh(at)^2 as d(t) can be a completely different path than x(t). This means that there is no reason to assume d(t) has a maximum of h. Finally, the last thing is just logic. I have iterated this in my last post, but it just doesnt make sense for d(t) to have a maximum of h because the only force in that system is the pushing force, which should only be able to push. This is where the x(t) that I chose runs into problems because not only can the force be negative at certain distances (which would result in pulling, and therefore does not fit with steelpushing in the books) it blows up to infinity at infinit distance which also does not fit the books.

I think there was a misunderstanding, Fₚ(d(t)) is the force only of the steelpush. We specifically subtracted the force of gravity from the F(x(t)) (force equation we derived from x(t)) to get that. I will post the defininitiions of all the functions I am using right below this, I would especially focus on x(t), d(t), F(x(t)) and Fₚ(d(t)) so we can discuss this while having a common set of defined functions. The final equation I am trying to solve for is Fₚ(d(t)) which is also the equation I am taking the limit of in this case. This equation should approach 0 as d(t) gets larger because this is purely the force of the steelpush as a function of distance (not in the presence of gravity) and that should approach 0 as d(t) gets infinitly large. I dont believe d(t) is limited by h in the same way x(t) is either due to the fact that x(t) is in the presence of gravity and d(t) is not. There may be a way to prove this mathmatically, but at leasy logically the lim t->inf of d(t) should be inf because steelpushes can by definition only push and there are no other forces because by definition d(t) is the motion of the steelpusher only under the influence of the steelpush therefore there is only ever be a postive therefore the velocity will always be increasing with respect to time therefore the distance will also always be increasing with respect to time and therefore the lim t->inf d(t) = inf. I belive the majority of the confusion has come from unclear communication about what each function means as far as I can tell. If this is not the case and we were interprating the functions the same way then I would welcome some clarification on your statments! As an additional thing, before I solve for d(t) I do actually need to solve for x(t) which I am so close to (due to the fact that I started with x(t) = htanh(at)^2) but this is missing something very important being the initial conditions, because that is only true if x(0) and x'(0) = 0 so I dont have a general solution to the homogeneous equation as of yet let alone d(t). If you have any way to help me solve x"(t) = 2a²(h - 4x(t) + (3/h)x(t)²) or you have some technique for adding initial conditions when I have the specific solution when x(0) and x'(0) = 0 that would be super helpful!

Awesome! Well I am still super happy to be talking to someone who has considerable experience with physics so thanks for humoring my wild thoughts! Ohhhh, I did not know that you could extrapolate out from the homogenous equation like that. I have not formally learnt how to solve differential equation as of yet, so my techniques are often very scattered. I usually end up just using the laplace transform for almost anything cause it applies to a lot of stuff, so I did not really notice you could do this. I will give it a shot, thank you! So, I end up with a final force equation Fₚ(d(t)) and I am wanting that to approach 0 as d(t) becomes larger because at large distances it should logically follow that the force is ≈ 0. With Fₚ(d(t)) = 2ma²(h - 4d(t) + (3/h)d(t)²) + mg I just noticed that with respect to d(t) Fₚ(d(t)) looks like a second degree polynomial and with that postivie d(t)² term it looks like that equation will blow up to infinity at extremly large distances which is a problem considering we want the force to decrease with distance. Am I misinterpreting something here, or is there an error in this work? Does it not matter that the force would be extremely large for high values of d(t)? Another thing, this might be bad notation on my part, but h is not a function of time or anything else, its not the current height of the allomancer it is defined as being the final height they would reach if they are pushing at full strength directly upwards in scadrials gravity. It is an initial paramter that comes from how strong the push is, just like how a is a paramter that defines how quickly the allomancer reaches that final height h. I used these values because they seemed like the easiest values to work out in readings and are fairly simple to fit into the skeleton structure of the original tanh(t)^2 with h multiplied on the outside and a being multiplied on the inside of the tanh(t)^2. So taking the limit as h approaches infinity is equivelent to asking how the function behaves as the allomancers push strength approaches infinity (though you would have to increase it in a very particular way to not also get a change in a as that is another paramter that depends on the strength of the push.) The current height of the allomancer in this example would be represented by x(t) or d(t) (depending on whether you are in gravity or not) as the allomancer is always pushing straight up away from the metal. I hope that clear some things up, though if you believe I am still approaching this wrong I am happy to be proven wrong. I would absoltely love it if my solution worked!

