# Calculations

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## Calculating Destructive Capacity

In order to determine a character's Destructive Capacity, we must first look through the character's feats, and determine how much energy was exerted to preform such a feat. Sometimes the destructive capacity of the feat can be determined easily with no need of a calculation, but most of the times it isn't as simple. Here we will explain some of the methods we use in order to calculate feats.

### __Destruction/Creation of Planets and Planetoids__

For feats that involve the Destruction or the creation of planets or planetoids, we use a term called Gravitational Binding Energy, more information about the subject on the GBE page.

### __Meteors and Kinetic Energy__

In order to determine the energy of meteors, we generally use Kinetic energy, as it is the most reliable way to gauge the energy a meteor poses if we are given the required Details.

The equation used:

Ek=0.5*M*V^2

Terms:

Ek=Kinetic Energy

M=mass

V=velocity=speed

There are several speed values that we can use without the need for proof:

If the meteor in question was determined to have come from outer space (or outside of our atmosphere) we will use the value of minimum impact velocity, which is the minimum value of speed an object needs to enter the earths atmosphere, that value is 11,000 m/s.

If the meteor in question was shown to be Ablated, but didn't come from space, we will use Ablation speeds, which are the minimum speed an object needs to move in order for it to be ablated by its own friction with the atmosphere, the values range from 2000-4000 m/s.

A reasonable high end for meteors that come from outer space is 17,000 m/s, as it is the speed value of most of the meteors that have entered earths atmosphere.

Note: if the meteor in question wasn't ablated and didn't come from outer space, we will use Potential-Gravitational Energy, as it does not require the use of speed. That is unless we can mathematically find a speed for the meteor.

__Potential Gravitational Energy: Energy of falling Objects and Energy to lift Objects__

Gravitational potential energy is energy an object possesses because of its position in a gravitational field. If an object from a high position falls towards the ground the kinetic energy the object gets from falling is equal to the difference in its gravitational potential energy before and after the fall, provided no other forces such as air resistance act upon it. Vice verse the energy necessary to lift an object to a certain height is also equal to the change in gravitational potential energy of the object before and after being lifted.

__In cases close to the ground__

In cases where an object is lifted or falls relatively close to the surface of the earth the difference in gravitational potential energy can be calculated using the simple formula

E_{p} = M*g*h

where

- M = mass of the object in kg
- g = the gravitational acceleration. For earth this is about 9.81 m/s
^{2} - h = how high the object was lifted / how far down the object fell in meter
- E
_{p}= gravitational potential energy difference in joules.

__In cases far away from the ground__

In cases where an object is lifted very high or falls from very high up the upper formula can not be used any more. Instead the following formula should be used:

E_{p} = |(G*M*m)/r_{1} - (G*M*m)/r_{2}|

Where

- G is the gravitational constant, which is 6.674*10
^{-11}N*m^{2}/kg^{2} - M is the mass of the planet in kg, in case of earth 5.972*10
^{24}kg - m is the mass of the object falling in kg
- r
_{2}is the distance between the center of mass of the planet and the center of mass of the object after the fall / before being lifted (in meters). So usually it is radius of the planet (in case of earth 6371000 m) + how far the object is away from the ground after the fall / before being lifted. - r
_{1}is the distance between the center of mass of the planet and the center of mass of the object before the fall / after being lifted (in meters). So usually it is radius of the planet (in case of earth 6371000 m) + how far the object is away from the ground before the fall / after being lifted. - E
_{p}is the gravitational potential energy difference in joules.

__Terminal velocity__

Air resistance is mostly relevant for the case of falling objects and even then can be ignored for most heavy objects like meteors or big constructs.

However for lightweight objects, like for example humans, it becomes relevant.

The air resistance opposes gravity in pulling the object down and due to that slows down the fall of the object, reducing the kinetic energy it gains while dropping.

If an object falls long enough it will reach its terminal velocity and not get any faster than that (provided it wasn't faster to begin with or influenced by other forces than gravity).

Because of that the kinetic energy an object has at terminal velocity often forms a limit to how much energy an object can gain through falling, which means that if our calculated change in potential energy is greater than that we have to consider it as a too high estimate for the energy when reaching the ground.

