Muons are elementary particles that are formed high in the atmosphere by the interactions of cosmic rays with atomic nuclei up there. Muons are radioactive and have average lifetimes of about two-millionths of a second. Even though they travel at almost the speed of light, very few should be detected at sea level after traveling through the atmosphere—at least according to classical physics. Laboratory measurements, however, show that muons in great number do reach Earth’s surface. What is the explanation?

Solution 25E Introduction The phenomenon can be explained in two different ways, though the both explanations are equivalent. One explanation is the length contraction from the muon frame of reference and the other explanation is time dilation from the laboratory frame of reference. Explanation 1 The muon particles are travelling at a speed very close to light with respect to the laboratory frame of reference. So from the laboratory frame of reference, the lifetime of the muon will be dilated, that is the lifetime of the muon will appear longer that the proper lifetime. Now, since muons are taking the longer time to decay, the probability of finding them on the earth’s surface will be higher than as expected. Explanation 2 Now, let us consider the situation from the muon's frame of reference. From muon's frame of reference, the earth and hence the laboratory is moving at a speed very close to the speed of light. Hence, hence the distance between the muon and earth will appear smaller due to length contraction. Since the distance appears smaller now, a greater number of muons can reach the earth’s surface before decay. This explains why muons reach earth’s surface at a greater number than it is expected.