You have to calculate, first, the speed under the normal frame, the observer one. Second, you have to use the formula of time dilation derived for speeds close to the speed of light.
1) Observer frame
V = d/ t => t = d / v = [1.05 mm 1m/10^3mm] / (0.992c)
c = 3.010^8m/s
= > t = 1.0510^ -3 m / (0.992310^8 m/s) = 3.328 10^ -12 s
2) Particle frame
t observer = t particle / √[1 - (v/c)^2] equation from special relativity theory
=> t particle = t observer √[1 - (v/c)^2 ]
substituting the values
t particle = 3.328 10^ -12 s √ [ 1 - (0.992)^2 ] = 0.42 10^ -12 s =
= 4.2 10^-13 s
Answer: The lifetime of the particle on its own frame is 4.2 10^-13 s
