The half-life of a muon at rest is 1.5 μs. Muons that have...

The half-life of a muon at rest is $1.5 \mu$ s. Muons that have been accelerated to a very high speed and are then held in a circular storage ring have a half-life of $7.5 \mu$ s. a. What is the speed of the muons in the storage ring? b. What is the total energy of a muon in the storage ring? The mass of a muon is 207 times the mass of an electron.

The half-life of a muon at rest is $1.5 \mu$ s. Muons that have been accelerated to a very high speed and are then held in a circular storage ring have a half-life of $7.5 \mu$ s.
a. What is the speed of the muons in the storage ring?
b. What is the total energy of a muon in the storage ring? The mass of a muon is 207 times the mass of an electron.

Key Concepts

Rest Mass

Rest mass is the intrinsic mass of a particle when it is at rest, and it is a fundamental property used in calculations involving relativistic energy. The rest mass provides the baseline energy (via E = mc²) from which additional energy contributions due to motion (incorporated through the Lorentz factor) are calculated.

Relativistic Total Energy

Relativistic total energy is the sum of the rest energy and the kinetic energy of a particle moving at speeds close to the speed of light. It is given by E = ?mc², where m is the rest mass and ? is the Lorentz factor. This formulation shows how the energy of a moving particle increases with speed due to relativistic effects.

Time Dilation

Time dilation is a consequence of special relativity where the elapsed time between events is not absolute but depends on the observer's frame of reference. In particular, a moving clock will appear to tick slower compared to one at rest, which is why particles like muons have longer observed half-lives when they are moving at high speeds relative to an observer.

Lorentz Factor (?)

The Lorentz factor, denoted by ?, is a measure that quantifies the effects of time dilation, length contraction, and relativistic mass increase experienced by an object moving at a significant fraction of the speed of light. It is defined as ? = 1/?(1 - v²/c²), where v is the object's speed and c is the speed of light, and it is central to calculations in special relativity.

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