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Proposed Physics Model for Steelpushing: Elastic Collisions


digitalbusker

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Me likes. It explains some of the nuances a lot better than my inverse-square law does. So basically, so long as your Allomantic strength remains the same, the amount of kinetic energy you can push/pull with is the same, regardless of your weight/mass. So when Wax taps iron and Pushes, the total energy applied is the same as when he stores, except a lot more of the energy is applied to what he is Pushing on. Awesomeness!

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Would this mean that a Mistborn using Iron/Steel to "fly" would basically be following a spring PE equation following Hooke's Law mixed with a gravitational PE equation? So, say Vin wanted to continue flying, either using a Allomantic highway or her horseshoe trick.

 

The equation for spring potential energy is PE= 1/2 (kx2) where K=F/x or N/m

 

Let's say Vin weighs 120 lbs [54.5 kg] which equates to ~534.5 N and she's decided to start burning steel having fallen to about 5 metres above the ground when she used to be at 10 metres at the top of her last jump. The first half of the equation goes like this:

 

PE= 1/2 (534.5/5)(52) so PE= 1, 336.25 N. This would be the potential energy produced by Vin's allomantic force applied via steelpush. The other half of the equation involves the potential energy due to her falling because of Scadrial's gravity, which for the sake of simplicity is going to be assumed as roughly the same as Earth's.

 

PEgrav= mgh so PEgrav= (54.5 kg)(9.8 m/s2)(5m)= 2, 670.5 N. Therefore, when we total it all up, the amount of energy being applied to the coin Vin is pushing into the ground while burning Steel is 4, 006.75 N.

 

I have no idea if I understood this theory correctly, so tell me if I'm wrong, but I think this is a good approximation.

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I think you messed up your units in places. Energy should be in joules not newtons. Also, you're assuming that her springy ness is directly proportional to her mass, which is the complete opposite of what this theory is stating. It's also possible that I have no idea what I'm talking about and should just shut up now. I was never good at elastic collisions...

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Elastic collisions is what I was looking for - that is almost precisely how the velocity is split when steelpushing in the novels.

The larger thing stays still when the smaller one moves, the person doesn't experience the same force pushing them whenever they steelpush - and for those Coinshots that exhibit fine control of their push strength... They can vary the energy put into the system rather than the force applied to objects.

I really should have seen that sooner.

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There is a well defined physics model that deals with two objects interacting in a way that changes their velocities to a degree that's inversely proportional to their masses: elastic collisions. If we reframe this situation in terms of an elastic collision, then the allomantic strength term becomes not force but kinetic energy. A given allomancer's push strength is the amount of kinetic energy they can add to the system. What happens to the velocities of the components of the system is then subject to conservation of (the new, higher) momentum. This preserves the behavior we see in the books.

 

(Actually a person's Steelpush strength would be delta-kinetic energy per unit time, but integrating over time is left as an exercise for the reader.)

 

If you'd like a pseudo-concrete framework for thinking about this, imagine that the blue lines you see while burning Steel are tangible to the things at either end of them. Normally they are happy to simply change length to remain in contact with their endpoints, but when Pushed or Pulled, an Allomancer pours some Investiture into them and makes them grow or shrink with some amount of power. The line doesn't care where its center winds up, or how long or short it winds up being, it just knows how hard to push, and the end that resists less will be moved farther than the other.

 

So to bring this back around to the question of whether your Push/Pull strength depends on your mass: Yes and No. There's some innate Allomantic Strength term that's independent of your mass, but in practice the things you can do with Allomantic iron or Allomantic steel are influenced by both your strength and your mass. (When you're trying to move yourself it's better to be lighter, when you're trying to move other things it's better to be heavier.)

 

After reading the first fight between Elend and the Inquisitor in Hero of Ages, I'd like to put a question upon whether this is how Allomantic Strength alters the model.

You have stated that it allows you to add more energy at a given instant... But from the scene, which I'll quote here, that isn't quite right.

 

 

Two Allomancers of near-similar weight, shoving against each other. They would both be thrown back - the Inquisitor to attack Vin, Elend into a pile of koloss.

Except, the Inquisitor didn't anticipate Elend's Allomantic strength. How could it? Elend did stumble, but the Inquisitor  was thrown away with a sudden, violent Push.

It seems to me that Elend's Allomantic strength, although it may have increased the amount of energy given to the system, seems to allow him to act as a better anchor. It effectively counts him as a greater mass than he is.

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I'm not really good at physics - I was at highschool, but then i did choose arts at Uni and forgot everything :P - but I'll try to convey my opinion on this the best way.

 

Initially I liked digitalbusker's idea/theory a lot, it did explain so much in the correct way, but there's at least one thing I can't explain with this method:

 

Staying up in the air while pushing some random metal down below. If you are inputting kinetic energy (movement) how do you explain reaching an equilibrium like this? A position with no movement.

 

I guess you can somehow convert the kinetic energy into potential energy while keeping the total mechanical energy untouched. I just don't see how.

