DrPhysics

14 hours ago, DoomslugTD said:

I just know that in this model the force of gravity is constant regardless of the motion of the particle, and that the force of the steelpush is always the same at a certain distance regardless of the motion of the particle as well

This is where you went wrong in trying to solve for forces. Your model doesn't assume these things. It assumes that x(t) = h · tanh(at)². Could that be a result of those assumptions? Sure. But it isn't built on those assumptions. As I said before, that doesn't mean your model is useless or you should throw it out. It just means that it won't work if you try to generalize it to situations where motion might not be x(t) = h · tanh(at)².

11 hours ago, DoomslugTD said:

Why does limiting x do have a maximum of h limit d? We have real world examples of systems that dont tend to infinity in the presence of gravity but would if they were not. Like if you a negativly charged plate fixed to the ground and a postivly charged ball above it they charges would push away to a certain distance before gravity became stronger then the colobmn force and it settled at a final height, but if you removed gravity from this equation the ball would accelerate from the plate but never stop therefore tending towards infinity.

Every one of these systems is built starting from first principles using forces, and the description of motion comes second, so you can tweak it by adding/removing forces and seeing how that impacts motion.

Your model is built on a description of motion, and you are adding in forces second. So, you can change motion and see how that impacts forces, but you can't change forces to see how that affects motion. 

What you are doing is equivalent to what happened with the Rayleigh-Jeans law and the Ultraviolet catastrophe.  Short version: They came up with an equation that fit pretty well to the spectrum of a blackbody. It was an empirical formula designed to fit what we saw (like your motion equation). But, when we tried to use it to describe the thermodynamics/statistical mechanics of what was going on, it broke at infinity (specifically, every blackbody would emit an infinite amount of energy in Ultra-Violet and shorter wavelengths). It worked well to predict spectrum, but it couldn't be used on more fundamental ideas. Later on, Plank constructed what we use now from thermodynamic principles, and his solution became part of what inspired Einstein to describe the photon. But before then, the Rayleigh-Jeans law was still very useful in describing the radiation we see coming out of black bodies.

Thermodynamics creates the radiation, so you can use thermodynamics to describe the radiation, but you can't use something that purely fits the radiation to describe thermodynamics.

Similarly in your case, forces cause changes in motion, so you can use forces to describe that changing motion, but you can't use something that purely fits motion to describe forces. That doesn't mean what you have is useless, it just means it has limits. 

12 hours ago, DoomslugTD said:

Finally, the last thing is just logic. I have iterated this in my last post, but it just doesnt make sense for d(t) to have a maximum of h because the only force in that system is the pushing force, which should only be able to push. This is where the x(t) that I chose runs into problems because not only can the force be negative at certain distances (which would result in pulling, and therefore does not fit with steelpushing in the books) it blows up to infinity at infinit distance which also does not fit the books. 

Exactly this. x(t) can potentially describe a push while under gravitational pull, with a max of h. Anything you derive using your x(t) will have that same limitation. Even if you find a general and particular solution for your differential equation, it will still be bound by h because it was bounded by h when you created it. It only blows up at infinity if you don't let h go to infinity as well. 

1 hour ago, DoomslugTD said:

I dont believe d(t) is limited by h in the same way x(t) is either due to the fact that x(t) is in the presence of gravity and d(t) is not.

1 hour ago, DoomslugTD said:

due to the fact that I started with x(t) = htanh(at)^2)

By.choosing to start with this, you limited the function (both x and d) such that their maximum value is h. Basing your differential equations on x forces this to be true.

So either you constrain x and d to be less than h, or you throw it all out and start over.

Or another way to look at it is this: you can't build a model on the assumption of how something works on near-surface gravity then hope that goes away by just subtracting off gravity. You can see that in your force differential equation. When you take the limit as d goes to h you are left with mg, which is how hard they would need to Push to hold themselves at a constant height.

1 hour ago, DoomslugTD said:

I end up with a final force equation Fₚ(d(t)) and I am wanting that to approach 0 as d(t) becomes larger because at large distances it should logically follow that the force is ≈ 0. With Fₚ(d(t)) = 2ma²(h - 4d(t) + (3/h)d(t)²) + mg I just noticed that with respect to d(t) Fₚ(d(t)) looks like a second degree polynomial and with that postivie d(t)² term it looks like that equation will blow up to infinity at extremly large distances

OK. I understand now. Don't forget that h is your max height, so d(t) must always by less than or equal to h (minus whatever the correcting factor is that includes gravity). If you let d(t) go to infinity, h also goes to infinity. And when they are both infinity, the force equation becomes mg, which is what we expect. 

On 10/4/2024 at 12:46 PM, kpShadowFox said:

I was wondering if you could post the actual code used to run the sim?

Sure. I couldn't find the one I played with to figure out the parameters, but here is the one that built the animation. The math all happens in the first half. The second half spits out each animation frame as an svg file. I used a separate piece of software to combine those into a gif.

