Muon decay question in special relativity Watch

sc4rface
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Hi,

I have 2 questions. First one, which I have spent so much time trying to find a simple answer to or an explanation yet to no avail. How do we decide what to use as proper time and what to use as 'time' when doing special relativity questions. All answers seem to go down a complicated mathematical route.

2) Certain questions talk about muons travelling in laboratories which I get stuck on like this one:

A certain type of sub-atomic particle has a half-life of 18 ns when at rest. A beam of these particles travelling at a speed of 0.995c is produced in an accelerator.

(i) Calculate the half-life of these particles in the laboratory frame of reference.

For which I got the correct answer of 1.8x10^-7 s.


(ii) Calculate the time taken by these particles to travel a distance of 108 m in the laboratory at a speed of 0.995c and hence show that the intensity of the beam is reduced to 25% of its original value over this distance.

Any answers would be highly appreciated.

Thank you so much.
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Wobbly_Giraffe
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How many marks is part II worth?

I'm not overly confident in the theory but it does fit in to the answer so hey ho here we go:

To calculate the time I used Time = Distance/Speed which gives a value of 3.618*10^-7 s

Over this time period approximately two half lives have occured - 3.618*10^-7/1.8*10^-7=2.01

Therefore 1/2^2=1/4 or 25% so there are 25% of the original particles left.

This means the amount of energy transfered in said time frame must have also been reduced to 25% (since there are only 25% of the particles) therefore the intensity is also reduced to 25%

Hope that helps
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Eimmanuel
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(Original post by sc4rface)


I have 2 questions. First one, which I have spent so much time trying to find a simple answer to or an explanation yet to no avail. How do we decide what to use as proper time and what to use as 'time' when doing special relativity questions. All answers seem to go down a complicated mathematical route. ….
I believe you heard about what is “time dilation”. The non-proper time interval would always be longer than proper time interval.

The proper time interval would be the time interval measured by an observer for whom the two events take place at the same position or the observer is at rest with respect to the events.

Take the muon question as an example.
A certain type of sub-atomic particle has a half-life of 18 ns when at rest. A beam of these particles travelling at a speed of 0.995c is produced in an accelerator.

(i) Calculate the half-life of these particles in the laboratory frame of reference.


18 ns is the proper time. When the muon moves at the speed of 0.995c and we attach an observer to it, the observer would be at rest with the moving muon and the observer would measure the half-life to be 18 ns which is the proper time.

While the observer who is at rest to the laboratory frame would see the muon moving and measure the half-life to be “dilated” and the laboratory frame’s observer would “say that the moving muon’s clock runs lower”. …


(Original post by sc4rface)


(ii) Calculate the time taken by these particles to travel a distance of 108 m in the laboratory at a speed of 0.995c and hence show that the intensity of the beam is reduced to 25% of its original value over this distance.
As for this part, you can refer to Wobbly_Giraffe’s answer and you can “interpret” the intensity of the beam as the number of muons per second per unit area reaching the detector.

Maybe a “more visual way” of seeing what does 2 half-lives pass mean is :

100% → 1st half-life → 50% → 2nd half-life → 25% remaining
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sc4rface
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Thanks a lot for your answer that was helpful. So what exactly does the laboratory 'frame of reference mean'. How should I decide what is To(proper time) and what is T(time) in questions like this. Is the moving body always the proper time, even though we refer to it as the stationary observer.

Thanks
(Original post by Eimmanuel)
I believe you heard about what is “time dilation”. The non-proper time interval would always be longer than proper time interval.

The proper time interval would be the time interval measured by an observer for whom the two events take place at the same position or the observer is at rest with respect to the events.

Take the muon question as an example.
A certain type of sub-atomic particle has a half-life of 18 ns when at rest. A beam of these particles travelling at a speed of 0.995c is produced in an accelerator.

(i) Calculate the half-life of these particles in the laboratory frame of reference.


18 ns is the proper time. When the muon moves at the speed of 0.995c and we attach an observer to it, the observer would be at rest with the moving muon and the observer would measure the half-life to be 18 ns which is the proper time.

While the observer who is at rest to the laboratory frame would see the muon moving and measure the half-life to be “dilated” and the laboratory frame’s observer would “say that the moving muon’s clock runs lower”. …




As for this part, you can refer to Wobbly_Giraffe’s answer and you can “interpret” the intensity of the beam as the number of muons per second per unit area reaching the detector.

Maybe a “more visual way” of seeing what does 2 half-lives pass mean is :

100% → 1st half-life → 50% → 2nd half-life → 25% remaining
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Eimmanuel
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(Original post by sc4rface)
Thanks a lot for your answer that was helpful. So what exactly does the laboratory 'frame of reference mean'. ….
Laboratory frame of reference means a stationary frame of reference with to the moving frame of reference.



(Original post by sc4rface)
….How should I decide what is To(proper time) and what is T(time) in questions like this. Is the moving body always the proper time, even though we refer to it as the stationary observer.
Thanks
I really hope that the answer is so simple. Maybe it is quite "easy", if you look at the following description.

Proper time interval Δt0 to describe the time interval between two events that occur at the same point in a particular reference frame.

Time interval Δt involves events that occur at different space points in the frame of reference.

Have a look at the following example –Example 37.3.
https://books.google.com.sg/books?id...00.600&f=false
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