- #1

- 74

- 2

- Homework Statement
- The height of the earth's atmosphere from top to ground is 10 km. Due to collisions of cosmic rays with the earths atmosphere, muons are produced at the top of the atmosphere

at a velocity of 0.999c where the speed of light is [tex]c=3.00*10^8 m/s[\tex]. At rest a muon decays in [tex]2*10^-6[\tex] sec.

i) Does the muon reach the ground?

If it does reach the ground

ii) How would an observer on the ground explain it?

iii) How would an observer moving with the muon explain it?

- Relevant Equations
- $$\Delta t=\frac{1}{\sqrt{1-\frac{v^2}{c^2}}}\Delta\tau$$

$$\Delta l'=\sqrt{1-\frac{v^2}{c^2}}\Delta l$$

i) The muon reaches the ground

ii)

To a ground observer, the decay time is dilated

$$\Delta t_d=\frac{1}{\sqrt{1-\frac{v^2}{c^2}}}\Delta\tau_d>\Delta \tau_d$$

The time for the muon to reach the ground is

$$\Delta t_g=\frac{10 km}{0.999c}< \Delta t_d$$

which is why it reaches the ground.

iii)

From the muon's frame, the ground comes toward it with a velocity [itex]v=0.999c[\itex], and hence the distance to the ground is length contracted

by

$$\Delta l'=\sqrt{1-\frac{v^2}{c^2}}10 km$$

so

$$\frac{\Delta l'}{0.999c}=\frac{\sqrt{1-\frac{v^2}{c^2}}10 km}{0.999c}<2\times 10^{-6} s $$

So the muon would reach the ground.

Does this sufficiently answer the above questions? Any feedback would be greatly appreciated. Thanks.

ii)

To a ground observer, the decay time is dilated

$$\Delta t_d=\frac{1}{\sqrt{1-\frac{v^2}{c^2}}}\Delta\tau_d>\Delta \tau_d$$

The time for the muon to reach the ground is

$$\Delta t_g=\frac{10 km}{0.999c}< \Delta t_d$$

which is why it reaches the ground.

iii)

From the muon's frame, the ground comes toward it with a velocity [itex]v=0.999c[\itex], and hence the distance to the ground is length contracted

by

$$\Delta l'=\sqrt{1-\frac{v^2}{c^2}}10 km$$

so

$$\frac{\Delta l'}{0.999c}=\frac{\sqrt{1-\frac{v^2}{c^2}}10 km}{0.999c}<2\times 10^{-6} s $$

So the muon would reach the ground.

Does this sufficiently answer the above questions? Any feedback would be greatly appreciated. Thanks.