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In episode 1 of Railgun, Misaka's railgun up-ends a good chunk of what appears to be an olympic-sized swimming pool. Later in the same episide, it is revealed that her railgun travels at a velocity of 1030 m/s.

enter image description here

However, the numbers don't add up.

Suppose Misaka uses 10 gram coins. At 1030 m/s, that coin has this much energy:

Energy = 1/2 m v^2
       = 1/2 (0.01 kg) (1030 m/s)^2 = 5304.5 kg (m/s)^2
       = 5304.5 Joules

An olympic-sized swimming pool has 2,500,000 kg of water. Based on the image above, that cone probably travels a good 100 meters into the air.

So let's say for benefit of the doubt, that only 10% of the water is lifted 100 meters into the air.

Energy = m g h
       = (0.01 * 2500000 kg) (9.8 m/s^2) (100 m) = 2.45 * 10^8 kg (m/s)^2
       = 2.45 * 10^8 Joules

The coin needs 2.45 * 10^8 Joules of energy to uplift the swimming pool as shown above. But her coin can only provide 5304.5 Joules. There is a difference of about 4 orders of magnitude.


Okay... Before anyone dismisses this as just another case of Anime physics, let's look at what else could produce so much energy:

  1. The spin of the coin is not captured in the net velocity of 1030 m/s. But given the moment of inertia of a little coin, it would need a tremendous (relativistic?) amount of spin to carry 10^8 Joules.
  2. The charge of the coin is unspecified in the Anime. Perhaps Misaka somehow polarizes the charge in the coin and somehow makes it release upon contact with the target.
  3. The mass-energy (E = mc^2) of the coin is 9 * 10^14 Joules. Is this what she did?

So the question is: Is there any official explanation for where the coin gets so much energy? Or are we left to dismiss this as another case of Anime physics?

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4  
I'm surprised by the lack of comments saying that I'm absolutely insane to try to explain Anime Physics. :P –  Mysticial Jan 31 '13 at 20:18
2  
You are, but we'll live. –  Madara Uchiha Feb 1 '13 at 8:34
    
And if anyone is interested in the chat discussion about the physics here: chat.stackexchange.com/transcript/message/7951592#7951592 –  Mysticial Feb 3 '13 at 21:48
    
How have you calculated the height of the water cone? Souldn't you try to figure out the volume of the cone instead of introducing unknown variables like the size of the pool, the 10% of the lifted water? –  chirale Apr 29 '13 at 22:40
1  
@chirale Well... it's Anime. :) If anything, it's a probably more of a hollowish cone where most of the water is on outside. Not sure what type of explosion could possibly produce such a distribution of water though. –  Mysticial Apr 29 '13 at 22:51

5 Answers 5

While subjective, it was my strong impression from the start that projectile component is indirect (and possibly borderline irrelevant) to her ability.

  1. she can produce a lot of energy
  2. energy goes where projectile goes
  3. however it is not established that projectile carries the energy

Projectile might be just be beacon or focus type of component, possibly purely psychological one (with technique being opposite of just randomly shooting out bursts of uncontrolled electricity).

I remember her shooting large projectile in latter episode, however might still apply.

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+1, this is a great explanation IMO. There might've been a scene where people find a coin of hers, but I recall that many firings of her railgun had the coin become a beam, I assume melting it in the process (if anything remains of it at all). –  Fadeway Feb 3 '13 at 6:45
    
A way to verify this would be to see if she ever uses anything non-metal as the projectile. If her Railgun is actually a Railgun in the scientific sense, it would have to act on something that can be accelerated with electromagnetism, so a metal of some kind. If she can use non-metallic items, then its not actually a railgun and this explanation would be very convincing. –  Lawton Aug 7 '13 at 2:39

I argue that it is a case of Anime Physics by refuting the alternative possibilties you suggested.

  1. You're right, carrying such an amount of energy as moment of inertia is impossible. Even Misaka's superpowers are a world away from relativistic speeds.
  2. The energy cannot be stored as charge in the coin, as it would keep dissipating as "lightning".
  3. The energy can't have come from mass energy; besides the lethal radiation that would result, to release mass energy without using antimatter would be to release nuclear energy. Nuclear energy can only be released under extreme pressure (nuclear reactions in bombs are initiated by compressing uranium with the blast of a smaller bomb). If someone could find an example of Misaka finding her coin after she shot it, that clearly refutes the mass energy argument.

