Relativity : Muons traveling Qns

[Delzac]

Homework Statement

Muons move in circular orbits at a speed of 0.9994c in a storage ring which has a radius of 500m. How many trips around the storage ring do we expect the muons to make before they decay? How many trips around the storage ring do the muons think they have made? Are there any contradictions?
[ The question didn't give the by what time will muons decay, so i did a little search and took 2.2*10^-6 sec as the time before the decay.)

Homework Equations

$$L = L_0/\gamma$$
$$\Delta T = \gamma \Delta T_0$$

The Attempt at a Solution

(i) v = 0.9994c, r=500m, C=2*pi*500=1000*pi, $$\Delta T_0 = 2.2*10^-6$$

Therefore using $$\Delta T = \gamma \Delta T_0$$,
$$\Delta T = 6.35*10^-5$$
Distance moved = (0.9994)(3.0*10^8)(6.35*10^-5)=19,038.57m
Trips around storage ring = 19,038.57/1000pi = 6.06 times.

(ii)
Distance moved in the frame of muons = 0.9994c*2.2*10^-6 = 659.604
Trips around the ring as in frame of muons = 659.604/1000pi = 0.20996

(iii)
There are not contradiction since in the frame of muon, the ring or the distance traveled by it has been length contracted. Therefore, the muon actually traveled more than it seem to have traveled.

In the frame of an observer however the muon although should decay in 2.2micro seconds didn't due to time dilation. Thus with this apparent increase in time, the muon can cover more distance and hence have more trips around the rings.
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Can anyone check my answer for part i ii and iii. I believe i got i and ii correct. But most importantly is my explanation in iii in anyway wrong or unclear or lacking?

Any help will be greatly appreciated

 

[Delzac]

Ermm...any help?

 

[m2003]

ThanksYour calculations for part (i) and (ii) look correct. However, for part (iii), your explanation is a bit unclear. Here are some suggestions for improvement:

- Start by restating the question in your own words. For example, you could say something like "The question asks us to consider the number of trips around the storage ring that muons make before they decay, and how this is affected by the principles of relativity."
- Then, explain the key concept of time dilation. You could say something like "According to the theory of relativity, time passes at different rates for objects moving at different speeds. This means that the time experienced by the muons, who are moving at a very high speed, is different from the time experienced by an observer who is stationary relative to the storage ring."
- Next, explain how this affects the number of trips around the storage ring. You could say something like "Due to time dilation, the muons experience time passing more slowly than the observer. This means that while the observer sees the muons making 6.06 trips around the ring, the muons themselves think they have only made 0.20996 trips."
- Finally, conclude by addressing the question of contradictions. You could say something like "This may seem contradictory, as the muons appear to have made both more and less trips around the ring. However, this is simply a result of the differences in how time is perceived by the muons and the observer, and is consistent with the principles of relativity."