An unstable subatomic particle is observed to have an average lifetime of 4 x 10

Question

An unstable subatomic particle is observed to have an average lifetime of 4 x 10-6 s under laboratory conditions where the speed of the particles (many of the same type are observed) is fairly small: v<<c. The same particle is observed to emerge from high energy collisions in the upper atmosphere and reach detectors on a mountaintop 6000 m below.
a) The scientists on the mountaintop estimate the lifetime of the particle by assuming it can go no faster than c. What value do they calculate? Are they measuring/estimating a “proper time”? Why or why not? (The term “proper” has nothing to do with accuracy.)
b) What about the measurement in the laboratory? Was that a proper time? Why or why not?
c) It’s the same particle (this can be determined by other means such as the use of magnetic fields). Why are the two measurements of the lifetime different? i.e. How does relativity explain this?
d) The ratio of the two time measurements can be used to estimate the factor

without
determining the speed. Show the algebra and find the average value of

for these
particles. If you were at rest with an average particle the distance to the mountaintop
wouldn’t be 6000 m, it would be …?

Explanation / Answer

The questiom is incomplete, you will have to write the velocity so that the precise values can be told.

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