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:
Δtj = γijΔti
Δti = time in a rest frame
Δtj = 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
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