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Chapter 28: Problem 1

ssm A particle known as a pion lives for a short time before breaking apart into other particles. Suppose that a pion is moving at a speed of \(0.990 c\), and an observer who is stationary in a laboratory measures the pion's lifetime to be \(3.5 \times 10^{-8}\) s. (a) What is the lifetime according to a hypothetical person who is riding along with the pion? (b) According to this hypothetical person, how far does the laboratory move before the pion breaks apart?

Short Answer

Expert verified
a) 2.48 脳 10鈦烩伖 s; b) 7.37 m

Step by step solution

01

Identify Given Values and Required Quantities

We are given the speed of the pion as \( v = 0.990c \), and the laboratory frame lifetime \( t' = 3.5 \times 10^{-8} \) seconds. We need to find the pion's lifetime in its own rest frame \( t \) (Part a) and the distance traveled by the laboratory frame in the pion's rest frame (Part b).
02

Calculate the Proper Lifetime (Part a)

The proper lifetime \( t \) is the time experienced by an observer moving with the pion. We use the time dilation formula: \[ t = \frac{t'}{\sqrt{1 - \left(\frac{v}{c}\right)^2}} \]Substitute \( v = 0.990c \) and \( t' = 3.5 \times 10^{-8} \) s:\[ t = \frac{3.5 \times 10^{-8}}{\sqrt{1 - (0.990)^2}} = \frac{3.5 \times 10^{-8}}{\sqrt{1 - 0.9801}} \]\[ t = \frac{3.5 \times 10^{-8}}{\sqrt{0.0199}} \]\[ t \approx 2.48 \times 10^{-9} \text{ s} \]So, the lifetime according to the pion is approximately \( 2.48 \times 10^{-9} \) seconds.
03

Calculate the Distance (Part b)

In the pion's rest frame, the laboratory moves with speed \( v = 0.990c \) for the proper time \( t \). The distance \( d \) traveled by the laboratory in this time is given by:\[ d = v \cdot t \]Substitute \( v = 0.990c \) and \( t = 2.48 \times 10^{-9} \) s:\[ d = 0.990c \times 2.48 \times 10^{-9} \]Assume \( c = 3.00 \times 10^{8} \) m/s, then:\[ d = 0.990 \times 3.00 \times 10^{8} \times 2.48 \times 10^{-9} \]\[ d \approx 0.990 \times 744 \times 10^{-1} \]\[ d \approx 736.56 \times 10^{-1} \]\[ d \approx 7.37 \text{ m} \]Thus, the laboratory moves approximately 7.37 meters before the pion decays.

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Key Concepts

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

Special Relativity
Special relativity is a theory proposed by Albert Einstein that transforms our understanding of time and space. At its core, it suggests that the laws of physics are the same for all non-accelerating observers. It also introduces the idea that the speed of light is constant in all frames of reference. One of its intriguing outcomes is time dilation.
  • Special relativity combines space and time into a single continuum called space-time.
  • In relativity, time can "stretch" depending on the observer's velocity relative to the speed of light.
  • It fundamentally changes how we measure time and distance when bodies move at significant fractions of the speed of light.
Understanding these concepts is essential to tackle problems involving high-speed particles, such as the pions discussed in our exercise.
Pion Lifetime
Pions are subatomic particles that exist fleetingly before decaying into other particles. The lifetime of a pion is the duration it exists before undergoing this transformation: typically in the magnitude of several billionths of a second. This lifetime is affected by the pion's speed due to special relativity. The faster the pion moves, the longer it appears to last for a stationary observer.
  • Pions are important in studying particle physics and the interactions that occur at high energies.
  • Its lifetime, as measured in labs, experiences time dilation; it appears longer than the actual lifetime experienced by the pion itself.
This idea of observing different lifespans is a crucial part of understanding relativistic effects on moving entities.
Proper Time
Proper time is a term used in relativity to define the time measured by an observer traveling with the object in question. It's the shortest time interval between two events, like the start and end of a pion's life. Proper time helps in understanding time dilation, as it's the reference time against which the dilated or "stretched" time is measured from a different frame.
  • Proper time reflects the time duration as experienced in the pion's own rest frame.
  • The contrast between proper time and observer time highlights the effects of high-speed travel on time perception.
Through proper time, we can compute how much an observer's clock differs when moving rapidly compared to a stationary one.
Speed of Light
The speed of light, denoted by \( c \), is approximately \( 3.00 \times 10^{8} \) meters per second in a vacuum. It is the ultimate speed limit in our universe and features prominently in Einstein's equations of special relativity. The consistent speed of light serves as a fundamental aspect of time dilation calculations.
  • Nothing can travel faster than light, making it a constant in all frames of reference.
  • The pioneering ideas of relativity pivot around this constant, influencing mass, time, and space perceptions.
By analyzing particle motion near light speed, experiments affirm relativity's predictions, such as the nature of time experienced by fast-moving pions.

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Most popular questions from this chapter

ssM A spacecraft approaching the earth launches an exploration vehicle. After the launch, an observer on earth sees the spacecraft approaching at a speed of \(0.50 \mathrm{c}\) and the exploration vehicle approaching at a speed of \(0.70 \mathrm{c}\). What is the speed of the exploration vehicle relative to the spaceship?

A Martian leaves Mars in a spaceship that is heading to Venus. On the way, the spaceship passes earth with a speed \(v=0.80 \mathrm{c}\) relative to it. Assume that the three planets do not move relative to each other during the trip. The distance between Mars and Venus is \(1.20 \times 10^{11} \mathrm{m},\) as measured by a person on earth. (a) What does the Martian measure for the distance between Mars and Venus? (b) What is the time of the trip (in seconds) as measured by the Martian?

Ssm How fast must a meter stick be moving if its length is observed to shrink to one-half of a meter?

There are many astonishing consequences of special relativity, two of which are time dilation and length contraction. Problem 50 reviews these important concepts in the context of a golf game in a hypothetical world where the speed of light is only a little faster than that of a golf cart. Other important consequences of special relativity are the equivalence of mass and energy, and the dependence of kinetic energy on the total energy and on the rest energy. Problem 51 serves as a review of the roles played by mass and energy in special relativity. Imagine playing golf in a world where the speed of light is only \(c=3.40 \mathrm{m} / \mathrm{s} .\) Golfer \(\mathrm{A}\) drives a ball down a flat horizontal fairway for a distance that he measures as \(75.0 \mathrm{m}\). Golfer \(\mathrm{B}\), riding in a cart, happens to pass by just as the ball is hit (see the figure). Golfer A stands at the tee and watches while golfer \(\mathrm{B}\) moves down the fairway toward the ball at a constant speed of \(2.80 \mathrm{m} / \mathrm{s} .\) Concepts: (i) Who measures the proper length of the drive, and who measures the contracted length? (ii) Who measures the proper time interval, and who measures the dilated time interval? Calculations: (a) How far is the ball hit according to golfer \(\mathrm{B} ?\) (b) According to each golfer, how much time does it take golfer \(B\) to reach the ball?

The distance from earth to the center of our galaxy is about 23000 ly \(\left(1 \mathrm{ly}=1\right.\) light-year \(\left.=9.47 \times 10^{15} \mathrm{m}\right),\) as measured by an earth-based observer. A spaceship is to make this journey at a speed of \(0.9990 \mathrm{c}\). According to a clock on board the spaceship, how long will it take to make the trip? Express your answer in years \(\left(1 \mathrm{yr}=3.16 \times 10^{7} \mathrm{s}\right)\)
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