Question

A pion is an elementary particle that, on average, disintegrates $2.6\times {10}^{-8}\mathrm{s}$ after creation in a frame at rest relative to the pion. An experimenter finds that pions created in the laboratory travel 13 m on average before disintegrating. How $f$ ast are the pions traveling through the lab?

Short answer

The speed of the pions traveling through the lab is $5\times {10}^7$ m/s.

Step-by-step solution

Calculate the time need for a pion to travel 13m at rest

First, convert the given pion life, $2.6\times {10}^{-8}$ s, into a time rate that can be used to determine the time it would take the pion to travel a distance of 13m at rest. This is calculated by dividing the distance the pion travels by its average rest lifespan, which gives the time in the frame moving with the pions. We represent the distance as $d$ and the lifespan as $T$. Therefore $d/T=13m/2.6\times {10}^{-8}$ s.

Calculate the pions' speed

Given that speed (v) is distance divided by time, after finding the time required for pions to travel 13m in step 1, we can substitute the same into the speed formula $v=d/t$ to find the pion speed. So $v=d/t=13m/(13m/2.6\times {10}^{-8}$ s.

Key concepts

Elementary Particles

Elementary particles are the smallest known building blocks of the universe. Unlike atoms, which have complex structures consisting of a nucleus and orbiting electrons, elementary particles are not composed of other particles; they are fundamental in nature. The Standard Model of particle physics categorizes these particles into quarks, leptons, and bosons. Within this framework, a pion is a type of meson, which is a hadron created by the strong force interactions between quarks and antiquarks.

Pions play a significant role in explaining the forces that hold the atomic nucleus together. There are three types of pions: positive, negative, and neutral, each characterized by different charges and mass properties. Pions are unstable particles, which means they decay into lighter particles quite rapidly after their creation, typically on the order of nanoseconds. This rapid disintegration makes studying pions challenging but also provides insights into the fundamental forces and symmetries of the universe.

Relativistic Effects

Relativistic effects come into play when particles are traveling at speeds close to the speed of light, denoted by the symbol 'c'. At these high velocities, the rules of Newtonian mechanics no longer apply, and we must look to Einstein's theory of relativity to describe the motion and interactions of particles. In this context, we often find that time, mass, and length seem to change when measured from different frames of reference.

For pions traveling through a lab, relativistic effects will manifest if their speed is a significant fraction of the speed of light. This can result in phenomena such as length contraction and time dilation, which we'll discuss more in the next section. Relativistic effects are crucial for understanding how particles like pions behave at high speeds and also for ensuring that experimental measurements align with theoretical predictions.

Time Dilation

Time dilation is a relativistic effect whereby time, as measured by a clock moving relative to an observer, appears to pass more slowly when compared to a clock at rest with respect to the observer. This concept is a cornerstone of Einstein's theory of special relativity and has profound implications for high-velocity particles, such as pions in our exercise.

In the context of the exercise, time dilation explains why pions, which disintegrate after an average lifespan of approximately 26 nanoseconds when at rest, can travel a distance of 13 meters in the laboratory frame before decaying. The high speed causes time in the moving (pion) frame to stretch out, or dilate, relative to the time in the laboratory frame. When experiments such as these are performed, and pions are found to travel further than they should, given their brief existence, it confirms the effects predicted by relativity. Additionally, it highlights the importance of considering both the velocity of particles and their relativistic effects when conducting high-energy physics experiments.