# Explain why a muon created in the upper atmosphere by cosmic-ray collisions can be detected at Earth's surface, even though classical Newtonian mechanics predicts it should decay long before reaching the ground.

> IB DP Physics Higher Level (2023 syllabus) — A.5 Galilean and special relativity (HL only) · Explain · 4 marks

> Muons are created at an altitude of approximately 15 km when cosmic rays collide with atmospheric nuclei. In their own rest frame, muons have a mean lifetime of about 2.2 μs. At speeds close to c, classical mechanics predicts they travel only about 660 m before decaying — far less than 15 km.

## Mark scheme (4 marks)

1. In the Earth's reference frame, the muon's mean lifetime is extended (time dilation) by the Lorentz factor γ > 1, so the muon lives long enough to reach the surface.
2. In the muon's own rest frame, the distance from the upper atmosphere to the surface is length-contracted by a factor of 1/γ, making it short enough for the muon to cross within its proper lifetime.
3. Both frames give the same physical outcome (muon reaches the surface), demonstrating that special relativity is self-consistent: time dilation in one frame is equivalent to length contraction in the other.
4. Classical (Galilean) mechanics does not apply because the muon travels at a speed approaching c (v ≈ 0.998c), so the Lorentz factor γ ≫ 1 and relativistic effects are significant.

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