Running the math on Steel Compounders

[Steelrunner]

There's been a fair amount of discussion about steel compounding on these forums as one of the most exciting forms of Twinborn. The general consensus (with support from Word of Brandon: wob.coppermind.net/events/93-odysseycon-2016/#e2695 ) is that a steel compounder could feasibly run at incredible speeds while being magically protected from many of the consequences for an extended period of time.

So I decided to do the math on what it would look like for a steel compounder running at 10x normal speed (well within what Bleeder does in Shadows of Self, I think). Here's what I found:

Imagine our Steelborn throwing a ball. Now, an ordinary person throwing a ball has their arm swing through a distance (call it 1 m) in a certain time and releases with a certain final velocity (call it 25 m/s for fairly ordinary humans - thats ~50 mph). Now, our Steelborn is moving 10x as fast, which I take to mean he takes 1/10 the time from start of throw to release. That implies he has 10x the release velocity (also what we expected). But he accelerated a ball to 10x the speed in 1/10 of the time of an ordinary human and acceleration = velocity/time (for constant acceleration of an object from rest, a decent approximation here), which means the Steelborn's ball has 100x the acceleration. F = ma, so either the ball had 1/100th of the mass, or our Steelborn applied 100x the force.

Option 1: Changing strength

Let's assume the ball's mass didn't change for now. Our Steelborn needs to apply 100x the force, which means our Steelborn needs to be 100x as strong as normal.

It gets better. Our Steelborn has 100x the strength of an ordinary man and can run 10x as fast. Power = Force x Velocity, which means our Steelborn has 1000x the power output. If an ordinary human can output 300 Watts, our Steelborn can output 300 kW, or 400 horsepower.

Of course, with 100x the strength, our Steelborn will need his tendons, ligaments, bones, etc. reinforced by 100x to avoid his enhanced strength tearing himself apart (see WoB above). Human skin normally has about 20MPa tensile strength, so a 100x increase puts us close to Kevlar's 3000 MPa tensile strength.

His maneuverability with A-steel remains unaffected, though. There's no reason to assume his Steelpushes have enhanced strength, so he can't Push a coin any faster or Push himself off a skyscraper any quicker. (F = ma.)

But the truth is, he hardly needs this. With 10x throwing speed, an ordinary 25 m/s pitch turns into a 250 m/s projectile (9mm pistol rounds travel at ~300 m/s, or ~1000 fps). With 20 bullets a throw and assuming he can get off 5 throws/sec, that means he is putting out the equivalent of 100 pistol rounds/second. (World War II machine guns fired ~10 rounds/sec.) He'd have to be aware of the recoil, though: he's not any heavier to absorb it!

It's not all upside, though. Even with 10x the top speed and 100x the strength, our Steelborn is limited by the friction between his feet and the ground, which hasn't changed. Unless he can push off of something, he can't accelerate (or turn) at anything more than about 10 m/s^2. If he's moving at 10x an ordinary run speed of 4 m/s, and remembering that turning radius = velocity^2/acceleration on turns, we have turning radius = (40 m/s)^2/10 m/s^2 = 16 m. No hurtling around tables for him.

Actually, I don't think he would run at all. With 100x the strength, each footfall would propel our Steelborn high into the air. I suspect he would move like astronauts walking on the moon instead.

Option 2: Changing mass + gravity

But all that doesn't match up with what Bleeder does in Shadows of Self. So instead of assuming that our Steelborn gets stronger, let's assume that our Steelborn's mass decreases to 1/100th of normal instead. 50 kg -> 0.5 kg. Additionally, the acceleration due to gravity must increase by 100x to leave the total force due to gravity unchanged to prevent him from bouncing high into the air with each step.