Finally, the Myhtbusters proved in this video that a coin travelling at 3 times the speed of sound (around the speed of a bullet) merely dents concrete.

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According to the anime and manga, Mikoto has a "muzzle velocity" of 1030 m/s: Pew Pew

In contrast, the US Navy's railgun has a muzzle velocity of 2520 m/s (~5600 mph or ~7.5x the speed of sound), with a muzzle energy of 10.64 megajoules (10.64 million joules). Which is probably comparable to the amount of energy in an average-sized car moving at 250 mph.

Comparatively, the an AK-47 rifle has a muzzle velocity of 715m/s (~1600 mph or ~2x the speed of sound), with a muzzle energy of ~2010 joules (assuming 7.62x39mm cartridges are used, but may vary depending on the type of ammunition).

While not very impressive stat-wise as far railguns go, since actual railguns are capable of achieving speeds of possibly more than ~5000 m/s. Take note that Mikoto is a middle-schooler that can fire eight shots of these per minute, which is on par with "typical" railguns.

Assuming the coin comparable to the size and weight of a U.S. quarter, one can assume that the coin she uses has a weight of about five grams. Using this formula we can get he muzzle energy:

Energy = 0.5*(mass)(velocity)2 = 0.5*(0.005kg)(1030m/s)2 = ~2652.25 joules

So Mikoto outputs a bit more damage than an semi-automatic assault rifle.

But that's not what we're seeing now is it?

I wonder what the insurance premiums are?

Not quite. But what could cause such a difference in output?

According to Page 1, Chapter 4 of the Railgun manga, she manipulates electromagnetism to achieve her results. This seems plausible since if the magnetic field were sufficiently focused, she could theoretically accelerate only the coin and/or nearby objects.

Now lets take a look at the pool test again, from the pool cleaning episode (S1, ep. 2) we have an approximate estimate of the dimensions of the pool:

Kuroko measurement!

Lets measure things in terms of Kuroko's height (~152cm)!

Wide shot!

Seeing as the pool doesn't taper or either end, lets assume that the pool is about 14 Kurokos long or ~21.28 m (yes it does seem kinda small), and about 11 Kurokos wide, or ~16.72 m, by the lines on the pool floor and a bit less than ~0.9 Kuroko, or lets say 1.36m deep.

As for the displaced water, we could attempt to integrate the volume of the plume of water, assuming it's half and half air, find it's weight, etc. But we'll be lazy here and assume that 1/1000th of the pool water volume was dispersed into the air when Mikoto fires her railgun. The pool would have a volume:

Volume = (1.36 m)*(21.28 m)*(16.72 m) = ~486.73 m3 of water

Since the "center of mass" the plume of water seems to be comparable be near the top of the gymnasium-type building, going be an eye estimate of the door compared to the building lets assume that it's 10 m tall. In order to calculate the energy needed to achieve something like this we do:

(Energy it takes to lift an object) = (mass of object)*(acceleration due to gravity)*(lift altitude).

In this scenario let us cut a bunch of corners and assume that all the energy of the shot goes into lifting the water (by which ignore the energy spent warming the water up, creating the sounds loud explosion, any dramatic winds effect), then we have

Energy = (1/1000)*(~486.73 m3)*(1000 kg/m3 water)*(9.8 m/s2)*(10m) = ~47699.54 joules

If we plug it backwards into the kinetic energy equation:

√[(~47699.54 J)*2 / (0.005 kg)] = ~4368.04 m/s

So the muzzle velocity of her railgun would be ~4368.04 m/s.

One can only assume that, since we must respect the canon value of 1030 m/s, that perhaps the damage caused by Mikoto ability is due to her manipulation of electromagnetic caused as the coin moves through the air or some other factor... But then again, what do we know about the physics of this world where Science and Magic co-exist?

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When I get the time, I'll need to run through the math to see where that 4 orders of magnitude discrepancy went. –  Mysticial Mar 28 at 0:48
    
Made a miscalculation that affected the depth. Should be fixed now. –  Krazer Mar 28 at 2:09
    
@Krazer by the looks of it, seems like a semi-olympic pool, 25m x 12.5m x 1.36m –  Mindwin Mar 29 at 4:57

Electricity can move the water:

The wiki says that Misaka can generate 1 billion volts.
(even though some sources states 5 billion volts, lets be MODEST <grin>)

If she charges the coin, the rapid transfer of charge from the coin to the water would cause repulsion between the coin and the surrounding water, propelling the water away from the pool. The shockwaves would reflect on the borders and the bottom of the pool, pushing the surface water up.

