How is Misaka's railgun so destructive?


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?




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?




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?

