Towards a mechanical description of Steelpushing

[luminos]

So, I love Sanderson's rules based magic systems, and really appreciate the way these systems make the story more interesting. In particular, they make things more interesting by adding intelligible restrictions to the use of magic: it has to follow established rules. Now, I know that some handwaving and suspension of disbelief is still necessary, but the math nerd in me wants to see how far I can push the explanations without doing that.

This thread is an attempt to quantify how steelpushes work in terms of mathematical equations dealing with regular physics. If anyone has already done this, I'd love to see their work.

Anyways, here is the procedure:

1.) Identify the basic properties of Steelpushing, as described in the books.

2.) Create a general formula to describe how Steelpushing works in terms of mass, acceleration, etc.

3.) Solve for the constants in the formula(s) based on specific examples in the books.

4.) Describe how this formula(s) would apply in other examples in the book, and see if the theoretical results give a rough match to the description.

General thoughts: Its been a while since I've read the original trilogy (although I've read alloy), so I might be hazy on some of the details. I think Steelpushing is basically exerting a force that is proportional to the pusher's weight. It also seems to inversely vary with distance from the object being pushed, but I might be making up or misremembering this.

The basic goal for me based on a formula using these facts is to solve for the proportionality constant. I want to figure out what proportion of the pushers weight is needed to achieve some of the "jumps" that Vin or Wax achieve.

What will be super helpful is any observations about the way pushing actually works in the books. Also welcome are guestimates about some numbers; things like how high you think a push can get a person off the ground. Although anything at all relating to this topic is welcome.

I'm tired, so I'll stop this post here, and come back to actually try to work some of this out later.

[ReaderAt2046]

Basic rules as I understand them:

Steelpushing generates a force of intensity X. As per the Third Law of Motion, this generates an equal push on the metal and the Allomancer. By the Second Law of Motion, this means that the acceleration of the metal relative to the accleration of the Allomancer is iversely proportionate to their respective weights. Mathematically: a_allomancer/a_metal = m_metal/m_allomancer.

[Eri]

I believe there's a power limit, not only force limit for an allomancer. (power as in P=dW/dt, not as ‘strength’.)

It is possible to ‘fly’ – move oneself upwards; but thrown coins are dodgeable without Atium. First I've assumed just a constant force, as you do, and I've tried to calculate that force. And it made no sense – either you should be unable to overcome gravity on your body, or the coins would shoot insanely fast.

And without a limit on power Pushing yourself against a coin flying in the air would work as well as against one lying on the ground. I'm not sure, why, but my husband says so, and he'd studied physics.

Edited March 7, 2012 by Eri

[Kurkistan]

I believe there's a power limit, not only force limit for an allomancer. (power as in P=dW/dt, not as ‘strength’.)

It is possible to ‘fly’ – move oneself upwards; but thrown coins are dodgeable without Atium. First I've assumed just a constant force, as you do, and I've tried to calculate that force. And it made no sense – either you should be unable to overcome gravity on your body, or the coins would shoot insanely fast.

And without a limit on power Pushing yourself against a coin flying in the air would work as well as against one lying on the ground. I'm not sure, why, but my husband says so, and he'd studied physics.

I applied my vast lack of knowledge about the meaning of energy and physics in general to a similar thread awhile back, and my conclusion as to why you couldn't maintain flight off of a movable object was because the amount of energy a Coinshot could put into a push at any given time was dependent upon distance between the Coinshot and the anchor, perhaps modeled by an inverse exponential scale, akin to gravity.

[Zas678]

Something else to take into account is the connection to Preservation. The more connection you have to Preservation, the more effective your Push is. Also, there should be a spot in the equation for how much you are flaring the power.

[Ironeyes]

Anybody taking friction into account? I always wondered how they managed to push on coins obliquely when the coins rested on a smooth surface, ie paving stones. Wouldn't that just shoot the coin sideways across the ground?

[ulyssessword]

My first draft of an equation:

f = force (newtons)

Met = Amount of metal burned (milligrams per second)

eff = efficiency of metal burning (allomantic magic amount (AMA units) per milligram per second)

v = velocity between allomancer and metal (positive for moving closer, negative for moving away, nothing for moving sideways)

C_v = Allomantic Velocity

C = Allomantic constant

M: Mass of the allomancer (kilograms)

m: mass of the metal object (kilograms)

d = distance between the Allomancer and targeted metal (metres)

f = (Met * eff) * ((v + C_v) / C_v) * (C * (M + m)) / (d²)

Now for an example:

Met = 1 mg/s

eff = 100 AMA/(mg/s)

v = 100 m/s (towards)

C_v = 1000 m/s (constant?)