Not sure if this is the right place to post this, but it was my best guess. For anything I like I am always tempted to make a game out of it and the cosmere is no exception. I have done many different attempts, with my most common being a sort of ttrpg, though that is something I will no longer be pursuing for obvious reasons. I will spare the details but I wanted to try making a tactical war game based on the cosmere as a project (warhammer ish) and I wanted to create this post to be able to ask a question whenever I think of it to assess what people thought about certain game mechanics. I will spare you the details of how the game works so far for now, though if you would like to know I am happy to share. My first question is for those who have experience playing war games. I know in most war games individual infantry units usually only have one hp, wound charactersitic or whatever else they call it. I love having precise control over things though, so I am very tempted to make the basic infantry have more like 3. I know this causes a lot more bookeeping, but I am not really the person to judge how annoying that is because I am the type of person who is more then happy to sit for 5 minutes crunching numbers for a game. Thoughts? I had also had the idea to have 3d printed rings (if I ever get around to actually making tokens or minis for this, which knowing me might not happen) that can go over minis with little slots you can clip small colored tokens into to keep track of any changing things like HP or exhaustian (as that is another mechanic I want to introuce but am hesistant of becoming cumbersome.)

@DrPhysics as a note before jumping into the rest of the responses, I believe you were double posting (making more than one post before someone responds) which I got in trouble for in this exact thread earlier. I was told that if you have more things to add you can edit your most recent post and add the things there, and if you make a new quote it will still alert the person just like if you created a new post. Just thought I would let you know I have made some edits to the original post to make things clearer and fix a couple mistakes you mentioned. To make sure we are talking about the same thing here is a list of the updated equation and what they mean. x(t) = distance as a function of time in the presence of gravity. (The motion we see in the book that I am basing this experiment on.) x'(t) = velocity as a function of time in the presence of gravity. x"(t) = accelerations as a function of time in the presence of gravity. x"(x(t)) = acceleration as a function of the specific distance x(t) in the presence of gravity. (It is there its just in the same line as x"(t).) d(t) = distance as a function of time with no gravity. (I have not solved for this (yet) because I am unsure how to solve the differential equation for d"(t).) d'(t) = velocity as a function of time with no gravity. (I have not solved for this (yet) because I am unsure how to solve the differential equation for d"(t).) d"(t) = accelerations as a function of time with no gravity. (I have not solved for this explicitly but we can write it in terms of d(t).) d"(d(t)) = acceleration as a function of general distance with no gravity. (I did not show this explicitly but its pretty easy to derive from the force equation. This is the differential equation you would need to solve in order to find d(t)) F(t) = total force as a function of time in the presence of gravity. F(x(t)) = total force as a function of the specific distance x(t) in the presence of gravity. Fₚ(t) = force provided by the steelpush as a function of time (this one depends on whether or not there is gravity or other external forces and requires us to solve for d(t) to figure out.) Fₚ(x(t)) = force provided by the steelpush as a function of the specific distance x(t) with no gravity. Fₚ(d(t)) = force provided by the steelpush as a function of general distance with no gravity. (This is the equation we are trying to solve for.) YES, sorry! Thats a blunder on my part. It is meant to be a t. In my edits I was taking the lim d->inf of f(d(t) (originally it was in terms of x but due to the edits some terminology was changed, but it was always in terms of distance) not lim t->t of x(t). This has left me a little confused about what you are talking about in that sentance. As for the rest of this paragraph I am a little unclear on what you are trying to say. We dont actually know what lim t->inf Fₚ(t) is because we only have Fₚ(d(t)) in terms of d(t) not explicitly in terms of t. We want that limit to also be 0 preferably, but like I said my edit was not focused on the lim t->inf it was focused on lim d->inf because we also know that the force function should tend to 0 as distance goes to infinity and we have the function explcility in terms of d(t). This is where we get the problem with the limits of my original htanh(at)^2 motion equation because the Lim d->∞( Fₚ(d(t))) where Fₚ(d(t)) = 2ma²(h - 4d(t) + (3/h)d(t)²) + mg is inf. This is obviously inccorect as the force from a steelpush should not blow up to infinity as the distance goes to infinity, but if we take htanh(at)^2 as our equation of motion in the presence of gravity that is what we end up with. (You can go through the original post again to see how I got the final form of Fₚ(d(t)) from htanh(at)^2. Essentially the equation of motion htanh(at)^2 satisfies requirments 1-6 but does not satisfy 7-8 which were the requirments I