# Mean lifetime of pi(+) mesons - Special Relativity

1. Nov 29, 2012

### lmstaples

1. The problem statement, all variables and given/known data

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

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

(iii) What distance would be travelled 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 Alvager and co-workers in 1964, gamma ray photons from the decay
of neutral $\pi^{+}$ mesons travelling 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?

3. The attempt at a solution

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

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

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

(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?

2. Nov 29, 2012

### TSny

Hello, Imstaples.

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

3. Nov 29, 2012

### lmstaples

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

4. Nov 29, 2012

### TSny

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

5. Nov 29, 2012

### 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)

Last edited: Nov 29, 2012
6. Nov 29, 2012

### 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

7. Nov 29, 2012

### lmstaples

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