E = mc² calculator
Pick a mass and the calculator tells you how much energy it is equivalent to under Einstein's most famous equation, and compares it with familiar benchmarks: a kilowatt-hour, a ton of TNT or the Hiroshima bomb.
Key formula
E = m · c² · c² ≈ 8.99·10¹⁶ m²/s²
Try it
How to use the simulator
E = mc² tells you how much energy is equivalent to a given mass. Because c² is huge (about 9·10¹⁶ m²/s²), even a single gram stands for an enormous amount. The calculator turns the mass you pick into joules and compares it with familiar benchmarks: a kWh, a ton of TNT, the Hiroshima bomb, the Tsar Bomba and one second of spin drive output.
Controls
- Mass slider: a log scale from a single astrophage to several kilograms.
- Mass buttons: shortcuts for 1 astrophage (20 pg), 1 μg, 1 g and 1 kg.
Things to try
- Pick 1 gram: about 9·10¹³ J, roughly 1.4 times the energy of the Hiroshima bomb (about 6.3·10¹³ J).
- Pick a single astrophage and see what it compares to. The energy is minuscule, but so is the cell.
- Keep in mind the calculator assumes all the mass becomes energy. No known reaction does that except matter-antimatter annihilation; uranium fission converts less than 0.1%.
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What the equation says
In 1905 Einstein derived from special relativity that mass and energy are two faces of the same thing. An object at rest has an energy equal to its mass times the speed of light squared. Because c² is a gigantic number, a small mass is equivalent to an enormous energy.
One kilogram of anything (water, iron or cotton) is equivalent to about 9·10¹⁶ joules, roughly the electricity the whole of Spain uses in a little over a month.
How much mass really gets converted
The equation gives the total equivalent energy, but almost no reaction releases all of it. When gasoline burns, the products weigh less than the reactants by about a billionth, far too little to weigh. Uranium fission turns about 0.09% of the mass into energy, and hydrogen fusing into helium about 0.7%.
Only matter-antimatter annihilation converts 100%. Through fusion the Sun turns about 4 million tonnes of mass into energy every second, and it still has fuel for billions of years.
Worked example
Example: one gram of mass
- E = 0.001 kg · (3·10⁸ m/s)² = 9·10¹³ J.
- A ton of TNT releases 4.18·10⁹ J, so one gram is equivalent to about 21,500 tons of TNT (21.5 kilotons).
- The Hiroshima bomb released about 15 kilotons (6.3·10¹³ J): one gram fully converted gives 1.4 times that energy. Less than a gram of mass was turned into energy in that bomb.
- In kWh: 9·10¹³ / 3.6·10⁶ ≈ 25 million kWh, the yearly electricity use of roughly 7,000 homes.
In Project Hail Mary
In the novel, a fictional microorganism stores energy in a way that measurably increases its mass, by exactly the amount E = mc² predicts. That match is what lets the story's scientists understand what is going on.
Frequently asked questions
Does a hot object weigh more than a cold one?
Yes, though the difference is unmeasurable in everyday life. Heating a liter of water from 0 to 100 °C adds about 4.7 nanograms of mass.
Why c squared?
It is what comes out of combining conservation of energy and momentum with relativity. The units demand it too: kg · (m/s)² is joules.
Can mass be turned into energy at will?
No. Nuclear reactions release only a small part, and antimatter is extremely expensive to make: with current technology, producing a gram would cost vastly more than the energy it would give back.