Editing Special relativity (section)

==== Relativistic analysis ====
[[File:Relativistic elastic collision of equal mass particles.svg|thumb|Figure 6–3. Relativistic elastic collision between a moving particle incident upon an equal mass stationary particle]]
Consider the elastic collision scenario in Fig.&nbsp;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
{{NumBlk2||<math>\quad\quad \vec{p} = \gamma m \vec{v} \quad \text{and} \quad E = \gamma m c^2 </math>  |6-4}}

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:
{{NumBlk2||<math>\quad\quad \gamma_1 m \vec{v_1} + 0 = \gamma_2 m \vec{v_2}  +  \gamma_3 m \vec{v_3} </math>  |6-5}}

Likewise, the sum of the total relativistic energies of the incoming particle and the stationary particle (which initially has total energy mc<sup>2</sup>) equals the sum of the total energies of the emergent particles:
{{NumBlk2||<math>\quad\quad \gamma_1 m c^2 + m c^2 = \gamma_2 m c^2 + \gamma_3 m c^2 </math>  |6-6}}

Breaking down ({{EquationNote|6-5}}) into its components, replacing <math>v</math> with the dimensionless {{tmath|1= \beta }}, and factoring out common terms from ({{EquationNote|6-5}}) and ({{EquationNote|6-6}}) yields the following:<ref name="Champion_1932" group="p"/>
{{NumBlk2||<math>\quad\quad \beta_1 \gamma_1 = \beta_2 \gamma_2 \cos{\theta} +  \beta_3 \gamma_3 \cos{\phi} </math>  |6-7}}
{{NumBlk2||<math>\quad\quad \beta_2 \gamma_2 \sin{\theta} = \beta_3 \gamma_3 \sin{\phi} </math>  |6-8}}
{{NumBlk2||<math>\quad\quad \gamma_1 + 1 = \gamma_2 + \gamma_3 </math>  |6-9}}

From these we obtain the following relationships:<ref name="Champion_1932" group="p">{{cite journal |last1=Champion |first1=Frank Clive |title=On some close collisions of fast β-particles with electrons, photographed by the expansion method. |journal=Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character |year=1932 |volume=136 |issue=830 |pages=630–637 |publisher=The Royal Society Publishing |doi=10.1098/rspa.1932.0108 |bibcode=1932RSPSA.136..630C |s2cid=123018629 |doi-access=free }}</ref>
{{NumBlk2||<math>\quad\quad \beta_2 = \frac{\beta_1 \sin{\phi}}{ \{ \beta_1^2 \sin^2{\phi} +  \sin^2(\phi + \theta )/\gamma_1^2 \}^{1/2} }    </math>  |6-10}}
{{NumBlk2||<math>\quad\quad \beta_3 = \frac{\beta_1 \sin{\theta}}{ \{ \beta_1^2 \sin^2{\theta} +  \sin^2(\phi + \theta )/\gamma_1^2 \}^{1/2} }    </math>  |6-11}}
{{NumBlk2||<math>\quad\quad \cos{(\phi + \theta)} = \frac{ (\gamma_1 - 1) \sin{\theta} \cos{\theta} }{ \{ (\gamma_1 + 1)^2 \sin^2 \theta + 4 \cos^2 \theta  \}^{1/2} }    </math>  |6-12}}
For the symmetrical case in which <math> \phi = \theta</math> and {{tmath|1= \beta_2 = \beta_3 }}, ({{EquationNote|6-12}}) takes on the simpler form:<ref name="Champion_1932" group="p"/>
{{NumBlk2||<math>\quad\quad \cos{\theta} = \frac{\beta_1}{ \{ 2/\gamma_1 + 3 \beta_1^2 - 2 \}^{1/2} }    </math>  |6-13}}