The Impossible Physics of Allomancy

[+Oltux72]

1 minute ago, Scion of the Mists said:

Why would the coins be "rammed into the ground"?  It's in a static equilibrium - all the forces are perfectly balanced so it doesn't move.  

No, they are not. Gravity still exists. The coin should fall down and then two allomancers have a coin below them and are pushing on it. In the case of two allomancers pushing something else must also happen. Yes, it is true that that weakens my argument. Something we do not understand is happening.

 

1 minute ago, Scion of the Mists said:

Your argument was that the force has to be roughly constant with a steep drop off near maximum range because otherwise "pushing duels" wouldn't make sense (with "pushing duels" being defined as two Allomancers pushing on the same object, but one overpowering the other, resulting in the object reaching one of the participants).  However, there's no indication that these "pushing duels" exist in the books (at least as you're defining them).  

Kelsier tells her not to get into a pushing duell unless she is firmly grounded. Hence they must exist.

[Scion of the Mists]

14 minutes ago, Oltux72 said:

No, they are not. Gravity still exists. The coin should fall down and then two allomancers have a coin below them and are pushing on it. In the case of two allomancers pushing something else must also happen. Yes, it is true that that weakens my argument. Something we do not understand is happening.

All three forces can be balanced: the Steelpush forces are slightly upward, which counters the force of gravity.  

 

18 minutes ago, Oltux72 said:

Kelsier tells her not to get into a pushing duell unless she is firmly grounded. Hence they must exist.

This is what he says to her: "I would, however, recommend that you avoid Push-matches with people who weigh more than you."  He says this directly after the section that I quoted, so that's clearly what he's referring to as a "Push-match."  Which is counter to your definition.  

[+Artemos]

1 minute ago, Scion of the Mists said:

All three forces can be balanced: the Steelpush forces are slightly upward, which counters the force of gravity.  

 

The precision necessary for this is not going to be found in Steelpushing. Allomancers don't have that much control over their push strengths, so they will almost always overshoot or undershoot the net acceleration needed to perfectly counter gravity. Especially in three dimensions.

I modeled five different scenarios of a pushing duel. Gravity is disabled in all of these because the object will always immediately fly out of the perfectly straight line between two allomancers in real life.

 

 

Simulation properties/definitions:

• Gravity is ignored.
• The cubes on the left and right are "allomancer Left" and "allomancer Right," respectively.
  • They are both anchored.
  • Both allomancers are pushing with their maximum possible force at every given time, calculated with the "F is proportional to e^-r" relationship mentioned earlier.
• The sphere at the center is the magnetic object that Left and Right are pushing on.
  • The object begins with zero velocity.
• The "equilibrium point" is the position where the push from Left and Right on the sphere are equal. The Net Force there is 0.

Scenario 1: Allomancer Left and Right are of equal strengths. The object begins touching allomancer Left.

• The equilibrium point is at the center.
• The object oscillates between the two allomancers, much like a spring. At first, Allomancer Left has a stronger push. As the object moves to the right, allomancer Right has a stronger push. This repeats indefinitely, as neither allomancer gains the upper hand.

Scenario 2: Allomancer Left and Right are of equal strengths. The object begins halfway between allomancer Left and the center.

• The equilibrium point is at the center.
• The object again oscillates, but with a weaker amplitude than in Scenario 1. Again, this is very similar to a spring. Because the object started farther away from allomancer Left, Left's maximum push is weaker. Likewise for Right.

Scenario 3: Allomancer Left and Right are of equal strengths. The object begins at the center.

• Both Left and Right exert equal forces on the object, and it is thus stuck perfectly at the center.
• This is caused by the object's initial position being at the equlibrium point.

Scenario 4: Allomancer Right is stronger than Left. The object begins at the center.

• Right is stronger than Left, so the equilibrium point is to the left of the center.
• Again, the object oscillates around the equilibrium point.
• If the object began at the equilibrium point, it would not move, similar to Scenario 3.

Scenario 5: Allomancer Left is stronger than Right. The object begins touching allomancer Left.

• The equilibrium point is to the right of the center.
• Left's sustained push is strong enough to push the object into Right.

Overall, pushing duels are only effective in this simulation if one allomancer is stronger than another. Only when the object starts at the equilibrium point does is stay still. Duels will play out differently if:

• The object had some momentum to begin with
• The allomancers flare their metals or otherwise push "harder"
• A different force-distance relationship is used, or force is calculated differently elsewhere

And, of course, this is all happening without gravity. With gravity, both allomancers quickly lose control of the object.

[Pagerunner]

I see you've all been quite busy while I've been away! Lots of good stuff. Let me see if I can get me caught up.

On 11/26/2018 at 4:17 PM, Oltux72 said:

Your model 1 seems correct. The allomancer determines a force. AFAICT your reason dismissing model #1 is flawed.

Quoting your pdf:

"This is the plainest explanation, but as poor Vin discovers, it is not a good fit, because that model
decouples the Allomancer and the coin."

Exactly. This is due to the normal force. It makes the second term go away. Which is exactly what is observed. As soon as the object an allomancer pushes against is securely anchored, it no longer matters. You are absolutely correct in complaining about the discontinuity. It arises from a model that assumes a full normal force as soon as the coin hits the ground. In reality there will be some deformation. But fundamentally if you introduce a new force (in form of the normal force) a discontinuity is not a problem. It is expected.

But I think you are conflating two things. That the allomancer determines a force does not give him an arbitrary range of forces available.

As far as we can tell, it certainly depends on

 

1. choice of the allomancer (experienced allomancers can push weakly and all of them can flare their steel)
2. inherent strength of the allomancer
3. distance to the object pushed
4. degree of investiture in the object pushed
5. mass of the object pushed

And maybe time is also a factor. Hence an allomancer determines (if he is trained well) a force between 0 and the upper limit given by factors 2) - 5)I

I'm not quite following you on the first point. In that model, the behavior (acceleration) of the allomancer is dependent upon three things: the force of the push, the mass of the allomancer, and the force of gravity. The behavior of the coin is not present in the equation (thus the term "decoupled"), and the presence or lack thereof of any normal force on the coin would not affect the behavior of the allomancer. With the introduction of a new force (in the form of the normal force), a discontinuity is expected in the behavior of the coin. But by nature of the model, there should be none in the behavior of the allomancer. Which necessitates the later models.

For the second point, I will acknowledge that all those are factors in the force. (And I think I have mentioned them except #4 elsewhere in the thread.) But in the steelpushing examples shown above, the odd behavior happens at a point in the steelpush where none of those particular parameters are changing. None of them can reconcile the differences I identified at the beginning of my writeup. Which is why I'm looking to a sixth parameter, the relative velocity of the allomancer and the target, as also affecting the strength.

On 12/5/2018 at 9:39 AM, Quantus said:

Can you add the physics/behavioral assumptions to the write-up, for those of us that havent reread Era1 in a while?  I want to go through it with Wax's Crasher trick in mind. I vaguely recall some less than intuitive workings surrounding the Conservations before/during/after Impact events, and Ive never really tried to wrap my head around them in Higgs terms. 

I think everything for the scope of this topic's analysis of steelpushing is in the two excerpts I have at the beginning of the writeup. I haven't looked too hard into energy/momentum conservation of how Wax plays with forces. In terms of the steelpushing side of things, I think it'd mostly tie down to what Kelsier tells Vin about heavier people being able to push harder. Wax gets to cheat and make himself really heavy.

On 12/6/2018 at 7:01 AM, Croaker's Apprentice said:

Although, at the end of the day, his heroes (and villains) can do whatever the X-Men can do 

Please note I intended that much more as a joke than a criticism ! 

