Collisions and conservation laws
Collisions and conservation laws

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Collisions and conservation laws

7.1 Relativistic momentum

According to Einstein’s theory the relativistic momentum of a particle with mass m and velocity bold v is given by

bold p = fraction m bold v over square root 1 minus fraction v squared over c squared end end end
Equation label: (12)

where v is the speed of the particle and c is the speed of light in a vacuum. The speed of light in a vacuum, c = 3.0 times 10 super 8 mathrm mathrm superscript minus 1 end, plays an important role throughout special relativity. Among other things it represents an upper limit to the speed of any particle.

Equation 12 implies that the momentum of a particle increases more rapidly with increasing speed than the Newtonian relation (bold p = m bold v) predicts. This is shown in Figure 6, where the behaviour of the Newtonian and relativistic definitions of momentum magnitude are compared. You can see the good agreement at low speed, but you can also see the increasing discrepancy as the speed increases. Note that the relativistic definition does not extend beyond v = c. This reflects the fact that in special relativity it is impossible to accelerate a particle with mass to the speed of light, as you will soon see.

Figure 6 The magnitude of the momentum of a particle of mass m plotted against the particle’s speed v according to Newtonian mechanics and special relativity. The Newtonian relation closely approximates that of relativity for values of v that are small compared with the speed of light, c.

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