Special Relativity, calculating velocity of Kl0 meson with reference frames

In summary, the kaons travel 45m in time t in the lab frame before decaying. Their speed in the lab frame is gamma=1/√(1-v^2/c^2).
  • #1
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Homework Statement



K mesons (“kaons”) are unstable particles composed of a quark and an antiquark. They can be produced copiously in energetic collisions between stable particles at accelerator laboratories. Soon after they are produced, kaons decay to lighter particles. One type of kaon, the KL0, has a lifetime of 5 × 10–8 seconds in its own rest frame. Now imagine that a beam of fast KL0 mesons is produced at a national laboratory. The average distance the kaons travel before decaying in flight is found to be 45 meters in the lab frame.
(a) What is the speed of the kaons in the lab frame? Use the value c = 3 × 108 m/sec in your calculation.
(b) What is the kaons’ gamma factor in the lab frame?
(c) Isaac Newton didn’t know about special relativity. If you told Newton that a kaon with a lifetime of 5 × 10–8 sec was zipping through your lab at the speed you calculated in part (a), how far would Newton expect the kaon to travel before decaying?


Homework Equations



So from my understanding, the time given is the t' (the time in the meson's reference frame), while the length given (L) is from the lab's reference frame. If this is true, when calculating velocity, don't you need either both t' and L', or t and L to solve for some velocity?

The Attempt at a Solution



From the given variables that was all I could do. If I threw out my conceptual understanding of the problem and just did v=L/t you get 3 times the speed of light. Kind of lost on where to go from there.
 
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  • #2
You can convert t' to t; the conversion will depend on velocity, unknown so far. Then L/t = v, which you can treat as an equation for the velocity.
 
  • #3
t' is indeed 5x10^-8.

We know the kaons travel 45m in time t in the lab frame.
So 45=vt

and t=gamma t'.

So 45=gamma x t' x v.

Now solve for v.
 
  • #4
apelling said:
t' is indeed 5x10^-8.

We know the kaons travel 45m in time t in the lab frame.
So 45=vt

and t=gamma t'.

So 45=gamma x t' x v.

Now solve for v.

Hm, but gamma requires v too? I also find this confusing.
 
  • #5
Thanks for the help voko & apelling. Kaldanis, I worked it through this way and it seemed to work...

t'=5x10^-8s
L=45m

So.. we know that t (time in the lab frame) must be longer than t' because the Kaon's clock ticks slower due to its high velocity, so..

t=[itex]\gamma[/itex]t'

v=L/t

v=L/([itex]\gamma[/itex]t')

[itex]\gamma[/itex]=1/√(1-v^2/c^2)

v=L√(1-v^2/c^2)/t'

v^2t'^2=L^2(1-v^2/c^2)

v^2t'^2=L^2-L^2v^2/c^2

v^2t'^2 + L^2v^2/c^2=L^2

v^2(t'^2 + L^2/c^2)=L^2

v=L/√(t'^2+L^2/c^2)

v=2.846 x 10^8 m/s

...
Thanks again for the help!



[itex]\gamma[/itex]
 

FAQ: Special Relativity, calculating velocity of Kl0 meson with reference frames

1. What is Special Relativity?

Special Relativity is a theory proposed by Albert Einstein in 1905 that explains the relationship between space and time. It states that the laws of physics are the same for all observers in uniform motion, and the speed of light is constant for all observers.

2. How does Special Relativity affect the calculation of velocity for Kl0 meson?

Special Relativity affects the calculation of velocity for Kl0 meson by taking into account the relative motion of the observer and the meson. The velocity of the meson will appear different to different observers depending on their relative motion.

3. What is the difference between inertial and non-inertial reference frames in Special Relativity?

Inertial reference frames are frames of reference in which an object remains at rest or moves with constant velocity unless acted upon by an external force. Non-inertial reference frames are frames of reference in which an object experiences acceleration. Special Relativity only applies to inertial reference frames.

4. How do you calculate the velocity of Kl0 meson in Special Relativity?

The calculation of velocity for Kl0 meson in Special Relativity involves using the Lorentz transformation equations to convert the velocity measured by one observer to the velocity measured by another observer in a different inertial reference frame.

5. Can Special Relativity be applied to objects moving at speeds close to the speed of light?

Yes, Special Relativity can be applied to objects moving at speeds close to the speed of light. In fact, the theory was developed specifically to explain the behavior of objects moving at high speeds, such as the Kl0 meson.

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