- #1

L_ucifer

- 12

- 0

Misplaced Homework Thread

**TL;DR Summary:**I got this question on a quiz for a Coursera course on special relativity, and I'm confused about the answer. I've detailed my thinking below any help on where I went wrong would be greatly appreciated.

Question.

Muons are unstable fundamental particles. In its own rest frame, the average lifetime of a muon is t0=2×10^−6s. Thus, if initially we have No muons, after time t, the number of remaining muons is N=No e^−t/t0.

Due to collision between cosmic rays and the earth's atmosphere, No=300,000 muons are created per square meter per minute. Suppose all the muons are created at a height of 10km from the earth surface, moving towards the center of the earth at v=0.98c and can travel through the atmosphere without interactions.

Let's now compare the predictions of Newtonian mechanics and relativity. In Newtonian mechanics, we would on average detect N

_{Newtonian}muons per square meter per minute on the earth surface. In reality, relativity is important and we can detect N muons per square meter per minute.

a. N

_{Newtonian}=1, N=0.01

b. N

_{Newtonian}=1, N=10,000

c. N

_{Newtonian}=0.01, N=10,000

d. N

_{Newtonian}=1, N=1

e. N

_{Newtonian}=0.01, N=1

f. N

_{Newtonian}=0.01, N=0.01Answer.

N

_{Newtonian}is fairly easy to find. We just find the time taken for the muons to reach the surface (distance / speed ) and then put this time in the decay equation. The answer we get is 0.01.

For N, the correct logic is you need to find the relative time of a muon's lifetime. So the equation is N=No e^−t/(t0 * gamma) and t is the same. This means the correct answer is c.

My logic for N is instead of finding the relative time of a muon's lifespan, we find the relative time for the muon to reach the surface. So t = distance/speed * gamma. I'm not sure why this is wrong and why we find the relative time for a muon's lifetime instead?

EDIT:

Here's the overall equation for the correct solution:

No = 300,000

t0 = 2 * 10^-6 s

t = distance / speed = 10,000 / 0.98c

N = No * e ^ -t/(t0 * gamma)

Here's the overall equation for my solution (which is wrong):

No = 300,000

t0 = 2 * 10^-6 s

t = distance / speed = 10,000 / 0.98c

N = No * e^-(t * gamma)/t0.

Last edited: