High-energy particles are observed in laboratories by photographing the tracks t

Question

High-energy particles are observed in laboratories by photographing the tracks they leave in certain detectors. The length of the track depends of the speed of the particle and its lifetime. A particle moving at 0.995c leaves a track 1.25 mm long. What is the proper lifetime of the particle?

The answer shows that

However, what is stopping me from finding

This although when calculated is not the same as the original answer. By taking this approach should I be using the relativistic velocity u instead? If so then how does

Explanation / Answer

we assume the lab as the rest frame

in the rest frame , T= 1.25 x 10-3 /0.995 c = 1.25 x 10-3 /(0.995 x 3 x 108 ) = 4.2 x 10-12 sec

using the formula

T' = T/ sqrt (1 - V2/c2) = (4.2 x 10-12 ) / sqrt(1 - (0.995c)2/c2) = 4.21 x 10-11

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