Calculate the energy equivalent of a given mass using Einstein's famous equation E = mc², where c is the speed of light. Demonstrates the enormous energy contained in even small amounts of matter.
E = mc² is the most famous equation in physics. Derived by Albert Einstein in 1905 as a consequence of special relativity, it states that mass and energy are interchangeable forms of the same underlying quantity. The coefficient is the speed of light squared — about 9 × 10¹⁶ m²/s² — meaning even a tiny mass carries an astonishing amount of energy.
This single relationship explains where the Sun's energy comes from (mass loss in fusion), how nuclear power works (fission releases mass-energy), the principle behind the atomic bomb, and the origin of mass in particle physics. It also reveals that all mass — your body, this calculator, every atom — is essentially "frozen energy" in incredibly concentrated form.
Practical context: 1 kilogram of any matter, if fully converted to energy, releases ~9 × 10¹⁶ joules — equivalent to 21.5 megatons of TNT, roughly 1,500 Hiroshima-scale weapons. Of course, full conversion is extraordinarily difficult — matter-antimatter annihilation comes closest, but in practice we extract only fractions of a percent of mass-energy in nuclear fission (~0.1%) or fusion (~0.7%).
Common applications: nuclear engineering, particle physics, cosmology, astrophysics (stellar energy production), and the conceptual basis for understanding why mass exists. The calculator gives the equivalent energy for any input mass.
**Scenario:** Hiroshima bomb released ~15 kilotons TNT energy. How much mass was converted? **Calculation:** E = 15,000 × 4.184 × 10⁹ J = 6.28 × 10¹³ J. m = E/c² = 6.28 × 10¹³ / 9 × 10¹⁶ = 7 × 10⁻⁴ kg = 0.7 g. **Result:** Less than one gram of matter became energy. Only ~0.7% of the ~60 kg of uranium fuel actually fissioned, and only ~0.1% of that mass converted — so ~60 g of mass "left" the bomb in pure energy form.
**Scenario:** The Sun radiates 3.85 × 10²⁶ W. Mass loss rate? **Calculation:** Per second, ΔE = 3.85 × 10²⁶ J. Δm = ΔE / c² = 3.85 × 10²⁶ / 9 × 10¹⁶ = 4.28 × 10⁹ kg. **Result:** Sun loses ~4.3 million tonnes of mass *every second* as energy radiates away. Over 4.6 billion years: ~6 × 10²⁶ kg lost — about 0.03% of the Sun's mass. The proton-proton fusion chain (4H → He) accounts for this.
**Scenario:** A 1 kg bag of sugar. Energy if mass fully converted? **Calculation:** E = 1 × 9 × 10¹⁶ = 9 × 10¹⁶ J. **Result:** ~9 × 10¹⁶ J — about 21.5 megatons TNT, roughly the largest hydrogen bomb ever built (Tsar Bomba ~50 Mt). For comparison, eating the sugar yields ~16 MJ chemical energy — only 0.0000000002% of its mass-energy. Chemistry barely touches the mass-energy reservoir.
**Use E = mc² for:**
- **Nuclear engineering**: fission, fusion, mass defect calculations. - **Particle physics**: rest energy, threshold energies for particle creation. - **Astrophysics**: stellar energy production, mass loss. - **Cosmology**: energy budget of the universe. - **Conceptual understanding**: mass and energy as a single quantity.
**Why "full conversion" is rare:**
The equation gives the total energy *equivalent* of mass. Most physical processes convert only a tiny fraction:
- **Chemical** (combustion, batteries): ~10⁻¹⁰ to 10⁻⁸ of mass. - **Nuclear fission** (U-235): ~10⁻³ of mass. - **Nuclear fusion** (H → He): ~7 × 10⁻³ of mass. - **Matter-antimatter**: 1.0 (full conversion). - **Black hole accretion**: up to ~0.4 (most efficient natural).
**Particle physics applications:**
- **Threshold energies**: creating new particles requires energy ≥ rest mass. - **Decay products**: total decay energy = Δmc² (Q-value). - **Collider design**: LHC delivers 13 TeV total energy → can create particles up to ~13 TeV mass.
**Cosmology:**
- **Total energy of universe**: includes dark matter (~27%), dark energy (~68%), ordinary matter (~5%). - **Big Bang nucleosynthesis**: 3 minutes of fusion turned ~25% of hydrogen into helium-4. - **Heat death scenarios**: depend on long-term mass-energy distribution.
**Mass defect example: deuterium**
Mass of proton + neutron: 938.272 + 939.565 = 1,877.837 MeV/c². Mass of deuterium (²H): 1,875.613 MeV/c². Defect: 2.224 MeV = binding energy.
Form one deuterium → release 2.224 MeV. Fusion of D + T → ⁴He + n releases 17.6 MeV.
**Practical nuclear energy:**
- **Fission reactor**: 200 MeV per U-235 fission. 1 kg U-235 → ~24,000 MWh thermal energy. - **Fusion (ITER, future)**: 17.6 MeV per D-T fusion. 1 kg D-T mix → ~340,000 MWh.
For comparison: 1 kg gasoline → ~12 kWh combustion. Nuclear is ~10⁶ times more energy-dense.
**Antimatter as fuel (theoretical):**
- 1 kg antimatter + 1 kg matter = 1.8 × 10¹⁷ J ≈ 43 megatons TNT. - Annihilation produces gamma rays (511 keV for e⁺/e⁻) or pions (proton-antiproton). - Storage requires magnetic trapping in vacuum. - Production: extremely inefficient — most expensive substance ever (~$100 trillion/g).
**Conservation laws:**
- **Mass + energy** is conserved (not mass alone or energy alone separately for nuclear/particle physics). - **Mass-energy** + **momentum**: conserved as a 4-vector. - **Charge, baryon number, lepton number**: separately conserved.
**Software:**
- **NIST Atomic Mass Tables**: precise masses for binding energy calculations. - **MCNP, Geant4**: nuclear reaction simulation. - **PYTHIA, MadGraph**: particle physics event generation.
**Pitfalls:**
- **Confusing rest mass with relativistic mass**: modern convention is invariant rest mass m₀. - **Mixing units**: SI vs natural units (eV, MeV, GeV). - **Assuming free energy**: real conversion is far from 100% efficient. - **Sign conventions**: kinetic energy = total − rest energy. - **Forgetting c is huge**: factor ~10¹⁶ amplifies tiny masses into huge energies.
Calculate kinetic energy using KE = 1/2 mv².
Calculate escape velocity using v = √(2GM/r).
Calculate electrical power using P = VI, P = I²R, or P = W/t.
Calculate decibels from power or intensity ratios.
Calculate the speed of sound in air at different temperatures.
Calculate force, displacement, and elastic potential energy using Hooke's Law F = kx.
Energy
8.988e+13 J
Energy (kWh)
2.497e+7 kWh
TNT Equiv.
21.5 kt
| Parameter | Value |
|---|---|
| Mass | 0.001 kg (1.000000 g) |
| Speed of Light (c) | 299,792,458 m/s |
| c² | 8.987552e+16 m²/s² |
| Energy (J) | 8.987552e+13 J |
| Energy (kWh) | 2.4965e+7 kWh |
| Energy (GWh) | 2.4965e+1 GWh |
| TNT Equivalent (kt) | 21.48 kilotons |
| TNT Equivalent (Mt) | 0.0215 megatons |
| Formula | E = mc² |