Special Relativity - Relativistic Addition of Velocities

An atom moving at speed 0.3c emits an electron along the same direction with speed 0.6c in the internal rest frame of the atom. The speed of the electron in the lab frame equal to

A. 0.25c
B. 0.51c
C. 0.66c
D. 0.76c
E. 0.90c

(GR9677 #37)

Solution:

Relativistic Addition of Velocities:

v = speed of the moving observer
u' = speed of object in the moving observer
u = speed of object in the moving observer relative to the rest observer

Given:
v = speed of atom = 0.3c
u' = speed of electron in the rest frame of the atom = 0.6c

u = speed of electron in the lab frame
= (0.6c + 0.3c) / [1 + (0.6)(0.3)]
= 0.9c /1.18 
= 90/118 
= 0.762c

Special Relativity - Relativistic Speed

The half life of π⁺ meson at rest is 2.5 × 10⁻⁸ second. A beam of  π⁺ mesons is generated at a point 15 meters from a detector. Only ½ of the π⁺mesons live to reach the detector. The speed of the π⁺ mesons is

A.¹/₂ c
B. √²/₅ c
C. ²/_√5 c
D. c
E. 2c

(GR9677 #48)

Solution:

v = v₀/γ 

with γ = (1 − ^v2/_c²)^−1/2

Given:

t₀ =  2.5 × 10⁻⁸s
L₀ = 15 m (measured in lab)

v₀ = L₀/t₀ = 15/(2.5 × 10⁻⁸) = 3/5 × 10⁹ =  (10/5)(3 × 10⁸) = 2c 

v₀/c = 2

v = v₀(1 − ^v2/_c²)^1/2
v² = v₀²(1 − ^v2/_c²) = v₀² − v²v₀²/c² = v₀² − 4v²
v² +  4v² = v₀²
5v² = 4c²
v = ²/_√5 c

Answer: C

Special Relativity - Speed of Photon

A π⁰ meson (rest-mass energy 135 MeV) is moving with velocity 0.8c  in the laboratory rest frame when it decays into two photons γ₁ and γ₂. In the π⁰ rest frame, γ₁ is emitted forward and γ₂ is emitted backward relative to the direction of flight. The velocity of γ₂ in the laboratory rest frame is

A. −1.0c 
B. −0.2c 
C. +0.8c 
D. +1.0c 
E. +1.8c 

(GR9277 #37)

Solution:

(B), (C), (E) are FALSE
Photon moves with the speed of light (1.0c)

Since γ₂ is emitted backward, the velocity of γ₂ is −1.0c

Answer: A

Special Relativity - Relativistic Speed

If a charged pion that decay in 10⁻⁸ second in its own rest frame is to travel 30 meters in the laboratory before decaying, the pion’s speed must be most nearly

A. 0.43 × 10⁸ m/s
B. 2.84 × 10⁸ m/s
C. 2.90 × 10⁸ m/s
D. 2.98 × 10⁸ m/s
E. 3.00 × 10⁸ m/s

(GR0177 #33)

Solution:

v = v₀/γ 
with γ = 1/(1 − ^v2/_c²)^½ 

Given:
t₀ = 10⁻⁸ second 
L₀ = 30 meters
v₀ = L₀/t₀ = 30 × 10⁸ = 10(3 × 10⁸) = 10c

v = v₀(1 − ^v2/_c²)^½
v² = v₀²(1 − ^v2/_c²) = (v₀² − v²v₀²/c²) = 100c² − 100v²
101v² = 100c²
v = (100/101)c = 0.99c = 0.99(3 × 10⁸) ≈ 2.98 × 10⁸ m/s

Answer: D

Special Relativity - Relativistic Addition of Velocity

A tube of water is traveling at ½c relative to the lab frame when a beam of light traveling in the same direction as the tube enters it. What is the speed of light in the water relative to the lab frame? (The index of refraction of water is 4/3)

A. ¹/₂ c
B. ²/₃ c
C. ⁵/₆ c
D. ¹⁰/₁₁ c
E. c

(GR0177 #80)

Solution:

Relativistic Addition of Velocities:



v = speed of the moving observer
u' = speed of object in the moving observer
u = speed of object in the moving observer relative to the rest observer

Given:
v = speed of water = ¹/₂ c
Index of refraction of water, n = ^c/_u'  = ⁴/₃
u' = speed of light in the water = ³/₄ c

u = speed of light in the water relative to the lab frame
= (¹/₂ c + ³/₄ c) / [1 + (¹/₂)(³/₄)]
= (⁵/₄ c)/(¹¹/8)
 = (⁵/₄ c)(⁸/11)
 = ¹⁰/₁₁ c

Answer: D