AMU 2026 science PYQ — An unstable particle has a proper lifetime of . It moves through … | Mathem Solvex | Mathem Solvex
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AMU 2026 — science PYQ
AMU | science | 2026
An unstable particle has a proper lifetime of 2μs. It moves through a laboratory with a speed 0.98C. Approximately how far does the particle travel in the laboratory frame before decaying?
Choose the correct answer:
A.
0.6 km
B.
0.6 km
C.
3 km
(Correct Answer)
D.
9 km
Correct Answer:
3 km
Explanation
1. Given Data
Proper lifetime of the particle (Δt0) = 2μs=2×10−6 s
Speed of the particle (v) = 0.98C
Speed of light (C) ≈3×108 m/s
2. Concept of Time Dilation
According to the Special Theory of Relativity, a clock moving relative to an observer ticks slower. The lifetime of the particle measured in the laboratory frame (Δt) is dilated (extended) and is given by the formula:
Δt=1−C2v2Δt0
3. Step-by-Step Calculation
Step A: Calculate the Lorentz Factor (γ)
γ=1−(0.98)21
γ=1−0.96041
γ=0.03961
γ≈0.1991≈5.025
Step B: Calculate the Dilated Lifetime (Δt)
Δt=5.025×2×10−6 s
Δt≈10.05×10−6 s
Step C: Calculate the Distance Traveled in the Lab Frame (d)
The distance traveled before decaying is the speed multiplied by this dilated time interval:
d=v×Δt
d=(0.98×3×108 m/s)×(10.05×10−6 s)
d≈(2.94×108)×(10.05×10−6)
d≈29.547×102 m
d≈2955 m≈3 km
Correct Answer
The correct option is (c) 3 km.
Explanation
1. Given Data
Proper lifetime of the particle (Δt0) = 2μs=2×10−6 s
Speed of the particle (v) = 0.98C
Speed of light (C) ≈3×108 m/s
2. Concept of Time Dilation
According to the Special Theory of Relativity, a clock moving relative to an observer ticks slower. The lifetime of the particle measured in the laboratory frame (Δt) is dilated (extended) and is given by the formula:
Δt=1−C2v2Δt0
3. Step-by-Step Calculation
Step A: Calculate the Lorentz Factor (γ)
γ=1−(0.98)21
γ=1−0.96041
γ=0.03961
γ≈0.1991≈5.025
Step B: Calculate the Dilated Lifetime (Δt)
Δt=5.025×2×10−6 s
Δt≈10.05×10−6 s
Step C: Calculate the Distance Traveled in the Lab Frame (d)
The distance traveled before decaying is the speed multiplied by this dilated time interval:
