Question

An unstable high-energy particle is created in the laboratory, and it moves at a speed of 0.989c. Relative to a stationary reference frame fixed to the laboratory, the particle travels a distance of 1.27 x 10-3 m before disintegrating. What is (a) the proper distance and (b) the distance measured by a hypothetical person traveling with the particle? Determine the particle's (c) proper lifetime and (d) its dilated lifetime.

Answer 1

Apologies for first two part!!

C)

You have two equations and two unknowns.

With:
t = lifetime measured in lab
τ = proper lifetime
x = travel distance in lab fram


t = γτ
τ² = t² - x²/c²

combining the two equations gives:

τ = x / (c√(γ²-1)) = x / (c√(1/(1-v²/c²) -1)) = (1.27*10^-3 m) / (c√(1/(1-(0.989c)²/c²) -1)) = 1.024 * 10^-12 seconds
D)

The dilated lifetime is just γτ:

t = γτ = (1.27*10^-3 m) / (c√(1-(0.989c)²/c²) √(1/(1-(0.989c)²/c²) -1)) = 4.358 * 10^-12 seconds.

Calculations maybe faulty, logic is correct.

Hope this helps :)