Physics 106b: Assignment 1, 1/13/17
Due 7pm Friday, 20 January in the “Ph106 In Box” in East Bridge mailbox.
Relativity Basics
Reading
Hand and Finch Ch. 12 or Taylor Ch. 15. In §12.6-10 HF discuss a physical approach to the
properties of energy and momentum in special relativity, using explicit calculations of these
quantities for electromagnetic radiation. The arguments used should be familiar to you if
you took Ph1c. The geometric approach to 4-vectors is discussed more fully in Spacetime
Physics by Taylor and Wheeler. The discussion in Goldstein et al. is considerably more
advanced.
Problems
In these questions, units are used such that the speed of light c = 1. There are five questions.
1. Penny through the hole: Consider the setup shown in the S inertial frame of reference
in the figure.
z
x
(-4/5,0)
(0,0)
A penny is moving with x-velocity ux = 3/5 and z-velocity uz = 1/5. At time t = 0 it
just fits in, and then passes through an elliptical hole in the xy plane with diameter in
the x-direction of length 4/5, which is smaller than the diameter of the penny in its rest
frame. The frame S 0 moves in the +x direction relative to S with the speed v = 3/5, so
that in this frame the penny moves in the z 0 -direction, and the measured length of the
hole is less than the diameter of the penny. The origins of the two frames coincide at
t = 0 and the origin in the S frame is taken as the front edge of the hole.
Draw pictures analogous to the one above but in the S 0 frame, showing the penny and the
hole, and indicating the coordinates of the ends of both, at the time t0 = 0 when the right
end of the penny coincides with the right edge of the hole, and the time t0 at which the
left end of the penny coincides with the left edge of the hole. Describe how the (larger)
penny passes through the (smaller) hole from the perspective of the S 0 frame of reference.
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2. Light, a bullet, and a ruler: In frame S, a ray of light and the path of a bullet with
speed u run along the edge of a stationary ruler at an angle θ to the x-axis in the xy
plane. Now look at the angles θ0 relative to the x0 -axis observed in the frame S 0 moving
with speed v along the x-axis of S.
(a) What is the angle θr0 that the ruler is measured to have?
(b) What is the angle θb0 of the path of the bullet?
(c) What is the angle θl0 of the light ray?
(d) Are the three angles θr0 , θb0 , θl0 equal? Do the bullet and light ray run along the edge
of the ruler in the S 0 frame? Explain your answer in terms of frame independent
events.
3. Inertial frame specified by 4-velocity: A particular frame of reference S can be
characterized by its 4-velocity v. Similarly the motion of a massive particle is characterized
by its 4-velocity u (or equivalently its 4-momentum p = mu with m the mass). Properties
of the particle measured in S can be conveniently calculated in terms of the two 4-vectors
u (or p) and v.
(a) What are the components of v in the frame S?
(b) How is the relative (3-)velocity of the frame S relative to some other frame S 0 related
to components of v?
(c) Find the energy E of the particle in the frame S in terms of p and v.
(d) Show that the usual (3-)velocity ~u of the particle measured in frame S is given by
u
− v in the frame S.
the spacelike components of the 4-vector u·v
(e) Find the magnitude of the momentum |~p| of the particle in S in terms of p and v.
Note: In parts (c) and (e) your answer should be expressed in terms of scalar products
and magnitudes of 4-vectors, not components in some frame.
4. Relativistic rocket: A rocket ship is accelerated by ejecting mass at a constant speed
V 1 relative to the rocket. We want to find the velocity of the rocket in our inertial
frame from which the rocket was launched from rest, ignoring gravity.
(a) In the instantaneous rest frame S 0 of the rocket use momentum conservation to show
that the velocity increment du0 given by ejecting mass |dm| so that the rocket mass
changes to m + dm (dm is negative), is given by
m
du0
= −V.
dm
(1)
(b) Now transform to our frame in which the speed of the rocket is u to find the corresponding equation for m du/dm.
(c) Integrate the equation to find the speed u as a function of the mass m, the initial
mass m0 and V . What is the asymptotic speed for m m0 ?
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(d) Find the corresponding equation for Newtonian mechanics, and make a plot of u
against m0 /m over a large range to compare the two results for V = 0.2.
5. Pion production from the microwave background: Consider the reaction between
a nucleon N and the background microwave radiation approximated as a sea of photons
with energy Eγ = 3K ≡ 2.5 × 10−10 MeV
γ+N →N +π
with the neutron mass mN = 940MeV, and pion mass mπ = 140MeV.
(a) Find an equation giving the threshold energy EN of the nucleon for this reaction to
occur in terms of the given parameters, assuming a head on collision (photon and
nucleon moving in opposite directions).
(b) Evaluate EN in MeV to one significant figure using the approximation EN mN .
Hints: One approach is to evaluate EN from (pγ +pN )2 . The threshold energy corresponds
to all the particles produced being at rest in the center of momentum frame.
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