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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 episode, it is
revealed that her railgun travels at a velocity of _**1030 m/s**_.

![enter image description here](https://i.stack.imgur.com/pifXO.jpg)

## However, the numbers don't add up.

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

[code]

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

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.

[code]

    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
    
[/code]

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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How is Misaka's railgun so destructive?


I argue that it is a case of Anime Physics by refuting the alternative
possibilities 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"](http://en.wikipedia.org/wiki/Spark_gap).
  3. The energy couldn't have come from mass energy. Besides the resulting lethal radiation, to release mass energy without using antimatter would mean releasing 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 will clearly refute the mass energy argument.

Finally, the Mythbusters proved in [this
video](http://www.youtube.com/watch?v=PHxvMLoKRWg&feature=g-vrec) that a coin
travelling at 3 times the speed of sound (around the speed of a bullet) merely
dents concrete.


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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. The energy goes where projectile goes
  3. However, it is not established that the projectile _carries_ the energy

The projectile might be just a 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 later episodes, however, the
speculation above might still apply.


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

![Pew Pew](https://i.stack.imgur.com/K7BbB.jpg)

In contrast, the [US Navy's
railgun](http://www.navy.mil/submit/display.asp?story_id=34718) 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](http://en.wikipedia.org/wiki/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?](https://i.stack.imgur.com/OOkC8.jpg)

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

According to [Page 1, Chapter 4](https://i.stack.imgur.com/dsMCu.jpg) 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!](https://i.stack.imgur.com/XuiXv.jpg)

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

![Wide shot!](https://i.stack.imgur.com/9ystI.jpg)

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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## Electricity can move the water:

[The wiki says that Misaka can generate 1 billion
volts.](http://toarumajutsunoindex.wikia.com/wiki/Misaka_Mikoto)  
( _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](http://en.wikipedia.org/wiki/Capacitance)

![enter image description here](https://i.stack.imgur.com/0e9Fy.gif)

The radius of a quarter is 13mm

![enter image description here](https://i.stack.imgur.com/qbJ2J.gif)

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

![enter image description here](https://i.stack.imgur.com/1kIkl.gif)

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](https://i.stack.imgur.com/byk5M.gif) ![enter
image description here](https://i.stack.imgur.com/KgqOL.gif)

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](https://i.stack.imgur.com/o0vlb.gif) ![enter image description
here](https://i.stack.imgur.com/Rj6gD.gif)

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

![enter image description here](https://i.stack.imgur.com/ksEb5.gif)![enter
image description here](https://i.stack.imgur.com/BB5pu.gif)

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

> [Krazer](https://anime.stackexchange.com/users/63/krazer) said:  
>  energy spent warming the water up, creating the sounds loud explosion, any
> dramatic winds effect

![enter image description here](https://i.stack.imgur.com/E4DlO.gif) ![enter
image description here](https://i.stack.imgur.com/9FpE6.gif)

And that's ![enter image description
here](https://i.stack.imgur.com/FyCAG.gif) that accounts for ![enter image
description here](https://i.stack.imgur.com/pOFjy.gif)

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.

Also, if you take the events of later episodes, when she uses other items than
a coin

> A giant robot claw and later on a whole giant robot

you can see that the amount of stored energy is greater, and so is the
destructive power. Makes sense, because the capacitance of those items is
bigger than a coin's.

## Many thanks to [Wolfram Alpha](http://www.wolframalpha.com/) 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 here](https://i.stack.imgur.com/v1XFU.png)As 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](http://en.wikipedia.org/wiki/Tacoma_Narrows_Bridges) 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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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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It is also worth noting that (insofar as I recall) the 'psychic' powers of the
type Misaka has are loosely based on the ever-popular, fantastic, double-
misinterpretation of the Schrödinger's-cat thought experiment: that perception
influences reality, and thus altering perception in the correct manner should
alter reality. (Or something like that. There was surrounding techno-babble,
but that seemed to be the gist of it.)

Thus, an alternate though less-interesting explanation of the disproportionate
effects of her railgun would be something along the lines of: Misaka's own
perception of how powerful the railgun should be is wrong, _resulting in
exaggerated effects_.


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It is a case of anime physics as other posters have alluded to but not your
average anime physics, it is said in certain parts of the manga and the sister
series index that the way esper powers work is by manipulating a reality
warping field to fit the style of their powers. therefore misaka's railgun
works because it has sufficient reality warping energy and math behind it


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You guys are all forgetting the basic fact that the 1030 m/s measurement was
taken while firing into water. The effects of drag on the coin fired into
water are far greater in magnitude than the effects of drag while fired
through air. Air is almost one thousand times less dense than water. If we
perform the equation for drag, we come up with a force magnitude on the coin
of 18,466 Newtons. [18466=.5*1000kg/m^3*(1030m/s)^2*.82*0.000042455m^2]

.82 is the drag coefficient for a long cylinder like an arcade coin,
1000kg/m^3 is the density of water, and .000042455m^2 is the cross sectional
area of the coin fired.

if we perform the equation in reverse to find out the velocity of the coin in
air, we are left with a velocity of 29,428m/s.

Sounds a hell of a lot more like a proper rail gun, doesn't it?


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'A Certain Magical Index' has 50 odd light novels, 13 volumes of manga, 2
seasons of anime, a movie and a couple of video games.

'A Certain Scientific Railgun' has 2 light novels, 11 volumes of manga, 2
seasons of anime, an OVA, and a video game of its own.

In none of these sources (that I can recall, anyway) is Misaka's railgun trick
implied to be anything other than an arcade token (not a coin, which is why it
is even ferrous to begin with) travelling three times the speed of sound.

And there is one other weird property of the railgun attack - Touma, the boy
with the anti-magic fist, can catch it. (As seen in chapter 7, volume 1 of the
Railgun manga.)

Since Touma can dispel it, that means that there is still something
supernatural about the coin, even while it is melting. That supernatural
property will be why it has a pushing force well above what it should have on
momentum alone.

I have no evidence for my next point, but I believe that Misaka's magnetic
field is simply continuing to push the coin well after it leaves her hand.
That would explain both oddities nicely.


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