Why Are We Bombarded by Muons? Muons are unstable subatomic particles that decay to electrons with a mean lifetime of 2.2 μs. They are produced when cosmic rays bombard the upper atmosphere about 10 km above the earth's surface, and they travel very close to the speed of light. The problem we want to address is why we see any of them at the earth's surface. (a) What is the greatest distance a muon could travel during its 2.2 μs lifetime? (b) According to your answer in part (a), it would seem that muons could never make it to the ground. But the 2.2 μs lifetime is measured in the frame of the muon, and muons are moving very fast. At a speed of 0.999c, what is the mean lifetime of a muon as measured by an observer at rest on the earth? How far would the muon travel in this time? Does this result explain why we find muons in cosmic rays? (c) From the point of view of the muon, it still lives for only 2.2 μs, so how does it make it to the ground? What is the thickness of the 10 km of atmosphere through which the muon must travel, as measured by the muon? Is it now clear how the muon is able to reach the ground?

Verified step by step guidance

1. Step 1: For part (a), calculate the greatest distance a muon could travel during its 2.2 ms lifetime in its own frame of reference. Use the formula for distance: $d=vt$, where $v$ is the speed of the muon (close to the speed of light, $c$) and $t$ is the lifetime of the muon (2.2 ms). Substitute $v=c$ and $t=2.2\times 10⁻3$ seconds to find the maximum distance.
2. Step 2: For part (b), consider the effects of time dilation due to the muon's high speed (0.999c). Use the time dilation formula: $t_{\mathrm{obs}}=t_{\mathrm{muon}}/\sqrt{1-v^2/c^2}$, where $t_{\mathrm{obs}}$ is the observed lifetime on Earth, $t_{\mathrm{muon}}$ is the muon's proper lifetime (2.2 ms), and $v=0.999c$. Calculate the dilated lifetime and then use the formula $d=v{t}_{\mathrm{obs}}$ to find the distance traveled in this time.
3. Step 3: For part (c), consider the muon's perspective. Due to length contraction, the thickness of the atmosphere (10 km) is reduced. Use the length contraction formula: $L_{\mathrm{muon}}=L_{\mathrm{earth}}\sqrt{1-v^2/c^2}$, where $L_{\mathrm{earth}}$ is the thickness of the atmosphere (10 km) and $v=0.999c$. Calculate the contracted thickness of the atmosphere as seen by the muon.
4. Step 4: Compare the contracted thickness of the atmosphere (from the muon's perspective) to the maximum distance the muon can travel in its proper lifetime (2.2 ms). This explains how the muon is able to reach the Earth's surface despite its short lifetime in its own frame of reference.
5. Step 5: Summarize the results. From the Earth's frame, time dilation allows the muon to live longer and travel farther. From the muon's frame, length contraction reduces the thickness of the atmosphere, making it possible for the muon to reach the ground. Both perspectives are consistent with the principles of special relativity and explain why we observe muons at the Earth's surface.

Key Concepts

Muon Lifetime and Decay

Muons are unstable particles with a mean lifetime of 2.2 microseconds (ms) before they decay into electrons. This lifetime is a measure of how long a muon exists before it transforms into other particles. Understanding this concept is crucial for analyzing how far a muon can travel before decaying, especially when considering its speed and the effects of relativistic physics.

Relativity and Time Dilation

According to Einstein's theory of relativity, time is experienced differently for observers in different frames of reference. For a muon traveling at speeds close to the speed of light (0.999c), time appears to pass more slowly compared to an observer on Earth. This phenomenon, known as time dilation, allows muons to exist longer from the perspective of an Earth observer, enabling them to travel greater distances than their mean lifetime would suggest.

Length Contraction

Length contraction is another relativistic effect where the length of an object moving at relativistic speeds is measured to be shorter in the direction of motion from the perspective of a stationary observer. For the muon, the thickness of the atmosphere it must traverse appears less than 10 km when measured in its own frame of reference. This contraction helps explain how muons can reach the Earth's surface despite their short lifetime.