Assignment 9, solutions

Problem 1:

If the mass of the neutrino is mc2 = 0.03 eV, by how much does the speed of a 1 MeV neutrino differ from the speed of light, i.e. what is (c - v)/c?

Problem 2:

In inertial frame O a rod of length L is oriented along the x-axis and moving with velocity u in the positive y direction.  This rod is then viewed from an inertial reference frame O' moving with velocity v in the positive x direction.
(a)  What is the velocity of the rod in O'?
(b)  What is the length of the rod in O'?
(c)  What angle does the rod make with respect to the x' axis?

Problem 3:

imageA relativistic particle is launched at the origin (0,0) with initial momentum p(0) = (px(0), py(0)),  with px(0) > 0 and py(0) > 0, and is subject to a constant force pointing in the negative y direction.
(a)  Solve the equations of motion for x(t) and y(t).
(b)  Determine the time T at which the particle reaches the x-axis again (i.e. y(T) = 0).
(c)  Find the trajectory of the particle, i.e. y = y(x).

NOTE: Give all answers for the laboratory frame.

Problem 4:

Two distant galaxies A and B are observed by us on the earth to be receding in opposite directions, the speed of recession of each being 0.75c.  Will an observer on A be able to see galaxy B?  If so what will he say that the speed of recession of B is?

Problem 5:

In the following, two examples of a decaying system are presented, where the decay products travel with velocities comparable to the speed of light c.
1. Two electrons are ejected simultaneously in opposite directions from an atom.  Each electron has a speed as measured by a laboratory observer of 0.5 c.  What is the speed of one electron as seen from the rest frame of the other electron
(a)  in the non-relativistic approach?
(b)  in the relativistic approach?
Distinguish carefully the velocities in the respective frames.

2. The neutral pi meson, πo, has a rest mass of 135 MeV/c2.  It decays into two photons (γ rays) of equal energy and opposite direction in its rest frame.  In the laboratory frame the πo is moving with a total energy 25% larger than its rest energy.
(a)  What are the energies of the γ rays, as measured in the laboratory, if the decay process causes them to be emitted in opposite directions along the pion's original line of motion?
(b)  What is the velocity of each γ ray as observed by the other?