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

Nuclear & Particle Physics - Radioactive

A radioactive nucleus decays, with the activity shown in the graph above. The half-life of the nucleus is

A. 2 min
B. 7 min
C. 11 min
D. 18 min
E. 23 min

(GR9277 #26)

Solution:

From the graph: at t = 0, N ≈ 6 × 10³

Half life, N_½ = 3 × 10³ → t = 7

Answer: B

Special Relativity - Time Dilation

Tau leptons are observed to have an average half-life of Δt₁ in the frame S₁ in which the leptons are at rest. In an inertial frame S₂, which is moving at a speed v₁₂ relative to S₁, the leptons are observed to have an average half-life of Δt₂. In another inertial reference frame S₃, which is moving at a speed v₁₃ relative to S₁ and v₂₃ relative to S₂, the leptons have an observed half-life of Δt₃. Which of the following is a correct relationship among two of the half lives, Δt₁, Δt₂, and Δt₃.

A.

B.

C.

D.

E.

(GR9277 #38)

Solution:

Time Dilation:

Δt_j =  γ_ijΔt_i



Δt_i = time in a rest frame
Δt_j = time in a moving frame

In this problem, Δt₁ = time in a rest frame
Time dilation only relates to time in a rest frame Δt₁ with time in a moving frame, Δt₂ or Δt₃

The only right answers:
Δt₂ =  γ₁₂Δt₁
Δt₃ =  γ₁₃Δt₁

(C) and (D) are FALSE (no Δt₁ in the equation)
(E) is FALSE (relates v₂₃ with Δt₁ and Δt₂)

(A) FALSE
Δt₂ = Δt₁ [1 − (v₁₂)²/c²]^½ = Δt₁ (1/γ₁₂) → Δt₁ = γ₁₂Δt₂ 

(B) TRUE
Δt₁ = Δt₃ [1 − (v₁₃)²/c²]^½ = Δt₃ (1/γ₁₃) → Δt₃ = γ₁₃Δt₁ 

Answer: B

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

Nuclear & Particle Physics - Radioactive

A sample of radioactive nuclei of a certain element can decay only by -emission and -emission. If the half-life for -emission is 24 minutes and that for -emission is 36 minutes, the half-life for sample is

A. 30 minutes
B. 24 minutes
C. 20.8 minutes
D. 14.4 minutes
E. 6 minutes

(GR0177 #66)

Solution:

Total half-time for 2 or more processes:



For the sample with and emissions:

 

Answer: D