The terminal velocity of an object can be calculated like described here, but, for example due to the drag coefficient being depended on the fall speed of the object, it is in practice often difficult to calculate. For a human terminal velocity is approximately 53 m/s close to the ground.

An example of terminal velocity being relevant for the result of a calculation can be found here.

### __Change in Temperature and Vaporization or Melting Energy__

This would be one way of determining attacks that make use of fire, ice, etc. If something is melted, ignited, or frozen we can determine destructive capacity for one or two steps:

First:

- We determine the energy given or taken in changing the temperature, the equation for this is E=m*c*ΔT
- E is the energy
- m is mass
- c is specific heat capacity, this varies between materials and is based on how much energy is required to get that material to a certain heat (if you put stone and steel over the same fire they will recieve energy at the same rate but the steel will heat up faster)
- ΔT is the change in temperature; the starting temperature will usually be a reasonable room temperature unless there is a reason why the temperature would begin hot or cold in the calculation, the final temperature will usually be either melting point, freezing point, boiling point, or (if a burnable material is present) the auto-ignition temperature or burning material; this is measured in kelvin.

Second:

- We determine the energy given or taken in changing the state of matter (if the feat we are calculating did not involve a state change i.e. something being set on fire does not include a change of state, then we do simply perform this second step).
- If the feat involved a change between solid and liquid (in either direction) we must multiply the mass of the material with the material's "heat of fusion".
- If the feat involved a change between liquid and gas (in either direction) we must multiply the mass of the material with the material's "enthalpy of vaporisation", this varies based on atmospheric pressure.

This type of feat will most often involve water:

Specific heat capacity of water is 4181J/kg°K

Heat of fusion for water is 334.16J/g

Here a few pre-calculated example values (starting with the materials at 20°C):

- Melting Granite: 4358.9475 J/cm^3
- Melting Cement: 12.232.65 J/cm^3
- Melting Glass: 2494 J/cm^3
- Freezing Water: 418 J/cm^3
- Vaporization of water: 2575 J/cm^3
- Vaporization of titanium: 49079.7 J/cc

(Values taken from here and here)

### __Heat, Radiation and Nuclear-like Explosions__

When dealing with a feat that is heat related only, we can still manage to find Destructive capacity, by finding the given temperature of the object in question, estimate or calculate its surface area and emissivity, and apply the values to the following Calculator:

http://www.endmemo.com/physics/radenergy.php

While it does calculate joule/second, it is still reliable enough for us, though it should be noted that if the values inserted aren't extremely high, the results would likely be underwhelming. It is usually assumed that only 1 second of the calculated Joules per second value contributes to the attack potency, or less if the object only had that heat for less than one second.

When dealing with an explosion that doesn't leave a crater behind, we can use the following Calculator:

http://www.stardestroyer.net/Empire/Science/Nuke.html

Although it is tricky to use, we need to find the radius of the explosion in question, and insert the megaton value into the calculator until we get the same value for the "Air Blast radius (near total fatalities)" as the radius that we have scaled.

## Volume, Mass, Destruction Values and TNT table

Many feats cause visible destruction after they are preformed (such as leaving a crater, destroying a mountain/meteor). To measure these feats we need to know the Volume of the matter that was destroyed in the attack.

Volume is a space composed of three perpendicular dimensions, if its a mountain, it has length, width, and height. If it is a crater, it has length, width, and depth.

Here is a page with classic formulas of classic geometrical shapes that maybe useful when calculating volume:

http://www.basic-mathematics.com/volume-formulas.html

For example, sometimes, when trying to find the volume of a mountain, we may use the formula of a cone to give us a rough estimation of its size.

Usually after we have found the volume, we may get the final measurement to be in cubic meters (m^3) or sometimes in cubic Kilometers (Km^3). At this point, in order to get the Energy measurement for the feat, we need to convert the volume from whatever unit we are using, into cubic Centimeters (Cm^3 or cc).

