Three groups of experimenters measure the average decay time...

Three groups of experimenters measure the average decay time for a specific type of radioactive particle. Group 1 accelerates the particles to $0.5 c,$ moving from left to right, and then measures the decay time in the beam, obtaining a result of 20 ms. Group 2 accelerates the particles to $-0.5 c,$ from right to left, and then measures the decay time in the beam. Group 3 keeps the particles at rest in a container and measures their decay time. Which of the following is true about these measurements? a) Group 2 measures a decay time of $20 \mathrm{~ms}$. b) Group 2 measures a decay time less than $20 \mathrm{~ms}$. c) Group 3 measures a decay time of $20 \mathrm{~ms}$. d) Both (a) and (c) are true. e) Both (b) and (c) are true.

Three groups of experimenters measure the average decay time for a specific type of radioactive particle. Group 1 accelerates the particles to $0.5 c,$ moving from left to right, and then measures the decay time in the beam, obtaining a result of 20 ms. Group 2 accelerates the particles to $-0.5 c,$ from right to left, and then measures the decay time in the beam. Group 3 keeps the particles at rest in a container and measures their decay time. Which of the following is true about these measurements?
a) Group 2 measures a decay time of $20 \mathrm{~ms}$.
b) Group 2 measures a decay time less than $20 \mathrm{~ms}$.
c) Group 3 measures a decay time of $20 \mathrm{~ms}$.
d) Both (a) and (c) are true.
e) Both (b) and (c) are true.

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Key Concepts

Special Relativity

This concept describes the physics of objects moving at speeds close to the speed of light, where time, length, and simultaneity become relative. It provides the framework for understanding how measurements differ between observers in different inertial frames.

Time Dilation

Time dilation is a phenomenon predicted by special relativity where a clock moving relative to an observer ticks slower than a clock at rest with respect to that observer. This means that the decay time of a moving particle, as measured in the laboratory frame, appears longer compared to the proper decay time measured in the particle's own rest frame.

Proper Time

Proper time is the time interval measured by an observer at rest relative to the event being measured. In the context of particle decay, it is the true lifetime of the particle as measured in its own rest frame, and is always shorter than or equal to the time measured by an observer for whom the particle is moving.

Lorentz Factor

The Lorentz factor quantifies the amount of time dilation (and length contraction) experienced by an object moving at a significant fraction of the speed of light. It is derived from the relative speed between the observer and the moving object and is critical for calculating how much longer the decay time appears in a moving frame compared to the proper time.

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