Here's the code:

import numpy as np
import matplotlib.pyplot as plt
import pandas as pd
#Figure out some horseshoe basics, assuming Vin moves with constant horizontal velocity
v0 = 35 #80 to 120 is 35- 55 m/s
h= 10 #Vin's height above ground
#vs = np.arange(10,12,1)
#hs = np.arange(10,36,1)
#time_grid = np.zeros((vs.shape[0],hs.shape[0]))



m = 0.5 #Horseshoe mass kilograms




pull_angle = 35 #Angle where Vin starts pulling
pull_angle = np.radians(pull_angle)
pull_force_scale = 2e4 #similar to coulombs constant
g=9.8


#for i,v0 in enumerate(vs):
#    for j,h in enumerate(hs):
r_vin = np.array([0.0,h])
v_vin = np.array([v0,0.0])
dt = 0.00001
t=0

v=np.array([0.0,0.0])
r = np.array([-h*np.tan(pull_angle),0])
r_vin_arr = [r_vin.copy()]
v_vin_arr = [v_vin.copy()]
v_array = [v.copy()]
r_array = [r.copy()]
a_array = []
pull=True
release_angle = -35
release_slope = np.tan(np.radians(release_angle))
print_break=False
while r[1]>=0:
    t+=dt
    dr = r-r_vin
    R = np.linalg.norm(dr)
    if dr[0]>0 and r[1]<(release_slope*dr[0]+r_vin[1]):
        if not print_break:
            print(dr,release_slope, release_slope*dr[0]+r_vin[1])
            print_break=True
        a=np.array([0,-g])
    else:
        a_mag = pull_force_scale/R**2
        a_mag = np.min([200*g,a_mag]) #Don't let Vin's acceleration break 2 g's
        a= np.array([a_mag*(-dr[0]/R),a_mag*(-dr[1]/R)-g])
    a_vin=np.array([0,0])
    v+=a*dt
    r+=v*dt
    v_vin+= a_vin*dt
    r_vin+=v_vin*dt

    #plt.plot(r[0],r[1],'r.')
    v_array.append(v.copy())
    
    
    
    r_array.append(r.copy())
    a_array.append(a.copy())
    r_vin_arr.append(r_vin.copy())
    v_vin_arr.append(v_vin.copy())
    if (t*2)%1==0:
        print('At time {}s'.format(t))
    if t>3:
        break

#time_grid[i,j]=t
print('t: {:.4f}; ({},{})'.format(t,v0,h))
r_array = np.array(r_array)
plt.plot(r_array[::10,0],r_array[::10,1])




t_array = np.arange(0,t+dt,dt)
r_array = np.array(r_array)
v_array = np.array(v_array)
a_array = np.array(a_array)
r_vin_arr = np.array(r_vin_arr)
v_vin_arr = np.array(v_vin_arr)
dx = r_array[:,0]-r_vin_arr[:,0]
x_remain = dx[-1]-dx[0]
t_remain = x_remain/v0
t_rem_arr = np.arange(t,t+t_remain+dt,dt)
r_remain = np.array([0*t_rem_arr,0*t_rem_arr]).T
r_remain[:,0] = r[0]
t_array = np.arange(0,t+t_remain+dt,dt)
r_array = np.concatenate([np.array(r_array),r_remain])
r_arr_vin = np.array([0*t_array,0*t_array]).T
r_arr_vin[:,1]=h
r_arr_vin[:,0]=v0*t_array




a_mag = np.sqrt(a_array[:,0]**2+a_array[:,1]**2)
#build svgs for animation

#Put all data together in dataframe for convenience
data=pd.DataFrame(np.concatenate([t_array.reshape(t_array.shape[0],1),r_array[:-1,:],r_arr_vin],axis=1),columns=['t','hs_x','hs_y','vin_x','vin_y'])
data['dx'] = data['hs_x']-data['vin_x']
#load in svg front matter
front_matter = '''<?xml version="1.0" encoding="UTF-8"?>
<!-- Created with Inkscape (http://www.inkscape.org/) -->
<svg width="25mm" height="16mm" version="1.0" xml:space="preserve" xmlns="http://www.w3.org/2000/svg" xmlns:cc="http://creativecommons.org/ns#" xmlns:dc="http://purl.org/dc/elements/1.1/" xmlns:rdf="http://www.w3.org/1999/02/22-rdf-syntax-ns#">
	<metadata>
		<rdf:RDF>
			<cc:Work rdf:about="">
				<dc:format>image/svg+xml</dc:format>
				<dc:type rdf:resource="http://purl.org/dc/dcmitype/StillImage"/>
			</cc:Work>
		</rdf:RDF>
	</metadata>

    
	<rect width="94.488" height="60.472" fill="#fff"/><g transform="translate(575.37 192.65)">

<g stroke-width=".24567">
{}
</g>


		<circle id="CenterPoint" cx="-509.23" cy="-173.75" r=".35838" display="none" fill="#fb0000" stroke="#000" stroke-width=".22812"/>
		<rect x="-575.37" y="-135.95" width="94.488" height="3.842" fill="#006d1e"/>
		<g fill="#696969">



