A.5 · IB Physics
Muons and length contraction
Release 300 muons at height H and watch how many reach the ground without relativity, with time dilation in the Earth frame, and with length contraction in the muon frame.
γ = 1 / √(1 − v²/c²)
Earth frame: t = H / v, N/N₀ = (½)^(t / γt½)
Muon frame: L = H / γ, t′ = L / v, N/N₀ = (½)^(t′ / t½)
Proper half-life t½ = 1.52 μs
Earth frame: t = H / v, N/N₀ = (½)^(t / γt½)
Muon frame: L = H / γ, t′ = L / v, N/N₀ = (½)^(t′ / t½)
Proper half-life t½ = 1.52 μs
- Speed
- γ
- Earth time t = H/v
- Half-life, Earth frame
- Atmosphere, muon frame L
- Muon time t′ = L/v
- Half-lives elapsed
- Surviving, no relativity
- Surviving, relativity
Try this
- At v = 0.9900c and H = 10 km, compare the two surviving fractions and the number of half-lives elapsed in each case.
- Set v = 0.9990c and check that t′ = L/v equals t/γ, so both frames predict the same surviving fraction.
- Double H to 20 km and predict the relativistic surviving fraction before pressing Reset and watching.
Common error: mixing frames — use the dilated half-life with the full height H in the Earth frame, or the proper half-life with the contracted thickness H/γ in the muon frame, never a contracted distance together with a dilated time.