Relativistic Mechanics Momentum and energy

Momentum p =  mv Momentum is conserved in all interactions.

Total Energy E =  mc 2 Total energy is conserved in all interactions.

Rest Energy E =  mc 2 If v = 0 then  = 1, E = mc 2 Rest energy is mc 2 Kinetic energy is (  –1)mc 2

Mass is Energy Or, E = K + mc 2 Particle masses often given as energies –More correctly, as rest energy/c 2 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/c 2 Proton MeV/c 2 Neutron MeV/c 2

Correspondence At small  : Momentum  mv  mv Energy  mc 2 = (1–  2 ) –1/2 mc 2 Binomal approximation (1+x) n  1+nx for small x So (1–  2 ) –1/2  1 + (–1/2)(–  2 ) = 1 +  2 /2  mc 2  mc 2 + 1/2 mv 2 Is this true? Let’s check:

Convenient Formula E 2 = (mc 2 ) 2 + (pc) 2 Derivation: show R side = (  mc 2 ) 2

A massless photon p = h/ E = hf = hc/ h =  10 –36 J·s (Planck constant)  mc 2 incalculable:  =  and m = 0 But E 2 = (mc 2 ) 2 + (pc) 2 works: –E 2 = 0 + (hc/ ) 2 = (hf) 2 –E = hf