Pion's mean free path in a lake. exercIse question

[tasos]

Homework Statement

Charged pion with average life time $$\tau=10^{-8} sec$$, and mean free path in the water$$\ell=100m$$ falls prependicular to a lake (depth of lake is at $$\ell_0 =30m$$ with velocity $$V=0.9999c$$

What of the next is correct?

1). The particle isn't gona touch the bottom of the lake
2.)The particle have at least 60% probability to touch the lake
3.)The particle have a probability lower than 40% to touch the bottom of the lake

Homework Equations

The Attempt at a Solution

The first think i did is to calculate the Height of the lake "seen" by the pion
$$L=\ell_ 0 \sqrt{1+\frac{V^2}{c^2} } =0.42m$$

After that i calculate the time needs to touch the bottom of the lake

$$t=\frac{L}{V} =0.14 \times 10^{-8} $$

So if pion's life time is $$\tau=10^-8 sec$$ and it needs $$t=0.14 \times 10^{-8} $$, i say that it not gona touch the bottom of the lake.

BUT the excersice is giving me also the mean free path in the water$$\ell=100m$$
and i don't know how to use it, or if i need to use it.

Any sugestions? Thanx a lot.

 

[NascentOxygen]

I'm not able to help with particle physics, but ...

just looking at your conclusion, I can't see how you could arrive at it. The particle has a mean lifetime of 10^-8 s, and you say it would require just $\frac 1 7$ of this duration to traverse the lake's depth.

How do you conclude it probably won't reach the lake bed?

 

[tasos]

in pion's system from this results we conclude that is going to decay before it reach the bottom. This 1/7 you say its the problem. But given the average free path in the water ,i think i need to calculate some probability.
For example if i had a beam with $$N_0$$ pions, given the average path we no that $$N=N_0 e^{-\frac{t}{\tau}} $$ so we can see how many particles will survive for a given time t.
The same equation is for the path $$N=N_0 e^{-\frac{L}{\ell}}$$

 

[mfb]

in pion's system from this results we conclude that is going to decay before it reach the bottom.

Why? The flight time in the pion system is just 1/7 of its lifetime. Why do you expect all pions to decay so early?

Pions can decay or hit a nucleus in the water, in both cases the pion is not there any more. You'll have to check how likely both cases are to see if the pion makes it to the bottom.