C = 100 Nm²/AMAkg (constant)

M: 50 kg

m: 0.5 kg

d = 10 m

-- Start with the equation...

f = (Met * eff) * ((v + C_v) / C_v) * (C * (M + m)) / (d²)

-- Substitute in the values...

f = (1 mg/s * 100 AMA/(mg/s)) * ((100 m/s + 1000 m/s) / 1000 m/s) * (C * (50 kg + 0.5 kg)) / ((10 m)²)

-- Multiply together within the sets of numbers...

f = 100 AMA * 1.1 * (100 Nm²/AMAkg * 50.5 kg / 100 m²)

-- Multiply the sets together...

f = 110 AMA * 50.5 N/AMA

-- And finish!

f = 5555 N

I think I got everything that there is. You can push more strongly on objects coming towards you than going away, giving a max speed to pushes, both power and metal efficiency are covered, and distance scales the force the same way as gravity does. I might revise and clarify this later.

EDIT:

Basic rules as I understand them:

Steelpushing generates a force of intensity X. As per the Third Law of Motion, this generates an equal push on the metal and the Allomancer. By the Second Law of Motion, this means that the acceleration of the metal relative to the accleration of the Allomancer is iversely proportionate to their respective weights. Mathematically: a_allomancer/a_metal = m_metal/m_allomancer.

Absolutely right. My equation just tries to determine how strong the force is...

Edited March 8, 2012 by ulyssessword

[Odium's_Shard]

In my opinion, as a man of limited Physic value, what you just did was well beyond me and seemed pretty darned brilliant. Well done!

[Thor]

*snip*

Not bad, though I disagree with the masses being additive and I don't think velocity components need to be included (the force is not dependent on velocity, just distance, imo.)

Here is my shot at an equation. (Obviously based on gravity/EM force and there are no numbers, though it should be fairly easy to calculate if one desires.)

Γ = Purity of Metal (based on a value of 1 being 100%, maybe not exactly correlate to the actual metal purity as purities less than some value may not give any power), unitless

τ = Burn Rate (rather obvious), units are mass/time

σ = Allomantic Strength (based on connection to Preservation), unit is Allomantic Power (P is symbol, new unit)

µ = Metal Constant (constant based on metal, Steel has a positive value, Iron is negative), units of Allomantic Power * time * mass^2/length^3

m = Mass of Object (being pushed/pulled), units of mass

M = Mass of Allomancer (duh), units of mass

r = Displacement (between Object and Allomancer), units of length

Formula is (units should be correct...):

F = ΓτσmM/µr^2

This formula gives an attractive force for Iron (F will be negative for Iron) and repulsive for Steel, is inversely proportional to distance (power may need to be higher for the dropoff in strength we see), contains variables for burn rate, Allomantic Strength and the purity of the metal.

For more fun, the Potential Energy of an Allomancer's push/pull is:

U = -ΓτσmM/µr

[ReaderAt2046]

There's something important we've forgotten, which will help solve several conondrums.

The force of an Allomantic Push/Pull is proportionate to the mass of the metal being affected.

We have seen a hundred times throughout the books that the bigger a piece of metal, the better an anchor it is and the more force an Allomancer can exert on it. In view of this point, I venture to propound the following equation.

F_allomancy = k*m_object*R_b*[m_pure]/d

F_allomancy is the force exerted on the allomancer and the metal. It is positive for Pushes, negative for pulls. Units are Newtons for this example.

k is the allomantic constant, which represents the allomancer's strength in the given metal. It is positive for steel, negative for iron. Units are Newton-meter-seconds/kg².

m_object is the mass of the metal being pushed on, in kg.

R_b is the burn rate in kg per second.

[m_pure] is the purity of the metal, represented as a percentage of ideal purity. No units.

d is the distance in meters.

After this, the ordinary F=ma equation applies.

[ulyssessword]

Not bad, though I disagree with the masses being additive and I don't think velocity components need to be included (the force is not dependent on velocity, just distance, imo.)

