# E = mc² Calculator

Created by Davide Borchia
Last updated: Jul 08, 2022

Meet the most famous equation in physics with our E = mc² calculator: discover with us the relationship between mass and energy and how calculating from mass to energy gives us the answer both to mundane phenomena and devastating weapons.

• What is the equivalence of mass and energy;
• The history of the equation;
• How to calculate the mass to energy conversion;
• Some examples of this mass to energy calculations.

## The equivalence of mass and energy

Mass and energy are the same thing. No, seriously: the Universe liked them together, and this fundamental law rules the cosmos gives us light and warmth.

Humanity was mostly unaware of this deep underlying connection till 1905 when a young Alber Einstein concluded his annus mirabilis with a paper that finally, and without a trace of doubt, liked the two quantities. Here is the crucial sentence: "[…] if a body gives off the energy L in the form of radiation, its mass diminishes by L/V²".

At the time, Einstein wasn't aware of the implications of this small equation: in the course of the next twenty-something years, however, humans understood and embraced this law, and created the most destructive weapons, nuclear weapons, to use the immense energy to bring havoc on themselves.

## How to calculate the mass to energy conversion?

The unassuming equation that we use to convert mass to energy states:

$E = m\cdot c^2$

Where:

• $E$ is the energy;
• $m$ is the mass at rest; and
• $c$ is the speed of light.

To calculate E = mc², remember to use the correct measurement units. Remaining in the SI is a good choice: in this case, the energy would be expressed in joules, the mass in kilograms, and the speed in meters per second.

If you notice, the equation E = mc² resembles closely the equation for kinetic energy, $E = \tfrac{1}{2}\cdot m\cdot v^2$. This is not a fluke: all energy is fundamentally kinetic, thermal, chemical, and so on. Why do we lose the factor $\tfrac{1}{2}$? It is complicated!

## Some implications of the conversion of energy and mass

The factor $c^2$ in the calculations of E = mc² equation is huge, towering at a value of, approximately $9.0\cdot10^{16}\ \text{m}^2/\text{s}^2$. This makes the values calculated with the mass-energy equivalence prone to develop some interesting contrast in the orders of magnitude:

• A mass of $1\ \text{kg}$ equals to $89,875,517,873,681,764\ \text{J}$, in scientific notation, almost $90\cdot10^{15}\ \text{J}$. Using the megajoule as unit is almost necessary, and the gigajoule and petajoule would be even better choices.
• An energy of $4,184\ \text{J}$ (a kcal), equal to almost $0.05\ \text{ng}$ ($5\cdot10^{-11}\ \text{kg}$), the mass of a couple of blood cells.

The possibility to convert mass to energy lies underneath the concept of nuclear energy: here, a noticeable fraction of the mass of atoms in a reaction transforms into energy (because the modifications happen in the strongly bounded nucleus); both in nuclear fission and nuclear fusion, the result is a huge production of energy. The first time humans had control over it happened in Chicago, where Enrico Fermi powered the Chicago Pile-1, that on the 2nd December 1942, sustained the first controlled nuclear chain reaction. From there, we went explosive, and in 1945 the Trinity test opened the gates to the age of nuclear weapons.

The bomb detonated over the Japanese city of Hiroshima had a yield of about $15\ \text{kT}$, equalling $15$ thousand tons of TNT. Input this value in the E = mc² calculator: you will find that the resulting mass is $0.7\ \text{g}$: the conversion of the mass of a chickpea brought unimaginable devastation. Let that sink in.

From the point of view of nuclear fusion, the immense energy released by the conversion of a small mass during the rearrangement of nuclei gives us stars. Humans are currently trying to harness and control this reaction (in nuclear fusion reactors), but they are far from the objective. We excel, however, at releasing this energy in an uncontrolled and violent way: thermonuclear weapons are modern Damocles's swords on the heads of every one of us.

Davide Borchia Mass
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Energy
MJ
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