Problem

Part 1

Muons are generated by interactions of cosmic rays with the upper atmosphere of the Earth.

Determine the minimum energy needed for a muon to reach sea level. The thickness of the atmosphere is roughly . The lifetime in the muon’s rest frame is . The rest mass of a muon is .

Part 2

Determine the speed of the muons as a multiple of the speed of light .

Muons.jpg

Solution

Part 1

Let be the thickness of the atmosphere according to the muon's moving frame.

Hence,

.

To calculate , use the lifetime of the muon.

Divide the thickness of the atmosphere with the thickness according to the muon's frame to obtain the gamma factor.

Therefore the minimum energy needed for a muon to reach sea level is

or

.

Part 2

From part 1 it is found that ; therefore,

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