An electron and positron moving at equal and opposite velocities in the lab fram

Question

An electron and positron moving at equal and opposite velocities in the lab frame collide and annihilate, producing a pi^+ and a pi^_.The masses are given in the table below. Determine the minimum speed the electron and positron must each have in order to produce the pions. Calculate the speed of the positron in the electron's frame of reference. In the lab frame, the electron and positron each travel 1 km to the collision point. How far does the electron travel in the positron's frame of reference? What is the proper distance for this collision? How long do the electron and positron take to travel the 1 km in the lab frame? How long does the electron take to travel to the collision in the positron's frame? What is the proper time for this motion? If the pions leave the collision moving at 0.6c in the lab frame, calculate the speed the initial electron and positron must have had. How far does each pion travel in one half-life (1) in the lab frame, (2) in the pion's own frame, (3) in the other pion's frame. Which is the proper time? A positron traveling at 0.8c encounters a stationary electron and they annihilate producing a single photon. What is the total momentum in the electron's frame before collision? What is the momentum after collision and where is it? Calculate the total energy before and after collision. Relate the energy to the momentum of the photon using relativity. Calculate the wavelength and frequency for the photon using E = hf. The h is Planck's constant.

Explanation / Answer

D)   a) speed of electron = 0.67 c

      b) speed of positron = 0.8 c

      c) Far does electron travel = 0.852 km

            proper distance of collision = 0.512 km

     d)    long electron and positron travel =   24.2 ns

            long electron travel   =   25.13 ns

           proper time   =   12.324 ns

     e)   speed of electron =   0.685 c

            speed of positron = 0.457 c

     f)     lab frame =   0.672 km

            pion own frame = 0.765 km

            other pion frame = 0.567 km

E) a) total momentum =   9.653 * 10-27 kg.m/sec

     b)    total momentum after collision=   5.342 * 10-27 kg.m/sec

    c)    Total energy before   =   4.2 MeV

              After = 3.85 MeV

    d)   K2 + 2mc2K = (pc)2

      e)   wavelength =   523 nm

          frequency   = 5.736 * 1014 Hz

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