Muons are created by cosmic-ray collisions at an elevation h...

Muons are created by cosmic-ray collisions at an elevation $h$ (as measured in Earth's frame of reference) above Earth's surface and travel downward with a constant speed of $0.990 c$. During a time interval of $1.5 \mu \mathrm{s}$ in the rest frame of the muons, half of the muons present at the beginning of the interval decay. If one-fourth of the original muons reach Earth before decaying, about how big is the height $h$ ?

Key Concepts

Proper Lifetime

Proper lifetime refers to the time interval measured in the rest frame of an unstable particle, such as a muon, where the particle is at rest. It serves as the intrinsic lifetime of the particle independent of any relativistic effects and is a cornerstone concept when dealing with decay processes in special relativity.

Time Dilation

Time dilation is a relativistic phenomenon predicted by Einstein’s theory of special relativity, whereby a moving clock (or any time-dependent process) is observed to tick slower compared to a clock at rest. This effect is quantified by the Lorentz factor and is crucial for understanding why particles moving at speeds close to the speed of light appear to decay more slowly from the perspective of a stationary observer.

Lorentz Factor

The Lorentz factor, often denoted by gamma (?), is a dimensionless quantity defined as 1/?(1 - v²/c²). It quantifies the degree of time dilation and length contraction experienced by objects moving at relativistic speeds. The Lorentz factor plays a central role in converting proper time intervals to time intervals measured in another inertial frame.

Exponential Decay Law

The exponential decay law describes the statistical process by which unstable particles decay over time. The number of remaining particles decreases exponentially as a function of time, and the decay constant is related to the particle’s proper lifetime. This law is essential for predicting how many particles will survive a given time interval and is widely used in analyses of particle physics and radioactive decay processes.

Transcript

Please give Ace some feedback