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Chapter 37: Problem 5

The negative pion \(\left(\pi^{-}\right)\) is an unstable particle with an average lifetime of \(2.60 \times 10^{-8} \mathrm{s}\) (measured in the rest frame of the pion). (a) If the pion is made to travel at very high speed relative to a laboratory, its average lifetime is measured in the laboratory to be \(4.20 \times 10^{-7} \mathrm{s}\) . Calculate the speed of the pion expressed as a fraction of \(c .\) (b) What distance, measured in the laboratory, does the pion travel during its average lifetime?

Short Answer

Expert verified
The pion's speed is approximately 0.998c, and it travels roughly 125 meters.

Step by step solution

01

Identify Known Values

We are given two average lifetimes: the rest lifetime of the pion, \( t_0 = 2.60 \times 10^{-8} \text{s} \), and the dilated lifetime observed in the laboratory, \( t = 4.20 \times 10^{-7} \text{s} \). Additionally, we need to find the speed of the pion as a fraction of the speed of light, \( c \).
02

Use Time Dilation Formula

The time dilation formula is given by:\[t = \frac{t_0}{\sqrt{1 - \frac{v^2}{c^2}}}\]Where \( t \) is the dilated lifetime, \( t_0 \) is the proper lifetime, and \( v \) is the speed of the particle. Rearranging for \( v \):\[\sqrt{1 - \frac{v^2}{c^2}} = \frac{t_0}{t}\]
03

Solve for the Speed of the Pion

Square both sides to remove the square root:\[1 - \frac{v^2}{c^2} = \left(\frac{t_0}{t}\right)^2\]Rearrange to solve for \( \frac{v^2}{c^2} \):\[\frac{v^2}{c^2} = 1 - \left(\frac{t_0}{t}\right)^2\]Calculate:\[\frac{t_0}{t} = \frac{2.60 \times 10^{-8}}{4.20 \times 10^{-7}} \approx 0.0619\]\[\left(\frac{t_0}{t}\right)^2 = (0.0619)^2 \approx 0.00384\]\[\frac{v^2}{c^2} = 1 - 0.00384 = 0.99616\]\[\frac{v}{c} = \sqrt{0.99616} \approx 0.998\]
04

Calculate Distance in Laboratory

Now, calculate the distance traveled by the pion using the relation:\[d = v \times t\]Substitute \( v = 0.998c \) and \( t = 4.20 \times 10^{-7} \text{s} \):\[d = 0.998c \times 4.20 \times 10^{-7} \approx 1.25 \times 10^2 \text{ meters}\]
05

Final Results

The speed of the pion as a fraction of the speed of light is approximately \( 0.998c \), and the distance it travels in the laboratory is approximately \( 125 \text{ meters} \).

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Key Concepts

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

Relativistic Physics
In the realm of relativistic physics, things start behaving differently than we experience in everyday life. This field of physics describes phenomena that occur at very high speeds, close to the speed of light. One key aspect of relativistic physics is time dilation.
Time dilation happens because time is not absolute; it can change based on how fast an object is moving relative to an observer. In the exercise, we saw that the pion, a subatomic particle, had a lifetime that appeared longer when measured in a laboratory than it did in its own rest frame.
  • When objects travel at speeds close to the speed of light, as the pion did, time dilation becomes significant.
  • This effect is described by Einstein's theory of special relativity and can be quantified using the time dilation formula.
  • It demonstrates how time can stretch or compress depending on relative motion.
This intriguing fact has profound implications, particularly in high-energy physics and cosmology, as it allows particles to survive longer than they would if stationary.
Particle Lifetime
The concept of particle lifetime is central to understanding particle physics and reactions involving unstable particles. The pion is an unstable particle and does not last forever. Its lifespan is defined by its average lifetime, which is the time span it remains in existence before decaying.
In the pion's rest frame, its lifetime is around 2.60 脳 10鈦烩伕 seconds, but this changes when it's observed moving at high speeds.
  • The lifetime of a particle is a probabilistic measure, meaning not every pion lives exactly this average time.
  • Time dilation affects how we measure this lifetime in different frames of reference.
  • As shown in the exercise, when the pion travels at speeds close to the speed of light, its lifetime extends to 4.20 脳 10鈦烩伔 seconds in the laboratory frame.
This increase reflects the time dilation effect, allowing particles traveling at relativistic speeds to persist longer as perceived by a stationary observer. Understanding particle lifetime helps scientists explore the behavior and interaction of fundamental particles.
Speed of Light
The speed of light, denoted by the letter \( c \), plays a vital role in relativity theory. It is the universal speed limit, set at approximately 299,792,458 meters per second. Nothing with mass can travel faster than light. This constraint impacts how we calculate relativistic effects like time dilation and length contraction.
When the pion is moving, its speed relative to the speed of light helps determine how much its lifetime gets dilated.
  • In our exercise, the pion's speed was calculated to be about 0.998 times the speed of light. This high velocity was pivotal to the significant increase in its observed lifetime from the lab frame.
  • As objects approach the speed of light, energy requirements and relativistic effects become more apparent.
  • This critical speed also anchors equations in special relativity, influencing how time and space are experienced by moving objects.
Much of modern physics is built on understanding how the speed of light interrelates with other natural phenomena, shaping the structure of the universe as studied by scientists today.

