Question

In its own frame the average pi meson lives 2.60 x 10^-8s. Suppose it moves at 0.985c relative to the Earth. According to us, on Earth, what is a pi meson's average lifetime? (b) according to us, what is the pi meson;s average travel distance before decaying? (c) If time dilation did not occur what would we see as the pi meson's average travel distance before decaying?

Answer 1

The π meson’s lifetime in its own frame is the proper time
interval, tₒ =2.6•10^−8 s. An earthbound observer measures a longer dilated time interval t . If β =0. 985,
t = tₒ/sqrt(1-β²) = 2.6•10^−8 /sqrt(1-0.985²) =1.5•10^-7 s.
The distance the π meson travels in the earthbound observer’s reference frame is the π-meson’s velocity multiplied by the time interval measured by the earthbound observer. (Note that this is the relative velocity)
d = v•t =0.985•c•1.598510^-7 = 0.985•3•10^8•1.5•10^-7 = 44.5 m,
Without time dilation, the distance traveled would just be the proper lifetime multiplied by the meson’s velocity:
s = 0.985• c•2.6•10^−8 =7.68 m.