$($b$)$ The three leptons $e,,$ and $$ may be assomed to have some weak interaction coupling constant. The $$ lepton $($mass $1748Me\frac{V}{c^2})$ and muon $($mass $105Me\frac{V}{c^2})$ have decay modes and branching ratios as follows:

$longrightarrowe+v\frac{+}{b}ar(v),(100\%)$

$longrightarrowe+v\frac{+}{b}ar(v),(17\%)$

$($i$)$ Given that the mean lifetime of the muon is $2\mathrm{.}2{10}^{-6},$ make an estimate of the mean lifetime of the $-$lepton.

$($ii$)$ If $-$lepton with the momentum $5Ge\frac{v^2}{c}$ are produced in an $e^{-}{e}^{+}$collider, calculate the mean flight path before and after in the Laboratory system$-[$ Mint: consider the time dialation$).$