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EXERCISE 24
Muons are subatomic particles that sometimes strike Earth's surface. One way they can be produced is via cosmic ray interactions in the upper atmosphere. They exist for only 0.0001 second before they decay. Let's say some muons are produced at an altitude of 100 km - that is, 100000 meters from Earth's surface. They head toward the surface at 98% the speed of light. (a) During the lifetime of a muon described above, what is the maximum distance light can travel? Since nothing travels faster than light, why would this calculation lead you to conclude that no un-decayed muons would ever reach Earth's surface from an altitude of 100 km? (b) However, un-decayed muons produced at an altitude of 100 km HAVE been detected at Earth's surface. The reason is that muons are subject to special relativistic time dilation and length contraction. Either point of view explains the paradox! Let's do time dilation first. During the time the muon experiences 0.0001 second, how much time would you, standing on Earth's surface, experience? (Remember that t' = t * sqrt(1 - v^2/c^2) ) (c) Using the "dilated time" of you observing the muon, how far could the muon go before it decays? So, would the muon decay before or after it reaches Earth's surface? (d) Now let's do length contraction. You, standing on Earth's surface, see the muon being produced 100 km away. How far away does Earth's surface appear to the muon? (Remember that L' = L * sqrt(1 - v^2/c^2) ) (e) Using the "contracted length" of the muon observing Earth, how much time would it take for the muon to get to Earth's surface? Will it decay before or after it gets there? |