Answer to Question #107488 in Mechanics | Relativity for Mariane Lim

Question

A. Suppose a cosmic ray colliding with a nucleus in the Earth’s upper atmosphere produces a muon that has a velocity v=0.950c . The muon then travels at constant velocity and lives 1.52 ms as measured in the muon’s frame of reference. (You can imagine this as the muon’s internal clock.) How long does the muon live as measured by an Earth-bound observer?

A muon in the Earth’s atmosphere lives longer as measured by an Earth-bound observer than measured by the muon’s
internal clock.

Answer 1

Solution:

We use the next equation

$\Delta{t}=\gamma\Delta{t_0};$

So, now we need find the $\gamma$. For this we use the next equation

$\gamma={\frac 1 {\sqrt {1}- {\frac {v^2} {c^2}}}};$

$\gamma={\frac 1 {\sqrt {1}- {\frac {(0.950 c)^2} {c^2}}}};$

$\gamma=3.20;$

Now we can find $\Delta{t}$

$\Delta{t}=3.20 \times 1.52(\mu s);$

$\Delta{t}=4.87(\mu s).$

Answer: 4.87 $\mu s$