I'm not sure what the relationship between masses is, but I'm fairly sure that it isn't a straight multiplication, and addition seemed like the only other simple answer. If Allomantic force scaled with mass multiplicatively as you proposed, an Allomancer that was fixed to a point could accelerate any object at the same rate, regardless of its mass.

As for Allomantic pushes/pulls depending on velocity, I thought it was required because of the behaviour of dropped coins while "flying". If the force was equal regardless of velocity, the push would accelerate at about 200 000 m/s² (5g coin vs 50 kg Allomancer, Allomancer pushing strongly enough to leap upwards at 10 m/s²), and the Allomancer would feel the push normally.

[luminos]

I just wanted to say that I am superappreciative of everyone who's worked on this problem, and that I haven't forgotten about starting this thread. I'll put in my own attempts at an equation sometime next week, but I'll probably borrow heavily from the other contributors.

[Shivertongue]

It's simple.

F = S_crEw [i^t] = [i^t]^s M_aG^ic

There. I contributed.

You crazy kids, getting all your high-falutin' math mixed up in my fantasy. Why, in MY day, when something worked by magic, we just accepted that it was magic! And we liked it!

And we had to walk our dragons UPHILL to Mage College, BOTH WAYS, on Pluto! And we were happy about it!

[Aiken Frost]

It's simple.

F = S_crEw [i^t] = [i^t]^s M_aG^ic

There. I contributed.

Huahuaha, awesome!

[lil_literalist]

Aha! A topic geared just for my talents! It seems like Thor and RA2046 are on the right track, but I'd like to take a crack at this myself. First of all, I would like to gather some existing knowledge, discuss what it means, and then move on to a formula. I'm going to refer to anything that you're pushing or pulling on as an anchor, even if it's something like a coin.

Observations

1. The thickness of the blue lines seems to have some relation to how strong the force is; the thicker the lines, the stronger the force could be if you pushed/pulled.

2. Lines grow thinner as anchors get farther away.

3. Lines of small anchors are thinner than large anchors.

4. An allomancer can vary the strength of his pushes/pulls (although not necessarily with much control) through flaring metals.

5. If a metal is not pure, it may be ineffective.

6. The pushes/pulls of some allomancers are stronger than others.

Significance:

2. As many have suggested, I agree that this follows an inverse-square relationship. Otherwise, blue lines might be sprouting from everywhere with a large metal object, and not just the places nearby. For instance, if this were not an inverse-square relationship, you would be able to see the spires of Kredik Shaw anywhere in Luthadel.

3. The force is dependent upon the mass of the anchor. (As a side note, I suspect that it would be more rational to say that the force is the sum of all of the forces between the metal molecules and the allomancer. In other words, just as you are attracted to every single atom gravitationally, an allomancer is pushing off of every metal molecule in the object at once. So for example, if you had a square kilometer of iron plate beneath you, it wouldn't provide nearly the same force as the same amount of iron shaped into a compact object like a sphere, right next to you.) For the purpose of this post, we will assume that the anchor is a single point in space.

4. There is a variable of effort involved. Its relationship to the amount of metal being burned per unit time is unknown, but is possibly linearly-related.

5. According to Kelsier, if a metal is not absolutely pure, it can still work. However, the more unpure it is, the less safe. This suggests an exponential relationship, possibly more than a simple square.

6. An innate difference, which will just be a unique constant for each allomancer. (Though it would be interesting to try and find how this varies by diluting of allomancy from the original allomancers to the nobility of Vin's and Kelsier's time).

One last notable thing is that there is nothing to suggest that the mass of the allomancer has anything to do with the force. If Zane is hovering over a coin on the ground and suddenly catches a person falling from the sky, the force of his push will not increase; he will simply descend until the coin gets closer to him and the force of his push again equals out the combined gravitational pull of both persons. If you argue that it's the body mass of the allomancer, think about this: if an overweight lurcher went on a crash diet, would the blue lines get thinner? Does this mean that fat coinshots have more force behind their pushes than skinny ones? No, it just means that because of their body mass, they are affected less by the force that pushes back on them.

Definition of variables

F = The Force on either anchor or allomancer (ignoring sign conventions for now)

A = Innate Ability. This is a measure of how strong the allomancer is. (Higher for people like the Lord Ruler, lower for your everyday allomancer.)

