An unstable particle called the pion has a mean lifetime of 25 ns in its own rest frame. A beam

step 1 okay. Question. I was asking also the main lifetime for the pipes. We No question, it is this ki

An unstable particle called the pion has a mean lifetime of...

An unstable particle called the pion has a mean lifetime of $25 \mathrm{~ns}$ in its own rest frame. A beam of pions travels through the laboratory at a speed of $0.60 \mathrm{c} .$ (a) What is the mean lifetime of the pions as measured in the laboratory frame? (b) How far does a pion travel (as measured by laboratory observers) during this time?

Key Concepts

Time Dilation

In special relativity, time dilation is the phenomenon where a clock moving relative to an observer ticks slower compared to a clock at rest in the observer's frame. This effect becomes significant when the relative velocity approaches the speed of light, and it explains why a process, such as the decay of an unstable particle, appears to take longer when observed from a laboratory frame compared to its rest frame.

Lorentz Factor

The Lorentz factor, denoted as gamma (?), quantifies the degree of time dilation and length contraction experienced by objects moving at relativistic speeds. It is calculated using the formula ? = 1/?(1 - v²/c²), where v is the velocity of the object and c is the speed of light. This factor is crucial in transforming physical quantities from the proper frame to any other inertial frame.

Proper Lifetime

The proper lifetime of an unstable particle is the lifetime measured in the particle's own rest frame, where it is not moving. This measurement is intrinsic to the particle and does not account for the relativistic effects observed in frames where the particle is in motion.

Relativistic Distance Calculation

When an object is moving at speeds close to that of light, the distance it travels as measured in a different inertial frame (such as a laboratory frame) must account for the effects of time dilation. By multiplying the dilated lifetime by the object's velocity, one can determine the distance traveled in the laboratory frame, reflecting the interplay between time and space in relativistic motion.

Transcript

Please give Ace some feedback