Muons are made above a sea-level detector and fly straight down

PHYSICS · Multi-part · Fermi University Physics · Relativity & Gravitation · Particle Physics & Quantum Fields · Particles, interactions and experiment

Particle Physics, Ch Scales, units and the four-momentum

Problem

Muons are made

15.0\,\mathrm{km}

above a sea-level detector and fly straight down. Take

c\tau = 659\,\mathrm{m}

and

m_\mu c^2 = 105.66\,\mathrm{MeV}

, and suppose the muons lose no energy in the air, so the fraction surviving a ground-frame distance

x

e^{-x/L}

with

L = \beta\gamma c\tau

. ((a)) Find the smallest energy

E

for which at least half the muons reach the detector undecayed. ((b)) At these energies

\beta\gamma

is very nearly proportional to

E

. Use this to show, without a new calculation of

E

, that muons of twice the energy found in (a) survive with a fraction of

1/\sqrt{2}

, and explain why doubling the energy does not double the surviving fraction. ((c)) Find the smallest energy for which

90

per cent survive, and compare it with your answer to (a).

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