A muon formed high in the Earth's atmosphere travels at...

A muon formed high in the Earth's atmosphere travels at speed $v=0.990 c$ for a distance of $4.60 \mathrm{~km}$ before it decays into an electron, a neutrino, and an antineutrino $\left(\mu^{-} \rightarrow \mathrm{e}^{-}+\nu+\bar{\nu}\right)$. (a) How long does the muon live, as measured in its reference frame? (b) How far does the muon travel, as measured in its frame?

Key Concepts

Proper Time

Proper time refers to the time interval measured by an observer for whom the event occurs at a single location, such as the lifetime of an unstable particle measured in its rest frame. This is the intrinsic time interval between events, and it is not affected by relative motion between different inertial frames.

Time Dilation

Time dilation is a phenomenon in special relativity where a moving clock is observed to tick slower compared to a clock at rest in the observer's frame. This effect explains why the lifetime of fast-moving particles can appear longer when measured from a stationary reference frame than in the particle’s proper frame.

Length Contraction

Length contraction is the relativistic effect in which the length of an object moving at a significant fraction of the speed of light is measured to be shorter along the direction of motion in the stationary observer's frame. This concept is crucial for determining distances as observed from different inertial frames, particularly in scenarios involving high-speed particles.

Frames of Reference

Frames of reference are coordinate systems used to describe the physical properties of events such as time intervals and distances. In special relativity, the observations of time and space can differ between frames moving at relative velocities, making it essential to distinguish between the rest frame of an object and the frame in which the object is observed to be moving.

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