# Special Relativity Muon Distance Travelled Question

• ecneicS
In summary: The atmosphere is contracted in the muon's frame, and the length contracted distance is the distance through the atmosphere that the muon sees.In summary, muons are unstable subatomic particles that are produced by cosmic rays and travel at nearly the speed of light. They have a mean lifetime of 2.2 microseconds, but still manage to reach the ground from an altitude of 12.3km. Using time dilation and length contraction, it is possible to calculate the thickness of the atmosphere through which the muon must travel, as measured by the muon.
ecneicS

## Homework Statement

uons are unstable subatomic particles that decay to electrons with a mean lifetime of 2.2 microseconds. They are produced when cosmic rays bombard the upper atmosphere about 12.3km above the Earth's surface, and they travel very close to the speed of light. It is known that these muons reach the ground. From the point of view of the muon, it still lives for only 2.2 microseconds, so how does it make it to the ground? What is the thickness of the 12.3 km of atmosphere through which the muon must travel, as measured by the muon? Assume speed of muon is .999c

## Homework Equations

time dilation, length contraction

## The Attempt at a Solution

I am not so much asking for a solution as I am asking why what I have done is wrong.

The scenario is that the muon is flying towards Earth at a really high speed, and we are given its lifetime in respect to the muon's frame of reference.

So to me it makes sense to think of the following: That the muon is at rest and that the atmosphere and the Earth are moving with the same speed towards the muon that the muon was moving towards the Earth in the previous frame of reference.

By doing this and assuming that the muon does actually hit the earth, it follows logically (in my head) that because we have the velocity of the Earth in respect to the muon's reference frame, and that we have the lifetime of the muon in respect to the muon's reference frame, that we can solve for the height of the atmosphere in respect to the muon's reference frame.

V=d/t
d=Vt
d=(0.999c)*(2.2*10^-6)
d=660m = 0.66km

Please, if you can, point out the error in what I've done or tell me if you agree with my result.

ecneicS said:
So to me it makes sense to think of the following: That the muon is at rest and that the atmosphere and the Earth are moving with the same speed towards the muon that the muon was moving towards the Earth in the previous frame of reference.
Good.

By doing this and assuming that the muon does actually hit the earth, it follows logically (in my head) that because we have the velocity of the Earth in respect to the muon's reference frame, and that we have the lifetime of the muon in respect to the muon's reference frame, that we can solve for the height of the atmosphere in respect to the muon's reference frame.
Don't assume that the muon just barely makes it to the earth. The travel time to pass through the atmosphere will be less than the muon's lifetime.

## 1. What is the concept of "Special Relativity Muon Distance Travelled"?

The concept of "Special Relativity Muon Distance Travelled" is based on the theory of Special Relativity, which states that the laws of physics are the same for all observers in uniform motion. This theory predicts that objects moving at high speeds will experience time dilation, meaning that time will pass slower for the moving object compared to a stationary observer. This is relevant to muons, which are subatomic particles that are created in Earth's upper atmosphere and have a very short lifespan. Due to their high speed, muons travel a longer distance before decaying than they would if they were at rest, as observed from Earth.

## 2. How does the distance travelled by muons support the theory of Special Relativity?

The distance travelled by muons supports the theory of Special Relativity because it demonstrates the time dilation effect predicted by the theory. If the laws of physics were not the same for all observers, the muons would not travel a longer distance before decaying when moving at high speeds. This is strong evidence that supports the validity of the theory of Special Relativity.

## 3. How do scientists measure the distance travelled by muons?

Scientists measure the distance travelled by muons by using detectors that are placed at different heights and angles to the incident muon flux. The detectors are able to measure the number of muons that reach them, and by comparing the number of muons at different heights, scientists can calculate the distance travelled by the muons before decaying. This method is known as the "Muon Lifetime Experiment" and has been used to verify the predictions of Special Relativity.

## 4. What practical applications does the concept of "Special Relativity Muon Distance Travelled" have?

The concept of "Special Relativity Muon Distance Travelled" has several practical applications, including the development of particle accelerators and nuclear reactors. The understanding of time dilation and the effects of high-speed particles is crucial in the design and operation of these technologies. Additionally, the study of muons and their distance travelled has provided valuable insights into the Earth's upper atmosphere and the interactions between cosmic rays and our planet.

## 5. Are there any limitations to the concept of "Special Relativity Muon Distance Travelled"?

Like any scientific theory, the concept of "Special Relativity Muon Distance Travelled" has its limitations. One limitation is that it only applies to objects that are moving at speeds close to the speed of light. Additionally, the precision of measurements can be affected by external factors such as atmospheric conditions, which can introduce errors in the calculations. However, these limitations do not undermine the validity of the theory, as it has been extensively tested and proven to accurately predict the behavior of high-speed particles.

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