Here are some terms that will make said conversion easier:

1 km^3 = 1000000000 m^3

1 m^3 = 1000000 cm^3

### __Destruction Values__

From here, According to the method used within the Naruto Forums, there are different methods of Destruction (for lack of a better term) that require different levels of energy for every cubic centimeter of the volume that was destroyed during the feat:

**Fragmentation:** Applied when the matter that was destroyed was turned into fairly large and distinguishable pieces. The value is 8 joules per Cubic centimeter (j/cc).

**Violent Fragmentation:** Applied when the matter that was destroyed was turned into small but still distinguishable pieces. The value is 69 (j/cc).

**Pulverization:** Applied when the matter that was destroyed was turned to dust. We usually use this value when we see no remains of the matter that was destroyed in the aftermath of the attack. The value is 214.35 (j/cc).

**Vaporisation:** Applied when the matter that was destroyed was vaporised during the attack. Much like for Pulverization, we usually use this value when we see no remains of the matter that was destroyed in the attack, but in addition there has to be a considerable amount of visible vapor and/or character statements that imply vaporization, usually the latter. The value is 25700 (j/cc).

**Atomization:** Applied only if clearly stated. It describes the energy to separate all atoms in a chemical substance. The value is 30852.2 (j/cc).

**Subatomic Destruction:** Applied only if clearly stated. It describes the energy necessary to destroy all atoms in a substance, by separating the particles in their nucleus. Note that Protons and Neutrons still stay intact. Value is 5.403E13 (j/cc).

These values are only relevant for solid objects like rocks, buildings and mountains.

After we have determined both the method of destruction and the volume (in cubic centimeters), we multiply both Values to get the value of energy that was exerted for the feat, and thus we have the destructive capacity.

__Table of Destruction Values__

Fragmentation | Violent Fragmentation
| Pulverization
| Atomization
| Atomic Destruction
| |
---|---|---|---|---|---|

Concrete | 6 j/cc | 17-20 j/cc | 40 j/cc | 4.168E12 j/cc | |

Steel | 208 j/cc | 568.5 j/cc | 310-1000 j/cc | 59526.65 j/cc | 6.703412 j/cc |

Iron | 20 j/cc | 42.43 j/cc | 90 j/cc | 58401 j/cc | 6.6965E12 j/cc |

Glass | 0.75 j/cc | 1 j/cc | 1000 j/cc | ||

Ice | 0.5271 j/cc | 0.825 j/cc | 4.3919 j/cc | 51384.16 j/cc | 8.9363E12 j/cc |

Human body | 4.4 j/cc | 7.533 | 12.9 j/cc | 72416.33 | 1.114e13 j/kg |

Silver | 27669.25 j/cc | 8.983E12 j/cc | |||

Aluminium | 32241.5 j/cc | 2.172E12 j/cc | |||

Tin | 18693.24 j/cc | 6.058E12 j/cc | |||

Copper | 47784.97 j/cc | 7.11E12 j/cc | |||

Zinc | 13758.07 j/cc | 5.982E12 j/cc | |||

Silicon | 36484.37 j/cc | 1.895E12 j/cc | |||

Carbon | 156545 j/cc | 1.9414E12 j/cc | |||

Tungsten | 88098.07 j/cc | ||||

Gold | 36058.28 j/cc | 1.474E13 j/cc | |||

Calcium | 7116.124 j/cc | 1.276E12 j/cc | |||

Water | 51384.16 j/cc | 8.9363E12 j/cc | |||

Diamond | 210081 j/cc | ||||

Cement | 8 j/cc | 69 j/cc | 214 j/cc |

Values taken from here, here, here, here and here.

__Specific Heat Capacity/Latent Heat of Fusion and Vaporization__

Specific Heat Capacity
| Latent Heat (Fusion)
| Latent Heat (Vaporization)
| Melting Point
| Boiling Point
| |
---|---|---|---|---|---|

