How Fast Must Muons Travel to Reach a Distant Target Without Decaying?

[Elvis 123456789]

Homework Statement

Unstable particles cannot live very long. Their mean life time t is defined by N(t) = N0e−t/τ , i.e., after a time of t, the number of particles left is N0/e. (For muons, τ=2.2µs.) Due to time dilation and length contraction, unstable particles can still travel far if their speeds are high enough

Problem 4
a) For musons at rest, traveling at 0.6c, 0.8c, 0.9999c, plot the N(t)/N0 ratio as a function of time.

b) For muons traveling at 0.8c, derive N(L)/N0 as a function of L, where L is the distance travelled, measured in the Earth RF. You don’t need to plot these functions.

c) Assume you are designing an experiment using muons beams that are directed at a target in a neighboring city 50km away. You want to have at least half of the muons reach the target without decaying. What is the minimum speed of the muons you must have?

Homework Equations

N(t) = N0e−t/τ

t_e = t_m *γ

L_m = L_e/γ

t_e & L_e is the time and length measured in the Earth frame of reference

and t_m and L_m is the time and length measured in the muon's frame of reference

I did all the parts but I feel pretty unsure about it. I was hoping you guys could take a look and let me know if it seems ok. Thanks in advance!

The Attempt at a Solution

Parts a.) and b.) are in the attached image

c.) N(L_e) = N_0 * e^(-L_e/v*τ_e)

after some algebra I end up with the expression

v = [ (τ_m*ln(2)/L_e)^2 + 1/c^2]^-1/2

τ_m = 2.2 * 10^-6 s
L_e = 50*10^3 m

after plugging in these values I get

v = 299,987,443.2 m/s

 

[TSny]

Your work looks correct to me. You could express the answer for part (c) in terms of the speed of light c.

 

[Elvis 123456789]

Your work looks correct to me. You could express the answer for part (c) in terms of the speed of light c.

Thank you very much! If its not too much trouble, do you think you can take a look at my other post at

https://www.physicsforums.com/threads/another-relativistic-particle-decay-question.884105/

? These relativity questions just seem really ambiguous to me.