How Does Special Relativity Affect Pion Decay Measurements?

[BOAS]

Homework Statement

A beam of pion mesons travels down a tube at the CERN accelerator moving at v/c=0.95 with respect to the laboratory.

(i) Compute the γ factor for these pions.

(ii) The pion lifetime is 2.6 × 10⁻⁸ s, what is the lifetime measured at the laboratory?

(iii) The pions travel 50 m until they are dumped at the end of the beam pipe. If the beam contained 50000 pions, how many remain at the moment of dumping?

(iv) From the pion’s perspective, how much distance has the laboratory covered before the beam hits it? and how long did it take?

(v) What would be the answers to questions (iii) and (iv) ignoring time dilation?

(vi) What is the energy of these pions from the laboratory’s perspective?

Homework Equations

The Attempt at a Solution

I am confused by question (iii), but i'll show my working up to that point before explaining why I'm confused.

(i) $\gamma = \frac{1}{\sqrt{1 - \frac{v^{2}}{c^{2}}}} = 3.2$

(ii) The pion lifetime given in the question is the 'proper time' as measured by the pion, $\Delta t' = 2.6 \times 10^{-8}$s.

As measured from the lab frame, $\Delta t = \gamma \Delta t' = 8.3 \times 10^{-8}$s

(iii) and now I'm confused... I think the important time for pion decay is the rest frame for the pions. This situation is analagous to the astronaut on a long journey aging at a slower rate than Earth observers.

The question states that the pions travel 50 m, but I'm unsure of this relation to the proper length. The pions are not at rest with respect to the tube, so I think this 50m is the contracted length.

Using 50m and v=0.95c, the time taken to traverse the tube is 1.75x10^-7s

Solving the decay equation $N = N_{0}e^{-\frac{t}{\tau}}$ with this value and the 'proper time' for the pion, I find that there are 59 pions at the end.

Part (iv) makes me think I've gone wrong...

Please could somebody take a look?

Thanks!

 

[mfb]

The 50 m are in the laboratory frame.
The contracted length will be shorter.