Steelpush/Ironpull Formula

[Kelsier's Son]

Hi, I'm Kelsier's son, and I may have figured out the formula for Steelpushes and Ironpulls, but specifically Steelpushes. Allow me to explain.

This is all assuming the Cosmere abides by the normal laws of physics.

We know, thanks to Newton, that Force is equal to mass times acceleration, F = ma.

Well, what's acceleration? It's the change in velocity (Vf - Vi)/t

Thus, we get F = m((Vf - Vi)/t)

Now, where would we get the velocity? Well, when Vin or Wax pushes on an object, it's giving it kinetic energy, whose formula is KE = (1/2)mV^2

Solving for V, we get sqrt(m/(2KE))

Now, this energy isn't coming from nowhere. I theorize that a certain gram amount of steel/iron has a certain amount of Potential Energy, what I call Metallic Potential Energy, whose formula I have derived to be MPE = bMt, where b is the burn rate (half burn, flaring, etc.), M is the energy at b = 1, and t is time. 

We can set the MPE equal to the KE, so that MPE = KE, which we can substitute their respective equations: bMt = (1/2)mV^2. Solving for V, sqrt(m/(2(bMt)).

Now, we can take this velocity and plug it into the force equation from earlier Fs = m((sqrt(m/(2(bMt2) - sqrt(m/(2(bMt2) )/t2 -t1).

This gives us the Average force from Time one to Time two. If we want the instantaneous force, we have to get a little advanced.

Let's make Fs a function of time, Fs(t), where Fs(t) is our Force equation from earlier. Let's slap a Limit in there. Now, Fs(t) = m lim(t1->t2) ((sqrt(m/(2(bMt2) - sqrt(m/(2(bMt2) )/t2 -t1).

Well, wait, that's a derivative.

If we take advantage of that, and do a little bit of algebra, we get Fs(t) = t/(2sqrt(2bMt/m)) This my friends is my Theory of Steelpushes

I hope this makes sense. If it doesn't, I understand.

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Edited April 27, 2023 by Kelsier's Son

[Treamayne]

Welcome to the Forums. Nice math!

Also, we don't yet have a standard Investiture Unit of measure, but know that burned metal is pulling Investiture from the spiritual realm and the Investiture is what is converting into Kinetic Energy. It doesn't really effect your equations except for a possible added step where you would be converting bMt to IUs - but since they are equivalent it's not much of a factor at this point.

6 hours ago, Kelsier's Son said:

Ignore the quotes, I didn't realize what their inputs were until I pushed the button and couldn't figure out how to delete them

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Hope that helps.

 

Edited April 20, 2023 by Treamayne
Clarity

[alder24]

Good one. I like math, but I don't think it would be that easy. First, it's not just the burn rate, Allomantic strength is also a very important factor here, the stronger the Allomancer (like Elend compared to Vin), the more force he can exert within the same time and the same burn rate.

Is that the variable "M"? Why is energy M not E? M is for mass!  It confused me! If so, ignore the next paragraph. 

This could likely be resolved by adding another variable like "p", which would be Allomantic power of Allomancer, in most cases, their power would be on the same level, like Kelsier=Wax, Vin was said to be a bit stronger, likely she ingested Mists when she was younger, Inquisitors are on the other hand a bit weaker, because of their hemalurgic decay (something like 0.9p of Kelsier), but Elend or Rashek are towering over them with massively stronger Allomancy, Rashek could even push on metal traces and invested spikes. That's a lot of power.

But then there is Wax, who has Iron feruchemy and can change his mass, which just breaks equations of kinetic and potential energy. Like when Wax changes his mass mid flight, due to conservation of momentum (which is conserved), his speed changes, but also his potential and kinetic energy is change, and his speed, derived from kinetic energy equation, doesn't match up with his new real speed, derived from conservation of momentum. I’ve done math on this and that’s what I got. His example clearly shows us that our equations for energy are incomplete, as they don't take into account investiture, 3rd state of energy. This would likely break your equation as well, especially in case of Wax and Feruchemical Iron.

But that still looks like a good approximation of the force of steelpush, using our real physics. I think the best test of it would be to use math and calculate the force Vin used when pushing on top of Luthadel walls, it was said how tall they were, so it should be doable.

[Kelsier's Son]

It doesn't necessarily mean it breaks physics; it just means that Wax moves into the realm of Rocket Physics