332 6 Special Theory of Relativity
6.124 For the reactionp+p→p+Λ+K+calculate the threshold energy and
the invariant mass of the system at threshold energy. The rest energies of the
p,ΛandK+are respectively 938, 1,115 and 494 MeV
[University of London 1969]
6.125 TheΩ−has been produced in the reaction
K−+p→K^0 +K++Ω−
What is the minimum momentum of theK−in the Laboratory for this reac-
tion to proceed assuming that the target proton is at rest in the laboratory?
Assume that aΩ−is produced in the above reaction with this momentum
ofK−. What is the probability thatΩ−will travel 3 cm in the Lab before
decaying? You may ignore any likelihood of theΩ−interacting. The rest
energies ofK+,K−,K^0 ,P,Ω−are 494, 494, 498, 938, 1,675 MeV, respec-
tively. Lifetime ofΩ−= 1. 3 × 10 −^10 s.
[University of Bristol 1967]
6.126 Assuming that the nucleons in the nucleus behave as particles moving inde-
pendently and contained within a hard-walled box of volume (4/3)πR^3 where
Ris the nuclear radius, calculate the maximum Fermi momentum for a pro-
ton in 29 Cu^63 the nuclear radius being 5.17 fm.
A proton beam of kinetic energy 100 MeV (momentum 570.4 MeV/c) is
incident on a target 29 Cu^63. Would you expect any pions to be produced by
the reactionp+p→d+πfrom protons within the nucleus (neglect the
binding energy of nucleons in the copper, i.e assume a head-on collision with
a freely moving proton having maximum Fermi momentum and calculate the
total energy in the CMS of thep+pcollision). The binding energy of the
deuteron is 2.2 MeV
[University of Bristol 1969]
6.3 Solutions..................................................
6.3.1 LorentzTransformations............................
6.1t 1 =γ(t 1 ′+vx 1 ′/c^2 )
t 2 =γ(t 2 ′+vx 2 ′/c^2 )
t 2 −t 1 =γ(t 2 ′−t 1 ′)+γv(x 2 ′−x 1 ′)/c^2
= 0 +γν(x 2 ′−x 1 ′)/c^2
10 −^6 =γβ× 900 /c
γβ= 10 −^6 × 3 × 108 / 900 = 1 / 3
β= 0 .316. The velocity ofS′is 0.316cwith respect toSalong the positive
direction.