Question about time dilation

[MatinSAR]

Homework Statement

Muons are elementary particles with a (proper) lifetime of 2.2 μs. They are produced with very high speeds in the upper atmosphere when cosmic rays (high energy particles from space) collide with air molecules. Take the height L0 of the atmosphere to be 100 km in the reference frame of the Earth, and find the minimum speed that enables the muons to survive the journey to the surface of the Earth.

Relevant Equations

$\Delta t=\dfrac {\Delta {t'}}{\sqrt {1-u^2/c^2}}$

My try which was failed :

Observer at rest measures time $\Delta {t'}= 2.2 \mu s$
In the frame of reference of the Earth observer measures time $\Delta t=\dfrac {\Delta {t'}}{\sqrt {1-u^2/c^2}}$

I have two unknowns $u$ and $\Delta t$ so I cannot find $u$. Is there another equation that I don't remember?

Many thatnks.

 

[Orodruin]

How long does it take the muon to reach the ground if it is produced at height $L_0$ and travels at speed $u$?

 

[MatinSAR]

How long does it take the muon to reach the ground if it is produced at height $L_0$ and travels at speed $u$?

$L_0/u$
Can I use this for $\Delta t$ ?

 

[Orodruin]

$L_0/u$
Can I use this for $\Delta t$ ?

Is it the time the muon needs to survive for it to reach the ground?

 

[MatinSAR]

Is it the time the muon needs to survive for it to reach the ground?

According to a friend the time needed for muon survival is $L_0/c$ but I think it should be $L_0/u$ , I know I'm wrong but I don't know why.

 

[Orodruin]

According to a friend the time needed for muon survival is $L_0/c$ but I think it should be $L_0/u$ , I know I'm wrong but I don't know why.

How do you know you are wrong and not your friend?

 

[MatinSAR]

How do you know you are wrong and not your friend?

He's by far smarter + He said he checked the answer using his book.

 

[PeroK]

$L_0/u$
Can I use this for $\Delta t$ ?

You seem to be looking at this the wrong way round. I would start with:
$$\Delta t = \frac{L_0}{u}$$As the journey time, in the Earth reference frame, from the upper atmosphere to the surface, for a particle with speed $u$. Then I would look for an equation to relate that to the lifetime of the particle in its own rest frame.

Your approach seems to be to start with an equation and fish around for the quantities you need!

 

[Orodruin]

He's by far smarter + He said he checked the answer using his book.

Having checked the answer in the book doesn’t seem like a smart argument to me.

 

[MatinSAR]

Your approach seems to be to start with an equation and fish around for the quantities you need!

That's what I do to solve any problem ...

You seem to be looking at this the wrong way round. I would start with:
$$\Delta t = \frac{L_0}{u}$$As the journey time, in the Earth reference frame, from the upper atmosphere to the surface, for a particle with speed $u$. Then I would look for an equation to relate that to the lifetime of the particle in its own rest frame.

Thanks. I will think and I'll back in a while.

Having checked the answer in the book doesn’t seem like a smart argument to me.

Yes but he was right anyway ...

Thanks for the help @Orodruin and @PeroK ...

 

[Orodruin]

Yes but he was right anyway ...

No, he wasn’t.

 

[MatinSAR]

No, he wasn’t.

That's good news for me. So was that $L_0/u$ right?

 

[Orodruin]

That's good news for me. So was that $L_0/u$ right?

Yes. I assume your friend only checked the numerical result. These are going to be very similar for both cases as you need $u$ to be pretty close to $c$ to achieve the required amount of time dilation.

 

[MatinSAR]

Yes. I assume your friend only checked the numerical result. These are going to be very similar for both cases as you need $u$ to be pretty close to $c$ to achieve the required amount of time dilation.

Many many thanks @Orodruin ... So I should simply solve $\dfrac {L_0}{u}=\dfrac {\Delta {t'}}{\sqrt {1-u^2/c^2}}$ ?

 

[PeroK]

That's good news for me. So was that $L_0/u$ right?

That's not the answer. That's only the first part of the calculation.

 

[Orodruin]

Many many thanks @Orodruin ... So I should simply solve $\dfrac {L_0}{u}=\dfrac {\Delta {t'}}{\sqrt {1-u^2/c^2}}$ ?

Yes.

 

[MatinSAR]

Thank you for your help and time.

 

[Orodruin]

Note that it is a useful exercise to also check what the approximation $\Delta t = L_0/c$ results in and to understand why it gives a result which is pretty close to the correct result.

 

[MatinSAR]

Note that it is a useful exercise to also check what the approximation $\Delta t = L_0/c$ results in and to understand why it gives a result which is pretty close to the correct result.

Yes. I've understood your post earlier about it. It solved all my confusions ...

These are going to be very similar for both cases as you need $u$ to be pretty close to $c$ to achieve the required amount of time dilation.

So I understand why both answers are correct. Thanks.

 

[Orodruin]

So I understand why both answers are correct.

Numerically within expected errors. The assumption of $L_0/u$ is correct and will be a better match when $c\Delta t’$ is not much smaller than $L_0$.

 

[MatinSAR]

Numerically within expected errors. The assumption of $L_0/u$ is correct and will be a better match when $c\Delta t’$ is not much smaller than $L_0$.

Now I know many things to teach that friend. But with mentioning That where I learned them.

Wish you a good day @Orodruin and @PeroK ...

 

[Mister T]

That's what I do to solve any problem ...

Then you will forever find the subject matter tedious. You're treating your homework and exams like an exercise in applied math. Physics is a study of the phenomena. Focus on that and what it takes to reach an understanding. It will propel you to excellence. It is what your instructor wants. Otherwise you will always feel like the homework questions and exams are intentionally tricky.

 

[MatinSAR]

Then you will forever find the subject matter tedious. You're treating your homework and exams like an exercise in applied math. Physics is a study of the phenomena. Focus on that and what it takes to reach an understanding. It will propel you to excellence. It is what your instructor wants. Otherwise you will always feel like the homework questions and exams are intentionally tricky.

@Mister T Thanks for your suggestion. I'll try.