Question
Part A
If a muon is traveling at 70% of the speed of light, how long does it take to decay in the observer's rest frame (i.e., what is the observed lifetime of the muon)?
Part B
If a muon is traveling at 99.9% the speed of light, how long will it take to decay in the observer's rest frame (i.e., what is the observed lifetime of the muon)?
Part C
Suppose that a cosmic-ray proton crashes into a nitrogen molecule in the upper atmosphere, 45 km above the earth's surface, producing a pion that decays into a muon. Assume that the muon has a downward velocity of 99.9943% the speed of light.
How far would the muon travel before it decayed, if there were no time dilation?
Part D
Now, let us consider the effects of time dilation. How far would the muon travel, taking time dilation into account?
I'm sorry that this is such a long problem. I will give Lifesaver if you please help!!!!
Explanation / Answer
(a) = 2.2*10^-6 s velocity v = 0.7 * c ==> v /c = 0.7 according to relativity time deliation ' = / sqrt ( 1 - (v/c)2) = 3.081*10-6 s (b) if velocity of moun v = 0.999 * c ==> v/ c = 0.999 ' = / sqrt ( 1 - (v/c)2) = 49.20*10-6 s ----------------------------------- (c) If there were no time delaiation effect life time of the moun = 2.2*10^-6 s velocity of the moun v = 0.999943*c = 0.999943*3*10^8 m/s = 2.999829*10^8 m/s distance travelled by the moun d = velocity * time = 2.999829*10^8 m/s )( 2.2*10^-6 s ) = 6.5996238*10^2 m ------------------------------------ (d) According to time delation v = 0.999943*c ==> v / c = 0.999943 ' = / sqrt ( 1 - (v/c)2) = 206. 05*10^-6 s d = velocity * time = ( 2.999829*10^8 m/s )( 206.05*10^-6 s ) = 618.11476545*10^2 m ' = / sqrt ( 1 - (v/c)2) = 49.20*10-6 s ----------------------------------- (c) If there were no time delaiation effect life time of the moun = 2.2*10^-6 s velocity of the moun v = 0.999943*c = 0.999943*3*10^8 m/s = 2.999829*10^8 m/s distance travelled by the moun d = velocity * time = 2.999829*10^8 m/s )( 2.2*10^-6 s ) = 6.5996238*10^2 m ------------------------------------ (d) According to time delation v = 0.999943*c ==> v / c = 0.999943 ' = / sqrt ( 1 - (v/c)2) = 206. 05*10^-6 s d = velocity * time = ( 2.999829*10^8 m/s )( 206.05*10^-6 s ) = 618.11476545*10^2 m