What I love about B's stories (in reverse order of importance) are-

1. The kinetic, super-cool imagery of his action set- pieces/ stunt- filled frenetic- balletic fight scenes

2. The humor

3. The moral wisdom

4.The great characters and emotional heft and power of their sagas

I don't really care how the pseudo-science of Cosmere super-powers works; not being either a physicist or an RPG rules- lawyer, I don't feel the need to 'lift the hood' to see how they work, especially because I don't think they would stand the scrutiny (and nor should they).

The 'true magic' of the stories we love lies in the interface between character journey/development, plot foreshadowing, an element of mystery, and dramatic resolution. I agree that fantasy magic having some delineation and rules is a good thing, but probably the most important thing about magical rules is that they *exist to be broken* (although as I say, the breaking must be properly earned). 

BS himself is hardly above employing a "Deus ex Machina" resolution where warranted - or should we say "Deus ex Caligo" ? - at the end of "Final Empire".

"Magic Before System" as they say...

Ironically enough, just the other day I saw a TV show where someone was tying Nightcrawler's limitations of teleportation to things like the different rotational velocities you have at different latitudes and longitudes.

But steelpushing is a little bit different. It's explicitly laid out to the reader with physics in mind; Newton's laws, every action has an equal and opposite reaction. There are other harder scientific elements at play in Mistborn specifically: the mystery of cadmium's relationship to FTL travel, the way Wax manipulates his weight. There are things to be said for creating rules to break them, but creating rules that are promptly ignored is a different beast entirely. Most of the models I present, I didn't come up with; they're points of view that other people have had, and they've come into conflict as people discussing the books with one another can't figure out how steelpushing works. It's not lifting the hood for the sake of seeing how it works, it's those funky complicated shapes where different perspectives show you different words. (Like the NYES sculture https://goo.gl/images/vSY1RD). I picked the word "Impossible" in the title of this thread because, to a lot of people, it was apparently inconsistent and didn't make sense. I'm going into this thread trying to come up with something that doesn't have to hand wave past a scenario or two, because, as you said, none of those examples have "properly earned" breaking the steelpushing physics rules to us.

On 12/7/2018 at 4:26 PM, Oltux72 said:

The really hard part is giving even a sensible formula for the maximum strength an allomancer can push with. In particular the relationship with distance is odd. For almost any physical influence we find that the strength drops off with the square off the distance. Allomancy clearly does not follow that rule. We have three observations hard to reconcile.

 

1. pushing duells make sense
2. pushing up at full power from a close anchor does not kill you
3. pushing with full force against a close anchor pushes you up swiftly.
4. there is a maximum height you can support yourself at from a given anchor and that is below the sensing range

That means in effect that the force cannot drop off much until you get relatively close to maximum range. If it depended strongly on range, the weaker allomancer could push back anything as soon as it got close enough.

I do like a relationship with distance that doesn't go to infinity at close range. I haven't dug too much into specifics, mostly because of a lack of quantifiable data, but I like the general idea of a hockey stick; distinct y-intercept, slow slope, then transition to steel slope, then x-intercept.

But as has been said already (and I'm gonna restate because I feel like it), equilibrium in a pushing duel is consistent with observed behavior. Because, from what we've seen, the goal isn't to strike your opponent, but to launch them. Which means the forces they apply to themselves need to be out of balance, not the forces on the coin. It's kind of like tug-of-war. The goal isn't to pull on the rope the hardest; it's to pull on the ground the hardest. A ten-year-old in cleats can beat me if I'm wearing roller blades. (Certain members of my family will say that I will lose even if I am also wearing cleats, but let's not let that get in the way of a good illustration.)

On 12/10/2018 at 11:26 AM, Oltux72 said:

No, they are not. Gravity still exists. The coin should fall down and then two allomancers have a coin below them and are pushing on it. In the case of two allomancers pushing something else must also happen. Yes, it is true that that weakens my argument. Something we do not understand is happening.

On 12/10/2018 at 0:24 PM, Artemos said:

The precision necessary for this is not going to be found in Steelpushing. Allomancers don't have that much control over their push strengths, so they will almost always overshoot or undershoot the net acceleration needed to perfectly counter gravity. Especially in three dimensions.

The precision to keep a coin in the air lodged between two Allomancers seems to me to be analogous to balancing directly atop a coin. If you're a centimeter off, then timber it should be. But there is enough variance allowed in the vectors to make adjustments and keep yourself centered. If you let the coin drop to the ground, you can't apply force to launch an adversary, so at least one party will attempt to keep the coin suspended.

On 12/10/2018 at 0:24 PM, Artemos said:

I modeled five different scenarios of a pushing duel. Gravity is disabled in all of these because the object will always immediately fly out of the perfectly straight line between two allomancers in real life.

<snip>

Scenario 1: Allomancer Left and Right are of equal strengths. The object begins touching allomancer Left.

• The equilibrium point is at the center.
• The object oscillates between the two allomancers, much like a spring. At first, Allomancer Left has a stronger push. As the object moves to the right, allomancer Right has a stronger push. This repeats indefinitely, as neither allomancer gains the upper hand.

Scenario 2: Allomancer Left and Right are of equal strengths. The object begins halfway between allomancer Left and the center.

• The equilibrium point is at the center.
• The object again oscillates, but with a weaker amplitude than in Scenario 1. Again, this is very similar to a spring. Because the object started farther away from allomancer Left, Left's maximum push is weaker. Likewise for Right.

Scenario 3: Allomancer Left and Right are of equal strengths. The object begins at the center.

• Both Left and Right exert equal forces on the object, and it is thus stuck perfectly at the center.
• This is caused by the object's initial position being at the equlibrium point.

Scenario 4: Allomancer Right is stronger than Left. The object begins at the center.

• Right is stronger than Left, so the equilibrium point is to the left of the center.
• Again, the object oscillates around the equilibrium point.
• If the object began at the equilibrium point, it would not move, similar to Scenario 3.

Scenario 5: Allomancer Left is stronger than Right. The object begins touching allomancer Left.

• The equilibrium point is to the right of the center.
• Left's sustained push is strong enough to push the object into Right.

Overall, pushing duels are only effective in this simulation if one allomancer is stronger than another. Only when the object starts at the equilibrium point does is stay still. Duels will play out differently if:

• The object had some momentum to begin with
• The allomancers flare their metals or otherwise push "harder"
• A different force-distance relationship is used, or force is calculated differently elsewhere

And, of course, this is all happening without gravity. With gravity, both allomancers quickly lose control of the object.

Now this is really interesting. Going back to what I said before, I don't think striking an opponent is what's at stake; it's dislodging them, so fixing the two allomancers isn't going to be a good way to determine the steelpushing resolution. But let's say it does fit for a time, until someone runs out of strength. That's still quite illustrative, since we've seen the coin reach equilibrium without oscillation. So we've got a hiccup to all of our models.

Is this using the proportional-to-distance-squared model? No other factors, aside from the proportionality constant being allomancer-dependent? In that case, I'm not surprised it behaves much like a spring - it's going to be pretty much exactly the ideal spring equations, which yield infinite oscillation. An undamped, oscillating system. You need some sort of damping to the equation - friction in springs is a good one to understand. Overdamped systems will oscillate, but slow down. Overdamped systems will approach the equilibrium point and not oscillate. (Doors will automatic closing mechanisms [not fully automated Star Trek doors, but the pneumatic elbow things] will be overdamped; you don't slam the door shut, you ease it closed.)  So, we need something that will function like friction in slowing down the movement of our "spring." To keep it from responding too quickly - our really saggy spring that doesn't overshoot when you release tension on it, and doesn't completely close the bathroom door all the way. (I'm not referring to a specific scenario at my old college, no, why would you think that?)

