The following problems must be turned in by students enrolled in PHY 105 and those who plan on enrolling in PHY 106 next semester. (For those not in PHY105, the grade will not be recorded.) Reading, K&K Chapters 11,12,13 with particular emphasis on 11.3-11.5, 12.2-12.4, 13.1-13.3. Knight's Chapter 36 contains additional explanations.
Problem 11.1 Two atomic clocks are synchronized. One is
placed aboard a commercial jet airplane and is flown around the world,
following a circumference of the earth.
(a)When the jet returns to its starting point, what is the difference in the clock readings? Give your answer in microseconds. Look up or estimate any necessary physical parameters, and make reasonable approximations.
(b)A jet-bound observer might claim that the Earth-bound clock should run slower. Explain briefly why there is no paradox (this is related to the "twin paradox").
(c)What is the distance travelled according to those on the jet?
(d)What is the radius of the path according to those on the jet?
Problem 11.2 (K&K 12.5, reworded) An observer sees two spaceships flying apart, each with speed 0.99c relative to the observer. What is the speed of one spaceship as viewed by the other?
Problem 11.3 A laser beam is aimed at angle θ above the horizontal axis (the x-axis) as measured in the rest frame of the laser. An observer in a different reference frame sees the laser moving at very high velocity, v, along the x-axis in the observer's frame. What is the laser beam's angle above the horizontal in this observer's frame? (You may leave your answer in terms of β=v/c and the Lorentz factor, γ, as well as v and θ.)
(Hint: this problem involves two spatial dimensions and one temporal dimension. It might be useful to think in terms of 4-vectors. Consider a pulse of light from the laser. Define two events to be the locations of the laser pulse at different times.)
Note: the inverse of this effect, in which the apparent direction of starlight detected by telescopes on Earth depends on the motion of the Earth, is called "stellar aberration," and was discovered as early as 1727.
Problem 11.4 Consider each of the following processes. If the process can occur, what can you say about the photon(s)? If the process cannot occur, why not?
Consider only constraints from relativity; you may ignore such things as particle charge and spin.
(Hint: think about the center-of-mass frame of the massive particle(s) in each process.)
Problem 11.5 A particle of mass m and relativistic speed v collides and sticks to a stationary particle of mass M. What is the final speed of the composite particle? What is the mass of the composite particle? How much kinetic energy was converted into mass? Compare the low-speed limits with the Newtonian answers.
(Notes: (1) Kinetic energy is total relativistic energy minus the rest energy, i.e., K=E-mc2. (2) Whenever we say "mass" we mean "rest mass". (3) Give all answers in the frame in which M was initially at rest.)
Problem 11.6 K&K 13.1
Problem 11.7 K&K 13.4
Problem 11.6 For fun though optional!
(but not part of the PS grade). Many physicists believe that every particle
in the universe, if you look closely enough, is actually an oscillating
piece of something called a relativistic string. A relativistic string is a
substance with constant energy per length, rho, so to stretch it an amount
delta x takes an amount of work equal to (rho)(delta x).
Two particles of mass m are connected by a relativistic string with energy per unit length rho. At time t=0 they are pulled a distance L apart and released from rest and begin to move towards each other. What is the distance between the particles as a function of time? Sketch it on a spacetime diagram. Hint: set up a differential equation that relates the energy at time t and at time t+dt.