'''
end_matter = '''
		</g>
	</g>
{}
	<g>
		<text x="6.1777692" y="3.0096824" fill="#000000" font-size="2.6667px" stroke="#000000" stroke-width=".24567" xml:space="preserve">
			<tspan x="6.1777692" y="3.0096824" font-size="2.6667px">Pull</tspan>
		</text>
		<text x="6.1777692" y="5.9401627" fill="#000000" font-size="2.6667px" stroke="#000000" stroke-width=".24567" xml:space="preserve">
			<tspan x="6.1777692" y="5.9401627" font-size="2.6667px">Push</tspan>
		</text>
		<path d="m5.1481 1.9009-3.0097 1e-6" stroke="#b93f04" stroke-width=".62362"/>
		<path d="m5.1481 4.9105-3.0097 1e-6" fill="#0b5794" stroke="#0b5794" stroke-width=".62362"/>
	</g>
</svg>
'''
vin_string = """		<g transform="translate({} 192.65)">
		<path d="m-508.77-175.77c-0.072-7.8e-4 -0.1274 0.0142-0.25193 0.0587-0.1885 0.0673-0.19408 0.0679-0.45349 0.0468-0.1788-0.0145-0.27465-0.0136-0.30028 3e-3 -0.045 0.029-0.0624 0.1335-0.0255 0.15303 0.0168 9e-3 0.0275-2e-3 0.0301-0.0309 4e-3 -0.0404 0.0139-0.0432 0.10725-0.0288 0.0568 9e-3 0.1576 0.0294 0.22401 0.0459l0.12075 0.0299-0.0643 0.0248c-0.038 0.0146-0.14983 0.02-0.27327 0.0132-0.22569-0.0125-0.37029 0.0175-0.45068 0.0973-0.0556-0.0301-0.19518-0.0868-0.22063-0.0867-5e-3 1e-5 -0.0232-5e-3 -0.0406-0.0115-0.0444-0.0162-0.0733-0.0229-0.13535-0.0312-0.0542-7e-3 -0.1013-0.0197-0.16458-0.0431-0.042-0.0155-0.0818-0.0269-0.0944-0.0269-7e-3 0-0.0103 4e-3 -0.0123 0.0136l-3e-3 0.0136-0.0141-0.0104c-8e-3 -6e-3 -0.0302-0.0204-0.0498-0.0325-0.0197-0.0121-0.0481-0.0314-0.0632-0.0428-0.0151-0.0115-0.0368-0.0263-0.0484-0.033-0.0307-0.0178-0.10457-0.0429-0.15489-0.0528-0.0149-3e-3 -0.0514-0.0121-0.0812-0.0204-0.0673-0.0187-0.076-0.02-0.15109-0.0225-0.0609-2e-3 -0.17304 7e-3 -0.20973 0.0163-0.0476 0.0124-0.0832 0.024-0.10421 0.0338-0.0127 6e-3 -0.0349 0.0148-0.0494 0.0198s-0.0433 0.0192-0.0639 0.0316-0.0385 0.0225-0.0398 0.0225c-1e-3 0-0.0103 5e-3 -0.0201 0.0113-0.0585 0.0369-0.15298 0.0381-0.21789 3e-3 -0.0119-6e-3 -0.0373-0.0204-0.0564-0.031s-0.0418-0.0227-0.0505-0.0271c-9e-3 -4e-3 -0.0193-0.011-0.0235-0.0148-0.0109-0.01-0.0864-0.0469-0.11767-0.0578-0.017-6e-3 -0.0369-0.01-0.0569-0.0118-0.0199-2e-3 -0.0399-1e-3 -0.0571 1e-3 -0.0631 0.0104-0.15828 0.0588-0.20246 0.10303-0.0262 0.0262-0.055 0.0657-0.0725 0.0992-0.0249 0.0478-0.026 0.0483-0.0489 0.0227-0.0167-0.0187-0.021-0.0213-0.0315-0.0187-0.0295 7e-3 -0.0396 0.0487-0.0161 0.0661 0.0189 0.014 0.0362 0.0118 0.0551-7e-3 0.0186-0.0186 0.0213-0.0197 0.0213-9e-3 0 4e-3 -7e-3 0.0136-0.0159 0.0206-9e-3 7e-3 -0.0148 0.0144-0.0136 0.0165 1e-3 2e-3 -6.8e-4 0.01-4e-3 0.0166-6e-3 0.012-6e-3 0.0134 3e-3 0.0201 0.0103 8e-3 0.0468 0.0175 0.0655 0.0175 0.0125 0 0.0424-0.0126 0.0477-0.0201 2e-3 -3e-3 2e-3 -7e-3 1e-3 -0.0102-1e-3 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4.5e-4 6e-3 9e-3 0.01 0.0192s9e-3 0.019 0.0111 0.0197c9e-3 3e-3 0.0375-0.0422 0.0339-0.0545-8.1e-4 -3e-3 -1e-3 -0.0138-9.8e-4 -0.0245 3.9e-4 -0.0151 2e-3 -0.0209 6e-3 -0.0256l3e-3 -3e-3zm0.29869 0.2095c1e-3 5.2e-4 1e-3 2e-3 7.3e-4 3e-3 -8.3e-4 2e-3 -4e-3 4e-3 -7e-3 5e-3 -3e-3 8e-5 -6e-3 4e-3 -8e-3 8e-3s-5e-3 0.0105-9e-3 0.0134c-6e-3 5e-3 -6e-3 5e-3 -4e-3 -4e-3 1e-3 -5e-3 5e-3 -0.0127 9e-3 -0.0169 6e-3 -7e-3 0.0146-0.0105 0.0176-9e-3zm-0.34306 0.10352c3e-3 -1e-3 0.0112 0.0128 0.01 0.0169-2e-3 5e-3 -6e-3 1e-3 -9e-3 -7e-3 -2e-3 -5e-3 -2e-3 -9e-3 -6.1e-4 -9e-3zm0.0219 0.0868c4e-3 1e-3 6e-3 0.0517 3e-3 0.0616-2e-3 5e-3 -4e-3 7e-3 -7e-3 6e-3 -2e-3 -9.1e-4 -3e-3 -0.012-2e-3 -0.0293h-1e-5c1e-3 -0.0153 2e-3 -0.0306 3e-3 -0.0338 1.9e-4 -3e-3 2e-3 -5e-3 3e-3 -5e-3z" stroke-width=".0045108"/>
        </g>
        """
shoe_path = '<path d="m-509.23 -173.75 L{} {}" stroke="#{}"/>\n'