I've already gone all in on the relative velocity term, so I'll keep running with that. As the coin is moving away from one allomancer, it will weaken them. (To state more rigoriously: the positive relative velocity causes a weakening effect IN ADDITION to the weakening effect of the increasing relative displacement.) And as the coin is moving towards the other allomancer, it will strengthen them. (Or at least remain at their base strength) That force imbalance will effectively slow down the movement of the coin more than otherwise predicted... it acts in the same direction as friction would on a spring. Without crunching the numbers, I can't be sure, but it's at least possible that properly tuned relative velocity effects could dampen our system and prevent oscillation.

[Jofwu]

You can't really look too closely at the whole Push-duel situation. It definitely reveals a little bit about what Brandon is thinking, but the situation is ultimately unrealistic. The situation is unstable, unless you allow for Allomancers to shift their Pushing vector to intentionally keep the coin between them. Otherwise the coin gets Pushed in some direction normal to the Pushes.

Great simulation @Artemos! Does it work out roughly the same if you use 1/r^2 or some other formulation with respect to distance? Having trouble thinking that through at the moment.

On 11/26/2018 at 5:17 PM, Oltux72 said:

Exactly. This is due to the normal force. It makes the second term go away. Which is exactly what is observed. As soon as the object an allomancer pushes against is securely anchored, it no longer matters. You are absolutely correct in complaining about the discontinuity. It arises from a model that assumes a full normal force as soon as the coin hits the ground. In reality there will be some deformation. But fundamentally if you introduce a new force (in form of the normal force) a discontinuity is not a problem. It is expected.

Oltux72, the text states that the Allomancer experiences a discontinuity in the force/acceleration they experience (little to nothing when the coin is free vs. something significant when it becomes anchored).

Model 1 does not produce this discontinuity for the Allomancer, which is why it doesn't work. If you draw a free body diagram of an Allomancer pushing with a set force, the behavior of the coin is irrelevant. The horizontal Allomantic force is known and constant, whether the coin is free or anchored.

You seem to be suggesting that the normal force on the coin (when it suddenly becomes anchored) is "transferred" to the Allomancer. This would explain the discontinuity if the model worked this way. But this isn't how model 1 is described. The problem with this interpretation is that, if you look at the forces on the coin-Allomancer system, there is an unbalanced force. It breaks Newton's 3rd law. Pagerunner effectively addressed this in Model 4 and dismissed it. This is one of the solutions Artemos has used in his game, so he might be able to speak more on that if he cares to. It's useful and simple for the sake of making a game work, but it breaks a lot of physics and thus isn't a fantastic solution.

 

14 hours ago, Pagerunner said:

Is this using the proportional-to-distance-squared model? No other factors, aside from the proportionality constant being allomancer-dependent? In that case, I'm not surprised it behaves much like a spring - it's going to be pretty much exactly the ideal spring equations, which yield infinite oscillation.

He's using a force proportional to e^-r. (so, relatively hockey-stickish)

I think behavior like this is inevitable without damping. For the book you just have to blame the air I suppose. Or assume the coin began relatively near the center? Maybe one of the Allomancer's flares their metal in defense before it reaches the equilibrium point in order to slow it down closer to that point than it would have otherwise? Or perhaps it's the function of relatively velocity that does it.

[+Artemos]

17 hours ago, Pagerunner said:

I don't think striking an opponent is what's at stake; it's dislodging them, so fixing the two allomancers isn't going to be a good way to determine the steelpushing resolution.

Very true, I hadn't thought of that.

17 hours ago, Pagerunner said:

Is this using the proportional-to-distance-squared model?

No, the previous simulation used the "F is proportional to e^-distance" relationship that Jofwu thought of. I also don't like the inverse-square model because of it giving an infinite force at close ranges.

I changed the simulation to use the relative velocity term, and, sure enough, it gave a pretty satisfying critically-damped effect to the system.

Here's a couple more simulations, showing different combinations of velocity relationships, distance relationships, and Allomancers being anchored. None of the targets have gravity, nor is anything in the sims affected by the "Allomantic Normal Force" I've theorized about. This keeps it simpler.

The targets all have a mass of 1kg.

Simulations are spoilers rather than embeds to not bloat the page. Click the videos to see the HD versions.

Simulation 1

Spoiler

 

• All Allomancers are perfectly anchored
• Pair 4-Right and Pair 5-Left are Stronger
• Distance relationship: e^-r
• Velocity relationship: None

A similar simulation as the one in the previous post. The target for Pair 5 collides with Allomancer Right.

Simulation 2 - Same as previous, but with the velocity term

Spoiler

 

• All Allomancers are perfectly anchored
• Distance relationship: e^-r
• Velocity relationship: e^-v, where v = magnitude of the relative velocity between Allomancer and target. Because this factor is the magnitude of the relative velocity, it will be smaller (and thus, the force will be smaller) if the target is moving closer or further from the Allomancer.

Not particularly interesting. Because the velocity term works in both directions, it just makes everything a bit slower.

Simulation 3 - Same as previous, but with a modified velocity term

Spoiler

 

• All Allomancers are perfectly anchored
• Distance relationship: e^-r
• Velocity relationship: e^-v, where v = relative velocity between Allomancer and target if they are moving towards each other. If the Allomancer and target are getting closer, this term will be smaller. If they're getting further apart, this term will not affect the force (it'll equal 1)

This is the interesting one. Using Pagerunner's analogy, the velocity term works like friction or a drag force, always resisting the target's forward movement.

Simulation 4 - Same as previous, but with a modified velocity term

Spoiler

 

• All Allomancers are perfectly anchored
• Distance relationship: e^-r
• Velocity relationship: e^-v, where v = relative velocity between Allomancer and target if they are moving away from each other.

The opposite of Simulation 3. Unsurprisingly, this system is extremely unstable with what I want to call "reverse friction" (even if that's not really correct...)

Simulation 5 - Same as Simulation 3, but with a different distance term

Spoiler

 

• All Allomancers are perfectly anchored
• Distance relationship: 1/r^2
• Velocity relationship: e^-v, where v = relative velocity between Allomancer and target if they are moving towards each other.

Initially, the distance factor is extremely very high for Pair 1 because the target is so close to Allomancer Left. Most pairs are able to reach equilibrium quickly because both the velocity term and distance term help to balance out the forces when the target is closest to the center. In my mind, this is what Pushing duels between anchored targets feel like in the books.

By this point, I'll say that I've been slightly deceptive regarding the distance relationship. The term is actually equal to e ^ -r/D, where r = the distance between the Allomancer and target, and D = an arbitrary constant. This constant significantly changes the behavior of the system. In the above examples, D = 16. Here's the same simulation as Simulation 3, but with D = 1:

Simulation 6

Spoiler

 

This gives more of an under-damped system.

and D = 32:

Simulation 7

Spoiler

 

Like in Simulation 3, this one's critically damped, but the targets end closer to the Allomancers.

Comparing between distance relationships is difficult, since e^-r/D is highly dependent on the constant factor.