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I'm not really good at physics - I was at highschool, but then i did choose arts at Uni and forgot everything :P - but I'll try to convey my opinion on this the best way.

 

Initially I liked digitalbusker's idea/theory a lot, it did explain so much in the correct way, but there's at least one thing I can't explain with this method:

 

Staying up in the air while pushing some random metal down below. If you are inputting kinetic energy (movement) how do you explain reaching an equilibrium like this? A position with no movement.

 

I guess you can somehow convert the kinetic energy into potential energy while keeping the total mechanical energy untouched. I just don't see how.

This was why I originally came up with my inverse-square formula, at a certain distance the force of the Push and gravity are equal, hence the floating.

 

I'd guess the energy one can Push with in the elastic-collision model is also limited by range.

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This was why I originally came up with my inverse-square formula, at a certain distance the force of the Push and gravity are equal, hence the floating.

 

I'd guess the energy one can Push with in the elastic-collision model is also limited by range.

 

I would definitely be restricting push the amount of energy one can put into the push by range. Inverse square law is what I would be going for as well - just because that's how things work with normal physics... (At least with sound/light/etc. but then again, they are putting energy out in all directions - where as pushing on a single object is just one).

 

I'm not really good at physics - I was at highschool, but then i did choose arts at Uni and forgot everything :P - but I'll try to convey my opinion on this the best way.

 

Initially I liked digitalbusker's idea/theory a lot, it did explain so much in the correct way, but there's at least one thing I can't explain with this method:

 

Staying up in the air while pushing some random metal down below. If you are inputting kinetic energy (movement) how do you explain reaching an equilibrium like this? A position with no movement.

 

I guess you can somehow convert the kinetic energy into potential energy while keeping the total mechanical energy untouched. I just don't see how.

Maybe at that distance, a lot of the energy from the push is being lost, and when someone is floating, they're simply feeding no energy to the system (but any lower and they will be pushing it back up). This could also explain Zane floating near the ground if really talented Allomancers were capable of limiting the energy the put into the system.

An alternative is that the system IS converting gravitational potential into kinetic, but the Allomantic steel is adding just as much energy through the Allomantic steel link between the two objects and the energy kinda cancels itself out.

Think two sound waves created by two sources doing completely the opposite at a point - resulting in no sound...

 

But yeah, I get what you're saying, they're all hard and fast applications...

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  • 6 months later...

So I am necroing this thread because I have a question and you all clearly have far more advanced knowledge in physics than I could ever hope to hold, so I am hoping you all could help me. Based on your calculations, if we were to assume some variables, such that it is Vin doing the push, she is pushing an average copper penny, lets say she weights 120 pounds, she is shooting a target a distance away (I suck at gauging distances, so maybe an average room length apart?)and she is flaring steel, is there anyway to calculate the kinetic energy in the coinshot? And then with all the same variables, calculate the kinetic energy with a dularium assist? Also assuming Vin maintains the push during its entire trajectory, would that affect the calculations at all when the coin hits the target?

 

Bonus points if the formula could be shown, and maybe explained to me so I could learn a bit more on the subject and be able to apply it in the future myself as well  :)

Edited by Pathfinder
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  • 1 month later...

I always saw Pushing and Pulling as more of an inelastic collision. When Vin Pushes on a coin, it isn't like throwing the coin, and having no contact with it after it leaves her hand. She continues to push on it, and when the coin hits something, say a wall, the coin and the wall stay 'stuck' together while Vin goes flying. I'm kinda picturing Vin and the coin as one object, rather than two, up until the actual collision. If the coin hits something that will move, it is propelled by the coin, not just the momentum of the coin.

 

I'm in the middle of class and therefore have nothing to back me up right now, but I just wanted to throw the idea out there.

 

NOTE: this was written by Lord Pifferdoo's girlfriend, Lady Pifferdoo, who will have a profile eventually. 

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So I am necroing this thread because I have a question and you all clearly have far more advanced knowledge in physics than I could ever hope to hold, so I am hoping you all could help me. Based on your calculations, if we were to assume some variables, such that it is Vin doing the push, she is pushing an average copper penny, lets say she weights 120 pounds, she is shooting a target a distance away (I suck at gauging distances, so maybe an average room length apart?)and she is flaring steel, is there anyway to calculate the kinetic energy in the coinshot? And then with all the same variables, calculate the kinetic energy with a dularium assist? Also assuming Vin maintains the push during its entire trajectory, would that affect the calculations at all when the coin hits the target?

 

Bonus points if the formula could be shown, and maybe explained to me so I could learn a bit more on the subject and be able to apply it in the future myself as well  :)

I just did something like this in another thread, for final velocity, but kinetic energy is just one equation removed from that: Ek=(1/2)mv2, i think.  basically the one thing we don't have is the equation that determines the strength of a push.  Without that we only have guesses as to what the force number should be (though we can hazard some educated guesses based on how steelpushing appears to work).  And yes, maintaining the push would constantly increase the acceleration which would increase the final velocity, which would increase the energy.

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