Calculating Pion Halflife: Gamma, Time Dilation, and Particle Decay

[blue2004STi]

Problem:

A group of 'pi' mesons (pions) is observed traveling at speed 0.8c in a particle physics laboratory. (a) What is the factor 'gamma' for the pions? (b) If the pions' proper half-life is 1.8x10^-8 s, what is their half-life as observed in the lab frame? (c) If there were initially 32,000 pions, how many will be left after they've traveled 36m? (d) What would the answer be to (c) if one ignored time dialation?

Equations:

Time dialation- delta T = (delta T')*gamma
gamma = 1/(sqrt(1-(beta)^2))
beta = velocity/speed of light

Attempt at solution:

a) No problem here... Just plug 'n chug... 1/sqrt(1-(.8c/c)) = 5/3

b) No problem here... delta T = (1.8x10^-8)(5/3)
delta T = 3x10^-8s

c) Here's where it begins... I started by calculating how much time was used going 37m in the labs frame which is 1.5x10^-7s. Then I calculated how much time was used in the pions frame... 1.5x10^-7(5/3) = 9x10^-8s. That's where I have no clue what to do next... I can't find any examples in the book or using google or in my notes for my class. Any ideas on an equation or maybe just a simple calculation I'm missing?

d) I can't figure out (c) so I don't know how to answer this question with any accuracy.

Thanks,

Matt

 

[edziura]

For part c, could you not solve the problem in the lab frame? In that frame:

Lab delta x = 36 m (I assuming this is a lab frame measurement)

Lab time of flight for 36 m = lab distance / lab speed = 36 m / 0.8 c

Lab half life = 3 (10^-8) seconds

A = Ao e^(-t / t 1/2) which is Ao e^(-5).

 

[blue2004STi]

I think that's what (d) is asking though...

Thanks though