Simulation 8 - Simulation 3, but with unanchored Allomancers and modified Allomancers

Spoiler

 

• All Allomancers are not anchored
• All targets begin in the center
• Distance relationship: e^-r
• Velocity relationship: e^-v, where v = relative velocity between Allomancer and target if they are moving towards each other.
• Allomancer Pair 1: Equal masses, Right is Stronger
• Allomancer Pair 2: Equal Strengths, Left is slightly more massive
• Allomancer Pair 3: Equal Strengths, Left is significantly more massive
• Allomancer Pair 4: Left is more massive, Right is Stronger
• Allomancer Pair 5: Right is less massive but also Stronger (Looking at you, Vin)

Alright, here's the fun one. In Pair 1, Right is stronger than Left. Right could push the target closer to Left, but both Right and Left lost their anchoring around the same time.

Pairs 2 and 3 had Left being heavier, which let it stay anchored and "win" the duel around the same time for both Pairs. You can see that the formula I use for calculating Allomantic Force is slightly dependent on mass because Pair 3-Left pushes the target slightly further than Pair 2-Left.

In Pair 4, Left was again heavier, but Right was stronger. The target got closer to Left, but Right was still pushed away

As Kelsier always said, it's a bad idea to get into a pushing contest when your enemy weighs more than you. In Part 5, Right is lighter but stronger. Right's anchoring broke first, but its force did manage to push Left afterwords.

3 hours ago, Jofwu said:

This is one of the solutions Artemos has used in his game, so he might be able to speak more on that if he cares to.

I might come back here later to talk about this, but I should, uh, eat breakfast lunch.

Edited January 17, 2019 by Artemos
Clarified target mass

[Jofwu]

38 minutes ago, Artemos said:

(Is there any way to have them minimized when the page is opened?)

Spoiler tags do that. Fine to use those for something like that.

[Pagerunner]

35 minutes ago, Artemos said:

I changed the simulation to use the relative velocity term, and, sure enough, it gave a pretty satisfying critically-damped effect to the system.

Here's a couple more simulations, showing different combinations of velocity relationships, distance relationships, and Allomancers being anchored. None of the targets have gravity, nor is anything in the sims affected by the "Allomantic Normal Force" I've theorized about. This keeps it simpler.

Hot diggity, Sims 3 and 8 are sweet. That's exactly what we want to see. What's the mass of the coin you're using in these sims, as compared to the allomancers?Can you throw together a simulation for the initial issue, shooting a coin at a wall, with the same relative velocity equations? See if we get the sudden lurch of force on the allomancer when the coin hits the wall?

2 hours ago, Jofwu said:

He's using a force proportional to e^-r. (so, relatively hockey-stickish)

I'm thinking of a hockey stick turned the other way. Exponential is a steep slope that shallows out. I'm thinking of something that starts shallow, and then takes a dive toward the end.

[Jofwu]

15 minutes ago, Pagerunner said:

I'm thinking of a hockey stick turned the other way. Exponential is a steep slope that shallows out. I'm thinking of something that starts shallow, and then takes a dive toward the end.

Ah, so something in the general form of  y = (1-x)^(1/n).

Bigger the n, sharper the hook.

So distance would matter relatively little out to some range, and then it would zero out quickly beyond that.

[+Artemos]

26 minutes ago, Pagerunner said:

What's the mass of the coin you're using in these sims, as compared to the allomancers?

I knew I'd mess up somewhere. I put that in a text box on screen but I think it "fell off" at some point.

Each target is 1kg. Most of the Allomancers are 60kg, as shown on the text boxes.

 

29 minutes ago, Pagerunner said:

Can you throw together a simulation for the initial issue, shooting a coin at a wall, with the same relative velocity equations? See if we get the sudden lurch of force on the allomancer when the coin hits the wall?

This'll be interesting. Will do.

[+Artemos]

As promised, here's some more.

Simulation 1 - Same physics as Simulation 3 in the above post.

Spoiler

 

This performs exactly as @Pagerunner expected. As the coin is moving away from the Allomancer, the force is negligible. When it hits the wall and stops moving, the force increases dramatically.

Simulation 2 - Instead of the velocity term, this simulation uses my Allomantic Normal Force strategy.

Spoiler

 

Once the coin is anchored against the wall, the force that the wall exerts on the coin is applied to the Allomancer.

Note that the time scales are different between the two simulations (20% and 4%) because Simulation 2 runs faster in real time than Simulation 1. Simulation 2 has significantly higher forces than those in Simulation 1. This is because the Allomantic Normal Force is something that is added to the original Force, whereas the velocity term is a factor between 0 and 1 that decreases the Force. Changing various constants could make them feel more equal, but that seemed unnecessary.

Edited January 17, 2019 by Artemos
Changed wording

[+Oltux72]

On 1/17/2019 at 3:55 AM, Pagerunner said:

I'm not quite following you on the first point. In that model, the behavior (acceleration) of the allomancer is dependent upon three things: the force of the push, the mass of the allomancer, and the force of gravity. The behavior of the coin is not present in the equation (thus the term "decoupled"), and the presence or lack thereof of any normal force on the coin would not affect the behavior of the allomancer.

That is the basic error in your reasoning.

Let me explain in detail and please excuse the length of the explanation:

Let's look at the case of an allomancer and a coin in a vacuum and in free fall. We have action equals reaction and F = m * a
Let's do the math:

m (allomancer) * a (allomancer) = m (coin) * a (coin)

a (allomancer) / a (coin) = m (coin) / m (allomancer) [that is exactly the same result you would get for electrostatic repulsion at any given moment, hence utterly ordinary]

The allomancer being much heavier than the coin, almost all of the acceleration the force generates is experienced by the coin.
Now the crucial the point here is that the masses in these equations are those the force acts upon, not just those directly affected by the magical force. Hence you need to include the mass of the clothing the allomancer is wearing, or if the coin is in a wallet and stays in it, the mass of the wallet on the coin's side. Now we consider the case of a coin securely anchored to the ground. Which mass is acted upon on the coin's side of the equation. Theoretically a whole planet. Compared to the mass of a planet, the mass of a coin irrelevant. And that is exactly what your model shows.
The coin will experience a normal force from being used to accelerate a planet. You can see that indeed as the normal force being transmitted to the allomancer. The result is the same.

On 1/17/2019 at 3:55 AM, Pagerunner said:

With the introduction of a new force (in the form of the normal force), a discontinuity is expected in the behavior of the coin. But by nature of the model, there should be none in the behavior of the allomancer. Which necessitates the later models.

No. The effective mass of the coin changes when it hits the ground. And that effect is felt by the pushing allomancer.

It is really the same physics that allow a swimmer's same leg muscles to accelerate him much more by pushing on the walls of the pool rather than making swimming motions.

[Jofwu]

2 hours ago, Oltux72 said:

The allomancer being much heavier than the coin, almost all of the acceleration the force generates is experienced by the coin.
Now the crucial the point here is that the masses in these equations are those the force acts upon, not just those directly affected by the magical force. Hence you need to include the mass of the clothing the allomancer is wearing, or if the coin is in a wallet and stays in it, the mass of the wallet on the coin's side. Now we consider the case of a coin securely anchored to the ground. Which mass is acted upon on the coin's side of the equation. Theoretically a whole planet. Compared to the mass of a planet, the mass of a coin irrelevant. And that is exactly what your model shows.
The coin will experience a normal force from being used to accelerate a planet. You can see that indeed as the normal force being transmitted to the allomancer. The result is the same.

Oltux72, you're conflating Pagerunner's "Model 1" and "Model 4".

What you're describing here is addressed in Model 4. It's the idea that we're dealing with some "effective mass". For the coin being Pushed in a vacuum you just have the mass of the coin, and when the coin is anchored the "effective mass" is... something much larger. You'll have to see Pagerunner's document concerning his critique of Model 4, as I don't remember what he says on the matter.