print('Writing images')

fps = 300
N_blur = 20
N_frames = int(t_array.max()*fps)
frame_skip =  int(1/fps/dt)
frame_skip_blur = int(frame_skip/N_blur)
#N_blur = int(frame_skip/frame_skip_blur)
blur_opacity = np.linspace(1.0,0.5,N_blur)
shoe_space = int(t_array.shape[0]/5)
for frame in range(N_frames):
     #track_vin = ''
     track_vin = vin_string.format(575.37)
     horseshoe_string=''
     shoe_paths=''
     with open('Figures/frame_{}.svg'.format(frame+1),'w') as f:
        
        x_vin = 0#v0*(frame*frame_skip)*dt*3.77952
        #track_vin+=vin_string.format(x_vin+504.37)
        
        for blur_step in range(N_blur):
            x_vin = v0*(frame*frame_skip-frame_skip_blur*blur_step)*dt*3.77952
            #track_vin+=vin_string.format(x_vin+575.37)
            #track_vin+=vin_string.format(x_vin+504.37)
            arr_idx = [((i*shoe_space+frame*frame_skip)-frame_skip_blur*blur_step) % t_array.shape[0] for i in range(5)]
            
        
            #Get temporary slice of data frame
            tmp=(data.iloc[arr_idx,:][['dx','hs_y','t']]*3.77952).copy()
            tmp['angles'] = (np.degrees(np.arctan2(tmp['hs_y']-h*3.77952,tmp['dx']))+360)%360
            tmp['colors'] = 'fff'
            tmp.loc[(tmp['angles']>=235)&(tmp['angles']<=270),'colors']='0b5794'
            tmp.loc[tmp['angles']<235,'colors']='b93f04'
            tmp.loc[(tmp['angles']>325),'colors'] = 'b93f04'
            tmp['dx']=(tmp['dx']-509.23)
            #tmp['dx']=(tmp['dx']-580.23)+x_vin
            tmp['hs_y']=-tmp['hs_y']-(135.95)
            for idx,row in tmp.iterrows():
                horseshoe_string+='<ellipse cx="{}" cy="{}" rx="1.5108" ry=".6221" fill-opacity="{}"/>\n'.format(row[0],row[1],blur_opacity[blur_step])
                if blur_step==0:
                    shoe_paths+=shoe_path.format(row[0],row[1],row['colors'])
                    #print(kluge)
                
            
        f.write(front_matter.format(shoe_paths))
        f.write(horseshoe_string)
        f.write(end_matter.format(track_vin))
        

20 hours ago, DoomslugTD said:

I believe you were double posting (making more than one post before someone responds) which I got in trouble for in this exact thread earlier. I was told that if you have more things to add you can edit your most recent post and add the things there, and if you make a new quote it will still alert the person just like if you created a new post. Just thought I would let you know

Thanks for the heads up. I'll do that in the future.

20 hours ago, DoomslugTD said:

I take it you are a proffesor?

Yes. I've been a professor for about 10 years now. As you go through your program, never forget that the only big difference between you and me is 20 years of experience. We all remember being that brand-new student who was trying to figure out what was going on.

20 hours ago, DoomslugTD said:

I am pretty sure x"(t) = 2ha² · sech(at)² · (sech(at)² - 2 tanh(at)²) I even just plugged the stuff into a derivative calculator and it confirmed it. Here is a link where you can look at it visually to intuitivily confirm its true https://www.desmos.com/calculator/lzrnge3drx. I then used the trig identity sech(x)² = 1 - tanh(x)² to get everything in terms of x(t)

You're right. I made a mistake (I hadn't squared tanh when I copied it to paper) and put an edit in the top, but left my original post in case you had already seen it. That said, it looks like in doing so I caused more confusion than anything. Next time I'll just fix the whole comment.

