Problems are listed in approximate order of difficulty. A single dot (•) indicates straightforward problems involving just one main concept and sometimes requiring no more than substitution of numbers in the appropriate formula. Two dots (••) identify problems that are slightly more challenging and usually involve more than one concept. Three dots (•••) indicate problems that are distinctly more challenging, either because they are intrinsically difficult or involve lengthy calculations. Needless to say, these distinctions are hard to draw and are only approximate.
•• Muons are subatomic particles that are produced several miles above the earth’s surface as a result of collisions of cosmic rays (charged particles, such as protons, that enter the earth’s atmosphere from space) with atoms in the atmosphere. These muons rain down more-or-less uniformly on the ground, although some of them decay on the way since the muon is unstable with a proper half-life of about 1.5 μs. (1 μs = 10−6 s.) In a certain experiment a muon detector is carried in a balloon to an altitude of 2000 m, and in the course of 1 hour it registers 650 muons traveling at 0.99c toward the earth. If an identical detector remains at sea level, how many muons would you expect it to register in 1 hour? (Remember that after n half-lives the number of muons surviving from an initial sample of N0 is N0/2n, and don’t forget about time dilation.) This was essentially the method used in the first tests of time dilation, starting in the 1940’s.
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