E = Effort. How strongly an allomancer is flaring metals. Normal pushes should be a fraction like 50%, while max flaring would be 100%. Duralumin allows percentages above 100%.

M = Mass of anchor.

P = Purity of metal, from 0% to 100% (although I'm not sure how you would burn a completely impure metal).

r = distance between the allomancer's center of mass and the anchor.

F = A*E*M*P^2/r^2

Comments on equation, and things to explore further

As I noted above on 4, the exponent of the purity is likely more than 2. If a metal is only 50% pure, then I would think that it should be pretty close to 0% effective, rather than 25% effective. The variable for purity needs to be given another thought. It might even involve something like a cos, where a little impurity is fine, but it quickly falls away.

It's possible that if you still argue for the body mass of the allomancer to be taken into account, that could be lumped in with innate ability A.

Can allomancers increase A through training or experience? Or are allomancers just as powerful when newly-snapped as when they're at their peak?

There should be a constant in there, to fix up all of the units. Since ability A is already an arbitrary variable that we don't understand or have units for, it would be fine to group it with that.

I chose to use Effort rather than a variable of how much metal an allomancer is burning (as others have done), as I felt that this is easier to understand. However, it might be useful to calculate this with the rate at which you burn metals. This might be especially true for burning duralumin, which burns all of your metals in a short burst. (Maybe the more duralumin you have, the shorter the burst, thus the greater the force?) Also, it's possible that 100% for one person might be different than 100% for another person. I really should revise this equation to reflect that, then.

The first equation used the velocity of the object relative to the allomancer as a variable. I feel that this is unfounded. (Though I will reexamine if supplied with evidence to the contrary.)

EDIT: Removed spoilers.

Edited April 23, 2012 by lil_literalist

[Gagylpus]

The force is dependent upon the mass of the anchor. (As a side note, I suspect that it would be more rational to say that the force is the sum of all of the forces between the metal molecules and the allomancer. In other words, just as you are attracted to every single atom gravitationally, an allomancer is pushing off of every metal molecule in the object at once. So for example, if you had a square kilometer of iron plate beneath you, it wouldn't provide nearly the same force as the same amount of iron shaped into a compact object like a sphere, right next to you.)

Yes. Actually, we can extend this to the allomancer as well. Every bit of matter in the allomancer's body is exerting a force on every bit of matter in the anchor. This is the only way for the center-of-mass rule (steelpushes and ironpulls always act through the allomancer's center of mass) to fall out of the equations naturally, without any handwavium.

This means that the basic allomantic force equations actually should not relate the force between the allomancer and the metal, but the (vector) differential bit of force between the differential bit of allomancer mass and the differential bit of metal mass. The differential force should be a scalar function f times the unit vector r/|r|, times the differential masses.

dF = f(|r|, burn rate, other variables)*dm_a*dm_m*r/|r|

Total force F = double integral of dF over allomancer body (domain of dm_a) and metal body (domain of dm_m)

(Each of those integrals, of course, is over a three-dimensional domain, so the end equation is six-tuple integral of a three-dimensional vector over very complicated domains. Not something that anyone wants to do by hand.)

To fit with what we see in the books, f should fall off with |r| (the distance between the allomancer bit of mass and the metal bit of mass) and increase with burn rate and innate allomantic strength. Also, the domain of the first integral is fixed to be over the whole body of the allomancer (although not mathematically fixed, since the allomancer can change their position and orientation). All the evidence of the books indicates the allomancer has no control over this. However, it does appear that they can control the target domain. Vin can pull on part of a spire of Kredik Shaw without being pulled towards the center of mass of all of Kredik Shaw, so a skilled allomancer can probably push on just part of a coin, causing it to spin as it flies away.

The one problem with this is that the books very frequently describe opposing pushes or pulls as squeezing or stretching the allomancer between them, when, if the allomantic force is a body force as the center-of-mass rule indicates, the opposing forces should just cancel out. There's a similar problem for metal subjected to opposing forces from two allomancers, such as the coin flattened by Vin and Kelsier in book 1. Although that might be explained by the two of them subconsciously pushing more on the side of the coin that is closer to them.

...and that was a bit longer than I expected. Haha.