Now, our Steelborn has the same strength and the same amount of traction (since he has the same weight as before) but 1/100th of the mass. F = ma, so that gives him 100x the acceleration. Hairpin turns are no problem: running at 40 m/s he has the same turning radius as an ordinary human running at 4 m/s (since turning radius = velocity^2/acceleration = 4^2/10 = (4x10)^2/(10x100)). Jumping is fine too: our Steelborn has 10x the takeoff velocity but 100x the downward acceleration. Jumping height = v^2/2g, so the effects cancel, leaving him able to **jump the same height as an ordinary human**. However, air time = 2*v/g, so he has 1/10th the air time, which is again what we expect.

His maneuverability with A-steel is enhanced too. There's no reason to assume his Pushes have changed strength, so F = ma says with 1/100th the mass, his Pushes give him 100x the acceleration. The increased gravitational acceleration won't cause problems because the total gravitational force is unchanged.

Additionally, he no longer needs Kevlar skin, as he hasn't gotten any stronger. Since his body has so little mass, his insane accelerations will result in the same forces on him as usual. He won't black out either.

Clearly, the mass reduction and gravity enhancement must extend to the Steelborn's clothes, guns, coins, etc. Otherwise, carrying a 1 kg pistol when he himself has a mass of only 0.5 kg would reduce his maneuverability to 1/3 of expected, but we see Bleeder think nothing of Steelrunning with pistols. The problems begin, however, when we consider him throwing something. What happens when coins leave his hands and return to normal mass? I see three scenarios:

1. Coins return to normal mass but velocity unchanged. They increase in both momentum (p = m*v) and energy (KE = 1/2m*v^2). This gives him tremendous firepower: he can throw coins at 10x ordinary speed and 10x the rate, exactly as described above. His pistol rounds travel at 10x speed too: muzzle velocity = sqrt(2*d*a), so since 1/100th the mass -> 100x the acceleration by F = ma, bullets have 10x the muzzle velocity. To top it off, he's immune to bullets: incoming bullets are reduced to 1/100th of their mass upon hitting him.

2. Coins are reduced to 1/100th speed. This satisfies Conservation of Momentum: p = m*v, so if mass increases by 100x to return to normal, velocity decreases to 1/100th. Since KE = 1/2m*v^2, their energy decreases to 1/100th (losing energy is more permissible than gaining it at least!). His coins are reduced from 250 m/s on leaving his hand to 2.5 m/s - 1/10th of an ordinary throw speed. This affects bullets too: they have 10x the muzzle velocity initially, but as soon as they stop touching him, they are reduced to 1/100th of the speed. The net effect is to reduce a 300 m/s pistol bullet to a 30 m/s one. Our Steelborn's problems get even worse when we consider bullets striking him: they are suddenly accelerated to 100x their velocity, satisfying Conservation of Momentum, but resulting in a 100x increase in bullet energy since KE = 1/2m*v^2.

3. Coins are reduced to 1/10th speed. This satisfies Conservation of Energy: KE = 1/2m*v^2, so a 100x increase in mass is balanced by 1/10 velocity, but doesn't satisfy Conservation of Momentum. His throwing speed is unaffected, since coins leave his hands 10x faster than normal, but then are reduced to 1/10 speed, or normal throwing velocity. The same applies to pistol bullets. Bullets striking him are reduced to 1/100th mass but have 10x the velocity, leaving their energy again unchanged.

Of all the options, I think 3 fits the best. Our Steelborn cannot throw or shoot at crazy speeds with F-steel - though he can still put out huge numbers of bullets and Push them to deadly speeds with the help of A-steel.

Now that we have established how he (roughly) interacts with objects, much of his power depends on how far this field extends. Does it affect a bike he rides? A car he drives? A train? I suspect the explanation invokes his Connection to an object and the object's own Identity, but I see no evidence for how.

The TL;DR
The best match with Shadows of Self is that you can't change your strength with F-steel: you obtain crazy speeds by reducing mass and increasing gravity while permitting your muscles to operate at higher speeds. (This pairs well with F-Iron: both manipulate mass.) Anything you throw is reduced in speed and increased in mass when it leaves your hands. This gives enhanced maneuverability and speed without crazy firepower.

