Muons are elementary particles produced in the upper atmosphere. They have a life time of 2.2 μs. Consider muons which are traveling vertically towards the earths surface at a speed of 0.998c. For an observer on earth, the height of the atmosphere above the surface of the earth is 10.4 km. Which of the following statements are true?a)The muons can never reach earth surfaceb)The apparent thickness of earth atmosphere in muons frame of reference is 0.96 kmc)The lifetime of muons in earth frame of reference is 34.8 μsd)Muons traveling at a speed greater than 0.998c reach the earths surfaceCorrect answer is option 'D'. Can you explain this answer?

Question

Muons are elementary particles produced in the upper atmosphere. They have a life time of 2.2 &mu;s. Consider muons which are traveling vertically towards the earths surface at a speed of 0.998c. For an observer on earth, the height of the atmosphere above the surface of the earth is 10.4 km. Which of the following statements are true?a)The muons can never reach earth surfaceb)The apparent thickness of earth atmosphere in muons frame of reference is 0.96 kmc)The lifetime of muons in earth frame of reference is 34.8 &mu;sd)Muons traveling at a speed greater than 0.998c reach the earths surfaceCorrect answer is option 'D'. Can you explain this answer?

Answer

Explanation:

Statement a: The muons can never reach the earth's surface.

This statement is false. Muons are produced in the upper atmosphere and they have a finite lifetime. However, due to time dilation effects predicted by special relativity, the muons experience time dilation and their lifetime appears longer for an observer on Earth. This allows them to reach the Earth's surface before decaying.

Statement b: The apparent thickness of the Earth's atmosphere in the muon's frame of reference is 0.96 km.

This statement is false. According to time dilation, the time experienced by the muons traveling at a high speed is dilated. This means that from the muon's perspective, time is passing slower compared to an observer on Earth. Therefore, the muons have a longer time to travel through the atmosphere, and the apparent thickness of the atmosphere in the muon's frame of reference is increased.

Statement c: The lifetime of muons in the Earth's frame of reference is 34.8 s.

This statement is true. The lifetime of muons is 2.2 s in their own rest frame. However, due to time dilation effects, the muons traveling at a high speed experience time dilation. The time dilation factor can be calculated using the formula:

γ = 1 / sqrt(1 - (v^2 / c^2))

Where γ is the time dilation factor, v is the velocity of the muons, and c is the speed of light.

For the given velocity of 0.998c, the time dilation factor is approximately 22.366. Therefore, the lifetime of muons in the Earth's frame of reference is 2.2 s * 22.366 ≈ 34.8 s.

Statement d: Muons traveling at a speed greater than 0.998c reach the Earth's surface.

This statement is true. Muons traveling at a speed greater than 0.998c would experience an even greater time dilation. This means that their apparent lifetime would be further extended, allowing them to reach the Earth's surface before decaying. Therefore, muons traveling at a speed greater than 0.998c can reach the Earth's surface.

Overall, the correct statements are c) The lifetime of muons in the Earth's frame of reference is 34.8 s and d) Muons traveling at a speed greater than 0.998c reach the Earth's surface.

Answer

A muon is an unstable subatomic particle, of the same class as an electron, but with a mass around 200 times greater. Muons are very short lived (2.2 microseconds). They are a very good example for real life observations of the special theory of relativity They can be created in particle accelerators but are created naturally when cosmic rays coming in from other space hit various atomic nuclei, atoms in the atmosphere, resulting in a spray of particles, some which being muons. Muons have an average velocity of 0.998c.

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