Another relativistic particle decay question

In summary: However, due to time dilation and length contraction, unstable particles can still travel far if their speeds are high enough.
  • #1
Elvis 123456789
158
6

Homework Statement


Unstable particles cannot live very long. Their mean life time t is defined by N(t) = N0e−t/τ , i.e., after a time of t, the number of particles left is N0/e. (For muons, τ=2.2µs.) Due to time dilation and length contraction, unstable particles can still travel far if their speeds are high enough.

For some particles, the mean life time is so small that it is more convenient to define τ using the quatity cτ (c is the speed of light). For example, the particle Λ has a cτ measured to be 7.89cm.

a) If the Λ is traveling at 0.5c, how many of L are left after traveling 7.89cm?

b) How far would the Λ’s have travelled, if only 50% of them are left?

c) (Extra) Derive the general expression of N(L)/N0 for the Λ particles, as a function of L (distance travelled) and the speed v (arbitrary, not just always 0.5c) of the Λ particles.

Homework Equations


N(t) = N0e−t/τ

t_e = t_Λ *γ

L_Λ = L_e/γ

t_e & L_e is the time and length measured in the Earth frame of reference

and t_Λ and L_Λ is the time and length measured in the lambda particle frame of reference

I did all the parts but I feel pretty unsure about it. These relativity questions just feel really ambiguous to me. I was hoping you guys could take a look and let me know if it seems ok. Thanks in advance!

The Attempt at a Solution



Parts a.) , b.), and c.) are in the attached image
[/B]
I assumed that the cτ = 7.89 cm is in the particle's FR

and for part a that the 7.89cm traveled was in the earth/lab FR
 

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  • #2
Your answers look correct to me.

But note that it would be nice to express your equations in terms of the defined quantity ##c \tau_\Lambda##. Thus you can write ##N = N_0 \exp \left(- \frac{L}{v \gamma \tau_\Lambda} \right)## as ##N = N_0 \exp \left(- \frac{L}{(v/c) \gamma (c \tau_\Lambda )} \right)##. Then you can just use the given value for ##c \tau_\Lambda ## in the calculation for part (a) without having to find ##v## in m/s or ##\tau_\Lambda## in seconds.

For part (c), I think they want an equation for the ratio ##N/N_0##, which just requires a little change in what you wrote. They might prefer the equation to be written in terms of the the quantity ##c \tau_\Lambda ##. But, maybe not.
 
Last edited:
  • #3
Elvis 123456789 said:
Due to time dilation and length contraction, unstable particles can still travel far if their speeds are high enough.
If there were no time dilation or length contraction, unstable particles would still travel far if their speeds are high enough.
 

Related to Another relativistic particle decay question

What is relativistic particle decay?

Relativistic particle decay refers to the process by which a particle with a high energy decays into other particles, often resulting in the release of energy.

Why is relativistic particle decay important in scientific research?

Studying relativistic particle decay can provide insights into the fundamental laws of physics, such as the conservation of energy and momentum. It can also help us understand the structure of matter and the behavior of particles at high energies.

How is relativistic particle decay different from regular particle decay?

The main difference is that relativistic particle decay involves particles with high energies, which means they are traveling at speeds close to the speed of light. This can lead to the production of more particles and a larger release of energy compared to regular particle decay.

What are some real-world applications of studying relativistic particle decay?

Relativistic particle decay is important in many fields, such as particle physics, astrophysics, and nuclear medicine. It can help us understand the formation of the universe, the behavior of stars, and the development of new medical technologies.

Are there any potential dangers associated with studying relativistic particle decay?

While studying relativistic particle decay can provide valuable insights, it also involves high-energy particles that can be potentially harmful. Scientists take precautions to minimize any risks and ensure the safety of themselves and others in the surrounding area.

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