91Ó°ÊÓ

Time interval:

$\mathbf{t}\mathbf{=}\frac{{\mathbf{t}}_{\mathbf{0}}}{\sqrt{\mathbf{1}\mathbf{-}{\displaystyle \frac{{\mathbf{v}}^{\mathbf{2}}}{{\mathbf{c}}^{\mathbf{2}}}}}}$ (1)

Length:

$\mathit{l}\mathbf{=}{\mathit{l}}_{\mathbf{0}}\sqrt{\mathbf{1}\mathbf{-}\frac{{\mathbf{v}}^{\mathbf{2}}}{{\mathbf{c}}^{\mathbf{2}}}}$ (2)

Gamma:

$\mathit{γ}\mathbf{=}\sqrt{\mathbf{1}\mathbf{-}\frac{{\mathbf{v}}^{\mathbf{2}}}{{\mathbf{c}}^{\mathbf{2}}}}$ (3)

Given: mean lifetime${\mathbf{Ï„}}_{\mathbf{0}}\mathbf{=}\mathbf{2}\mathbf{.}\mathbf{2}\mathbf{μ²õ}$

thickness${\mathit{l}}_{\mathbf{0}}\mathbf{=}\mathbf{10}\mathit{k}\mathit{m}$

speed$\mathit{v}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{999}\mathit{c}$

The distance a muon can travel is given as,

${\mathit{s}}_{\mathbf{0}}\mathbf{=}\mathit{v}{\mathit{τ}}_{\mathbf{0}}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{999}\mathit{c}\mathbf{*}\mathbf{2}\mathbf{.}\mathbf{2}\mathit{μ}\mathit{s}\mathbf{=}\mathbf{659}\mathbf{.}\mathbf{34}\mathit{m}\mathbf{≈}\mathbf{0}\mathbf{.}\mathbf{66}\mathit{k}\mathit{m}$

Calculate the value of gamma,

$\mathbf{γ}\mathbf{=}\sqrt{\mathbf{1}\mathbf{-}\frac{\mathbf{0}\mathbf{.}\mathbf{999}{\mathbf{c}}^{\mathbf{2}}}{{\mathbf{c}}^{\mathbf{2}}}}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{04471}$

The mean lifetime can be calculated as,

$\mathit{τ}\mathbf{=}\frac{{\mathbf{τ}}_{\mathbf{0}}}{\mathbf{γ}}\mathbf{=}\frac{\mathbf{2}\mathbf{.}\mathbf{2}}{\mathbf{0}\mathbf{.}\mathbf{04471}}\mathbf{=}\mathbf{4}\mathbf{.}\mathbf{92}\mathbf{*}{\mathbf{10}}^{\mathbf{-}\mathbf{5}}\mathit{s}$

Now, determine the distance travelled

$\mathrm{s}=\mathrm{\pm ¹Ï„}=0.999\mathrm{c}*4.92*{10}^{-5}=14745.24\mathrm{m}≈14.8\mathrm{km}$

The length contraction occurs for the objects travelling at speed close to the speed of light. Since out muon is travelling at nearly the speed of light, the thickness of the atmosphere is measured as:

$\mathrm{l}={\mathrm{l}}_0\mathrm{γ}=10*0.04471=447.1\mathrm{m}$

As calculated in part (a), the distance the muon can travel from its frame of reference is 0.66km. Compare the result to the result of the above part, the distance it travels in its frame of reference is less than that and hence, it can make it to the ground.

Thus, the greatest distance a muon can travel is 0.66km . The mean lifetime of a muon as measured by an observer at rest on the earth is $\mathbf{4}\mathbf{.}\mathbf{92}\mathbf{*}{\mathbf{10}}^{\mathbf{-}\mathbf{5}}\mathit{s}$. And the muon travels in this time. The thickness of the 10 km of atmosphere through which the muon must travel is 447.1m.