Incoming high-energy cosmic-ray protons strike Earth's upper atmosphere and collide with the nuclei of atmospheric atoms, producing a downward-directed shower of particles, including (among much else) the pions \(\pi^{+}, \pi^{-}\), and \(\pi^{0}\). The charged pions decay quickly into muons and neutrinos,

The muons are themselves unstable, with a half-life of 1.52 microseconds in their rest frame, decaying into electrons or positrons and additional neutrinos. Nearly all muons are created at altitudes of about \(15 \mathrm{~km}\) and more, and then those that have not yet decayed rain down upon the earth's surface. Consider muons with speeds \((0.995 \pm 0.001) c\), with their numbers measured on the ground and in a balloon-lofted experiment at altitude \(12 \mathrm{~km}\).

(a) How far would such muons descend toward the ground in one half-life if there were no time dilation?

(b) What fraction of these muons observed at \(12 \mathrm{~km}\) would reach the ground?

(c) Now take into account time dilation, in which the muon clocks run slow, extending their half-lives in the frame of the earth. Then what fraction of those observed at \(12 \mathrm{~km}\) would make it to the ground? (Such experiments supported the fact of time dilation.)

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