added in post after having this realization. I am sorry if I am not quite getting what you are saying, online forums are not the most condusive enviroment to well interpreted discussion sadly. I am pretty sure x"(t) = 2ha² · sech(at)² · (sech(at)² - 2 tanh(at)²) I even just plugged the stuff into a derivative calculator and it confirmed it. Here is a link where you can look at it visually to intuitivily confirm its true https://www.desmos.com/calculator/lzrnge3drx. I then used the trig identity sech(x)² = 1 - tanh(x)² to get everything in terms of x(t) (specifically so I could avoid using x'(t) as that goes against one of the stipulations). I double checked it in the same calculator link I gave you earlier so if you want to test that x"(t) = 2ha² · sech(at)² · (sech(at)² - 2 tanh(at)²) = 2a²(h - 4x(t) + (3/h)x(t)²) it is all there, just turn on f₂(t) and j(t) and seeing that they have the same value for all values of t. If you want to see my work for how I got it in terms of x(t) just send me a message, but I believe everything I did in that section was correct including getting the final solution of x"(t) as a polynomial in terms of x(t). As for the last point. Tanh does violate x'(0) = 0 but I used Tanh^2 which does not violate that, I think this is what you were correcting in your edit, but I just wanted to make sure. Honestly... so am I. I fully believe that there is more to be revealed, and there is a very high likelyhood someones perceptions effect how much force they output. The problem is I couldn't really do a fun math problem like this if I justified the weird motion like that. Basically this thought experiment goes under the assumption that perceptions do not effect the force because that is simply the only case for which we can currently do any math. We dont know for sure that it does or does not, so I will make what little progress I can in this niche assumption until more is revealed in hopes that this groundwork helps some people in the future. I acknoledge this wont be a definitive model for steelpushing as, like you said, we just dont have enough data, but a lot of real physics is based upon the idea of constraining ourselves to very niche and honeslty unrealistic assumptions in order to make progress towards a more general understanding. I am taking the same approach here and creating a model that fits a very narrow set of assumptions in order to progress towards a more general understaning. Ideally it will be revealed that everything wonderfully fits the crazy assumptions I made and my little project becomes a basis of understanding for steelpushing, but more likely I hope that it can be a stepping stone towards a fuller understanding by exploring a very narrow aspect of the greater whole if that makes any sense? Not disputing that, I am just specifiying the gravitational constant of scadrial to point out its particular to the planet. The whole reason I pointed out g and m originally is because when I finished and saw them there I was very confused why the force of a steelpush would depend on the gravitational constant of a planet, until I realized that I initially paramatized the function in terms of height. I wanted to note that in the original post in case any others came away with the same confusion I did at first when seeing those seemingly unreleated constants in the final form of the equation. But thank you for confirming scadrials gravitation constant is the same as earths, I suspected as much but did not really want to go searching through WOBs. I am a little confused at what you are saying, mostly because my mind has been jumping all over the place on this but I will reply with what i can. I think there is a really good argument to be made for a power limit, it makes some intuitive sense, but we just dont have any confirmation in the book that it exists as far as I know. I am completely fine though if a power limit emerges naturally from the force equation I provide, like for example if it means heavier people will stay near the source for longer thus doing more work I am fine with that, I just dont want to constrain my initial equation even more with things we dont have proof of in the book if that makes sense? I would be interested in seeing what equations might result from that constraint though. But I think I need a lot more clarity and detail about what you are saying here, because it does sound very interesting and promising, I just dont quite understand it. That is the distance function in the presence of gravity, I am interested in solving for d(t) which would be the distance function without gravity which would require solving the system d"(t) = 2a²(h - 4d(t) + (3/h)d(t)²) + g. Thank you! I have just started my first year as a physics major now. I was always interested in this kind of stuff so I just found what I could online, though that is getting harder and harder the more advanced it gets (hence finding it difficult to even know if there is a solution to the differential equation i put above.) I take it you are a proffesor? If so I am honored to have someone so knowledgable helping me with my little theory!

Ok good to know, do we have anybody that has made some estimates based on in book stuff? I like making games based on the things I like, and so I am currently trying to design a war hammer esc game for the cosmere for fun but I want to investiture usage to actually be vases on something. If there isn’t anything that’s fine though, thanks for the help!

