Modern Physics
Muon atmospheric survival
Muon-like β ≈ 0.995 — lab lifetime stretches enough to reach the ground.
Muon atmospheric survival — interactive Modern Physics simulation. Muon-like β ≈ 0.995 — lab lifetime stretches enough to reach the ground. Free browser-based virtual physics lab with live SI measurements and a guided tutorial.
Muon survival
Muon-like β ≈ 0.995 — lab lifetime stretches enough to reach the ground.
Cosmic-ray muons live ~2.2 μs at rest—too short to reach sea level classically. At β ≈ 0.995 lab clocks see dilated lifetime γτ, while the muon frame sees contracted atmosphere. Both descriptions agree on detection rate—a classic verification of relativity.
- Lab lifetime = γτ₀
- β ≈ 0.995
Investigation brief
Plan the question before you open the lab
The brief mirrors the prerendered page: driving question, competing predictions, variable roles, governing laws, setup, analysis and extension prompts remain visible and in this order.
Driving question
Muons created by cosmic rays high in the atmosphere have a proper lifetime of only about 2.2 microseconds — not nearly enough time, at nearly light speed, to reach the ground unaided. Yet muons are detected at sea level in large numbers. Does time dilation explain this?
Predictions to weigh
- No — muons must be created closer to the ground than expected.
- This has nothing to do with relativity.
- Yes — a fast-moving muon's decay clock runs slow from Earth's frame, extending its observed lifetime enough for it to reach the ground.
Variable roles
What you set:
- Speed fraction β = v/c
What you measure:
- Lorentz factor γ
- Dilated lifetime Δt (µs)
How the investigation runs
- Open the muon-time-dilation preset and press Reset. The muon's proper lifetime is fixed at 2.2 µs, speed β = 0.995.
- Enable the Lorentz factor and dilated-lifetime readouts.
- Set the muon's speed for each trial and record its dilated (Earth-frame) lifetime.
Governing equation
Muon Time Dilation — Δt = γ Δt₀
A muon moving at β = 0.995 (γ ≈ 10) has its 2.2 µs proper lifetime stretched to about 22 µs as observed from Earth — long enough, combined with its near-light speed, to traverse the atmosphere and reach detectors at sea level.
What the printable worksheet asks students to work out
- For one trial, compute γ = 1/√(1 − β²), then Δt = γ·(2.2 µs). Compare to the table. Confirm β = 0.995 gives roughly a tenfold lifetime extension.
- Without time dilation, even at nearly light speed a 2.2 µs proper lifetime lets a muon travel only about 660 m — far short of the ~15 km atmospheric height where most are created. Explain how the dilated lifetime you computed changes this picture.
Where this shows up beyond the lab
- From the muon's own reference frame, its lifetime is still just 2.2 µs — but the muon 'sees' the atmosphere length-contracted, so it doesn't need to travel as far. Explain why both explanations (Earth-frame time dilation, or muon-frame length contraction) predict the exact same experimental outcome: the muon reaches the ground.
- The 1941 Rossi-Hall experiment comparing muon counts at a mountaintop versus at sea level was one of the first direct experimental confirmations of special relativity's time dilation. Why might comparing counts at two different altitudes be a clever way to test this prediction?
- AP Physics 2 — Unit 15: Modern Physics
- IB Physics — A.5 Galilean and special relativity
- General High School Physics — Modern physics intro
- NGSS High School Physics — Wave-particle duality of light
- Welcome to Muon Time Dilation
- Select the relativity setup
- Press Play
- Muon survival in the lab
- Open the Properties panel
- You did it!
Open the interactive simulation to build the scene, press Play, and explore with live measurements and a guided tutorial.
Special Relativity: Time Dilation