Calculations

## 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 versa 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.

However, lifting should generally not be used to calculate Attack Potency unless it is a fast, explosive form of lift (for example: snatch, clean-and-jerk, etc.). This is based on the biomechanics behind how human type characters attack. Unlike a punch, a kick, or most other types of attacks, a lift is a slow sustained motion which allows for many more muscles fibers to be recruited into the movement more easily, generating much more energy than a fast movement used in combat. Lifting movements also allow the body's tendons to help out by storing the energy, then releasing it in a sudden burst, acting like a spring. If we use real world ratios, when the world's heaviest deadlift is compared to the world's most powerful punch, the deadlift has nearly 5 times more energy, demonstrating the disparity between the two types of movements. Similarly, if telekinesis (or any other ability of a similar nature) is used, the lifting must be performed in a timeframe capable of being used as an attack.

#### 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

Ep = M\*g\*h

where

* M = mass of the object in kg
* g = the gravitational acceleration. For earth this is about 9.81 m/s2
* h = how high the object was lifted / how far down the object fell in meter
* Ep = 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:

Where

* G is the gravitational constant, which is 6.674\*10-11 N\*m2/kg2
* M is the mass of the planet in kg, in case of earth 5.972\*1024kg
* m is the mass of the object falling in kg
* r2 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.
* r1 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.
* Ep 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 receive 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)

### Plasma

Plasma is one of the four fundamental states of matter, together with solid, liquid and gas.

A Plasma is an ionised gaseous substance, which is highly electrically conductive, to the point that long range electric and magnetic fields dominate the behavior of the matter.

The energy imbued into a plasma is highly variable. For example a candle flame is a form of plasma with very few energy and the matter that the core of the sun is made of is a plasma with very much energy.

As such the energy within a plasma or the energy necessary to create it can, in general, not be estimated without having further information.

Should the mass of the plasma, the material it is made of and the degree of ionisation be known, one can estimate the energy as follows:

1. Use the molar mass of the material the plasma is made of together with the mass to figure out the number of atoms in the plasma. So if you have 50 g of plasma, with your substance having a molar mass of 1.007 g/mol, then you have (50 g) / (1.007 g/mol) = 49.6524 mol of atoms = 6.022\*1023 \* 49.6524 atoms = 2.990067528\*1025 atoms. Be careful: the molar mass of a non-atomic substance, like water, gives the amount of mass per molecules, not atoms.
2. Next we want to calculate the amount of ionised atoms within the substance. If the degree of ionisation of the substance is x, then this is simply done by multiplying x with the amount of atoms calculated in the first step.
3. Now we can calculate the energy to turn it into plasma: For that we just have to multiply the amount of ionised atoms, with the Ionisation energy of the atoms in question.

Consider that the energy that was used to turn the material into the state before it was made into a plasma is not considered here. So if for example rock is turned into plasma, the energy to vaporize it can be put on top of this calculation result.

If there are other specific statements those can possibly be used to get results in an easier way.

For example plasma, for which high energy density physics applies, is defined to have at least an energy density of 1010 J/m3.

### 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

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

### Specific Heat Capacity/Latent Heat of Fusion and Vaporization

* 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

## Speed Calculations

To calculate the speed of a character or object the basic formula used is v = d/t, where v is the speed of the object, d is the distance the object moved and t is the amount of time it took the character/object to move that much.

Usually the time it took for an object to move the distance is calculate by dividing the distance another object moved during that time, through the speed of that object. That requires knowing the speed of the other object, of course.

More details, such as more in depth explanations on how to figure out the distance and the time as well as examples on how to calculate speed, can be found on the following pages:

* Projectile Dodging Feats
* Lightning Dodging Feats
* Laser/Light Beam Dodging Feats

### Slow Motion Calculations

Sometimes when calculating speed one might encounter scenes where time seems to move slowly from the perspective of a fast moving character. Fundamentally speed calculations can be performed in the same way as normal in such cases.

In other words one just has to figure out a timeframe through the movement of a reference object with known speed, measure how far the character moved during that timeframe and divide the distance through the length of the timeframe (v = d/t).

However, sometimes the time is not just slowed down, but appears outright frozen.

In this case, if a reference object with known speed can be visually confirmed to not have moved even 1 pixel (which requires the feat to happen in a visual media like a comic, movie or animation) one can figure out the timeframe, by saying that it must have been less than the timeframe that the object would have taken to move 1 pixel.

If that can not be confirmed different upper limits, with similar argumentation, can usually be confirmed. In cases with a moving camera/point of view it can be useful to compare the movement to an object that can be assumed to be static.

Especially for written feats another method can be relevant. In that one first wants to figure out how many times slower the time is from the real time and then figure out how fast a movement in that time really is, based on how fast it looks in the slowed down time.

So the formula would be (real speed of reference object / apparent speed of reference object) \* apparent speed of object of interest = real speed of object of interest

For example: If an object that really is 1000 m/s fast seems to move with only 10 m/s in slowed down time a character that seems to move with a human walking speed of 1.4 m/s in slowed time would move with (1000 m/s / 10 m/s) \* 1.4 m/s = 140 m/s.

Useful values for the apparent speed of movement would be:

* Human walking speed: 1.4 m/s
* Human jogging speed: 3.71 m/s
* Human running speed: 5-7.7 m/s

Sometimes even something like "time seems frozen" or that nothing moves is stated. Often this kind of statements are hyperboles. However, should that not be the case one may assume that the apparent speed of the reference object is less than or equal to 0.001 m/s.

Mind rule 7 regarding Cinematic Time, whenever calculating feats involving slowed time.

Do not assume that a character viewing the world in slow motion is caused by time dilation. The depiction usually doesn't match the real life phenomena and the effect is commonly caused by the character being able to process events very fast, so that they seem to happen in slow motion to them, instead.

## Calculators and other utilities

### Calculators

* Elipsoid calculator

* Projectile motion calculator

* Planetary parameter calculator

* Relativity calculator

* Asteroid destruction calculator

* Asteroid/Comet calculator

* Multipurpose calculator

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

* A calculator which can handle and calculate numbers up to 101010 (10 raised to the power of itself 1010 times)

* Optical lens calculator

* Tsunami speed calculator

* General advanced scientific calculator

* Integral calculator

* Derivative calculator

* Freefall calculator w/air resistance

* Another nuke calculator

### Other utilities

* List of scientific constants

* Debunking of common misconceptions regarding heat

* Explanation of explosives

* Various lists of notable constants, etc

## See also:

* Calculation Guide

* Calculations Introduction and References

* Mass-energy conversion feats

* Black Hole feats in fiction

* Lightning dodging feats

* Laser/Light Beam Dodging Feats

* Other Calculation Instructions