Exam-style question
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Using the time dilation equation, explain why a muon created high in the Earth's atmosphere can reach the surface before decaying, even though its rest lifetime is only 2.2 µs.
Model answer
What a good answer should say
- A muon moving at a speed close to c experiences time dilation, so its lifetime in the Earth's frame is Δt = γ Δτ, where Δτ = 2.2 µs is the proper lifetime.
- For γ ≈ 10, the dilated lifetime becomes about 22 µs, giving the muon enough time to travel the ~10 km distance to the surface before decaying.
- Thus, the relativistic increase in lifetime allows muons to be detected at sea level.
Explanation
Why this works
This answer shows application of the time dilation equation to a real-world scenario, linking the Lorentz factor to the increased lifetime and the distance travelled. It tests the student's ability to reason quantitatively about relativistic effects without performing a full calculation, which is appropriate for the learning objective.
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