Thoughts?

[Karger]

Welcome to the shard!  Great work.  I always assumed in the back of my mind was that what we were really getting was localized time manipulation.  That would explain this not happening to bleeder

23 minutes ago, Steelrunner said:

Actually, I don't think he would run at all. With 100x the strength, each footfall would propel our Steelborn high into the air. I suspect he would move like astronauts walking on the moon instead.

as well as the increased reaction time of the people involved.

Edited June 28, 2020 by Karger

[Halyo_Alex]

1 hour ago, Karger said:

the increased reaction time of the people involved.

Right, don't we see Bleeder look over and react to Wax and Wayne inside a Bendalloy bubble? So... That means she's being sped up in time, as if F-Steel is just a personal Bendalloy bubble.

[Karger]

14 minutes ago, Halyo_Alex said:

Right, don't we see Bleeder look over and react to Wax and Wayne inside a Bendalloy bubble? So... That means she's being sped up in time, as if F-Steel is just a personal Bendalloy bubble.

Basically.  It also explains why she had no trouble running even though her legs were moving way faster then the fastest human athlete.

[+Oltux72]

4 hours ago, Halyo_Alex said:

Right, don't we see Bleeder look over and react to Wax and Wayne inside a Bendalloy bubble? So... That means she's being sped up in time, as if F-Steel is just a personal Bendalloy bubble.

Partially. If that were the full explanation F-steel and F-zinc would be no different.

[GoWibble]

8 hours ago, Steelrunner said:

speeds while being magically protected from many of the consequences for an extended period of time.

Ummm. Kind of. Pick your evil. Burning up from wind or getting torn apart from G forces. It only protects you from the second. 

8 hours ago, Steelrunner said:

though he can still put out huge numbers of bullets and Push them to deadly speeds with the help of A-steel.

Is it just me, or do bullets already move at deadly speeds?

1 hour ago, Oltux72 said:

Partially. If that were the full explanation F-steel and F-zinc would be no different.

The main difference is that you can do something with your faster thinking speeds in the moment with C-Steel 

 

I think it looks good! The only thing that I noticed (apart from ^) is that somehow, TLR compounded his Allomancy, so you technically could have stronger pushes. Also that I can't wait to take physics. 

I tried to explain what I thought that the other compounding metals would do in this: 

(and no one responded ) But I kind of glossed over the physics because I: 1. Didn't know how to do it. and 2. Just went off of WoBs and what we saw in the book. 

[Chiberty]

13 hours ago, Steelrunner said:

His maneuverability with A-steel is enhanced too. There's no reason to assume his Pushes have changed strength, so F = ma says with 1/100th the mass, his Pushes give him 100x the acceleration. The increased gravitational acceleration won't cause problems because the total gravitational force is unchanged.

Actually, it seems that Push strength is proportional to mass. Kelsier mentions that Vin's Pushes are strong for someone her size, which indicates that smaller people have weaker Pushes, and in the scenes where Wax Pushes while his mass is increased, his Pushes appear to be much stronger.

[Karger]

39 minutes ago, ChickenLiberty said:

Actually, it seems that Push strength is proportional to mass.

f = ma Vin has a higher a then normal despite being smaller.

7 hours ago, Oltux72 said:

Partially. If that were the full explanation F-steel and F-zinc would be no different.

How is mental clarity even related?

[+Oltux72]

9 minutes ago, Karger said:

How is mental clarity even related?

It is mental speed. If F-Steel were just a personal movable time bubble it should accelerate all your activities including thinking.

[Halyo_Alex]

55 minutes ago, Oltux72 said:

It is mental speed. If F-Steel were just a personal movable time bubble it should accelerate all your activities including thinking.

F-Zinc doesn't speed up your physical body. F-Steel does.

Mental vs Physical speed. It's just that in order to not splat on walls you need to have enhanced reaction times, so F-Steel boosts your perception of time as well.

