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 Mc
2
= 2Mc
2
(b) Distance traveled on an average
d = γβct 0 = 2 ×
√
3
2
cτ =
√
3 cτ
(c) The sphere will shrink in the direction of motion but will not in the transverse direction. Consequently, its shape would appear as that of a spheroid
as shown in Fig. 6.5
Fig. 6.5
6.39 ν
=
ν−u
1−uν/c 2
put v = c
ν
=
c−u
1−uc/c 2 =
c−u
1−u/c
= c
6.3.3 Mass, Momentum, Energy
6.40 (a) m = m 0 γ
γ = τ/τ 0 = 6.6 × 10
−6
/2.2 × 10
−6
= 3
m = 3 × 207 = 621 m e
(b) T = (γ − 1)m 0 c
2
= (3 − 1)(207 × 0.51)
= 211 MeV
(c) Total energy E = mc
2
= 621m e c
2
= 621 × 0.511 = 3.173 MeV
p = β E/c
β = (γ
2
− 1)
1/2
/γ = (3
2
− 1)
1/2
/3 = 0.9428
p = (0.9428)(317.3)/c
= 299 MeV/c
6.41 E = m 0 c
2
= (1 × 10
−3 kg)(3 × 10
8 )
2
= 9 × 10
13 J
6.42 T = (γ − 1)m = m
γ = 2
β = (γ
2
− 1)
1/2
/γ = 0.866
The result is independent of the mass of the particle.