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Most popular questions from this chapter

(a) Consider the Galilean transformation along the \(x\) -direction: \(x^{\prime}=x-v t\) and \(t^{\prime}=t\) . In frame \(S\) the wave equation for electromagnetic waves in a vacuum is $$\frac{\partial^{2} E(x, t)}{\partial x^{2}}-\frac{1}{c^{2}} \frac{\partial^{2} E(x, t)}{\partial t^{2}}=0$$ where \(E\) represents the electric field in the wave. Show that by using the Galilean transformation the wave equation in frame \(S^{\prime}\) is found to be $$\left(1-\frac{v^{2}}{c^{2}}\right) \frac{\partial^{2} E\left(x^{\prime}, t^{\prime}\right)}{\partial x^{\prime 2}}+\frac{2 v}{c^{2}} \frac{\partial^{2} E\left(x^{\prime}, t^{\prime}\right)}{\partial x^{\prime} \partial t^{\prime}}-\frac{1}{c^{2}} \frac{\partial^{2} E\left(x^{\prime}, t^{\prime}\right)}{\partial t^{2}}=0$$ This has a different form than the wave equation in \(S\) . Hence the Galiean transformation violates the first relativity postulate that all physical laws have the same form in all inertial reference frames. (Hint: Express the derivatives \(\partial / \partial x\) and \(\partial / \partial t\) in terms of part (a), but use the Lorentz coordinate transformations, Eqs. (37.21), and show that in frame \(S^{\prime}\) the wave equation has the same form as in frame \(S\) : $$\frac{\partial^{2} E\left(x^{\prime}, t^{\prime}\right)}{\partial x^{2}}-\frac{1}{c^{2}} \frac{\partial^{2} E\left(x^{\prime}, t^{\prime}\right)}{\partial t^{\prime 2}}=0$$ Explain why this shows that the speed of light in vacuum is \(c\) in both frames \(S\) and \(S^{\prime} .\)

(a) How much work must be done on a particle with mass \(m\) to accelerate it (a) from rest to a speed of 0.090\(c\) and (b) from a speed of 0.900\(c\) to a speed of 0.990\(c ?\) (Express the answers in terms of \(m c^{2}-)(c)\) How do your answers in parts \((a)\) and \((b)\) compare?

A particle with mass \(m\) accelerated from rest by a constant force \(F\) will, according to Newtonian mechanics, continue to accelerate without bound; that is, as \(t \rightarrow \infty, v \rightarrow \infty .\) Show that according to relativistic mechanics, the particle's speed approaches \(c\) as \(t \rightarrow \infty\) . I Note: Auseful integralis \(\int\left(1-x^{2}\right)^{-3 / 2} d x=x / \sqrt{1-x^{2}} \cdot 1\)

Two events observed in a frame of reference Shave positions and times given by \(\left(x_{1}, t_{1}\right)\) and \(\left(x_{2}, t_{2}\right),\) respectively. (a) Frame \(S^{\prime}\) moves along the \(x\) -axis just fast enough that the two events occur at the same position in \(S^{\prime} .\) Show that in \(S^{\prime},\) the time interval \(\Delta t^{\prime}\) between the two events is given by $$\Delta t^{\prime}=\sqrt{(\Delta t)^{2}-\left(\frac{\Delta x}{c}\right)^{2}}$$ where \(\Delta x=x_{2}-x_{1}\) and \(\Delta t=t_{2}-t_{1}\) . Hence show that if \(\Delta x>c \Delta t,\) there is \(n o\) frame \(S^{\prime}\) in which the two events occur at the same point. The interval \(\Delta t^{\prime}\) is sometimes called the proper time interval for the events. Is this term appropriate? (b) Show that if \(\Delta x>c \Delta t,\) there is a different frame of reference \(S\) in which the two events occur simultaneously. Find the distance between the two events in \(S^{\prime} ;\) express your answer in terms of \(\Delta x, \Delta t,\) and \(c\). This distance is sometimes called a proper length. Is this term appropriate? (c) Two events are observed in a frame of reference \(S^{\prime}\) to occur simultancously at points separated by a distance of 2.50 \(\mathrm{m}\) . In a second frame \(S\) moving relative to \(S^{\prime}\) along the line joining the two points in \(S^{\prime},\) the two events appear to be separated by 5.00 \(\mathrm{m}\) . What is the time interval between the events as measured in \(S ?[\text { Hint: Apply the result obtained in part (b).1 }\)

After being produced in a collision between elementary particles, a positive pion \(\left(\pi^{+}\right)\) must travel down a \(1.20-\mathrm{km}\) -long thibe to reach an experimental area. A \(\pi^{+}\) particle has an average life-time (measured in its rest frame) of \(2.60 \times 10^{-8} \mathrm{s}\) ; the \(\pi^{+}\) we are considering has this lifetime. (a) How fast must the \(\pi^{+}\) travel if it is not to decay before it reaches the end of the tube? (Since u will be very close to \(c,\) write \(u=(1-\Delta) c\) and give your answer in terms of \(\Delta\) rather than \(u . )\) (b) The \(\pi^{+}\) has a rest energy of 139.6 \(\mathrm{MeV}\) . What is the total energy of the \(\pi^{+}\) at the speed calculated in part (a)?
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