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 anti neutrino $\left(\mu^{-} \rightarrow \mathrm{e}^{-}+\nu+\bar{\nu}\right) .$ (a) How long does the muon live, as measured in its reference frame? (b) How far does the Earth travel, as measured in the frame of the muon?

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 anti neutrino $\left(\mu^{-} \rightarrow \mathrm{e}^{-}+\nu+\bar{\nu}\right) .$ (a) How long does the muon live, as
measured in its reference frame? (b) How far does the Earth travel, as measured in the frame of the muon?

Key Concepts

Length Contraction

Length contraction is the phenomenon where objects moving at relativistic speeds appear shorter in the direction of motion when observed from a stationary reference frame. In high-speed scenarios, distances along the direction of motion are observed to be contracted, affecting measurements such as travel distances between events.

Time Dilation

Time dilation is a consequence of Special Relativity where time appears to slow down for an object moving relative to a stationary observer. This means that a moving clock ticks slower compared to a clock at rest, an effect that must be accounted for when comparing time measurements between different inertial frames.

Special Relativity

Special Relativity is the theory established by Einstein that describes how the laws of physics are invariant in all inertial frames of reference. It introduces concepts like time dilation and length contraction, which become significant at speeds close to the speed of light, and is essential for understanding how events are observed differently in different reference frames.

Proper Time

Proper time is the time interval measured by an observer for whom the events occur at the same spatial location, such as the lifetime of a particle in its own rest frame. It is the shortest measured time interval between two events and is used as the standard to compare with time intervals measured in other reference frames.

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