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The half-life of stationary muons is measured to be \(1.6\) microseconds. Half of any initial number of stationary muons decays in one half-life. Cosmic rays colliding with atoms in the upper atmosphere of the Earth create muons, some of which move downward toward the Earth's surface. The mean lifetime of high- speed muons in one such burst is measured to be 16 microseconds. (a) Find the speed of these muons relative to the Earth. (b) Moving at this speed, how far will the muons move in one half-life? (c) How far would this pulse move in one half-life if there were no relativistic time stretching? (d) In the relativistic case, how far will the pulse move in 10 half-lives? (e) An initial pulse consisting of \(10^{8}\) muons is created at a distance above the Earth's surface given in part (d). How many will remain at the Earth's surface? Assume that the pulse moves vertically downward and none are lost to collisions. (Ninety-nine percent of the Earth's atmosphere lies below \(40 \mathrm{~km}\) altitude.)

Step by step solution

01

- Relativistic Time Dilation Formula

Use the formula for time dilation: \ \( t' = \frac{t}{\sqrt{1 - \frac{v^2}{c^2}}} \), where t' is the given mean lifetime of the moving muons (16 microseconds), t is the rest lifetime (1.6 microseconds), v is the speed of the muons, and c is the speed of light.

02

- Solve for Speed

Rearrange the equation to solve for v: \ \[ \frac{t'}{t} = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}} \] \ \( 10 = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}} \) \ \( 1 - \frac{v^2}{c^2} = \frac{1}{100} \) \ \( v^2 = \frac{99}{100} c^2 \) \ Hence, \ \( v = \sqrt{\frac{99}{100}} c = 0.995 \cdot c \).

03

- Distance in One Half-Life (Relativistic)

Using the speed found (0.995c) and the half-life (1.6 microseconds): \ \( Distance = v \times t = 0.995c \times 1.6 \times 10^{-6} s = 0.995 \times 3 \times 10^{8} m/s \times 1.6 \times 10^{-6} s = 477 meters. \)

04

- Distance in One Half-Life (Non-Relativistic)

Using the rest half-life (1.6 microseconds): \ \( Distance = v \times t = 0.995c \times 1.6 \times 10^{-6} s = 0.995 \times 3 \times 10^{8} m/s \times 1.6 \times 10^{-6} s = 480 meters. \)

05

- Distance in 10 Half-Lives (Relativistic)

Using the speed found (0.995c) and the time for 10 half-lives (16 microseconds): \ \( Distance = v \times t = 0.995c \times 16 \times 10^{-6} s = 0.995 \times 3 \times 10^{8} m/s \times 16 \times 10^{-6} s = 4770 meters. \)

06

- Number of Muons Reaching Earth

The number of muons remaining after 1 half-life is halved, thus after 10 half-lives: \ \( N = 10^8 \times \left(\frac{1}{2}\right)^{10} = 97656.25. \)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

time dilation

Time dilation is one of the fascinating predictions of Einstein's theory of relativity. It describes how time appears to slow down for objects moving at speeds close to the speed of light relative to an observer. This effect is crucial for understanding how muons, which are created high in Earth's atmosphere, can reach the surface before decaying.
In our case, the measured mean lifetime of high-speed muons was significantly longer (16 microseconds) than that for stationary muons (1.6 microseconds). Using the time dilation formula, we find this relation:
\[ t' = \frac{t}{\sqrt{1 - \frac{v^2}{c^2}}} \] Here, t' is the dilated time, t the proper time, v the speed of the muons, and c the speed of light.
This phenomenon allows fast-moving muons, which would otherwise decay rapidly, to exist longer and reach the Earth's surface.

half-life

Half-life is the time taken for half of a given number of unstable particles to decay. For stationary muons, this is measured at 1.6 microseconds. The concept of half-life is helpful in understanding decay processes across various timescales and remains constant regardless of the number of initial particles.
In our exercise, we deal with moving muons created by cosmic rays, which have a relativistically extended mean lifetime of 16 microseconds. This calculated time is ten times longer than the rest lifetime, demonstrating the significant effect of time dilation.

relativity

Relativity, proposed by Einstein, revolutionized our understanding of space, time, and motion. One of its profound implications is that time is not absolute; it can dilate or contract depending on the relative velocity of observers.
In the problem of muon decay, relativity allows us to understand why muons, despite their short rest-lifetime, survive long enough to reach Earth's surface. By using the time dilation formula, we can determine that at high speeds, the muon's effective lifetime extends, allowing them to travel greater distances before decaying.

muons

Muons are subatomic particles similar to electrons but with a much greater mass. These particles are unstable and decay into electrons and neutrinos via weak interaction. Muons are produced through cosmic rays colliding with atoms in the atmosphere, which then travel downwards at substantial speeds.
Although muons decay quickly—within 1.6 microseconds at rest—due to relativistic effects, their observed lifetimes when moving can be much longer. This extended lifetime allows muons created high in the atmosphere to be detected at the Earth's surface, providing a practical demonstration of time dilation.

cosmic rays

Cosmic rays are highly energetic particles originating from outer space that strike the Earth constantly. When these rays collide with atoms in the upper atmosphere, they create showers of secondary particles, including muons.
These secondary muons travel towards the Earth's surface at near-light speeds. Despite their short half-life in a stationary state, the relativistic effects prolong their lifetimes as observed on Earth. The study of these muons helps scientists understand cosmic ray properties and demonstrates relativistic principles.