With that new x(t) notation, you have everything correct. 

20 hours ago, DoomslugTD said:

I am interested in solving for d(t) which would be the distance function without gravity which would require solving the system d"(t) = 2a²(h - 4d(t) + (3/h)d(t)²) + g.

Ans here's where the teacher will come out a little bit. What you are looking for is called a "particular solution". Since you already know that the solution to the homogeneous (the equals zero case) is x(t) = h · Tanh(at)², you assume that d(t) = h · Tanh(at)²+<some function of t>, and just take guesses at that something until you get somethings that works. Maybe start with d(t) = h · Tanh(at)²+At² (since you are left with just a constant in the second derivative). Then, you plug in initial conditions (like d(0)=0) and see if you can solve for what "A" is.

That said, some equations are unsolvable, so you'd have to use numerical methods instead, but it wouldn't give you an equation.

20 hours ago, DoomslugTD said:

 it was focused on lim d->inf because we also know that the force function should tend to 0 as distance goes to infinity and we have the function explcility in terms of d(t)

Some things to add context to my answer:

• This might be me misunderstanding your notation. When I read this statement, it looks like you are trying to take the limit of your d(t) function as the function itself goes to infinity. I'm going to answer assuming this limit, but if that is now what you meant my answer might not apply.
• Since we don't know d(t), I'll use x(t), but once we have d(t) that satisfies the equation, the same limits should hold true

If you want to see the behavior of an equation as you approach infinity, you can't just take the limit of the equation. Instead, you need to take the limit of individual parts of the equation and see how they behave. 

If you want to see how your equation behaves at really long distances, you need to be really careful about how you apply boundary conditions. If you simply take the limit of x''(t) as h goes to infinity, x''(t) does become infinite. However, that limit assumes I am starting on a mass and ending up and infinite distance away, so my acceleration should be infinite. Your constraint needs to be x''(t) is zero as both h and t go to infinity (I'm now really far from the planet and have been pushing a long time). In that case, x(infinity) = h, so:

 x''(t) = 2a²(h - 4x(t) + (3/h)x(t)²) =  2a²(h - 4h + (3/h)h²) =  2a²(h - 4h + 3h) = 0

Once you have a full solution to d(t), you should be able to let h go to infinity to get zero because you've eliminated gravity from the situation.

In short, two things to remember:

1. Don't panic about a limit not working when you don't have the full solution. 
2. Don't apply physical limits to an equation without considering the constraints you used when making the equation.

2 hours ago, Chirolina said:

I wonder if the leading shoe would require a quick transition from Pull (over her head) to Push (to slam it into the ground fast enough, instead of it just falling at normal gravity speed). 

It's going fast enough that you don't have to push it.  The pulling near her head turns it more than speeding it up/slowing it down.  So,  it just keeps all the momentum it had coming from behind,  and the pull just turns that towards the ground. 

7 hours ago, therunner said:

As such, I think estimating her travel speed to be in the 45-60 mph range fits the data in the book.

I dropped in an edit. 45 is too slow to manage 5 horseshoes well (according to the simulation), but 60 works just fine. That speed seems to fit all of the data.

Thanks for poking holes. 

On 9/29/2024 at 6:20 AM, therunner said:

WoA chapter 52. Vin comments that spikeway reduced ~1 hour carriage ride to ~10 minute trip.

Found it. The exact quote is:

Quote

What she really needed was a spikeway—a path marked by spikes driven in the ground that an Allomancer could push against, throwing themselves through the air again and again. On such an organized pathway, she'd once traveled from Luthadel to Fellise—an hour's carriage ride—in under ten minutes.

Another quote that gives us an estimate of speed is this one:

Quote

The wind became a roar as she Pushed herself faster and faster, steering her pathway to the south. 

There are several studies on wind loudness for cyclists. (Like this one, but it may be behind a paywall. University Faculty perks) A "roar" is hard to quantify, but concerts (including symphonies) and nightclubs come in the 110 dB range, which means she was probably going at least 45 mph (as a cyclist who has hit 60 mph going downhill, I'd lean more towards that being the minimum).  A lot of the language hints that she is going faster than ever before (like the roaring wind), so she's probably going faster than she normally wood on a spikeway (though not necessarily faster than you could go on a spikeway). 

That said, going that speed with the horseshoes probably doesn't line up travel times with how far away she was when she started the trip. 

On 9/29/2024 at 6:20 AM, therunner said:

WoA chapter 52. Vin comments that spikeway reduced ~1 hour carriage ride to ~10 minute trip.

Thanks. Now I know where to poke next. 

On 9/29/2024 at 6:20 AM, therunner said:

BTW what did you use to simulate?

A Python script I wrote. To be a little more specific, I used the Euler-Cromer method with a custom force function, as describe in the model details. If you want more specifics, I can try finding it again.

On 9/27/2024 at 12:44 AM, therunner said:

Mistborn travel ~6x as fast as carriage ride, and carriages moved between 5-10 mph

I'm curious where you get that from. It's a number I have seen before (the more data I get, the better the model). 

