Question

The pion \(\left(\pi^{+} \text {or } \pi^{-}\right)\) is an unstable particle that decays with a proper half-life of \(1.8 \times 10^{-8}\) s. (This is the half-life measured in the pion's rest frame.) (a) What is the pion's half-life measured in a frame \(\mathcal{S}\) where it is traveling at \(0.8 c ?\) (b). If 32,000 pions are created at the same place, all traveling at this same speed, how many will remain after they have traveled down an evacuated pipe of length \(d=36 \mathrm{m} ?\) Remember that after \(n\) half-lives, \(2^{-n}\) of the original particles survive. (c) What would the answer have been if you had ignored time dilation? (Naturally it is the answer (b) that agrees with experiment.)

Short answer

1000 pions remain; ignoring time dilation, only 112 would remain.

Step-by-step solution

Calculate Time Dilation

To find the half-life in the moving frame, we need to apply the time dilation formula. The formula is \[ t' = \frac{t_0}{\sqrt{1 - \frac{v^2}{c^2}}} \]where \( t_0 = 1.8 \times 10^{-8} \) s is the proper half-life, \( v = 0.8c \), and \( c \) is the speed of light. Calculate \( \sqrt{1 - \frac{(0.8c)^2}{c^2}} = \sqrt{1 - 0.64} = \sqrt{0.36} = 0.6 \). Therefore, \[ t' = \frac{1.8 \times 10^{-8}}{0.6} = 3.0 \times 10^{-8} \text{ s} \].This is the half-life of the pion measured in frame \( \mathcal{S} \).

Calculate Time To Travel Across Pipe

The time for the pions to travel the length of the pipe is given by \[ \text{Time} = \frac{\text{Distance}}{\text{Velocity}} = \frac{36 \text{ m}}{0.8c} \].Since \( c \approx 3 \times 10^8 \text{ m/s} \), this becomes \[ \text{Time} \approx \frac{36}{0.8 \times 3 \times 10^8} \approx 1.5 \times 10^{-7} \text{ s} \].

Number of Half-Lives Elapsed

Calculate how many half-lives pass during the time it takes to travel down the pipe. This is given by \[ n = \frac{\text{Time}}{t'} = \frac{1.5 \times 10^{-7}}{3.0 \times 10^{-8}} = 5 \].Therefore, 5 half-lives have passed during this time.

Calculate Remaining Pions

The number of surviving pions after \( n \) half-lives is determined by \[ N = 32000 \times 2^{-n} = 32000 \times 2^{-5} \].Calculate \( 2^{-5} = \frac{1}{32} \).Thus, \[ N = 32000 \times \frac{1}{32} = 1000 \].Therefore, 1000 pions remain after traveling the length of the pipe.

Ignoring Time Dilation for Pion Decay

Without accounting for time dilation, use the proper half-life to calculate \( n \) as follows: \[ n = \frac{1.5 \times 10^{-7}}{1.8 \times 10^{-8}} = 8.33 \].Then use \[ N = 32000 \times 2^{-8.33} \].Calculate \( 2^{-8.33} \approx 0.0035 \), thus \[ N \approx 32000 \times 0.0035 = 112 \].Only 112 pions would remain without considering time dilation.

Key concepts

Half-Life

The half-life is a concept in physics used to describe the time it takes for half of the particles in a sample to decay. It is a key idea in understanding processes that undergo exponential decay, such as radioactive substances or unstable particles like pions. For the pion, which is an unstable subatomic particle, the proper half-life is the time it takes in its rest frame for half of the pions to decay. Specifically, the rest frame is where the particle is at rest relative to an observer. In this particular case, the proper half-life for a pion is given as \[ t_0 = 1.8 \times 10^{-8} \text{ seconds}. \] This value means if you started with a certain number of pions, after - 1.8 x 10^-8 seconds,half of the original number would have decayed, assuming the pions are not moving relative to the observer.

Special Relativity

Special relativity, a theory developed by Albert Einstein, changes how we understand time and space, especially at high velocities close to the speed of light. A fundamental aspect of special relativity is the phenomenon of time dilation, where time can appear to "slow down" for objects moving at speeds comparable to the speed of light. Time dilation can be mathematically described using the formula: \[ t' = \frac{t_0}{\sqrt{1 - \frac{v^2}{c^2}}} \] where - \( t_0 \) is the proper time (time interval in the rest frame),- \( v \) is the velocity of the moving object,- \( c \) is the speed of light,- and \( t' \) is the dilated time observed in another frame moving relative to the rest frame.In the exercise, the pion travels at 0.8c relative to another observer. Consequently, its half-life in the moving frame (\( \mathcal{S} \)) is longer than in its rest frame due to time dilation, calculated to be:\[ t' = 3.0 \times 10^{-8} \text{ seconds}. \] This extension implies that processes, like the decay of particles, appear to take more time from the perspective of an observer moving relative to the particle.

Pion Decay

Pion decay is a phenomenon involving pions, which are particles that are part of the family of particles known as mesons. These particles are unstable and can transform or decay into other particles. In the context of the exercise, pions decay over time, and this decay process follows a statistical pattern often described by the concept of half-life.When pions are moving at high speed, as considered in the problem, the effect of special relativity must be considered. The decay rate is influenced by time dilation, leading to an extended lifespan of pions as observed in frames where the pions are in motion.Here's a summary of the process in the exercise:- The original count of pions: 32,000-\[ n = 5\] half-lives, as calculated from the distance covered and the time taken.- The remaining number of pions: - With time dilation considered: about 1,000 pions. - Without considering time dilation: only around 112 pions.Ignoring time dilation would result in underestimating the number of remaining pions, as seen in the comparison. Correct calculations confirm the experimental observations showing special relativity's effect on the decay process.