 Solution

By time dilation, the laboratory mean lifetime is $\tau =\gamma {\tau }_0$, where ${\tau }_0=2.6\times 10^{-8}\mathrm{s}$. The exponential decay law evaluated at travel time $d/v$ gives the surviving pion fractionUse the exact product $\gamma v=(400/0.94)c$ from part (a), rather than approximating the speed separately. With $c=2.99792458\times 10^{8}m/s$,Thus about $88.6\%$ emerge without decaying, and about $11.4\%$ decay inside the pipe. This assumes the given common longitudinal speed and neglects losses from other interactions.