How Fast Must the Muon Travel?

Physics /

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How Fast Must the Muon Travel?

Relativity / Time Dilation - 15-20 min

Muons have a proper mean lifetime of $\tau_0=2.2\,\mu\mathrm{s}$ , measured in their own rest frame. Consider many muons moving at the same constant speed $v$ in a straight line through an inertial laboratory, with no collisions or energy loss.

What fraction of the speed of light must $v$ be for their mean distance travelled before decay, measured in the laboratory, to be $d=10\,\mathrm{km}$ ? Use $c=3.00\times10^8\,\mathrm{m\,s^{-1}}$ and special relativity. The given lifetime is a mean lifetime, not a half-life.

Assumptions

Use special relativity and constant straight-line speed in an inertial laboratory frame. Neglect interactions, energy loss and gravity. Decay is random: the requested distance is the mean over many identical muons, not a guaranteed travel distance.