Enter values below, then submit to see muon lifetime, survival, distance, counts, graph, and downloadable reports here.
This calculator supports relativistic time dilation, decay survival, travel distance, lab-frame time, and particle-count estimates in one place.
| Scenario | Known Input | Observation | Initial Muons | Dilated Lifetime | Survival % | Remaining Muons |
|---|---|---|---|---|---|---|
| Cosmic-ray style example | γ = 10 | Distance = 5000 m | 100000 | 21.969811 μs | 46.628122% | 46628.12 |
| Short lab flight | β = 0.95 | Time = 5 μs | 50000 | 7.035675 μs | 49.132822% | 24566.41 |
γ = 1 / √(1 - β²)
β = v / c
KE = (γ - 1)mc²
τlab = γτ0
λ = 1 / τlab
t1/2 = ln(2) × τlab
N = N0e-t/τlab
Survival fraction = e-t/τlab
d = βct
Mean decay distance = βcτlab
Use proper lifetime for the muon rest frame. Use γ, β, or kinetic energy to move into the lab frame. Then apply the exponential decay law over the chosen observation time or distance.
- Select whether your known motion quantity is γ, β, or kinetic energy.
- Choose whether you want to analyze decay over time or over travel distance.
- Enter the proper lifetime, mass energy, and initial muon count.
- Fill in the active motion field and the active observation field.
- Press the calculate button to show the results section above the form.
- Review the graph, detailed table, and quick summary cards.
- Use the export buttons to save the result summary as CSV or PDF.
1) What does this calculator solve?
It estimates relativistic muon lifetime in the lab frame, travel distance, elapsed proper time, decay constant, half-life, survival fraction, and remaining particle count for a selected observation interval.
2) Why is the lab lifetime longer than the proper lifetime?
A moving muon experiences time dilation. Observers in the lab measure a longer lifetime by a factor of γ, while the muon’s own rest-frame lifetime stays at the proper value.
3) Should I enter β, γ, and energy together?
No. Choose one motion input mode only. The calculator uses the selected mode to derive the other motion values automatically and keeps the workflow simpler.
4) What is the survival fraction?
It is the fraction of muons expected to remain undecayed after the chosen lab-frame time. It comes from the exponential decay model, e-t/τlab.
5) When should I use distance mode?
Use distance mode when your experiment or problem gives a flight path length, detector spacing, or atmospheric travel distance instead of a direct time interval.
6) What does mean decay distance represent?
It is the average path length a muon covers before decaying in the lab frame. It equals βcτlab and helps compare detector placement or atmospheric penetration.
7) Can I change the default muon constants?
Yes. Both proper lifetime and muon mass energy are editable. That is useful for teaching, sensitivity checks, or testing alternative rounded values in worked examples.
8) What do the CSV and PDF buttons export?
They export the currently displayed summary metrics from the calculated result block, including lifetime, distance, survival, counts, and related decay quantities.