Playground
Bar chart comparing rest mass energy for various particles. Slider picks mass in kg; shows energy in J and MeV.
Variables
| Symbol | Name | SI | Dimension | Range |
|---|---|---|---|---|
| Energyoutput Rest energy equivalent of the mass | J | M·L²·T⁻² | 1e-15 – 100000000000000000000 | |
| Rest mass Invariant mass of the object | kg | M | 1e-30 – 1000 | |
| Speed of light Speed of light in vacuum (constant) | m/s | L·T⁻¹ | 299792458 – 299792458 |
Deep dive
Derivation
Starting from the relativistic momentum p = γmv and total energy E = γmc², compute E² − (pc)² = γ²m²c⁴(1 − v²/c²) = m²c⁴, giving E² = (pc)² + (mc²)². Setting p = 0 (rest frame) yields E = mc².
Experimental verification
Confirmed countless times: nuclear binding energies match the mass defect Δm·c² to <0.01%. Cockcroft–Walton (1932) split lithium-7 with protons and verified the energy released equals the mass deficit. Pair production e⁺e⁻ from photons (Anderson 1932) and PET scans rely on this daily.
Common misconceptions
- E = mc² describes only rest energy; the full energy of a moving particle includes kinetic energy via the Lorentz factor.
- Mass is not literally 'converted' to energy — rather, the system's invariant mass changes when energy leaves (e.g., a hot object weighs slightly more than a cold one).
- It does not say 'matter and energy are the same thing'; mass is one specific property and energy is another, related by c².
Real-world applications
- Nuclear power plants — uranium fission releases ~0.1% of fuel mass as energy.
- Stellar fusion — the Sun converts ~4 million tons of mass to energy every second.
- PET medical imaging — positron–electron annihilation produces 511 keV gamma rays.
- Particle accelerators — LHC creates new massive particles from collision kinetic energy.
Worked examples
Energy in 1 gram of mass
Given:
- m:
- 0.001
- c:
- 299792458
Find: E
Solution
E = mc² = 0.001 × (3×10⁸)² ≈ 9×10¹³ J ≈ 21 kilotons of TNT
Rest energy of an electron
Given:
- m:
- 9.1093837e-31
- c:
- 299792458
Find: E (in MeV)
Solution
E = mc² ≈ 8.187×10⁻¹⁴ J ≈ 0.511 MeV
Scenarios
What if…
- scenario:
- What if we could convert all of a 70 kg human's mass to energy?
- answer:
- E = 70 × c² ≈ 6.3×10¹⁸ J — about 1500 megatons of TNT, roughly 30× the largest nuclear weapon ever detonated.
- scenario:
- What if c were only 1000 m/s?
- answer:
- Mass-energy equivalence would be 9×10¹⁰ times weaker. Stars couldn't fuse, atoms couldn't bind, and chemistry would dwarf nuclear energy — the universe would be unrecognizable.
- scenario:
- What if a hot cup of coffee weighed itself?
- answer:
- It would be heavier by Δm = ΔE/c². Heating 250 g of water by 80 K adds ~84 kJ, increasing mass by ~9×10⁻¹³ kg — far below any scale's resolution but real.
Limiting cases
- condition:
- m → 0
- result:
- E → 0
- explanation:
- A massless object has no rest energy (photons carry only kinetic/momentum energy via E = pc).
- condition:
- v → c (relativistic)
- result:
- E → ∞
- explanation:
- Total energy E = γmc² diverges as a massive object approaches the speed of light — that is why c is unreachable.
- condition:
- v ≪ c
- result:
- E ≈ mc² + ½mv²
- explanation:
- Low-velocity expansion of γ recovers Newtonian kinetic energy on top of the rest energy.
Context
Albert Einstein · 1905
Derived in Einstein's 'Annus Mirabilis' paper on special relativity, showing that mass and energy are interchangeable.
Hook
How much energy is locked inside a single paperclip?
A paperclip has a mass of about 1 gram. If all of its mass were converted to energy, how much energy would be released?
Dimensions: [E] = [m]·[c]² → M·(L·T⁻¹)² = M·L²·T⁻² ✓ (energy)
Validity: Universally valid in special and general relativity for any closed system. Rest-mass form applies to objects in their rest frame; for moving particles use the full relation E² = (pc)² + (mc²)².