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

• tasos
In summary, a charged pion with an average lifetime of 10^-8 seconds and a mean free path of 100 meters in water falls perpendicular to a lake with a depth of 30 meters and a velocity of 0.9999c. Based on the given information, it is not likely that the pion will reach the bottom of the lake before decaying or colliding with a nucleus. Further calculations would be needed to determine the probability of either event occurring.
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

## 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.

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?

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 $$N=N_0 e^{-\frac{L}{\ell}}$$

tasos said:
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.

## 1. What is a pion's mean free path?

A pion's mean free path is the average distance that a pion can travel in a material before undergoing a scattering event.

## 2. How is a pion's mean free path measured?

A pion's mean free path can be measured by tracking the number of pions that are scattered or absorbed by a material at different thicknesses, and then using this data to calculate the average distance traveled by the pions.

## 3. Why is the mean free path of a pion important?

The mean free path of a pion is important because it provides information about the properties of the material it is traveling through. It can also be used to estimate the amount of energy that is deposited by the pion as it travels through the material.

## 4. How does the mean free path of a pion in a lake compare to other materials?

The mean free path of a pion in a lake is typically longer than in other materials, such as air or water, due to the higher density of the lake. This means that pions can travel further before undergoing a scattering event in a lake compared to these other materials.

## 5. How can the mean free path of a pion in a lake be affected by environmental factors?

The mean free path of a pion in a lake can be affected by various environmental factors, such as the water temperature, depth of the lake, and presence of other materials in the water. These factors can impact the scattering processes and therefore affect the pion's mean free path in the lake.

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