Mean lifetime of pi(+) mesons - Special Relativity

[lmstaples]

Homework Statement

The proper mean lifetime of a $\pi^{+}$ mesons is 26ns.
(a) (i) What is the mean lifetime of $\pi^{+}$ mesons traveling with $\beta=\frac{v}{c}=0.73$?

(ii) What distance is traveled at this velocity during one mean lifetime?

(iii) What distance would be traveled in the absence of time dilation?

(iv) How long would the laboratory-frame distance calculated in part (ii) appear to be in
the rest frame of the $\pi^{+}$ meson?

(b) In the experiment of Alvager and co-workers in 1964, gamma ray photons from the decay
of neutral $\pi^{+}$ mesons traveling at 0.99975c were found to travel at c in the samedirection as the $\pi^{+}$ mesons.

What was the speed of the photons in the rest frame of the $\pi^{+}$ mesons?

The Attempt at a Solution

(a) (i) Mean Lifetime = $\frac{26}{\sqrt{1-(0.73)^{2}}}=38ns$

(ii) Distance traveled = $(0.73)*(2.9979*10^{^})*(38*10^{-9})=8.32m$

(iii) Distance traveled = $(0.73)*(2.9979*10^{^})*(26*10^{-9})=5.69m$

(iv) Not sure how to do that?

(b) Because of the postulates of Special Relativity, surely the speed of the photons is just c (speed of light) as light travels the same speed in all reference/inertial frames? Is it a trick question or not?

 

[TSny]

Hello, Imstaples.

Your work looks good, including (b). For (iv) did you learn about "length contraction"?

 

[lmstaples]

Yeah! I have done about length contraction and time dilation ect

 

[TSny]

OK, can you relate what you've learned about length contraction to question (iv)?

 

[lmstaples]

I thought about it: can I ask whether the value of (iv) would be bigger or smaller than (ii) as when I did work it out I got something around 12m

Unless I have the frames round the wrong way and its is infact equal to (iii)

 

[TSny]

Imagine a stick in the lab frame extending from where the mesons are created to where they decay after one mean lifetime.

In which frame is this stick at rest? In which frame is it moving? Your notes on length contraction should specify how the "rest length" compares to the "moving length". Or see http://en.wikipedia.org/wiki/Length_contraction

 

[lmstaples]

Ah I see where I was going wrong... I think haha