Assignment 7, solutions

Problem 1:

Consider the decay Λ0 --> n + π0, followed by π0 --> 2γ.
(a)  Given the masses MΛ, Mn and Mπ, find the energy of the decay products n and π0 in the rest frame of the Λ0.
(b)  The two gamma rays from the decay of the π0 are observed to have equal energies in the rest frame of the Λ0.  Find the angle between the two gamma rays in this frame, in terms of the particle masses.


Problem 2:

As a rocket ship passes Earth at speed (3/5)c, clocks on Earth and on the ship are synchronized at t = 0.  The rocket ship sends a light signal back to Earth when its clock reads one hour.
(a)  According to Earth's clock, when was the signal sent?
(b)  According to Earth's clock, how long after the rocket passed did the signal arrive back on Earth?
(c)  According to the ship clock, when did the signal arrive back on Earth?


Problem 3:

An entrepreneur decides to operate a spacewash, which is normally at rest relative to its home planet.  Although the device is only 100 meters long, he advertises the spacewash as "the only one that simultaneously washes the front and back of a WizzFlizz 200TM."  According to the catalog, the WizzFlizz 200TM is 200 meters long.
(a)  An inspector arrived at spacewash to check how it operates.  What is the minimum speed v0 (in units of c) at which the WizzFlizz 200TM must go through the spacewash for the advertisement to be true?  (Assume that the wash is instantaneous.)
(b)  The pilot of the WizzFlizz 200TM observes the wash process from the spaceship.  At the minimum speed v0, how accurate does the pilot's clock need to be to see that the simultaneity claim is false?  (Assuming that it's true if observed from the spacewash.)
(c)   Suppose a spaceship is approaching the home planet with insufficient speed (4/5)v0.  The spacewash has its own engine that allows it to accelerate toward the ship.  To what speed should it accelerate (relative to its home planet) to perform the wash as promised in the advertisement?


Problem 4:

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 5:

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.