E = mc² Calculator
Enter a mass to get its rest energy in joules, kilowatt-hours, electronvolts and tonnes of TNT, or enter an energy to get the mass it corresponds to. A second mode takes a particle's speed or kinetic energy and returns its total and kinetic energy by the relativistic formulas. The working is shown, with a ruler that places your answer next to familiar energies. Free, no sign-up.
Rest mass. The menu includes u, MeV/c² and GeV/c² for particles.
Results
8.988 × 10¹³ J
Energy in joules
2.497 × 10⁷ kWh
Energy in kilowatt-hours
21.48 kt TNT
Energy as TNT
5.610 × 10³² eV
Energy in electronvolts
0.001000 kg
Mass in kilograms
The working
E = m c²
E = 0.001 × (2.9979 × 10⁸)² = 8.988 × 10¹³ J
E = mc² gives the energy equivalent of a mass at rest; it does not mean that all of it can be released. Chemical and nuclear reactions convert only a small fraction of a fuel's mass (about 0.7% for hydrogen fusion), and complete conversion needs matter meeting antimatter. The speed of light is exactly 299 792 458 m/s, and the TNT equivalent uses the convention of 4.184 GJ per tonne.
How to Use the E = mc² Calculator
- 1. Choose the mode. Energy from mass, or mass from energy applies E = mc² directly. A moving particle takes a mass and either a speed or a kinetic energy and gives the total energy, kinetic energy, Lorentz factor and speed.
- 2. Pick what to solve for and enter the rest. The box for the unknown shows the answer. Each box has a unit menu: masses run from micrograms to tonnes and include u, MeV/c² and GeV/c², and energies from joules to petajoules, kWh, eV to TeV, calories, BTU and tonnes of TNT. Scientific notation such as 9.109e-31 works.
- 3. Try an example. The menu loads the electron, the proton, a gram of matter, the mass lost in a 15 kt explosion, what the Sun radiates each second, and more.
- 4. Read the cards and the ruler. The energy is stated in four unit systems, and a logarithmic scale from 10³ J to 10²⁷ J marks where it sits among familiar yardsticks.
What E = mc² Says
Rest energy
E₀ = m c²
Total energy
E = γ m c²
Kinetic energy
K = (γ − 1) m c² ≈ ½ m v² (v ≪ c)
Einstein's 1905 paper “Does the inertia of a body depend upon its energy content?” showed that a body's mass is a measure of the energy it holds. The conversion factor is the square of the speed of light, c² = 8.98755 × 10¹⁶ m²/s²: a huge number, which is why even a tiny mass corresponds to an enormous energy. The speed of light is defined as exactly 299,792,458 m/s.
The equation applies both ways. When a nucleus fuses or splits, the products weigh slightly less than the starting nuclei, and that mass difference appears as energy. A system that gains energy, even by being heated or compressed, gains mass, though in everyday life the gain is far too small to weigh. For a moving particle, γ = 1/√(1 − v²/c²) multiplies the rest energy, and the part above rest energy is its kinetic energy. See OpenStax University Physics, section 5.9 on relativistic energy.
Worked Examples
- One gram. E = 0.001 kg × c² = 8.988 × 10¹³ J. That is 2.497 × 10⁷ kWh, or 24.97 GWh, and equals 21.48 kt of TNT.
- One kilogram. E = 8.988 × 10¹⁶ J, or 21.48 Mt of TNT, the yield of about 1,400 bombs the size of Hiroshima's.
- Particles. An electron has a rest energy of 0.511 MeV, a proton 938.272 MeV and one atomic mass unit 931.494 MeV. Particle physicists therefore quote masses in MeV/c² or GeV/c².
- A 15 kt explosion. An energy of 15 kt TNT is 6.276 × 10¹³ J, which corresponds to a mass of just 0.698 g. About that much mass disappeared in the Hiroshima bomb, out of the roughly 64 kg of uranium it carried.
- The Sun. Radiating 3.828 × 10²⁶ J every second, the Sun loses 4.259 × 10⁹ kg, about 4.3 million tonnes, every second. NASA's fact sheet gives the figure as 4,260 million kg/s.
- A kilowatt-hour. The 3.6 MJ in one kWh has a mass equivalent of 4.006 × 10⁻¹¹ kg: 0.04 micrograms, too small to detect on an ordinary laboratory balance.
- An electron with 1 MeV of kinetic energy. Its rest energy is 0.511 MeV, so γ = 1 + 1/0.511 = 2.957, v = 0.9411c and the total energy is 1.511 MeV. The classical formula ½mv² would give a speed of 1.98c, faster than light, which is why it fails.
- A proton at 0.5c. K = (γ − 1)mc² = 145.2 MeV, against ½mv² = 117.3 MeV classically: the relativistic value is 1.238 times larger. For a 1000 kg car at 100 km/h the two agree to better than one part in 10¹².
Mass is not “turned into” energy in a flame
When a fuel burns, its mass changes by about five parts in a hundred billion, so the products weigh almost exactly what the reactants did. Fission of uranium-235 converts about 0.1% of the mass, and hydrogen fusion in the Sun about 0.7%. Only matter meeting antimatter converts all of its mass to energy. E = mc² states how much energy a mass is worth, not how much any process can release.
Assumptions and Limits
- Rest mass. The calculator uses invariant (rest) mass. Older texts speak of a “relativistic mass” γm; modern usage keeps m fixed and lets energy carry the γ.
- TNT equivalent. One tonne of TNT is defined by convention as 4.184 GJ; real explosives vary slightly.
- Speeds near c. The speed shortfall c − v is computed from γ directly, so protons with TeV energies still show how far below c they travel (6.8 TeV: about 2.85 m/s).
- Energy is a bound, not a yield. The result is the mass–energy equivalent, not the energy any real process delivers.
Frequently Asked Questions
- What does E = mc² mean?
- It says that a body at rest has an energy E equal to its mass m times the speed of light squared. Because c² is about 9 × 10¹⁶ m²/s², one kilogram of mass is equivalent to 8.988 × 10¹⁶ joules, about 21 megatons of TNT.
- How do you convert mass to energy?
- Multiply the mass in kilograms by c² = 8.98755 × 10¹⁶ m²/s² to get joules. For example, 1 g = 0.001 kg gives 8.988 × 10¹³ J, which is about 25 gigawatt-hours. The calculator also converts the result to kWh, eV and TNT.
- How much mass is converted in a nuclear bomb?
- Surprisingly little. The roughly 15 kiloton Hiroshima bomb released about 6.3 × 10¹³ J, equivalent to the mass of a paperclip, about 0.7 g. The vast majority of the material in the bomb remained as matter in other forms.
- Why do particle physicists quote masses in MeV/c²?
- Because the energy equivalent is the natural size for a particle: an electron's rest energy is 0.511 MeV, so its mass is 0.511 MeV/c². One MeV/c² equals 1.78266 × 10⁻³⁰ kg, a unit that avoids writing numbers like 9.109 × 10⁻³¹ kg.
- What is the difference between rest energy and kinetic energy?
- Rest energy mc² is what a body holds when it is stationary. Kinetic energy is the extra energy of motion, (γ − 1)mc². Together they make the total energy γmc². At low speeds the kinetic part reduces to the familiar ½mv².
- Can a particle with mass reach the speed of light?
- No. As v approaches c, γ and the energy needed grow without limit. A 6.8 TeV proton at the Large Hadron Collider has γ of about 7,250 and still travels about 2.85 m/s slower than light.