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At heights of 10 to 60 kilometers above the surface of the Earth, cosmic rays continually strike nuclei of oxygen and nitrogen atoms in the upper reaches of the atmosphere and produce muons (elementary particles of a mass about 207 times that of the electron). Some of the muons move vertically downward with a speed nearly that of light. Follow one of the muons on its way down. In a given sample of muons, half will decay to other elementary particles in 1.5 microseconds (1.5 x 10-6 seconds), as measured in a reference frame in which they are at rest. Half of the remainder decay in the next 1.5 microseconds, and so on. Analyze the results of this decay as observed in two different frames. Idealize the rather complicated details of the actual experiment to the following roughly equivalent situation: All muons are produced at the same height (45 kilometers); all muons have the same speed; all travel straight down; none are lost to collisions with air molecules on the way down. • Approximately how long a time will it take these muons to reach the surface of the Earth, as measured in the Earth frame? . If the decay time were the same for Earth observers as for an ob- server travelling with the muons (that is, ignoring time dilation), approximately how many half lives would have passed by the time the muons reach the surface?

At heights of 10 to 60 kilometers above the surface of the Earth, cosmic rays continually strike nuclei of oxygen and nitrogen atoms in the upper reaches of the atmosphere and produce muons (elementary particles of a mass about 207 times that of the electron). Some of the muons move vertically downward with a speed nearly that of light. Follow one of the muons on its way down. In a given sample of muons, half will decay to other elementary particles in 1.5 microseconds (1.5 x 10-6 seconds), as measured in a reference frame in which they are at rest. Half of the remainder decay in the next 1.5 microseconds, and so on. Analyze the results of this decay as observed in two different frames. Idealize the rather complicated details of the actual experiment to the following roughly equivalent situation: All muons are produced at the same height (45 kilometers); all muons have the same speed; all travel straight down; none are lost to collisions with air molecules on the way down. • Approximately how long a time will it take these muons to reach the surface of the Earth, as measured in the Earth frame? . If the decay time were the same for Earth observers as for an ob- server travelling with the muons (that is, ignoring time dilation), approximately how many half lives would have passed by the time the muons reach the surface?

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At heights of 10 to 60 kilometers above the surface of the Earth, cosmic rays continually strike nuclei of oxygen and nitrogen atoms in the upper reaches of the atmosphere and produce muons (elementary particles of a mass about 207 times that of the electron). Some of the muons move vertically downward with a speed nearly that of light. Follow one of the muons on its way down. In a given sample of muons, half will decay to other elementary particles in 1.5 microseconds (1.5 × 10-6 seconds), as measured in a reference frame in which they are at rest. Half of the remainder decay in the next 1.5 microseconds, and so on. Analyze the results of this decay as observed in two different frames. Idealize the rather complicated details of the actual experiment to the following roughly equivalent situation: All muons are produced at the same height (45 kilometers); all muons have the same speed; all travel straight down; none are lost to collisions with air molecules on the way down. ● Approximately how long a time will it take these muons to reach the surface of the Earth, as measured in the Earth frame? • If the decay time were the same for Earth observers as for an ob- server travelling with the muons (that is, ignoring time dilation), approximately how many half lives would have passed by the time the muons reach the surface?
Transcribed Image Text:At heights of 10 to 60 kilometers above the surface of the Earth, cosmic rays continually strike nuclei of oxygen and nitrogen atoms in the upper reaches of the atmosphere and produce muons (elementary particles of a mass about 207 times that of the electron). Some of the muons move vertically downward with a speed nearly that of light. Follow one of the muons on its way down. In a given sample of muons, half will decay to other elementary particles in 1.5 microseconds (1.5 × 10-6 seconds), as measured in a reference frame in which they are at rest. Half of the remainder decay in the next 1.5 microseconds, and so on. Analyze the results of this decay as observed in two different frames. Idealize the rather complicated details of the actual experiment to the following roughly equivalent situation: All muons are produced at the same height (45 kilometers); all muons have the same speed; all travel straight down; none are lost to collisions with air molecules on the way down. ● Approximately how long a time will it take these muons to reach the surface of the Earth, as measured in the Earth frame? • If the decay time were the same for Earth observers as for an ob- server travelling with the muons (that is, ignoring time dilation), approximately how many half lives would have passed by the time the muons reach the surface?
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How can I calculate the following: 

  • Therefore what fraction of those created at a height of 45 kilo- meters would remain once they reach the surface if the Earth?

(Mass of muon = 207 me ; Speed of muon, v = 3x10^8m/s ; Distance traveled, h = 45km ; Half-life of muons, t(half) = 1.5x10^-6s)

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