[+Oltux72]

17 minutes ago, Halyo_Alex said:

F-Zinc doesn't speed up your physical body. F-Steel does.

Yes that was my point.

17 minutes ago, Halyo_Alex said:

Mental vs Physical speed. It's just that in order to not splat on walls you need to have enhanced reaction times, so F-Steel boosts your perception of time as well.

Yes, that much is clear. The problem is that this distinction is incompatible with F-Steel just speeding up time. A-Bendalloy and F-Steel must be based on different principles.

[Chiberty]

1 hour ago, Karger said:

f = ma Vin has a higher a then normal despite being smaller.

That equation isn't directly applicable to what I'm talking about. None of the variables in that equation refer to Push strength, and "a" stands for acceleration, so Vin having a higher "a" than normal doesn't make sense. I'm saying that Allomancer mass is a factor in Push force, which we don't know the equation for (although there are a bunch of theories).

Edited June 28, 2020 by ChickenLiberty

[Karger]

1 hour ago, Oltux72 said:

It is mental speed. If F-Steel were just a personal movable time bubble it should accelerate all your activities including thinking.

It does. 

Quote

Alteroden

With [Feruchemical] zinc, you get mental speed. How is that any different from [Feruchemical] steel, except without [physical] speed?

Brandon Sanderson

I think of the mental speed actually turning you into... Let's say you sped up your body, and you wanted to figure out some really complex equations.

Alteroden

So it lets you have intuitive leaps.

Brandon Sanderson

Right. It basically turns you into Ken Jennings. That's how I imagine it.

Kurkistan

So it's not like bullet time?

Brandon Sanderson

No... It'll bullet time a little bit, it certainly will, because you're thinking faster than everyone else, but it has applications beyond bullet timing. Bullet time is really--

Kurkistan

That’s steel’s thing?

Brandon Sanderson

That’s kind of steel's thing. They kind of overlap on that one, because the steel thing... But yeah. It's more like "I think fast, but my reaction speed is not sped up".

Shadows of Self Chicago signing (Oct. 12, 2015)

 

2 minutes ago, ChickenLiberty said:

That equation isn't directly applicable to what I'm talking about. None of the variables in that equation refer to Push strength, and "a" stands for acceleration, so Vin having a higher "a" than normal doesn't make sense

Sure it does.  It means that her pushes accelerate a piece of metal faster then normal.

[Chiberty]

Just now, Karger said:

Sure it does.  It means that her pushes accelerate a piece of metal faster then normal.

Yes, that is the correct application of that equation. However, I was talking about how Vin's lower mass would normally mean her Pushes are not as strong, which does not have to do with that equation.

[Karger]

6 hours ago, ChickenLiberty said:

Yes, that is the correct application of that equation. However, I was talking about how Vin's lower mass would normally mean her Pushes are not as strong, which does not have to do with that equation.

That was not my interpretation.  She has less mass(she is a petite lady) so she would not normally be able to output much force but she compensates by just being a super strong mistborn.

Edited June 29, 2020 by Karger

[Chiberty]

1 hour ago, Karger said:

That was not my interpretation.  She has less mass(she is a petite lady) so she would not normally be able to output much force but she compensates by just being a super strong mistborn.

That is the same thing that I was saying. I was just explaining that F=ma is not the equation that describes that relationship. The equation that does is unknown, although we do know most of the variables that are factors in it.

[18th Shard]

I don't have the book on me, but I seem to recall in the prologue Bleeder going into a locked room. If my memory is correct, how does that factor in to the OP's suggestions on the physics of steel (having to unlock a door with a key)? Any kind of mass manipulation gets hairy when you consider that opening a locked door with a heavy key requires you to transfer the effects to the key as well, which then becomes too light to turn the lock, so the effect has to pass into the door as well, etc.

I think localized time dilation of some kind or a Flash-like Speed Force aura is necessary to prevent tons of physics breaking.