Silver
| 230 | 104726.6 | 5327260 | 962 | 1950 |

Iron | 460 | 247112.54 | 6213627 | 1538 | 2861 |

Aluminum
| 870 | 396567.46 | 10859277 | 660.32 | 2519 |

Tin
| 240 | 58972 | 2442928 | 231.93 | 2602 |

Copper
| 390 | 206137 | 4720692 | 1084.62 | 2927 |

Zinc
| 380 | 112420 | 1820128 | 419.53 | 907 |

Silicon
| 710 | 1787113 | 12780350 | 1414 | 2900 |

Gold
| 130 | 63463 | 1675127 | 1064.18 | 2856 |

Calcium
| 630 | 213000 | 3867458 | 842 | 1484 |

Water
| 4186 | 333.7 | 2256.7 | 0 | 100 |

Granite
| 821.46 | 947657.98 | 6077872 | 1215 | ? |

Titanium
| 470 | 390666 | 8878768 | 1668 | 3287 |

Oxygen
| 919 | 13875 | 213125 | -218.3 | -182.9 |

Nitrogen
| 1040 | 25702 | 199190 | -210.1 | -195.8 |

- Specific heat capacity =
**Joules/Kg*K**(K and °C are interchangeable) - Latent heat =
**J/Kg** - Boiling/Melting point =
**(°C)**

__TNT Measurements__

Most of the time, when calculating Destructive capacity, we end up with extremely large values of energy that are very long to write, also, even if using Orders of magnitude to "shorten" the number, for most people, these large values of energy mean nothing and they cannot rank them easily. That is why we need to convert the numbers we get to TNT measurements, as it is a measuring system that is easier to understand for a wider diversity of people.

To understand the TNT measuring system, we must first explain how it works: 1 gram of TNT contains about 4184 joules of energy. Therefore we can say that every 4184 joules equals 1 gram of TNT, and from here we establish a measuring system:

- 1 gram of TNT = 4184 (j) = 4.184 * 10^3 (j)
- 1 kg of TNT = 1000 gram of TNT = 4184000 (j) = 4.184 * 10^6 (j)
- 1 ton TNT = 1000 kg of TNT = 4184000000 (j) = 4.184 * 10^9 (j)
- 1 kiloton of TNT = 1000 tons of TNT = 4.184 * 10^12 (j)
- 1 Megaton of TNT = 1000 Kilotons of TNT = 4.184 * 10^15 (j)
- 1 Gigaton of TNT = 1000 Megatons of TNT= 4.184 * 10^18 (j)
- 1 Teraton of TNT = 1000 Gigatons of TNT= 4.184 * 10^21 (j)
- 1 Petaton of TNT = 1000 Teratons of TNT= 4.184 * 10^24 (j)
- 1 Exaton of TNT = 1000 Petatons of TNT= 4.184 * 10^27 (j)
- 1 Zetaton of TNT = 1000 Exatons of TNT= 4.184 * 10^30 (j)
- 1 Yottaton of TNT = 1000 Zetatons of TNT= 4.184 * 10^33 (j)

Now, to convert the Energy value to a TNT measurement, we need to divide the energy value by 4.184, and then divide the result of that division by the highest order of magnitude (that is divisible by 3) that is lower/equal to the order of magnitude that was received after the first division, and that will tell you the TNT measurement. For example:

Where does 2*10^24 (j) register on the TNT Measurement system?

(2*10^24) / 4.184 = 4.78*10^23

(4.78*10^23) / 10^21= 478

This was divided by 10^21 because it was the highest order of magnitude that was both divisible by 3 and lower than the order of magnitude of the number that was received after dividing by 4.184, and since the value was divided by 10^21, the TNT measurement is in teratons, according to the chart above:

2 * 10^24 (j) = 478 Teratons.

### __Mass__

When you need to determine the mass of an object, you must first find its volume. Then you must estimate from what type of substance that the object is made out of, as every substance has its own density:

Mass = Volume * Density.

Here is a chart of densities of common materials:

Continental crust, stone and earth: 2700 Kg/m^3

Meteors: 3000-3700 Kg/m^3

Concrete: 2400 Kg/M^3

Water: 1000 kg/m^3

Clouds: 1.003 Kg/m^3

## Calculators and other utilities

### __Calculators__

Planetary parameter calculator

Asteroid destruction calculator

A calculator which can calculate figures up to thousands of digits of precision

General advanced scientific calculator

Freefal calculator w/air resistance

### __Other utilities__

List of scientific constants
Debunking of common misconceptions regarding heat

Various lists of notable constants, etc