Speaking for myself... If we run with this and assume that the Allomancer control's the magnitude of the force, then we go back to the problem of having no acceleration discontinuity. It doesn't matter of the mass of the coin suddenly becomes something much larger. The magnitude of the force is fixed, the mass of the Allomancer is fixed, so the magnitude of the acceleration never changes. As for the coin, we simply see the acceleration of the coin decrease inversely with the sudden mass increase.

Now, you're describing a case where the force DOES increase (again different from the premise of Model 1). You're relating this sudden force increase to the sudden "effective mass" increase. If we hold the acceleration of the coin constant, the force increases proportional to the effective mass increase. That would cause the acceleration of the Allomancer to increase an INSANE amount if the entire planet is taken as the effective mass, and we very obviously don't see this. In any case, we obviously don't see the acceleration of the coin held constant, so this isn't quite right.

The only way for this to make sense is if the acceleration and (effective) mass of the coin BOTH change by some degree. So say effective mass suddenly increases 100-fold and the acceleration of the Allomancer is observed to increase 10-fold. It would mean the acceleration of the coin decreased 10-fold. So we have original F=ma, and after anchoring we have (10F)=(100m)(a/10). You can make the numbers work doing this. But... what determines how much the effective mass and acceleration change by? You could maybe come up with some equation that spits out values for how the force changes

Philosophically, this also suggests that the magnitude of your Pushing force is a function of the "effective mass" of the thing you're pushing on. And this seems to be contradicted by several examples in the books. You don't push on some general mass. You push on metal. If you embed a coin inside a rock, the strength of your Pushing force doesn't increase with the mass of the rock. Evidence suggests you should be Pushing the coin with the same force, and it just accelerates slower because of the added dead weight. Does that make sense? Yes, the non-metallic, dead weight DOES play into the thing's acceleration. But to say it plays into the magnitude of the force gets really sticky.

 

[+Oltux72]

16 minutes ago, Jofwu said:

Oltux72, you're conflating Pagerunner's "Model 1" and "Model 4".

 

No, on the contrary. Model 4 supposes Newton's third law is inoperative. But the results we see derive from the 3rd law.

 

16 minutes ago, Jofwu said:

Speaking for myself... If we run with this and assume that the Allomancer control's the magnitude of the force, then we go back to the problem of having no acceleration discontinuity. It doesn't matter of the mass of the coin suddenly becomes something much larger. The magnitude of the force is fixed, the mass of the Allomancer is fixed, so the magnitude of the acceleration never changes. As for the coin, we simply see the acceleration of the coin decrease inversely with the sudden mass increase.

No, and it seems I am not very good at explaining this. So there is one total force, which the allomancer determines. If both sides were anchored, that is what they would feel and that's it. I assume so far we agree. I think you also agree with the free fall case of no side being anchored. But now you are mixing the physics of these cases in an incorrect way. That is an easy mistake because the force is applied to the coin uniformly, but to the allomancer at, as far as we can tell, the surface of his body.

The allomancer undergoes acceleration. And the amount of acceleration is not a matter of choice. It is determined from the force and the ratio of effective masses. Precisely because only the force is under free choice any alteration of the mass ratio will change the acceleration.

16 minutes ago, Jofwu said:

Now, you're describing a case where the force DOES increase (again different from the premise of Model 1). You're relating this sudden force increase to the sudden "effective mass" increase.

No, no, no. Again the acceleration increases. And it increases precisely because acceleration and mass are linked by Newton's laws and controlling only one of them (force) means that a change in an uncontrolled variable (mass) changes the third variable (acceleration)

16 minutes ago, Jofwu said:

If we hold the acceleration of the coin constant, the force increases proportional to the effective mass increase. That would cause the acceleration of the Allomancer to increase an INSANE amount if the entire planet is taken as the effective mass, and we very obviously don't see this. In any case, we obviously don't see the acceleration of the coin held constant, so this isn't quite right.

Correct. The acceleration of the coin drops to zero, or something very close to it.

16 minutes ago, Jofwu said:

The only way for this to make sense is if the acceleration and (effective) mass of the coin BOTH change by some degree. So say effective mass suddenly increases 100-fold and the acceleration of the Allomancer is observed to increase 10-fold. It would mean the acceleration of the coin decreased 10-fold. So we have original F=ma, and after anchoring we have (10F)=(100m)(a/10). You can make the numbers work doing this. But... what determines how much the effective mass and acceleration change by?

Newton's laws! You almost got it. The change of the ratio of accelerations is the inverse of the change of the ratios of effective masses.

But that is nothing extraordinary. That is the normal everyday consequence of Newton's laws. You would have the exact same result if you replaced allomancy by electrostatic repulsion.

 

16 minutes ago, Jofwu said:

Philosophically, this also suggests that the magnitude of your Pushing force is a function of the "effective mass" of the thing you're pushing on. And this seems to be contradicted by several examples in the books. You don't push on some general mass. You push on metal. If you embed a coin inside a rock, the strength of your Pushing force doesn't increase with the mass of the rock. Evidence suggests you should be Pushing the coin with the same force, and it just accelerates slower because of the added dead weight. Does that make sense? Yes, the non-metallic, dead weight DOES play into the thing's acceleration. But to say it plays into the magnitude of the force gets really sticky.

It does not. That is the exact result you get if the force is constant.

[Jofwu]

13 minutes ago, Oltux72 said:

No, on the contrary. Model 4 supposes Newton's third law is inoperative. But the results we see derive from the 3rd law.

The only way I can make sense of what you're saying is if that's basically the case, though maybe I just don't quite understand you.

 

Let's walk through the algebra to find where the disconnect is. We're in a vacuum and ignoring gravity.The coin has mass M and the Allomancer has mass 100M.

The Allomancer chooses to apply a Pusing force F to a coin. Per Newton's 3rd law, both coin and Allomancer experience this force. Per Newton's 2nd law, the coin has acceleration a1 = F/M and the Allomancer has acceleration a2 = F/(100M) = a1/100.

They both accelerate at these different rates until the coin strikes a very massive object of mass 1000000000M (that's 9 zeros) such as a wall attached to the ground. The Allomancer is in direct control of the force applies. The magnitude of the force doesn't change. The Allomancer continues to accelerate at a2 = F/(100M) = a1/100. The coin/wall has combined mass 1000000001M. They accelerate at a3 = F/(1000000001M) = a1/1000000001.

 

In the previous reply you equated the mass*acceleration of each object per Newton's 2nd law. So using the same symbols/values that I used above that's:

F = (a1)M = (a2)(100M) for the case when the coin is free, and:

F = (a3)(100000001M) = (a2)(100M) for the case when the coin becomes anchored.

This doesn't give you anything but a ratio of accelerations however. In the first case you get a2/a1 = 1/100. In the second case you get (approximately) a2/a3 = 10000000.

Where do you go from here? This is just ratios of acceleration. In the first case the coin is accelerating 100 times faster than the Allomancer. In the second case, the Allomancer is acceleration 10000000 times faster than the coin/wall. I solved the actual values of acceleration above. There's no change to the Allomancer's acceleration in this math. Just in the coin's acceleration.

 

You seem to be saying that the magnitude of the force is "shared" between the coin (or coin/wall) and the Allomancer. In that case you're describing something like:

F = (a1)M + (a2)(100M)

You're saying the force is fixed and the masses are fixed, so you end up with some kind of ratio on the accelerations. But you're still stuck with just a ratio. It's unclear to me how you determine the value of these accelerations. You could certainly input the Allomancer's acceleration as something very low and you'd get the coin accelerating much faster. And then when the M suddenly becomes 1000000001M and you assume a small acceleration for the coin/wall then the Allomancer's acceleration would be very large.