80 mph is the max theoretical speed and the one that gives the smoothest ride. She could go slower, but below 60 it gets hard to cycle all 5 horseshoes. (I did a lot of pen and paper work before building the sinulation).

10 hours ago, therunner said:

Well if it behaves like shear thickening fluid, then the impact force will be greater, due to it thickening.

Greater than hitting water, less than hitting dirt. The thing that you'd have to worry about is hitting solid rock. Though,  with all the ashfall breaking down into soil, most rocks would be pretty buried (unless you hit a spot where they had been dug up).

5 hours ago, therunner said:

pulls them from the ground, and lets them fly over her until they fall (chapter 52 for reference). Though maybe in HoA she enhanced the method a bit? I don't have that on hand.

She starts with a pull and then letting it fly over her head, but as she picks up speed it just says

Quote

Vin Pulled, then Pushed, then Pulled, then Pushed, moving with continual singlemindedness

The difference between letting go behind and letting go ahead is around a tenth of a second. That's the type of thing we don't really think about but will do because we feel it making us go faster. That,  and letting it go from behind resulted in paths that either hit her with the horseshoe or the horseshoe went to far ahead (once she got to speed) and she'd hit the ground (like she's does when she's trying to start).

6 hours ago, therunner said:

100 m/s for the horseshoe sounds like way too much speed, I don't think they would survive even dozen of such impacts, and they would be driven way too deep in muddy ground.

100 m/s into mud wouldn't do much for horseshoes. The metal will be fine.

Also, mud is a shear thickening fluid (like ooblek), so a short, hard impact really thickens it up. If it was very muddy, you would probably see them stick more from the sustained push that holds up Vin's weight than you would from a high speed impact. The impact would leave a decent mark, but the odds of it sticking in a significant way are low.

43 minutes ago, Argent said:

The idea that Vin is slamming chunks of metal into the ground with bullet speeds

Yeah, I like to imagine her coming into the valley. At first, you'd just see a cloud (think of all the ash that she would kick up) blazing towards you at 80 mph. Then you'd start to hear the thumping. Then finally, just in front of the billowing cloud, you would see Vin with her mistcloak streaming behind, centered in a wheel of metallic death.

I wish there were more scenes that were detailed enough that I could do the math and figure them out. However, then I'd be more likely to find inconsistencies, so maybe it's a good thing.

1 hour ago, alder24 said:

This isn't a good place to argue about this

Good point. If you'd like to discuss it further, lets take it somewhere else. Otherwise, I'll just add this last point.

1 hour ago, alder24 said:

but Kaladin is above the cloud layer, at night, with strong wind blowing all around him - he lacks a good reference point and nearly motionless would be perceived by him as truly motionless. I highly doubt it would take hours like you suggesting. 

You might be thinking of a different scene. This scene is from when he is first practicing with his powers. He's still below cloud level (the paragraph before talks about reaching toward the clouds but being afraid of running out of stormlight).

I'm teaching a class on this later today, so I shoved some numbers into the model I built for my students. If he had slowed to a slow jog/brisk walking pace before canceling the lashings, it would take a full 20 minutes before he would be slow enough to perceive it as motionless (abt 0.1 m/s. Humans are really good at sensing motion if we can see anything that might be a fixed point, including clouds, which would be motionless relative to the wind. 0.05 m/s would probably be more reasonable and would double all the stopping times).  Canceling at a fast jog would bump the time up to 35 minutes, then anything faster than that bumps the time up to 45 minutes.

So, it looks like hours was an exaggeration, but if he's worried enough about stormlight that he makes the flight back to camp take the length of a short conversation, I doubt he would wait 20-40 minutes to come eventually to a stop.

28 minutes ago, alder24 said:

I see no error here. This is the one example where you can't ignore air drag as that's the only force acting on Kaladin at that moment and it's enough to eventually stop him. It's not that weak to have no effect on Kaladin, it depends on the speed squared - the faster you move, the stronger drag is.

Since it depends on speed squared, it's also true that the slower you go the weaker it is. 

It would take hours for air drag to slow him down to what a human would perceive as stopped. 

On 8/26/2024 at 10:52 PM, DoomslugTD said:

Now this raises two big questions, what is the gravitational constant of scadrial and the mass of the person doing in the equation for the force provided by a steelpush

It has the same g as earth. In arcanum unbounded, it lists Scadrial's gravity as one cosmere standard, which is set to Yolen's gravity, which is equal to earth's (though I couldn't find the WOB claiming that Yolen=Earth gravity, I know I've read it before on previous research).

On 8/26/2024 at 10:52 PM, DoomslugTD said:

though people in the past have raised a valid point that it could be because people with more mass stay closer to the object they are pushing for longer, allowing for greater total force to be exerted, though that would mean people with more mass accelerate slower when steelpushing.

This is another fact that supports power as a limit. The slower velocity at the beginning lets you increase the force and lets you do more work (bigger force for the same displacement). If we were only force limited, you'd have the same max force and same displacement, so the same amount of work would be done. You'd have more momentum, but the same amount of energy. With a power limit, you'd end up with both more energy and momentum.

