Question

A certain particle has a lifetime of 1x10^-7s when measured at rest. How far does it go before decaying, its speed is 0.99c when it is created?

Answer 1

If Δτ is the lifetime of the particle in its own frame of reference, then according to Einstein's special relativity, the elapsed time from our point of view will be greater (time dilation effect).
The time measured in our reference frame will be:
\[\Delta{t} = \frac{1}{\sqrt{1 - (v^2/c^2)}}\;\Delta\tau \geqslant \Delta\tau\]
For v=0.99c, Δt=7.1Δτ, hence a flight-time of \[7.1\times10^{-7} s\] and a distance travelled of 210 m.