Muons are elementary particles that have a very short lifetime of about two millionth of a second (when at rest). They can be created in the upper atmosphere (at an altitude of say 10,000 meters) by cosmic rays, in which case they travel towards the surface of the Earth at very high speeds, say 99.9% of the speed of light. If we ignored special relativity, how far could they travel at this speed before they decay? You'll notice that even though they are very fast this distance is much less than 10,000 meters, and they would never reach the ground. This is not correct however, because we know that many of these particles do reach the ground, and experiments observe them all the time. How does special relativity resolve this apparent contradiction, and how far can they actually travel before the end of their life?
What they are calling the lifetime (2 microseconds) is actually about 1.28 half lives. There is no "end of life" number for a muon. Like all radioactivity, muons decay exponentially.
I have some other objections to this problem. The "99.9% of the speed of light" number is too high. 10,000 meters is not the upper atmosphere. It well below where planes fly; most of the muons are created much higher in the atmosphere. They are bandying numbers around that are wrong, and implying that muons has some sort of a fixed life. If they are trying to teach you about special relativity, they should use real numbers, and explain the time dilation equation
t' = t /sqrt[1 - (v/c)^2]
Radioactive decay and all physical and life processes go more slowly on moving bodies, when observed Earth-frame coordinates. If t is the true half life of a stationary muon, t' will be measured half life when the muon is moving at velocity v.
According to special relativity, the concept of time and space is not independent but is interconnected in what is known as "spacetime." Special relativity introduces the principle of time dilation, which states that time can appear to pass differently for objects moving relative to one another.
In the case of muons traveling at 99.9% of the speed of light, the time dilation effect becomes significant. From the perspective of muons, time appears to pass more slowly compared to an observer at rest on the Earth. Therefore, the two millionth of a second lifetime of muons, as measured in their own frame of reference, would actually be extended from the point of view of an Earth observer.
To determine the distance muons can travel before they decay, we can use the concept of time dilation. Let's consider the distance from the upper atmosphere to the surface of the Earth, which is approximately 10,000 meters.
The time dilation factor (gamma) can be calculated using the equation:
γ = 1 / √(1 - v²/c²)
where v is the velocity of the muons (0.999c) and c is the speed of light.
Substituting the values:
γ = 1 / √(1 - (0.999c)²/c²)
γ ≈ 22.4
This means that according to an observer on the Earth, the lifetime of muons will be extended by a factor of approximately 22.4 due to time dilation.
To find the distance muons can travel before they decay, we can multiply the distance from the upper atmosphere to the Earth's surface by the time dilation factor:
Distance = 10,000 meters * 22.4
Distance ≈ 224,000 meters
Therefore, taking special relativity into account, muons can travel a distance of about 224,000 meters before they decay. This explains why many muons reach the Earth's surface despite their short lifetime and high velocity.