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 · View as Markdown

Written & reviewed by James Millett — Biology (Imperial College London), PGCE Science (University of Cambridge).

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.

Model answer (4 marks)

In the Earth frame the muon’s mean lifetime is dilated by the Lorentz factor γ>1, so it survives long enough to travel the 15 km to the ground.
In the muon’s rest frame the 15 km distance is length‑contracted to 15 km/γ, which is short enough that the muon can cross it within its proper lifetime of 2.2 µs.
Both descriptions predict the same physical outcome – the muon reaches the surface – showing that special relativity is self‑consistent: time dilation in one frame equals length contraction in the other.
Classical (Galilean) mechanics fails because the muon’s speed is ≈0.998c, giving γ≫1 and making relativistic effects essential.

Examiner tips

  • Use the exact terms ‘time dilation’, ‘length contraction’, ‘Lorentz factor γ’ and ‘proper lifetime’.
  • Show the two frames explicitly and state that both give the same result.
  • Mention that v≈0.998c makes γ≫1, so relativity is required.

Common mistakes

  • Confusing the muon’s lifetime with the Earth‑frame lifetime; not mentioning the Lorentz factor.
  • Using Galilean time dilation or length contraction incorrectly.

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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