[lil_literalist]

I'm still not convinced that the force is dependent upon the mass of the allomancer. When burning pewter, at least, it's not. Vin showed that she could jump insanely high by just using pewter. The explanation for that was that all of the power that pewter gave her was packed into her tiny body. I see no reason why steel or iron should be different.

[Gagylpus]

I'm still not convinced that the force is dependent upon the mass of the allomancer. When burning pewter, at least, it's not. Vin showed that she could jump insanely high by just using pewter. The explanation for that was that all of the power that pewter gave her was packed into her tiny body. I see no reason why steel or iron should be different.

Pewter is different because it enhances the force produced by the muscles, rather than generating an entirely new force. Vin gets more out of pewter because there's less of her to accelerate with her strengthened muscles, and because with her higher innate allomantic strength she can get more strength out of the same amount of pewter. (And possibly because there's less of her for the pewter to enhance in the first place.)

That doesn't neccessarily mean that the force generated by steelpushes and ironpulls is proportional to the allomancer's mass for the same amount of metal burned. I would guess that the default force level is proportional to the allomancer's weight (since it is, essentially, using your weight as leverage against the push/pull) but that heavier allomancers have a higher base burn rate as well. Then the impulse, per unit mass of the anchor, per unit mass of iron/steel burned, is approximately constant and independent of the allomancer's mass.

Actually, thinking about it, it seems likely that the base burn rates of all the physical metals are higher for heavier allomancers. With pewter and tin there's more body for the metal to enhance, and with steel and iron theres more mass to generate allomantic forces from.

[lil_literalist]

Pewter is different because it enhances the force produced by the muscles, rather than generating an entirely new force. Vin gets more out of pewter because there's less of her to accelerate with her strengthened muscles, and because with her higher innate allomantic strength she can get more strength out of the same amount of pewter. (And possibly because there's less of her for the pewter to enhance in the first place.)

By "gets more out of pewter," you mean, jumps higher? Then yes, I agree. But a normal Thug would also be able to do some pretty impressive jumping stunts as well, if the amount of power gained was directly related to body mass. But if Vin can jump far higher than a heavier Thug can, this means that they are generating roughly the same force.

Now, I don't recall seeing evidence that Vin was significantly stronger than other allomancers (except in Bronze, of course). What made her special, apart from being a mistborn, was her skill in using her powers, and how quickly she learned them. So her innate ability with Pewter should have been roughly on par with a normal Thug. If Thugs were able to use their larger mass in order to burn more pewter to compensate for their heavier bodies, then seeing her jump several feet (AFB, and I don't recall the exact number) wouldn't have been as remarkable as it was.

That doesn't neccessarily mean that the force generated by steelpushes and ironpulls is proportional to the allomancer's mass for the same amount of metal burned. I would guess that the default force level is proportional to the allomancer's weight (since it is, essentially, using your weight as leverage against the push/pull) but that heavier allomancers have a higher base burn rate as well. Then the impulse, per unit mass of the anchor, per unit mass of iron/steel burned, is approximately constant and independent of the allomancer's mass.

Actually, thinking about it, it seems likely that the base burn rates of all the physical metals are higher for heavier allomancers. With pewter and tin there's more body for the metal to enhance, and with steel and iron theres more mass to generate allomantic forces from.

We don't see any indications in the books that the burn rates are different for different-sized people. In fact, I would bet that we would have heard something if it were true, especially with regards to Atium.

If mass were a factor in the equation, then two coinshots of different weights pushing themselves upwards off of coin would peak at roughly the same height. On the other hand, If those same coinshots pushed against the same coin in-between them (as Kelsier and Vin did), then the coin would be closer to the lighter allomancer than the heavier (if the coinshots were anchored like Vin and Kel). I'm AFB, but I believe that the coin that Vin and Kel smashed up was described as mid-way between them.

I think that I'm comfortable with your physics, just not with your basic assumptions. If we can't agree on how allomancy works, then I'm afraid that we'll just have to agree to disagree.

[Eric]

Definition of variables

F = The Force on either anchor or allomancer (ignoring sign conventions for now)

A = Innate Ability. This is a measure of how strong the allomancer is. (Higher for people like the Lord Ruler, lower for your everyday allomancer.)

E = Effort. How strongly an allomancer is flaring metals. Normal pushes should be a fraction like 50%, while max flaring would be 100%. Duralumin allows percentages above 100%.