On 8/26/2024 at 10:52 PM, DoomslugTD said:

If anyone knows of any way to solve the final equation of motion I presented above I would love to hear it.

Also, You've already solved it. x(t) = h · Tanh(at)²

And I just noticed the part where you say that you haven't attended University yet. That's an impressive piece of calculus that I wouldn't expect one of my physics majors to be able to do until they were in their third or fourth year of university studies.

This isn't a typo, but it is a physics error that has bugged me for years and would be easy to fix.

One page 612:

Quote

Curious, he Lashed himself downward to slow further, then dismissed all of his Lashings except one up and one down. He eventually came to a stop hanging in midair.

If he dismissed his lashing except for one up and one down, there would be nothing to slow him to a stop (air drag isn't that strong). It's newton's first law: it doesn't take a force to move, it takes a force to change how you are moving.

The fix would be for him to stop first, cancel the lashings, then hang in the air.

On 8/28/2024 at 4:36 AM, Treamayne said:

So, what if some factors (Flaring, Duralumin) affect dp/dt - but other factors (e.g. allomantic stength through Lerasium or hemalurgy) only apply to dp2/dt.

If the forces can push more on one object than the other then you can't say momentum is conserved and that equation isn't valid.

My guess with the push differentials is that connection has a mass-like effect. The stronger your connection to preservation, the harder you are to accelerate.

4 hours ago, DoomslugTD said:

This is a very good point, and something I have though about briefly, but is also something that almost entirely makes this thought experiment useless and leaves us where we started, with no idea what equation governs the forces in steelpushing other than guesses based on real life phenomina.

For the sake of full disclosure, I'm in this camp. There just isn't enough data to come up with a solid model. 

But,  it's always very exciting to be proven wrong, so please do.

On 8/26/2024 at 10:52 PM, DoomslugTD said:

x(t) = h · Tanh(ax)²

Edit: I'm an idiot. I just realized that i missed the squared on tanh. I still want to know if it should be x(t) = h · Tanh(at)². If that's the case,  then I can second the math. What you have is true.

You did make one mistake in your edits. The limit of x(t) as t goes to infinity is h, not infinity. And if you plug that into your force equation, you get that as t goes to infinity, p=g. If you take the limit of p as h goes to infinity (i.e. final height is infinite and time is infinite), you still get that p=g. So your equation handles the limits just fine.

Original Comment:

I interpreted it the first time that you were saying the distance dependence went like tanh, not time. I'm guessing that instead, the equation is supposed to be x(t) = h · Tanh(at)²? (if it is x, you have an unsolvable transcendental equation)

If so, x'(t) =  ah · Sech(at)² and x''(t) = 2ah · tanh(ax) · sech(ax)²

Somehow you ended up ahead by a derivative. Also, when you simplified x''(t), you plugged in both x(t) and x'(t) as x(t). 

What you should get is x''(t) = 2/h x(t) x'(t), which you can't just separate out a g from and get an effective force. 

Making x(t) proportional to tanh means you will get particular solutions for F that will follow hyperbolic trig functions, not a polynomial.

Tanh also violates the condition x'(0)=0, so you'll have to try a different function. 

I think you have a good start with the rules (so keep going), but your math still needs work.

On 8/26/2024 at 10:52 PM, DoomslugTD said:

because there is no indication that the strength of a steel push depends on time or any derivative of f(x) such as velocity or acceleration, as such I have constrained it to be only in terms of the distance from the metal and any constants. 

There is a lot of evidence that suggests the maximum force is also limited by the power that an allomancer can deliver. (See this discussion for that and other rules that I've seen)

Another point you haven't considered is the cognitive aspect of pushing. We don't know what part of the rules are there because they are part of the limits of pushing, or if it is just how the allomancer expects it to be (example: Wax seeing the bullet as multiple pieces, rather than one object because he started thinking of it as more than one object).

I think we have three hard limits:

1. A Coulomb-like force that depends on the amount of metal and the connectedness/"allomantic mass" of the allomancer. For most objects, the distance dependence would be like 1/r^2 because of geometry (everything is a point mass when you are far away), but you could get other ones with different shapes of metal.
2. A power limit (Fv) that is tied to how quickly the allomancer can tap investiture
3. A belief limit - my brain expects this to happen, so my push is matched accordingly (We do this all the time with our muscles - it's a safety feature so we don't tear them). This is the only way I can think of that explains Vin stopping (instead of bobbing up and down) at her max push while hovering over a coin.

The other important piece to consider for any physics model, is that you should have a "why" for a fit. 1/r^2 pops up so much because of how things expand in 3d space.  What physical property implies a tanh?

On 8/23/2024 at 3:27 PM, AffordableInvestiture said:

Would radiants using the surge of gravitation to lash themselves multiple times cause them to experience g-forces? I suppose stormlight would help them withstand the effects.

@alder24 did an excellent job explaining what we see in the books. I just noticed that he didn't address why we wouldn't expect them to feel g-force (or when physics predicts that they would).

