Special relativity
Relativistic analysis

Consider the elastic collision scenario in Fig. 6-3 between a moving particle colliding with an equal mass stationary particle. Unlike the Newtonian case, the angle between the two particles after collision is less than 90°, is dependent on the angle of scattering, and becomes smaller and smaller as the velocity of the incident particle approaches the speed of light:

The relativistic momentum and total relativistic energy of a particle are given by

Conservation of momentum dictates that the sum of the momenta of the incoming particle and the stationary particle (which initially has momentum = 0) equals the sum of the momenta of the emergent particles:

Likewise, the sum of the total relativistic energies of the incoming particle and the stationary particle (which initially has total energy mc) equals the sum of the total energies of the emergent particles:

Breaking down (6-5) into its components, replacing v {\displaystyle v} with the dimensionless β {\displaystyle \beta }, and factoring out common terms from (6-5) and (6-6) yields the following:

From these we obtain the following relationships:

For the symmetrical case in which ϕ = θ {\displaystyle \phi =\theta } and β 2 = β 3, {\displaystyle \beta _{2}=\beta _{3},} (6-12) takes on the simpler form:
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