Relativistic Mechanics

Momentum and energy

Momentum

p = mvMomentum is conserved in all interactions.

Total Energy

E = mc2

Total energy is conserved in all interactions.

Rest Energy

• E = mc2

• If v = 0 then = 1, E = mc2

• Rest energy is mc2

• Kinetic energy is (–1)mc2

Mass is Energy

• Or, E = K + mc2

• Particle masses often given as energies– More correctly, as rest energy/c2

• Customary unit: eV = electron·Volt– 1 elementary charge pushed through 1 V– Just like 1 J = (1 C)(1 V)– e = 1.60×10–19 C, so 1 eV = 1.60×10–19 J

Particle Masses

• Electron 511 keV/c2

• Proton 983.3 MeV/c2

• Neutron 939.6 MeV/c2

Correspondence

At small :

Momentum mv mv

Energy mc2 = (1–2)–1/2 mc2

Binomal approximation (1+x)n 1+nx for small x

So (1–2)–1/2 1 + (–1/2)(–2) = 1 + 2/2

mc2 mc2 + 1/2 mv2

Is this true? Let’s check:

Convenient Formula

E2 = (mc2)2 + (pc)2

• Derivation: show R side = (mc2)2

A massless photon

• p = h/• E = hf = hc/• h = 6.62610–36 J·s (Planck constant)

• mc2 incalculable: = and m = 0

• But E2 = (mc2)2 + (pc)2 works:– E2 = 0 + (hc/)2 = (hf)2

– E = hf