The mean lifetime of stationary muons is measured to be 2.2000 μs. The mean lifetime of high-spe

step 1 The mean lifetime of stationary muons is measured to be 2 .2 microseconds. The mean lifetime of

The mean lifetime of stationary muons is measured to be...

The mean lifetime of stationary muons is measured to be $2.2000 \mu \mathrm{s}$. The mean lifetime of high-speed muons in a burst of cosmic rays observed from Earth is measured to be $16.000 \mu \mathrm{s}$. To five significant figures, what is the speed parameter $\beta$ of these cosmic-ray muons relative to Earth?

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Key Concepts

Time Dilation

Time dilation is a fundamental concept in Special Relativity where a clock moving relative to an observer ticks slower compared to a clock at rest in the observer's frame. This effect explains why the observed duration of processes (such as the decay of particles) is longer in a moving frame relative to the proper lifetime measured in the particle’s rest frame.

Lorentz Factor (?)

The Lorentz factor, denoted by ?, is defined as ? = 1/?(1 - ?²), where ? is the speed of an object divided by the speed of light. It quantifies the amount of time dilation, length contraction, and relativistic mass increase that an object experiences as it moves at high speeds relative to an observer.

Dimensionless Speed Parameter (?)

The speed parameter ? is the ratio of an object's speed to the speed of light (? = v/c), providing a normalized measure of its velocity. This parameter is crucial in Special Relativity since it directly enters into the Lorentz factor, allowing one to assess how relativistic effects alter physical measurements such as time intervals and spatial dimensions.

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