But we're completely making up those accelerations. You need some other relationship defined here if you want to solve for them.

In any case, this is a violation of Newton's 3rd law. Forces aren't applied such that they are "shared" like this. In electrostatics or gravitational forces you have some force that's defined (F here). And that same force is applied to each, per F = m1*a1 = m2*a2. You don't get variables amounts of the force going to each of them. They both experience the same force.

[Pagerunner]

18 hours ago, Artemos said:

Simulation 2 - Instead of the velocity term, this simulation uses my Allomantic Normal Force strategy.

Once the coin is anchored against the wall, the force that the wall exerts on the coin is applied to the Allomancer.

I haven't been following the other thread; what is this Allomantic Normal Force? Is it a separate force between the Allomancer and an object in contact with the pushed target? (Which would essentially double the force of the steelpush, which doesn't seem strong enough.) Or is an additive term to the existing steelpush? (Which I don't think would fit the scenario of an immobile Allomancer and immobile target, since the force would feed back on itself iteratively.) 

3 hours ago, Oltux72 said:

That is the basic error in your reasoning.

Let me explain in detail and please excuse the length of the explanation:

Let's look at the case of an allomancer and a coin in a vacuum and in free fall. We have action equals reaction and F = m * a
Let's do the math:

m (allomancer) * a (allomancer) = m (coin) * a (coin)

a (allomancer) / a (coin) = m (coin) / m (allomancer) [that is exactly the same result you would get for electrostatic repulsion at any given moment, hence utterly ordinary]

The allomancer being much heavier than the coin, almost all of the acceleration the force generates is experienced by the coin.
Now the crucial the point here is that the masses in these equations are those the force acts upon, not just those directly affected by the magical force. Hence you need to include the mass of the clothing the allomancer is wearing, or if the coin is in a wallet and stays in it, the mass of the wallet on the coin's side. Now we consider the case of a coin securely anchored to the ground. Which mass is acted upon on the coin's side of the equation. Theoretically a whole planet. Compared to the mass of a planet, the mass of a coin irrelevant. And that is exactly what your model shows.
The coin will experience a normal force from being used to accelerate a planet. You can see that indeed as the normal force being transmitted to the allomancer. The result is the same.

No. The effective mass of the coin changes when it hits the ground. And that effect is felt by the pushing allomancer.

It is really the same physics that allow a swimmer's same leg muscles to accelerate him much more by pushing on the walls of the pool rather than making swimming motions.

@Jofwu's got the wrong model, what you're describing here is Model 2. (Or a slight variation; I said velocity, you're saying acceleration.) Your basis is the acceleration between the allomancer and the coin, and that determines the force. But there are two issues with that:

• Newton's second law is not F=ma. It is ΣF=ma. You can fudge past the forces on the coin, which just has the Allomantic force and the normal force, by calling it an "effective mass," because the force balance on the coin only has those two terms. But the tricky situations happen when the Allomancer has an additional force, as well, either gravity or a normal force of their own.
• Like I say in my writeup, there are situations where there is no acceleration, but there is still an Allomantic force. Which changes; you can get in a pushing duel stalemate, nobody's moving; push harder, there's more force, but still nobody's moving. That means the force must be dependent on something other than the acceleration.

As far as comparisons to a swimmer, that actually is Model 1. Pushing on the wall or the water, you are exhibiting the same force, and wind up with the same acceleration on the swimmer. The difference is that you are constrained by the extension of your arm. You're thinking in terms of total energy transferred, which is a good way of looking at things. You can calculate it as W=Fd (where distance is how far your center of mass moves before you are no longer able to extend your muscles), or you can do it by integrating your acceleration over the time you are extending your muscles (which will be longer when you are able to push against the wall, because it takes longer to reach full extension because only your body is moving and not your feet [from the reference frame of the wall]). In these scenarios, force is the independent variable, and acceleration is the dependent variable. But that doesn't match with Allomancy, because in the cases in question you are not limited by the extension of your "Allomantic arm." 

 

[+Artemos]

4 hours ago, Pagerunner said:

I haven't been following the other thread; what is this Allomantic Normal Force? Is it a separate force between the Allomancer and an object in contact with the pushed target?

It's the method I've been using for the game because I believe it to be true, it uses fairly intuitive physics concepts, and it gives satisfying gameplay. It conserves momentum and the like because it makes Pushes work like physically grabbing and pushing on something and experiencing a resistance (a normal force) from that object.

From the original post:

If you held a long vertical pole and pushed down on the ground, the ground would push back on you and the pole. If you tried to push the pole into the ground, the ground would resist, and you could climb upwards relative to the ground. Allomancy mimics this effect. When pushing on the coin, it is like you are physically connected to the coin. If something resists your push, you experience that resistance. It's just like you're literally pushing against the coin with your fists. If the coin's in the air, hardly anything happens. If the coin's on the ground, the ground resists.

By this theory, there's two components to the Net Forces that the target and the Allomancer experience - the Allomantic Force and the Allomantic Normal Force (ANF). The Allomantic Force is calculated from all the factors you'd expect, such as Allomantic strength, burn rate, and distance from target. It's definitely equal for both the Allomancer and target. If both the Allomancer and the target are completely unanchored, the ANF term will be 0 and the Net Force is going to equal the Allomantic Force. When either the Allomancer or target are anchored or partially anchored, the ANF will be nonzero and the Net Force will be equal to the portion of the Allomantic Force that is resisted. It behaves similar to normal force due to gravity; a feather, a brick, and a safe can sit on a table, which provides normal forces of different strengths depending on how much the objects are pushing down on it.

I made these pictures with my original post last year. I'm not sure if the scale of the vectors are right, but the pictures still illustrate it pretty well.

Quote

(Which would essentially double the force of the steelpush, which doesn't seem strong enough.)

Indeed, if the target is perfectly anchored, the ANF would be equal to the Allomantic Force, and the Net Force experienced by the Allomancer would be twice the Allomantic Force. Whether that's strong enough is subjective, and sometimes I agree that a factor of 2 is not enough to account for the difference in anchored/unanchored Pushes.

4 hours ago, Pagerunner said:

Or is an additive term to the existing steelpush? (Which I don't think would fit the scenario of an immobile Allomancer and immobile target, since the force would feed back on itself iteratively.) 

My interpretation is that you cannot be resisted more than you can push. The ANF is calculated from the Allomantic Force (your "actual" pushing strength), not the Net Force experienced by the Allomancer or target. If both you and your target are anchored, you'll experience a Net Force twice of that against an unanchored target. There would be no feedback loop in much the same way that pushing a pole into a wall causes no feedback loop.

Spoiler

The ships hung in the sky in much the same way that bricks don't.

Still, there are some flaws to it. Unless the Allomancer and target are both anchored in the exact same way, the ANF for the two won't be equal, and thus the Net Forces for the two won't be equal. I'm pretty sure this is how pushing on things works in real life, but I haven't yet taken the time to do a full mathematical analysis and see if everything perfectly cancels out. I'll have to write a math problem that asks and answers, "if I'm pushing on the coin, which is pushing on the wall, which is pushing on the earth, which moves the earth a little, does that conserve momentum in the closed system of the planet conserved..." etc. etc.

At least this model explains the discontinuity in the force experienced by an Allomancer when the coin hits the ground. The second the Allomantic Force is resisted, a normal force begins being transferred to the Allomancer.

