Mass-Energy Equivalence — E = mc²

Mass-energy equivalence E = mc² explained: what the world's most famous equation really says, why c is squared, and a worked example.

What it means

Mass is frozen energy. The equation says every kilogram of matter contains an almost incomprehensible reservoir of energy — c² is the conversion factor, and because c is 300 million, its square is astronomically large. In everyday chemistry the mass changes are too tiny to notice, but in nuclear reactions the 'missing' mass of the products versus the reactants appears as kinetic energy and radiation. The equation does not say mass 'turns into' energy as if they were different substances — it says they are the same thing measured in different units.

Variables

Assumptions

• Object at rest (rest energy — for moving objects use the full relativistic energy)
• Complete conversion of mass to energy
• Speeds where relativistic treatment applies (the classical KE formula fails here)

Derivation

Einstein's 1905 analysis of the Doppler shift of light emitted by a moving body showed that emitting energy $E$ reduces the body's mass by $E/c^2$. The invariant formulation is $E^2 = (pc)^2 + (mc^2)^2$, which reduces to $E = mc^2$ for a body at rest.

Worked example

Given:

• m = 1 g = 0.001 kg
• c = 3.00 × 10⁸ m/s

1. Write the equation: $E = mc^2$

2. Substitute: $E = 0.001 \times (3.00 \times 10^8)^2$

3. Square the speed of light: $(3.00 \times 10^8)^2 = 9.00 \times 10^{16}$

4. Multiply: $E = 0.001 \times 9.00 \times 10^{16} = 9.0 \times 10^{13}\ \mathrm{J}$