1000 Solved Problems in Modern Physics

324 6 Special Theory of Relativity (c) If the alphas are to further break into neutrons and protons, then show that the overall ...

6.2 Problems 325 6.58 A particleAdecays at rest viaA→B+C. Find the total energy ofBin termsofthemassesofA,BandC. 6.59 Calculate ...

326 6 Special Theory of Relativity 6.68 A proton of momentumplarge compared with its rest massM, collides with a proton inside a ...

6.2 Problems 327 (1−cosθ)/c^2 , whereEis the total initial electron’s energy in the lab system. State when this approximation is ...

328 6 Special Theory of Relativity are the sole products of disintegration of a heavier object, what was the rest mass, velocity ...

6.2 Problems 329 6.97 Assume the decayK^0 →π++π−. Calculate the mass of the primary particle if the momentum of each of the seco ...

330 6 Special Theory of Relativity 6.108 A linear accelerator produces a beam of excited carbon atoms of kinetic energy 120 MeV. ...

6.2 Problems 331 6.114 A positron–electron pair production can occur in the interaction of a gamma ray with electron, viaγ+e−→e− ...

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 ...

6.3 Solutions 333 6.2 The transformation matrix is Λ= ⎡ ⎢ ⎢ ⎣ γ 00 iβγ 0100 0010 −iβγ 00 γ ⎤ ⎥ ⎥ ⎦ Setγ=coshαandβ=tanhα, so that ...

334 6 Special Theory of Relativity 6.4θ∗= 90 ◦ γ=γcγ∗= 1. 667 × 1. 038 = 1. 73 β=(1. 732 −1)^1 /^2 / 1. 73 = 0. 816 tanθ=sinθ∗/( ...

6.3 Solutions 335 ∂φ/∂x=γ(∂φ/∂x′−βc∂φ/∂t′)(9) ∂φ/∂t=γ(∂φ/∂t′−βc∂φ/∂x′) (10) ∂^2 φ ∂x^2 =γ^2 ∂^2 φ ∂x′^2 +(γ^2 β^2 c^2 ) ∂^2 cφ ∂ ...

336 6 Special Theory of Relativity or tanθμ∗= 1 /βcγc (3) tanθμ=sinθμ∗/γc(cosθμ∗+βc/βμ∗)= 1 /γc^2 βc(1+ 1 /β∗μ)(4) (Sinceθμ∗=π−θ ...

6.3 Solutions 337 E 1 ∗=E 2 ∗=(1/4)[(E+^2 −p+^2 +E−^2 −p−^2 +2(E+E−−p+p−cosθ)] E 1 ∗=E 2 ∗=(1/4)[m^2 +m^2 +2(E+E−−p+p−cosθ)] =(1 ...

338 6 Special Theory of Relativity Fig. 6.4Decay of a charged unstable particle into three pions Therefore,P=[ ∑ p(x)^2 +( ∑ p(y ...

6.3 Solutions 339 6.20τ 0 =τ/γ=τ(1−β^2 )^1 /^2 = 2. 9 × 10 −^6 ×(1− 0. 62 )^1 /^2 = 2. 32 × 10 −^6 s 6.21 (a) Time,τ=τ 0 /(1−β^2 ...

340 6 Special Theory of Relativity Soct 1 =d+βct 1 Solving fort 1 , t 1 =d/c(1−β) Forβ= 0 .6,γ= 1. 25 t 1 = 5 × 108 × 103 / 3 × ...

6.3 Solutions 341 and at the counter C Ic=IAexp (− 2 d/γβcτ)(2) ∴Ic= I^2 B IA = (470)^2 1000 = 221 (b) Take logarithm on both si ...

342 6 Special Theory of Relativity 6.38 (a)β=v/c= √ 3 2 γ=(1−β^2 )−^1 /^2 =(1− 3 /4)−^1 /^2 = 2 Total energy of the particle E=γ ...

6.3 Solutions 343 6.43 (a)γ= 1 +T/m= 1 + 1 , 000 / 940 = 2. 064 β=(γ^2 −1)^1 /^2 /γ=[(2.064)^2 −1]^1 /^2 / 2. 064 = 0. 87 (b)γ= ...