Short answer: You can't feel gravitational forces directly. That's why astronauts are weightless in space even though gravity is holding them in orbit.

Longer answer: What do you mean we can't feel gravity directly? I feel it every time that I stand up.

There are three big pieces to this:

1. Every single piece of the earth pulls on every single tiny piece of you.
2. Gravitational forces only change strength on planetary scales (e.g. the Sun's gravitational field at Earth's orbit is much stronger than its field at Jupiter's orbit), so the amount that Earth pulls on a little piece of your toe is equal to the amount that it pulls on a little piece of your head.
3. Since the force is so uniform, our body has no sensors that can pick it up. When you feel your weight while standing you aren't feeling gravity, you feel how much tension you need to keep in each muscle in order to not fall down.

The same thing actually happens with pressure forces as well. Pressure increases as you go deeper into the water because each layer of water has to hold up all the layers of water above it. One way to test this it to take a water bottle, poke a hole in the side, and watch the water stream out. The water at the bottom of the bottle feels both atmospheric pressure (pushing on the top of the water) and the pressure due to the water above it. When you poke this hole, the higher-pressure water is exposed to atmospheric pressure and the water flows from high pressure (inside the water bottle) to low (outside).

Before all the water runs out, drop the bottle. You'll see that the water stops leaking out while it is falling. Since everything is falling together, the water at the bottom doesn't have to hold up the water above it anymore, everything stays at atmospheric pressure and without a pressure difference, the flow stops.

The negative impacts of g-forces (at least the blackout ones) on humans are primarily due to the pressure difference between our feet and our heads. In normal conditions, our hearts pump hard enough to overcome that difference and force our blood to go through our body. 

Now imagine we're in a cockpit turning up. To do that, the seat pushes hard on us to go up, but it only pushes on our surface. So, the part of me next to the seat has to push up the part just above it, and that has to push up on the part above it, etc. creating a big pressure difference. When that pressure difference gets big enough, our heart can't pump hard enough to raise the blood from our feet to our head, our brain loses oxygen, and we pass out. We can actually take much stronger g-forces going forward than we can going up because the distance from the back of our head to the front is much smaller than the distance from our feet to our head, which makes the pressure difference smaller for the same g-force because the stuff at the back of our head doesn't have to hold up as much weight. (People have maintained consciousness pulling 45 g's in the "eyeballs in" direction).

So back to lashings. If lashings simply redirected gravity, we'd be free falling and wouldn't get a pressure difference across our body, because each part is being pulled on equally, therefore, no g-forces. 

But, you would start to feel g-forces as wind resistance picks up. Falling at terminal velocity feels similar to lying on the floor. You can actually feel the weight of your body. So, if you did a double lashing, at first you wouldn't feel anything, but as you sped up, eventually you'd reach a point where you felt twice as heavy just because of the wind pushing you. Changing the lashings would let you change directions very quickly without feeling g-forces, but you'd hit limits on actual windspeed and could black out. Over long distances, you'd max out at around 1.5 lashings (face first), because that's the max a person can feel for an extended time (more than a couple of seconds). We put astronauts in a position that maximizes the amount of g's they can handle for a long time, and we still limit our rockets to 2 g's to keep our astronauts safe (which is why we can't use a rocket designed for satellites to carry people.)

6 hours ago, Heilven said:

Here I was happy about coming up with an explanation for why pushing a coin doesn't send you flying until the coin hits the ground.

You might want to take a look at this thread where we discuss it in depth.

https://www.17thshard.com/forums/topic/158274-steelpush-physics/

That said, it would be fun to see your model.

On 8/19/2024 at 3:34 PM, Dofurion said:

could you summarize what we understand by Adhesion and Abrasion in real life?

It's tough to draw too much inference from real-world physics on these ones.

Abrasion is essentially power over friction. Specifically, being able to get rid of friction. To take it to the absolute most basic level, when two objects rub against each other, there is a force (friction) parallel to that surface that opposes the direction of the slip. The force is due to two main components: 1: rough surfaces will have little parts that interlock and bounce and jostle against each other; 2: some surfaces will actually form weak electromagnetic bonds when brought into contact. (Note: since electromagnetic bonds are what make things solid, the rough parts bouncing and jostling against each other are also the result of electromagnetic forces).

Full lashings (the physical part of adhesion), make things stick. It appears to act similar to static electricity, but much stronger. So, once again, it would be the result of manipulating electromagnetic forces.

At first glance, it might look like adhesion and abrasion are opposites, with one increasing the force and the other decreasing, but that breaks down when you dig deeper. For example, adhesion is an attractive force, perpendicular to a surface (it pulls things together). Its opposite would need to be a repulsive force, also perpendicular to the surface. 

The opposite of abrasion would be a force that greatly increases friction, preventing slipping. But such a force wouldn't make it harder to pick up an object, just harder to slide it.

Both work by manipulating electromagnetic forces, but so do tension and cohesion.

Therefore, real-world physics doesn't give us any hints on how the two compare. We can only rely on what we observe in the books, and that is summarized pretty well in this article on the coppermind.