[Pagerunner]

4 hours ago, Artemos said:

Indeed, if the target is perfectly anchored, the ANF would be equal to the Allomantic Force, and the Net Force experienced by the Allomancer would be twice the Allomantic Force. Whether that's strong enough is subjective, and sometimes I agree that a factor of 2 is not enough to account for the difference in anchored/unanchored Pushes.

...

Still, there are some flaws to it. Unless the Allomancer and target are both anchored in the exact same way, the ANF for the two won't be equal, and thus the Net Forces for the two won't be equal. I'm pretty sure this is how pushing on things works in real life, but I haven't yet taken the time to do a full mathematical analysis and see if everything perfectly cancels out. I'll have to write a math problem that asks and answers, "if I'm pushing on the coin, which is pushing on the wall, which is pushing on the earth, which moves the earth a little, does that conserve momentum in the closed system of the planet conserved..." etc. etc.

At least this model explains the discontinuity in the force experienced by an Allomancer when the coin hits the ground. The second the Allomantic Force is resisted, a normal force begins being transferred to the Allomancer.

I was speaking a little hesitantly earlier, but doubling the allomantic push is definitely not strong enough. In that case, a coin hitting a wall would be the same as the allomancer launching a second coin, and we see them launching coins by the fistful. So the increase in force when it hits a wall needs to be magnitudes greater than the force pushing on a solitary coin.

Other than that, the principle starts off sound. You don't have to worry about the sum of forces on each object being identical; they don't have to be, that's why the allomancer has a nonzero acceleration while the coin is stationary. You don't have to worry about conservation of energy, since the steelpush is adding energy to the system. I'd say you don't even necessarily have to worry about conservation of momentum, since only certain kinds of interactions conserve momentum. The big one we're concerned with is: every action has an equal and opposite reaction. The Allomantic Normal Force would apply in one direction to the allomancer, and in the other direction to the object the coin is pushed up against. It would be like using your hand to push against a coin, and when the coin hits the wall you reach out with your other hand and push with that as well.

There's the practical issue I said before, that doubling the force is not enough to cause the effects we see. Maybe you say that the 'other arm' is stronger, that the ANF could be three, four, ten, twenty times the Allomantic Force. That's when things get a little hairier. You are not only pushing on a piece of metal, but on something else that the metal is touching. (Which doesn't sit well with me philosophically, but hey, let's roll with it for the sake of the example.) This doesn't let you push the coin into the wall any harder than you were already pushing it before you hit the wall; your swole 'other arm' is direcly on the wall, throwing you back, while your wimpy original arm is unchanged. So now, when you push on Vin's earring, the bulk of the force is transferred to Vin herself, not to the earring. Which is counterintuitive, and it also doesn't match what's in the books.

If puncturing with a steelpush is beyond the scope of your simulation engine, then you can still go for a souped-up Allomantic Normal Force. It's the first way I've seen to quantify something like an "effective mass" to modify the strength of the push in the first place. But the theory of it isn't quite up to the task of matching everything that's in the books.

[Jofwu]

4 hours ago, Artemos said:

It conserves momentum and the like because it makes Pushes work like physically grabbing and pushing on something and experiencing a resistance (a normal force) from that object.

It works for the game pretty well, but I think it does break conservation of energy? It wouldn't matter in the game I expect because the ground is fixed, rather than just another mass that can be pushed around.

Looking at the FBDs linked, but removing gravity for simplicity. First you just have the Allomantic force, equal and opposite on each thing.

In second scenario, the coin is pushed against the ground, so it experiences a reaction equal to the Allomantic force. Net force on the coin is zero. You have this force also transferred to the Allomancer. But where's that force coming from? With the coin, it's coming from the ground. So if we draw an FBD of the ground, you see a normal force from the coin on it. To satisfy Newton's 3rd law, with your idea that force would have to be doubled. Like we've introduced a connection between the Allomancer and the ground now, and he gets to Push on the ground as well as the coin.

So if Allomantic force is F, then the coin has net force of F-F=0, Allomancer has F+F=2F, and ground has -F-F=-2F. But if we look at the original case, Allomancer has F and coin has -F.

Now (presumably?) you're burning the same amount of metal in each case, but producing different forces. I guess you could wave it away by saying the extra energy comes from Preservation of course. Just doesn't sit right with me though. The whole idea that you have this connection suddenly with which to push on the ground seems like an unnecessary complication without much concrete evidence for the underlying idea? (just, trying to match observations with a fudge factor)

[+Oltux72]

14 hours ago, Jofwu said:

You're saying the force is fixed and the masses are fixed, so you end up with some kind of ratio on the accelerations. But you're still stuck with just a ratio. It's unclear to me how you determine the value of these accelerations.

Exactly. At that point you need to compute coming from the actual allomantic force the allomancer generates. You are using fixed here in two slightly different ways. The masses are given by the situation and measurable. Anchoring however, alternatively can be seen as effectively anchoring them. As Pagerunner remarked, you can also see the allomantic "link" as transmitting the normal force. Both ways of seeing it give the same result.

The force is set to a constant value by the allomancer generating the push.

 

14 hours ago, Jofwu said:

You could certainly input the Allomancer's acceleration as something very low and you'd get the coin accelerating much faster. And then when the M suddenly becomes 1000000001M and you assume a small acceleration for the coin/wall then the Allomancer's acceleration would be very large.

But we're completely making up those accelerations. You need some other relationship defined here if you want to solve for them.

In any case, this is a violation of Newton's 3rd law. Forces aren't applied such that they are "shared" like this. In electrostatics or gravitational forces you have some force that's defined (F here). And that same force is applied to each, per F = m1*a1 = m2*a2. You don't get variables amounts of the force going to each of them. They both experience the same force.

Yes, but not the same acceleration. That is the basic reason behind Kelsier's rule of thumb that if you push on something heavy you'll move, and if you push onto something light, it'll move. The whole point is that by anchoring something you get a transition from light to heavy. Electrostatics and gravity would show the exact same behavior (except that there is no repulsive force in gravity of course)

[+Artemos]

7 hours ago, Oltux72 said:

As Pagerunner remarked, you can also see the allomantic "link" as transmitting the normal force. Both ways of seeing it give the same result.

I'm not sure if @Pagerunner ever adhered to the theory that the normal force is transmitted to the Allomancer. That's something that I believed, but I'm changing my mind from the discussion here. I'm thinking of another idea inspired by Model 3 that I think I'll make a separate topic about at some point.

12 hours ago, Pagerunner said:

If puncturing with a steelpush is beyond the scope of your simulation engine, then you can still go for a souped-up Allomantic Normal Force. It's the first way I've seen to quantify something like an "effective mass" to modify the strength of the push in the first place. But the theory of it isn't quite up to the task of matching everything that's in the books.

Yeah, quantifying "effective mass" seems very sketchy, especially regarding partially anchored metals. Let's say you're an Allomancer standing next to a a metal block sitting on a high-friction surface. You Push on the block, which grinds against the ground, but does still move. It's not perfectly unanchored, since the block is resisted by friction, nor is it perfectly anchored, since it's still moving a bit. The strength of this particular Push would be higher than that on an unanchored target but smaller than that on a perfectly anchored target. The effective mass of the target would have to be somewhere between the mass of the target and the mass of the target + planet, which really doesn't sit right with me. I can't imagining calculating it without using the normal force, as you said.

Edited January 19, 2019 by Artemos
Accidentally hit submit before I meant to

[Jofwu]

On 1/19/2019 at 4:02 AM, Oltux72 said:

The whole point is that by anchoring something you get a transition from light to heavy.

