Question

1. Special Relativity (a) An observer sees a πon moving past with momentum ~p. In the rest frame of the πon, it decays into a µon and a νtrino in perpendicular directions to the direction in which the observer is moving. Find the velocity of the µon in the rest frame of the πon.

(b) Find the velocity of the µon in the frame of the observer. What angle does it make with the direction of the πon's motion? Repeat for the νtrino (which may be assumed to be massless).

(c) If the µon's average lifetime is τµ, then what is the average time taken it to decay relative to the observer?

Answer 1

Solution:-

(a) In the reference frame of the pion that decays into muon and muon neutrino. It decays into the direction perpendicular to the direction of the observer. Since it is a relativistic collision, the mass and energy are always conserved.

Let us assume the pion has a mass , mass of muon is , mass of neutrino is .

From the energy conservation , in the rest frame we get:

Conservation of momentum :

Together we get ,

and

The velocity of the muon in the perpendicular direction is :

(b) To find velocity and angle of the MUON in the frame of the observer :

is the angle with respect to the observer.

To find velocity and angle of the NEUTRINO in the frame of the observer :

(c) The time taken by the pion before decay observed by an observer at rest frame is given by :