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Chapter 1: Problem 27

The pion has an average lifetime of \(26.0 \mathrm{~ns}\) when at rest. For it to travel \(10.0 \mathrm{~m}\), how fast must it move?

Short Answer

Expert verified
The speed at which the pion must travel is approximately \( 3 x 10^{8} \frac{m}{s} \) or \( 1.08 x 10^{9} \frac{km}{hrs} \).

Step by step solution

01

Convert Proper Time to Seconds

The given lifetime of the pion is in nanoseconds. In order to be suitable for calculations, we need to convert it into seconds. There are \(1 x 10^{-9}\) s in 1 ns, thus, \(26.0 \) ns is \(26.0 x 10^{-9}\) seconds.
02

Determine the Time for the Pion to Travel 10.0m

Given that the pion is moving at its maximum speed close to the speed of light, the time \(t\) it will take for a light beam to travel 10m can be calculated by the relationship \( t = distance / speed\), where distance is 10.0m and speed is the speed of light \(c\) = \(3 x 10^{8} m/s\).
03

Solve for the Velocity v

Apply the time dilation equation to solve for the velocity \(v\). Substitute the proper time \(t_0\) and the time \(t\) into the time dilation equation and rearrange the equation to solve for \(v\).
04

Convert the Velocity back to Usual Units

Once we calculate the value of velocity \(v\), which will be in \(m/s\), we may want to convert it back to a form that is more commonly used such as kilometers per hour \( \frac{km}{hrs} \). The conversion is \(1 \frac{m}{s} = 3.6 \frac{km}{hrs}\).

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Key Concepts

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

Pion Lifetime
The lifetime of a pion, a subatomic particle, is an essential concept when studying particle physics and understanding fundamental interactions that occur at the quantum level. A pion's average lifetime, when stationary, is given as 26 nanoseconds (ns), which is incredibly brief. This time frame refers to the 'proper time', which is the time measured by an observer at rest relative to the particle.

To fully comprehend how the pion's lifetime can affect experiments or observations, especially when the particle is in motion, one needs to delve into the phenomenon of time dilation in special relativity.
Relativistic Velocity Calculation
Calculating relativistic velocities involves using the principles of Albert Einstein's theory of special relativity, which states that as objects move closer to the speed of light, time will appear to slow down for them relative to a stationary observer. This is related to Lorentz transformations that adjust the measurements of space and time in different inertial frames to explain this effect.

To calculate the velocity at which a pion must move to travel a certain distance before decaying, you must apply the time dilation equation, which is derived from these transformations. The equation relates time, speed, and the speed of light in a way that allows scientists to predict and confirm the behavior of particles traveling at significant fractions of the speed of light.
Speed of Light
The speed of light, denoted as 'c', is a fundamental physical constant important in many fields of physics, including special relativity. It has a value of approximately \(3 \times 10^8 \text{ m/s}\) and is considered the ultimate speed limit in the universe.

In the context of time dilation and pions, the speed of light becomes relevant when considering how fast the pions must move to cover a certain distance within their brief lifetime. It's important to note that no matter how fast the pions move, they cannot exceed the speed of light.
Time Conversion
Time conversion is an indispensable step in many physics problems, particularly when dealing with the extremely precise measurements required in particle physics. In the pion lifetime problem, the conversion from nanoseconds to seconds is crucial for coherence in units when applying the time dilation formula. Additionally, converting calculated speeds into more practical units, such as kilometers per hour, can make the results more relatable and easier to understand for day-to-day references.

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Most popular questions from this chapter

A physics professor on Farth gives an exam to her students who are on a spaceship traveling at speed \(v\) relative to Earth. The moment the ship passes the professor, she signals the start of the exam. If she wishes her students to have time \(T_{0}\) (spaceship time) to complete the exam, show that she should wait a time (Earth time) of $$ T=T_{0} \sqrt{\frac{1-v / c}{1+v / c}} $$ before sending a light signal telling them to stop. (Hint: Remember that it takes some time for the second light signal to travel from the professor to the students.)

Speed of light in a moving medium. The motion of a medium such as water influences the speed of light. This effect was first observed by Fizeau in 1851 . Consider a light beam passing through a horizontal column of water moving with a speed \(z\) (a) Show that if the beam travels in the same direction as the flow of water, the speed of light measured in the laboratory frame is given by $$ u=\frac{c}{n}\left(\frac{1+n v / c}{1+v / n c}\right) $$ where \(n\) is the index of refraction of the water. (Hint: Use the inverse Lorentz velocity transformation and note that the speed of light with respect to the moving frame is given by \(c / n\).) (b) Show that for \(v<

(a) How fast and in what direction must galaxy \(\mathrm{A}\) be moving if an absorption line found at \(550 \mathrm{~nm}\) (green) for a stationary galaxy is shifted to \(450 \mathrm{~nm}\) (blue) for A? (b) How fast and in what direction is galaxy B moving if it shows the same line shifted to \(700 \mathrm{~nm}\) (red)?

A billiard ball of mass \(0.3 \mathrm{~kg}\) moves with a speed of \(5 \mathrm{~m} / \mathrm{s}\) and collides elastically with a ball of mass \(0.2 \mathrm{~kg}\) moving in the opposite direction with a speed of \(3 \mathrm{~m} / \mathrm{s}\). Show that because momentum is conserved in the rest frame, it is also conserved in a frame of reference moving with a speed of \(2 \mathrm{~m} / \mathrm{s}\) in the direction of the \(\mathrm{sec}-\) ond ball.

An electron moves to the right with a speed of \(0.90 c\) relative to the laboratory frame. A proton moves to the right with a speed of \(0.70 c\) relative to the electron. Find the speed of the proton relative to the laboratory frame.
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