Question

One of the many interesting fundamental particles in nature is the muon ?. This particle acts very much like a heavy electron and has a mass of 106MeV/c2, compared to the electron's mass of just 0.511MeV/c2. (Here we are using E=mc2 to obtain the mass in units of energy and the speed of light c). Unlike the electron, the muon has a finite lifetime, after which it decays into an electron and two very light neutrinos ( ?). We'll ignore the neutrinos throughout this problem. If the muon is at rest the characteristic time that it takes to decay is about 2.2?s (??=2.2×10?6s). Most of the time though, particles such as muons are not at rest and, if moving relativistically, their lifetimes will be increased by time dilation. In this problem we will explore some of these relativistic effects. Let's begin by looking at some muons moving at various speeds relative to a stationary observer.

Part A If a muon is traveling at 70% the speed of light, how long will it take to decay in the observer's rest frame (i.e., what is the observed lifetime)? Express your answer in microseconds to two significant figures. View Available Hint(s) ?? =   ?s   SubmitPrevious Answers Incorrect; Try Again; 4 attempts remaining 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? Express your answer in microseconds to two significant figures. ?? = 49•10?649{\cdot}10^{-6}   ?s   SubmitPrevious AnswersRequest Answer Incorrect; Try Again; 5 attempts remaining Part C What is the total energy E of a muon traveling at 99.9% the speed of light? Recall that the rest energy of the muon is approximately 106MeV. Note that 1GeV=1000MeV. Express your answer in billions of electron volts to three significant figures. View Available Hint(s) E =   GeV   Submit Cosmic rays are constantly raining down on the earth from outer space. (Figure 1)The majority of these cosmic rays are protons, and when they crash into the upper atmosphere, they can convert into particles called pions ( ?), which subsequently decay into muons with a characteristic lifetime of ??=2.6×10?8s (in their rest frame). These muons can then continue down toward the earth until they too decay (into electrons, which are so light that they stop very quickly in the atmosphere). Let's look at how time dilation affects these cosmic rays. 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 pion has a velocity of 99.99% the speed of light and decays into a muon, which, owing to kinematics, has a downward velocity of 99.9943% the speed of light. Part D How far would the pion travel before it decayed, if there were no time dilation? Express your answer in meters to two significant figures. d? =   m   SubmitRequest Answer Part E How far would the muon travel before it decayed, if there were no time dilation? Express your answer in meters to three significant figures. d? =   m   SubmitRequest Answer Part F Now, let's consider the effects of time dilation. How far would the pion actually travel before decaying? Express your answer in meters to three significant figures. d? =   m   SubmitRequest Answer Part G How far would the muon travel, taking time dilation into account? Express your answer in kilometers to two significant figures. d? =   km   SubmitRequest Answer Provide Feedback

Answer 1