M = Mass of anchor.

P = Purity of metal, from 0% to 100% (although I'm not sure how you would burn a completely impure metal).

r = distance between the allomancer's center of mass and the anchor.

F = A*E*M*P^2/r^2

I'm thinking that this doesn't quite describe the MAG version, so I'm going to rewrite it a little to do so (adding actual units). As the MAG is secondary canon, I do not mean this as a challenge to the fine work you've done. I just want to see if I can (and whether it still works).

Force = Newtons (kg·m·s-2)

Innate Ability = I guess Power Rating from the MAG is the only thing we have. Minimum usable level is 2, maximum before hemalurgic enhancement (or, potentially, Nicrosil tapping) is 10.

Effort = Since your Ability is effectively less when not flaring, we have 0 for basic burn, +1 for flaring (per MAG, which doesn't seem right), and +D (Duralumin Power Rating) when using Duralumin. Easy shorthand, even though it isn't quite additive in that fashion.

Mass = kg

Distance = m

Purity = Total Molality (Mol/kg) of all impurities (solute, Mol) within the metal (solvent, kg). The higher the value, the more impure. Using this measure eliminates the oddity of a 0% pure metal working with the equation. It requires that it be a divisor instead of a dividend to properly scale the power, though. (Might be a necessary change to the general formula as well.)

F = [ ( A + E )* M ] / P^2 * m^2

Comments on equation, and things to explore further

As I noted above on 4, the exponent of the purity is likely more than 2. If a metal is only 50% pure, then I would think that it should be pretty close to 0% effective, rather than 25% effective. The variable for purity needs to be given another thought. It might even involve something like a cos, where a little impurity is fine, but it quickly falls away.

One of the reasons I like Molality. Since we're talking about very small masses of metal, even small amounts of impurities tend to result in very high Molality. One gram of metal with .060 grams of a sodium chloride impurity would be 1.03 Mol/kg, roughly a 6% loss in power from roughly a 6% impurity by mass. But .090 grams would be 1.54 Mol/kg, almost a 58% drop in power from only an 8% impurity by mass. Tiny amounts adding up really quickly. Just what the doctor ordered.

Can allomancers increase A through training or experience? Or are allomancers just as powerful when newly-snapped as when they're at their peak?

The MAG assumes you can become stronger with time. I would argue that Marsh's training dialogue with Vin suggests the same, but it isn't explicit. Her ability to do right off what took him practice to obtain, but not having all of his ability with it. Could just be skill, but I think Marsh increasing in power over time played a part in what he was capable of doing.

I chose to use Effort rather than a variable of how much metal an allomancer is burning (as others have done), as I felt that this is easier to understand. However, it might be useful to calculate this with the rate at which you burn metals. This might be especially true for burning duralumin, which burns all of your metals in a short burst. (Maybe the more duralumin you have, the shorter the burst, thus the greater the force?) Also, it's possible that 100% for one person might be different than 100% for another person. I really should revise this equation to reflect that, then.

I think my adjustments accomplished this last goal, but only if the MAG is accurate. I really don't think it is completely, but it's a good place to start.

There's a similar problem for metal subjected to opposing forces from two allomancers, such as the coin flattened by Vin and Kelsier in book 1. Although that might be explained by the two of them subconsciously pushing more on the side of the coin that is closer to them.

Couldn't that also be explained by the side closer to the Push being subjected to slightly greater force from that side, simply due to the slight difference in distance from the Allomancer? Vin is closer to Side A than Kell, so the force of her Push doesn't diminish as much as his does on that side, but on Side B, the situation is reversed. The molecules seek equilibrium, squishing the coin.

Edited April 25, 2012 by Eric

[lil_literalist]

I'm a little rusty on my chemistry (ok, remarkably rusty), but wouldn't a completely pure metal have a molality of 0? I realize that on Scadrial, they probably aren't going to be making completely pure metals. But then when you square it, it will get even smaller. And when you divide by a small number, the force will get bigger, which clearly isn't the case.

I like that unit of measurement, though. You just didn't evaluate the limit at both endpoints. Perhaps dividing by (1+P)^2 would provide a proper fix.

Also, I haven't encountered the MAG yet (although I was able to figure out that it meant the Mistborn Adventure Game).

