Mass-Energy Equivalence — One Gram of Mass (Worked Example)

Given

• m = 1.0 g = 0.0010 kg

• c = 2.998 × 10⁸ m/s

• 1 kt TNT = 4.184 × 10¹² J

Find

Energy E and its TNT equivalent.

Principle

Einstein's mass-energy equivalence: mass is a concentrated form of energy, with c² as the conversion factor.

Formula

Solution

1. Convert mass: $m = 1.0\ \mathrm{g} = 1.0 \times 10^{-3}\ \mathrm{kg}$

2. Apply: $E = (1.0 \times 10^{-3})(2.998 \times 10^8)^2$

3. Calculate: $E = 1.0 \times 10^{-3} \times 8.988 \times 10^{16} = 8.99 \times 10^{13}\ \mathrm{J}$

4. TNT equivalent: $8.99 \times 10^{13} / 4.184 \times 10^{12} = 21.5\ \mathrm{kilotons}$

Unit check

kg × (m/s)² = kg·m²/s² = J. Correct.

What this means physically

A paperclip's worth of mass holds the energy of a small nuclear weapon — roughly the Hiroshima yield. Real reactions convert only a fraction of a percent of mass, which is why nuclear energy still needs kilograms of fuel.