In a collision process a particle of mass m₂, at rest in the laboratory, is struck by a particle of mass m₁ momentum P_LAB and total energy E_LAB. In the collision the two initial particles are transformed into two others of mass m₃ and m₄. The con-figurations of the momentum vectors in the center of momentum (cm) frame (traditionally called the center-of-mass frame) and the laboratory frame are shown in the figure.

(a) Use invariant scalar products to show that the total energy W in the cm frame has its square given by

W² = m₁² + m₂² + 2m₂E_LAB

and that the cms 3-momentum p' is

P = m₂P_LAB/W

(b) Show that the Lorentz transformation parameters ?_cm and ?_cm describing the velocity of the cm frame in the laboratory are

?_cm = P_LAB/m₂ + E_LAB,? ? ? ? ? ? ???_cm = m₂ + E_LAB/W

(c) Show that the results of parts a and b reduce in the non-relativistic limit to the familiar expressions,

Transcribed Image Text:

Pa m2 m3 my my PLAB -p" my m4 m4 Laburatury frame em frame