Special Relativity particle decay.

[PsychonautQQ]

Homework Statement

An accelerator produces a beam of Un that travels to a detector located 100m away. The particles travel with a velocity of .866c, so in the laboratory frame it takes the particles .385*10^-6 seconds to get to the detector. By the time the particles get to the detector, half of the particles have decayed. What is the half life of Un? (note: half life as it would be measured in a frame moving with the particles)

Homework Equations

tpγ=t
N=Ni e^(-t/τ)

The Attempt at a Solution

so since there is 1/2 of the total particles left, i wrote the decay equation
1/2N=N e^(-t/τ)

dividing by N
1/2 = e^(-t/τ)
so
ln(1/2) = -t/τ
solving this for τ to figure out the half life in the laboratory frame.

now i have to set this number equal to the proper version of it τp multiplied by the gamme factor and solve for τp

τ = τpγ
τ/γ = τp = the half life of the particles according to their moving frame.
This gives me the wrong answer according to the back of my book (which is sometimes wrong) anyone know what I did wrong here?

 

[mfb]

This gives me the wrong answer according to the back of my book (which is sometimes wrong) anyone know what I did wrong here?

What did you get, and what is the solution in the book?

What do you get if you divide your answer by the book's answer? Is it some number you recognize?

I think you calculated the lifetime instead of the half life. There is no need to use the decay equation at all. You know that half of the particles decayed...