Physics 264L: Assignment 3
Made available: Wednesday, September 9, 2015
Due:
Monday, September 14, 2015, by 6 pm.
Problems
1. Problem 3.8 on pages 87 of French.
2. Problem 4-3 on page 120 of French.
3. Problem 4-5 on page 120 of French.
4. Problem 4-15 on page 122 of French.
5. Problem 4-17 on page 123 of French.
6. (a) A rod of length L0 lies at rest in the x′ y ′ plane of an inertial frame of reference S ′ with one end
at the origin x′ = y ′ = 0 and the other end at point (L0 cos(θ0 ), L0 sin(θ0 )), so the rod makes
an angle θ0 up from the x′ axis. If the frame S ′ moves with constant speed v in the positive x
direction of a frame S, show that the length and orientation of the rod are given by
[
]1/2
v2
L = L0 1 − 2 cos2 θ0
,
c
tan(θ) = γ tan(θ0 ).
(1)
(b) The rod is now replaced by a laser beam that is emitted at an angle θ0 up from the x′ axis of the
S ′ frame. Show that in the S frame, the laser beam makes an angle θ given by the expression:
cos(θ) =
cos(θ0 ) + β
.
1 + β cos(θ0 )
(2)
(c) Explain why Eq. (2) implies that a source of light that, in its rest frame, emits light equally in all
directions like a spherical light bulb or star, will radiate its light strongly in the forward direction
in a frame for which the light is moving with speed v close to c. (This is called the “relativistic
headlight effect”.) Then use Eq. (2) to calculate the relative speed difference (c − v)/c = 1 − v/c
of a spherical light source such that half of its radiation will be emitted in a narrow forward cone
subtending an angle of one degree.
7. Problem 5-1 on page 159 of French.
8. A rod of proper length L moves with speed u parallel to and along the x-axis of a frame S. Show that
the length L′ of the rod measured by an observer at rest in a frame S ′ moving with speed v along the
positive x axis is given by:
[( 2
)(
)]1/2
c − v 2 c2 − u2
′
L =L
.
(3)
c2 − uv
1
9. To the nearest integer, please give the time in hours that it took you to complete this assignment,
including reading in the text.
Also, if you got help with this assignment, please give the names of the people from whom you got
help (classmates, the TA, myself, etc).
10. Optional Challenge Problems
(a) Challenge 3.1:
Dear 264L student: my name is Dima Mafia-orivich Ivanov, I am a wealthy
Russian who badly wants to see what the future will be like in 20 years but I don’t have the
patience to wait 20 years. So I would like to offer you 50 million dollars US to design a time
machine on or near Earth that will let me spend up to one year in the time machine to time
travel 20 years into the future. (I don’t want to get into a rocket and travel far away and return,
in case there is an emergency with my health or with the family business.) As a first step, using
your knowledge of special relativity, please tell me in some detail whether it is feasible to build
such a machine with current technology and, if so, how to do so. Your design should not subject
me to accelerations greater than 1 g (I want to live comfortably for the one year I spend in the
time machine), and the machine has to be buildable with available technology. (I will give you
a 10-million dollar bonus if you can further figure out how to let me time travel just when I am
sleeping for 8 hours each night, since that time is wasted anyhow, and I would like to spend time
with my friends during the day.) Please keep the bottle of Stolichnaya Elit as a gift for considering
my proposal.
(b) Challenge 3.2: An observer sees two objects moving with velocity vectors v1 and v2 . Show that
the relative speed v of one object with respect to the other object is given by
2
2
(v1 − v2 ) − 12 (v1 × v2 )
c
.
v =
)2
(
•
1 − v1 2v2
c
2
(4)
(c) Challenge 3.3: Investigate whether the familiar law of reflection for a beam of light striking a
mirror at rest—that the angle of incidence θi equals the angle of reflection θr —holds if the mirror
is moving with some constant velocity in some direction. In particular, does the law of reflection
hold for a planar mirror that is moving with speed v in a direction perpendicular to its plane?
Does it hold if the mirror is moving with speed v in a direction parallel to its plane? If it does
not hold, how are θi and θr related?
(d) Challenge 3.4: A cart rolls on a table with speed v. A smaller cart rolls on the first cart in the
same direction with speed v relative to the first cart. A third cart rolls on the second cart in the
same direction with speed v relative to the second cart, and so on. Find an explicit expression for
the speed vn of the nth cart in the frame of the table, and determine the limit of vn as n → ∞.
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