The mass of the thing you Push is irrelevant to how much you are accelerated.

If I push a coin with 1 Newton and I have a mass of 1 kilogram, then I will accelerate at 1 m/s^2.
If I push a house with 1 Newton and I have a mass of 1 kilogram, then I will accelerate at 1 m/s^2.

If my mass doesn't change and I Push with a constant force then the acceleration is constant.

You brought up electrostatics and gravity, where a sudden increase in a charge or mass would cause the acceleration of some other thing (with a set charge/mass) to suddenly increase. That happens because the force increases. 

[+Oltux72]

On 1/20/2019 at 6:34 PM, Jofwu said:

The mass of the thing you Push is irrelevant to how much you are accelerated.

If I push a coin with 1 Newton and I have a mass of 1 kilogram, then I will accelerate at 1 m/s^2.
If I push a house with 1 Newton and I have a mass of 1 kilogram, then I will accelerate at 1 m/s^2.

If my mass doesn't change and I Push with a constant force then the acceleration is constant.

You brought up electrostatics and gravity, where a sudden increase in a charge or mass would cause the acceleration of some other thing (with a set charge/mass) to suddenly increase. That happens because the force increases. 

That makes sense and makes me suspect that I am a fool. I made an error in attribution.

Nevertheless from a physical viewpoint the behavior of an anchored coin makes sense. It derives from the conservation of impulse.

[Pagerunner]

@Oltux72, here's another way of looking at it. We need to solve a system of equations. There are a number of Givens (pieces of information that constrain the system in a specific case), Equations (relationships that constrain the system) and Variables (pieces of information we are solving for). And Constants. It is an algebraic principle that the number of Variables we have needs to equal the sum of number of Givens and the number of Equations. If you don't have enough Givens/Equations, then you can't solve the system. You need more information. On the other hand, if you have too many Givens/Equations, you still can't solve the system. You have too much information, and they can't all be true.

From the force balances you've laid out thus far, we have six variables in play:

1. The force of the push, before contact. (F1)
2. The force of the push, after contact. (F2)
3. The acceleration of the allomancer, before contact. (a1)
4. The acceleration of the coin, before contact. (a2)
5. The acceleration of the allomancer, after contact. (a3)
6. The acceleration of the coin, after contact. (a4)

We have four equations, the free body diagrams on the allomancer and the coin, both before and after contact. (Three constants among these equations, the masses of the objects in play. Let's say m1 is the allomancer, m2 is the coin, and m3 is the planet+coin.) In the simplified situation you've outlined above, with no other forces acting on any of the three objects (except the normal force between the planet and coin), these will work out to:

1. Allomancer, before: F1 = m1 * a1
2. Coin, before: F1 = m2 * a2
3. Allomancer, after: F2 = m1 * a3
4. Coin, after: F2 = m3 * a4

We can have, at most, one given. That will specify the specific situation we're in; i.e. how hard the Allomancer is pushing. We cannot specify a second given for this system; that's inconsistent with the text. That would be like an allomancer specifying both the initial force and the final force, or their initial acceleration and their final acceleration. That's not how we see it work; if they set the initial force, the final force is determined by the math, and it catches them by surprise. Our current system is unconstrained; that means, algebraically, we need another equation to completely describe the situation.

What possibilities do we have? Working with our existing six variables, we're pretty limited:

• We can say F1 = F2, a.k.a. my Model 1, which is how something like a magnet or pushing off of a wall would work. (Remember, I said before that when pushing off a wall, you're limited by how long you can push for, not how hard you can push.) This is an explicit contradiction of the text, though, since we know F1 does not equal F2.
• We can relate the initial accelerations to the final accelerations. This isn't quite my Model 2, but it runs into the same hangups. The acceleration of the allomancer changes upon contact, so a1 can't equal a3. The acceleration of the coin changes, so a2 can't equal a4. And while you could try saying that the relative acceleration remains constant (a1-a2 = a3-a4), that doesn't hold up in more complex situations. (As I lay out in my writeup.)
• We can say an allomancer is subconsciously changing the force based on what they expect the behavior to be, and we actually do have two givens. This is probably the most instinctual of understandings, but it relies on the allomancer operating on feedback they receive. They see or feel a coin hit a wall and respond accordingly. But this also breaks down in complicated situations, where the force changes even when an allomancer does not observe the object they are pushing on hit a wall. They have no feedback, so no reason to change how hard they are pushing, but we still see a change in all forces and accelerations.
• We can manufacture a new relationship between the force and the acceleration at any given instant, in addition to the Newton's Law equations. This could be based on the allomancer's acceleration, relating F1 to a1 and F2 to a3. Or it could be relating the relative accelerations, F1 to a1-a2 and F2 to a3-a4. But either way, we will be adding two new equations instead, resulting in an overly constrained system. So we make sure we add a new variable, the 'strength' of the push (from a scale of, say, 0 to 100%), which we incorporate into our new relationship. This strength will remain the same both before and after contact; it's our single constraint, our one given. You need to derive a relationship for any given instant between: force, strength, and at least one acceleration.

The last option is the one I'm going with, and it actually has a lot of flexibility. You can make the relationship more complex, with additional variables, as long as you add a new equation to the system for each new variable you add. I took out the acceleration, and made my force dependent on strength and the relative velocities. added four variables to the system: the velocities of the allomancer and the coin, both before and after contact. And my corresponding four equations are the kinematic relationships between acceleration and velocity in all four instances, v1 = ∫a1. We'll need some more constants to describe the system, boundary conditions to say how fast we're moving when we start. But by incorporating this new relationship, I'm able to solve the system: eleven variables, ten equations, and one given.  And the solution it gives makes sense and matches the text. (As evidenced by the simulations.)

I am definitely open to hearing other proposed equations. (And I am sincere in this.) Artemos had one, a little more roundabout, with his allomantic normal force. It lets you come up with a solution for the system of equations. But, like I stated earlier, that one doesn't match the text as well, with regards to the magnitude of the change from F1 to F2.

Can you think of a new specific relationship or phenomenon that we can evaluate? One that would match all the complex situations presented throughout the books as well or better than the relative velocity correction factor? The other variables you mentioned earlier, like mass of the allomancer and distance to the target, will also have an relationship to the force. But for the sake of the scenarios being discussed, they'll just be constants, so they won't help us algebraically constrain the problem.

[+Oltux72]

4 hours ago, Pagerunner said:

• We can manufacture a new relationship between the force and the acceleration at any given instant, in addition to the Newton's Law equations. This could be based on the allomancer's acceleration, relating F1 to a1 and F2 to a3. Or it could be relating the relative accelerations, F1 to a1-a2 and F2 to a3-a4. But either way, we will be adding two new equations instead, resulting in an overly constrained system. So we make sure we add a new variable, the 'strength' of the push (from a scale of, say, 0 to 100%), which we incorporate into our new relationship. This strength will remain the same both before and after contact; it's our single constraint, our one given. You need to derive a relationship for any given instant between: force, strength, and at least one acceleration.

It still looks to me like you can get away with one initial force and a single equation. Or a set of equations derived from each other.
One proposal. The allomancer sets a total force. In the static case this is just what both sides feel. If something is moved however, the interaction is governed by conservation of impulse. The constraint is compatible with the static case. In a static case impulse is obviously conserved.

We have to throw out action equals reaction, but we already threw out conservation of energy. So can we derive observed behavior from conservation of impulse? I think so. At least the push on something heavy and you are shifted, push on something light and it is shifted behavior would be the result. What do you think?