The added effort and ability does indeed make for easy shorthand, but it was designed that way, to be easily calculated. It's good to remember, though, that regulating the force of your pushes is pretty hard, so most allomancers just regulate the time that they spend pushing on something.

EDIT: Ok, I checked over my books this morning. The coin-pushing contest between Kelsier and Vin said that the coin hung in midair, "directly between" the two. That means that they were pushing with an equal force, which also means that unless Vin has far more of ability than Kelsier (which I highly doubt), their mass should not matter.

Also, I just realized that the (1+P)^2 doesn't work after all, since 1 is unit-less and P is not. However, P is really just a constant times a percentage of how pure the metal is, so it's really just arbitrary anyway. I will eventually do another equation to take into account some of the stuff that we've talked about, I promise!

Edited April 25, 2012 by lil_literalist

[Eric]

EDIT: Ok, I checked over my books this morning. The coin-pushing contest between Kelsier and Vin said that the coin hung in midair, "directly between" the two. That means that they were pushing with an equal force, which also means that unless Vin has far more of ability than Kelsier (which I highly doubt), their mass should not matter.

Doesn't Kelsier explicitly state that she must be way stronger than she should be to accomplish that? When talking to Sazed afterwards, iirc.

I'm a little rusty on my chemistry (ok, remarkably rusty), but wouldn't a completely pure metal have a molality of 0? I realize that on Scadrial, they probably aren't going to be making completely pure metals. But then when you square it, it will get even smaller. And when you divide by a small number, the force will get bigger, which clearly isn't the case.

I like that unit of measurement, though. You just didn't evaluate the limit at both endpoints. Perhaps dividing by (1+P)^2 would provide a proper fix.

...

Also, I just realized that the (1+P)^2 doesn't work after all, since 1 is unit-less and P is not. However, P is really just a constant times a percentage of how pure the metal is, so it's really just arbitrary anyway. I will eventually do another equation to take into account some of the stuff that we've talked about, I promise!

We could use (1 + P^2). That should work, I think. It won't fall off as quickly as (1+P)^2, but it solves the units and the scaling issue.

F = [ ( A + E )* M ] / [ ( 1 + P^2 ) * m^2 ]

I know the brackets aren't technically necessary, but they serve for clarity.

EDIT: Found the conversation between Saze and Kell. Page 183 in my paperback copy. Also, plugged in some numbers. Using the MAG terms might be useful, but the values themselves are rather too limited. We'll need more of a continuum, and the constant will probably have to be a multiplier in the dividend. Using MAG values, the Force winds up very low.

Moving away from the MAG stuff, Effort should likely be a multiplier from 0 (not burning) to 2 (highest flare). Duralumin ought to be around 4, and the usual burn a 1. I think I remember Vin putting one of her metals on a "low burn" at one point, hence wanting values below the baseline.

Since we have Mol in the equation but not in the result, and no time in the equation, we need to factor in the burn rate. Which is related to the Effort, so it will probably be our units of measure there (perhaps Mol/s).

EDIT: My participation in the Mechanical Properties of Atium thread have led me to a potentially important insight: does the isotope of the elemental metals matter for the purity rating? If so, consider that no one will have ever had truly pure metals in the series. Who knows just how much more power that might have given someone...?

Edited April 29, 2012 by Eric

[lil_literalist]

I doubt that it matters too much. Anyway, I've found proof that there is no connection between the force of a push and someone's mass.

"Allomancers draw strength from their metals," Ham said, sighing and putting his foot down. "Some can squeeze out more than others--but the real power comes from the metal itself, not from the person's body."

...

"So," Ham said, "an Allomancer doesn't have to be physically strong to be incredibly powerful. If Vin were a Feruchemist, it would be different--if you ever see Sazed increase his strength, his muscles will grow larger. But with Allomancy, all the strength comes directly from the metal."

[Eric]

That's in reference to pewter. The power for steel and iron does come from the metal itself, but the mass of the Allomancer in relation to that of the anchor determines the force imparted to each.

[lil_literalist]

No, Ham is very clear. He does not talk about the power that pewter gives, but Allomany as a whole. Also, it makes far more sense if each of the powers worked the same way. Why should iron and steel be coupled to mass when pewter isn't?

Regardless, the power is coming from the metal. Not the person. It doesn't matter what size or shape the Allomancer is, since the power does not come from the body, but from the metal.