How Do Muons Travel from the Atmosphere to Earth Despite Their Short Lifespan?

[gyani]

Muons have a "proper lifetime" = 2.2 micro seconds ("proper lifetime" = lifetime of a muon which is at rest).
These Muons are created in the upper atmosphere at a height of 4700m above the Earth's surface. So, a muon traveling at a speed of 0.99c can travel only a distance of 653.4 m if you do distance = speed * time (the non-relativistic calculation). So you would not expect to find any muons to survive and reach the surface of the earth.

But the muons manage to travel almost 4700m and are detected near the surface of the earth. So now we need to explain it using Special Theory of Relativity.

Let me try Time Dilation. According to me( I am an observer in the frame of rest on earth), the muon is traveling is traveling at a speed v=0.99c. So it'll will suffer length contraction, but that'll not help me in my calculation. The muon is supposed to decay after 2.2 microseconds (that's the characteristic feature of a muon). But i know that the time interval 't' as observed by me for the muon, becomes larger given by t= gamma* to where to= proper lifetime and t= observed lifetime. So if to=2.2microsecond, t=15.6 microseconds. I think I've understood it correctly so far. So a fast-moving muon appears to live for a longer time.

WHAT I DON'T UNDERSTAND IS WHEN THEY SAY THAT:
LENGTH CONTRACTION: Muon sees shorter length (by gamma = 7.1). Does that mean that we're now considering the Muon to be at rest and the surrounding space to be moving towards it (in a direction perpendicular to the surface of the earth) at a speed of 0.99c? So the space gets contracted from Lo to L Where L= 653.4m. So the Muon thinks that it has traveled ONLY 653.4m, but we will see that the muon has actually traveled a distance of Lo= 4631.8 m and thus has managed to reach the surface of the earth.
Oh! Now i seem to have understood it after all!:P

Am i correct?

 

[yogi]

What is meant is that you first figure out how long the muon lived by how many arrive at the Earth's surface - this time is greater than the time in the lab (2.2 usec). So once you know that the muons clock is running slow relative to Earth clocks you can use the velocity and figure out how the muon would interpret the distance that it traveled. In the muon's frame the decay clock runs at the same rate as always (whether or not the muon is moving - it still decays in the same time as the lab generated muon 2.2 usec). So the effective distance that the muon would calculate would be vt and the time used by the muon would be 2.2 usec. So the muon believes that the distance traveled from birth to decay is very short.

 

[R0man]

Your understanding is mostly correct. Let's break it down step by step:

1. Length Contraction: As you correctly mentioned, length contraction is a phenomenon predicted by the Special Theory of Relativity where an object in motion appears to have a shorter length when observed from a stationary frame of reference. In this case, the muon is moving at a speed of 0.99c, so it will experience length contraction by a factor of 7.1. This means that the muon will see the distance between the upper atmosphere and the surface of the earth as much shorter than what we observe from a stationary frame of reference.

2. Time Dilation: As you mentioned, time dilation is also a phenomenon predicted by the Special Theory of Relativity where time appears to pass slower for an object in motion when observed from a stationary frame of reference. In this case, the muon's observed lifetime will be longer due to its high speed. This means that the muon will have enough time to travel the shorter distance (as observed by the muon) and still survive until it reaches the surface of the earth.

3. Combining Length Contraction and Time Dilation: When we combine these two phenomena, we can see that from the muon's perspective, both the distance and time have been affected. The distance appears shorter due to length contraction, and the time appears longer due to time dilation. This allows the muon to travel a shorter distance in a longer time and still reach the surface of the earth.

Overall, your understanding is correct. Just keep in mind that length contraction and time dilation are two separate phenomena, but they both play a role in explaining the survival of muons traveling at high speeds.