Answer
A pion will move $~~55~m~~$ through the atmosphere.
Work Step by Step
We can find the distance that pions travel according to the pion reference frame:
$L = v~t$
$L = (0.99)(3.0\times 10^8~m/s)(26\times 10^{-9}~s)$
$L = 7.722~m$
We can find $\gamma$:
$\gamma = \frac{1}{\sqrt{1-\beta^2}}$
$\gamma = \frac{1}{\sqrt{1-0.99^2}}$
$\gamma = 7.0888$
We can find the distance the pion travels according to the Earth reference frame:
$L = \frac{L_0}{\gamma}$
$L_0 = \gamma~L$
$L_0 = (7.0888)(7.722~m)$
$L_0 = 55~m$
A pion will move $~~55~m~~$ through the atmosphere.
