Question

The pion (pi+ or pi– ) is an unstable particle that decays with a proper half-life of 1.8 x 10⁻⁸ s. (This is the half-life measured in the pion's rest frame.) (a) What is the pion's half-life measured in a frame S where it is traveling at 0.8c? (b) If 32,000 pions are created at the same place, all traveling at this same speed, how many will remain after they have traveled down an evacuated pipe of length d = 36 m? Remember that after n half-lives, 2⁻ⁿ of the original particles survive. (c) What would the answer have been if you had ignored time dilation? (Naturally it is the answer (b) that agrees with experiment.)

Answer 1

The proper half-life of the pion is 1.8 x 10^-8 s. To find its half-life in frame S where it is traveling at 0.8c, use time dilation formula. For the given initial number of pions traveling at 0.8c, calculate how many will remain after traveling down a pipe of length d = 36 m using the formula n = n0 * 2^(-t/T) and consider the half-life in frame S. If we ignore time dilation, the calculation would not agree with experimental results.

Step-by-step explanation:

(a) The proper half-life of the pion is given as 1.8 x 10^-8 s. To find its half-life in frame S where it is traveling at 0.8c, we can use time dilation. The formula for time dilation is t' = t / √(1 - (v^2/c^2)), where t' is the time in frame S, t is the proper time, v is the velocity, and c is the speed of light. Plugging in the values, we get t' = (1.8 x 10^-8 s) / √(1 - (0.8^2)). Solving this equation gives us the half-life in frame S.

(b) If 32,000 pions are created and they all travel at the same speed of 0.8c, we can use the formula n = n0 * 2^(-t/T), where n is the remaining number of pions, n0 is the initial number of pions, t is the time traveled, and T is the half-life in frame S. Plugging in the values, we get n = (32000) * 2^(-(36 m)/(half-life in frame S)). Solving this equation gives us the number of pions that remain after traveling down the pipe.

(c) If we had ignored time dilation, the answer would be different. The number of pions that remain after traveling down the pipe would be calculated using the proper half-life of 1.8 x 10^-8 s instead of the half-life in frame S. This calculation would not agree with experimental results and would not take into account the effects of time dilation.