Question

.. A group of TT mesons (pions) is observed traveling at speed 0.8c in a particle-physics laboratory a) What is the factor y for the pions? (b) If the pions proper half-life is 1.8 x 108 s, what is their half-life as observed in the lab frame? (c) If there were initial- ly 32,000 pions, how many will be left after they have traveled 36 m? (d) What would be the answer to (c) if one ignored time dilation? 1.26

Answer 1

$\gamma$ = 1/(1-v²/c²)^1/2 = 1/(1-0.8²)^1/2 = 5/3

b) T_1/2 (proper) = 1.8E-8 s

T_1/2 (lab) = $\gamma$T_1/2 (proper) = 5/3 *1.8E-8 s = 3.0E-8 s

c) time to travel 36 m in lab frame:

length in meson frame 36*3/5 = 21.6 m

time of travel = 21.6/ 0.8c = 9.0 E-8 s

number mesons remaining after = 32000*exp(-0.693*9.0/1.8)

= 1000

if we do not consider time dilation or length contraction then

time of travel = 36/0.8c = 15.0E-8 s

number of mesons = 32000*exp(-0.693*15.0/1.8)

= 99