I could not find any posts specifically mentioning this, but if there are sorry for the repeat. Do we know enough to quantize how much stormlight different abilities / healing takes or how much they leak for different oath levels in terms of sphere denominations or any other form of measurment we have? Thanks

That would be included in the "constants" that I talked about as that should not depend on time or distance so is effecitively a constant according to this example. In general this is a highly simplified example where we are assuming they are doing a full push the whole time (though I will touch on what you mentioned about the cognitive aspect is a second) and not changing it with time as this gives us the simplist example and is reasonably what you would expect vin to be doing when she pushes to her apex. We really dont have anything in the books that says it has to take the form of 1/r^2, though it is the obvious suggestion given most forces in real life work like this (though not all.) I detailed the biggest problem with this in my orignial post, but the biggest thing is that it does not produce the motion that has been indicated in the book such as the smooth stop at the highest point. (We will get to the cognitive aspect in a sec, but I was assuming a full push the whole time to get to this as an experiment.) Regardless of the equation you are correct that the formula will depend on geometry at close ish distances as long as we assume steelpushing on the continuous surface or volume of the metal, though in the example I am basing this off of she is pushing off a coin at a relatively large distance, so it should mostly act as a point mass, thus simplifing our thought experiment to finding the equation of force based on a point mass like I am doing. This is actually a good point that I thought of, because investitiure and energy can be directly transformed into eachother there definitely could be a hard limit on the energy the allomancer can add to their system per second, but we do not really know enough about the conversion rate or how allomancy accesses investiture to make this concrete of a statment. Namely two things. One we dont know the conversion effeciency of allomancy, though that would be a fun thing to eventually experimently produce (we dont know if some investiture is dispersed in the process, or other small forms of energy or mass are created in the process, though it is not entirely unreasonable to make an ideal situation where the effeciency is 100% as this whole situation is highly idealized anyways. We still run into the conversion ratio for investiture to energy which we could only figure out experimently which seems like it might be a very difficult task) The second thing is that we dont actually know if allomancy has an investiture per second limit (whether through the person or the magic itself) because its magic. Because the investiture isnt being drawn from the metal (if it was we could just say allomancy converts whatever investiture / mass thats in the metal into investiture and thus must have a limit) and instead the metal is a catylist for drawing from invesititure, allomancy could easily just take as much investiture is required at any given moment to produce the effect we see without being limited in investiture / second. Reading the post you showed, they kind of a just assumed there was a power limit to allomancy without justifying it. I suspect that is because all real world systems have it because you are not just pulling energy from a 3rd magical source, but it actually has to come from somewhere in the system. This is not the case in the cosmere as there is nothing in the books that says someone must pull a constant amount of investiture at any given time. They are litteraly pulling power from god to do things, so they dont really need to limit how much investiture they turn into energy. Essentially we dont know that allomancers can only tap a certain amount of investiture per second, that is just one possible theory, albiet one that seems somewhat reasonable. Ideally we would come up with a force system that also coresponds to a constant addition of energy to the system, but it is not a requirment as far as I know. It would be intresting to add the constraint to see if we could narrow down the possible equations for motion / force but it is just one method we can use to get to a solution. This is a very good point, and something I have though about briefly, but is also something that almost entirely makes this thought experiment useless and leaves us where we started, with no idea what equation governs the forces in steelpushing other than guesses based on real life phenomina. If we let someone subconciously change their push strength during their ascent (such as vin subconciously changing her push strength to not bob up and down) we introduce a variable that can be any function of time, and thus we end up with at least one (and probably more) family of infint equations that could describe the motion and corresponding forces in the books. This is entirely reasonable, but defeats the thought experiments purpose and we just dont have enough info to solve for the equation if we allow the push force to depend implicily on time, thus I have made the (potentialy wrong) assumption that the person is pushing at full force the whole time as they believe they do in the books. I would love to be able to solve for the equation allowing for a variable that implicilty depends on time, but it is impossible right now, so I wanted to see if I could solve for an equation that works for this simplified scenario (that also may be entirely true, we do not know yet if their subconcious is actually able to control the output of this particular power.) I should have specified this all in my initial post, but it is also only a "Potential physics equation for steelpushing". It is one possiblility given certain assumptions, and an entirely different conclusion (such as 1/r^2 or an infinit family of other equations) can be reached with other assumptions that I didnt make, such as allowing the equation to implicitly depend on time. So there is a little error here, though I do understand your point. The tanh is the equation for motion taking time as an input that I proposed, wheras 1/r^2 is the equation for force that takes distance as an input. X(t) versus F(x) so they will understanbly take very differnt forms. F(x) according to the tanh model for motion is a quadratic which feels much closer to being reasonable than the tanh you were comparing it to. If you are confused you can check out the orignial post again where I derive that quadratic. I also made an edit to the post a while ago mentioning how tanh doesnt actually work which I think you will find interesting, which actually has somethign to do with that quadratic force equation being unreasonable. In the end I do actually need a F(x) that converges to 0 at infinity such as e^-r or 1/r^2 that also produces motion similar to the tanh for reasons I listed in the original post, which will produce a result that feels more reasonable. In total I do get your point, it does feel odd to have force or motion equations that are not similar to what we have in real life, but within the confines of the cosmere, an equation using magic does not really need to have any physical properites that imply its motion, because the laws of the magic system are the thing that determine the motion and forces and can be entirely arbitrary as they are accidental or puprosful results of a god messing with things. It is very nice if we get an equation that is similar to what we have in real life (and like a mentioned earlier, it will probably have to be cause of constraints I listed in the original post near the end), but it is not a requirment like in real systems as far as I can tell, despite how unintuitive that feels. I do really appreciate all the input though, you raise very good points, and we could definitley come to many different conclusions if we took it in all these exciting directions. I would be very happy to discuss the equations that would come out of these assumptions if you would like to, I would just like to acknoledge that they are essentially answer a "different question" if we do.