You can see in the anime that all the blasts have some sideways movement, but the SECOND blast displayed has a very distinctive. sideways movement, hinting that the impact pushes the water a bit to the sides as well, as the electrically charged water moves away from the coin while the coin traverses the pool.

The issue of the charge dissipating is not a problem. She can AIM lightning, so we can assume her electromagnetic powers can also alter the breakdown voltage of the air around the coin (by either increasing the pressure around coin or making vacuum.

We need to look at the capacitance of the coin

enter image description here

The radius of a quarter is 13mm

enter image description here

With a potential of 1 billion volts, the charge of the coin is

enter image description here

Now, We can calculate the electric force between the charged coin and the charged water, and for brevity's sake, lets assume:

  1. half the charge has transferred to the water.
  2. the walls and bottom of the swimming pool are totally isolant and indestructible.
  3. the coin has hit the bottom once half the charge is transferred.
  4. The water is 1mm away from the coin

In this situation, The force between coin and water is calculated with coulomb's law:

enter image description here enter image description here

We are getting to the Megajoule Here.

Given the mass of 10% the pool's water, that force gives the water a momentary acceleration of: enter image description here enter image description here

Now, to lift the water 100 meters, we need to imprint the water a velocity of 44.3m/s

enter image description hereenter image description here

So the interaction time between coin and water before the remaining energy dissipates making:

Krazer said:
energy spent warming the water up, creating the sounds loud explosion, any dramatic winds effect

enter image description here enter image description here

And that's enter image description here that accounts for enter image description here

Even if you were to take into account the charge dissipation, the diminishing repulsion force between the water, and other small assumptions here, there is plenty of energy for everything.

There is plenty of energy everywhere to be used any way you want.

But I think this clearly explains where the energy to lift the water comes from.

Many thanks to Wolfram Alpha for the calculations and images.


More theories:

Electricity can keep accelerating the coin even after it left the "muzzle".

If she charges the coin, she can generate another charge of the same sign on herself after the coin leaves the "muzzle". So even if the coin leaves at a velocity of 1030m/s, she could accelerate it even after it was fired. But its not even necessary, because...

The coin's Spin and one of most destructive forces: Harmonics.

enter image description hereAs we can see from this wikipedia graph, once maximum resonance (1:1) is reached, the energy transfer increases dramatically. Wind Harmonics are enough to destroy a bridge by moving it like a violin string. If she can spin the coin so its frequency matches perfectly the harmonic frequency of the pool, massive energy could be transfered.

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I just noticed that the water would take 9 seconds to go up 100 meters at 44.3 m/s. Since the blast takes only a few frames, we can assume the contact time is a bit LONGER THAN 5ms, and the upward water just thins/evaporates after hitting 100m –  Mindwin Mar 29 at 4:50
    
Damn... I'm gonna have a lot of equations to run through when I get the time... :D:D:D –  Mysticial Mar 29 at 4:52
    
I should've saved the calculations links, now that I think of it. –  Mindwin Mar 29 at 4:54
    
Well, the pool is only one example, we often see her stop and blast cars up in the air, produce enough heat to melt two consecutive metal bars, and leave a deep scratch on the earth it travelled over, even without direct contact. It would seem as though there's another energy aside from the coin's sheer velocity. –  Madara Uchiha Mar 29 at 9:09
    
Also not to mention that in the Railgun series, she shown that she can propel items other than a coin (namely, a giant robot arm and an entire satellite) to similar velocities. –  Madara Uchiha Mar 29 at 9:11

Let's not forget in the episode she did the robot arm she did so because the person was sitting outside the range of her coins. At the end of the beam there was simply no coin left. She explained as she blocked, caught, and fired said arm that there is a reason she usually uses coins.

I suspect several factors combine to contribute to her destructive potential including her designated badass status, continued acceleration, inertia, spin, hypersonic shockwave, and possible conversion of matter to plasma.

Remember she also probably diverting energy to fight recoil and other destructive aspects of firing the railgun. That shielding is sufficient to stop directed multiple nuclear blasts capable of rapidly melting through concrete and steel despite Misaka being exhausted at the time. I'm sure this might be sufficient energy cause a secondary acceleration.

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