So bigger metals can simply be pushed farther away, but the force you generate from pushing on them is not greater than if you pushed on a smaller metal if you are in range of both? Excuse the math, but does that mean the value of the force function does not depend on the mass of the metal when in range but the force function discontinously cuts off quicker for smaller metals and slower for larger metals. For example if F(d) = 1/(x^2) (From the book it actually cannot equal this, but it's the function representing magnetism which is vaugely similar) you would get the same force at closer ranges regardless of the size of metal, but for a smaller metal the function would cut off at say... 100 meters, but for a larger metal it might cut off at 200 meters, but for those last 100 meters you get out of the big metal you would get a very small force as you are still incredibly far away? Mathmatically you could represent this as F= {x≤f(M): b/(x^2), 0} where M is the mass of the metal and a and b are scaling constants. I was always under the impression it would be more like this F = {x≤f(M): b/((x/f(M))^2), 0}. This equation would mean that if a metals mass doubled its range compared to a "normal" push, that every point within the range of your push would effectively be considered to be half as far away as in a "normal" push. (For example, if your produced 1 newton at 10 meters with a normal push I would expect you to produce 1 newton at 20 meters when pushing with double the range.) This however is just how I always assumed range would be represented and it could likely be the first way instead. "So, what if some factors (Flaring, Duralumin) affect dp/dt - but other factors (e.g. allomantic stength through Lerasium or hemalurgy) only apply to dp2/dt. That might explain why Vin's flares and Duralumin pushes affected her as much as the object being pushed, but Elend's push-war with the inquisitor was so one-sided. If it was an inquisitor with only one A-Steel spike, it would have been a push match between 0.9 strength against 2 strength multipliers (based on previous notional example)." (Sorry I could not figure out how to quote something in an edit and I did not want to double post) I would really love that as a system, and it could explain a lot, but it would have a couple big implication around the face that it would break Newtons 3rd law. If you only effect dp2/dt it means you are adding more force to the object you are pushing then to yourself, which means you are not having an equal and opposite reaction anymore, which is a core tenant of the magic system so far. Its not like it could not happen as investiture could defintely be considered an "outside force" on this system, making Newtons 3rd law irrelevent, but so far as we know it has always been indicated that investiture does not unevenly add forces to one side in this way during steelpushes and just acts as a way for the two objects to interact at a distance. (Essentially like physically pushing off something but at a distance.) It would explain some of the discrepencies found in the system, and would allow for some really interesting nuance though, so I like the idea!

Sorry, I misinterpreted, I thought you were saying both the mass of the allowanced and the metal only affect the system according to Newtons 3rd law, not just the metal. My bad! If the mass of the metal only effects the system through Newtons 3rd law, why can steelpushers push larger metal things from further then smaller metal things? Thanks!

Well it does definetely effect it in that way as it is a physics system (somewhat) but with all the examples we have of heavier people pushing harder it feels weird that mass not does not play into the strength of the actual push. The only way inertia (the way mass makes objects harder to move) effects the system is by making you stay closer to the metal for longer therefore producing more net force over the duration of the push. It would not however effect the amount of force you genereate "per second" (technically not how force works) at a given distance. From examples in the books it definitely feels like heavier people produce more force "per second" at a given distance, especially when we give the example of Wax leveling a building. If it were just because of inertia, you could get an identical (even more effective) outcome by bracing yourself against a wall by pushing, as you are effectively giving yourself the inertia of the wall and the entire earth depending on how strong the fondations are, although you would likely be squished (Depending on how the force of a steelpush is distributed across your body). As a final example, if the force of a push was not dependent on mass, heavier mistborn would not be able to "hang" as high when pushing up into the sky, as the force of gravity is stronger on heavier people, but the steelpushing force counteracting it would not be. I dont believe a height difference like this is mentioned in the books, which feels odd if it is the case.