V
Doppler effect:
There are two contributions to the Doppler effect: one due to the time dilation and the one due to
the movement of the source parallel to the line of sight.
The first effect will alw
Space-time displacement
Space and time form the space-time displacement 4-vector
X = (ct, x, y,z)
The coordinates of this 4-vector X transform following the Lorentzs transformations:
& x " = #x $ #%ct
Notes on the train and the tunnel paradox
Imagine a train and a tunnel that, when both a rest, have a length of L.
L
Now imagine that the train is moving toward the tunnel with a relative ve
Problem 2: Causality in the Klingon Empire
(a)
(b) We transform the three events from Kronos frame (unprimed) to
Federation negotiators frame (primed) using Lorentz transformations
x = (xct) and ct =
Thus
5
fY (y) = (y 1)3/2
2
with fY (y) = 0 otherwise.
For (2) we note that for z (1/2, 1)
1
Y
x
2
5
(y 1)3/2 dy
2
1/z
FZ (z) = P (1/Y z) = P
=
1/z
=
2
5
(y 1)3/2 dy
2
Applying the FTC, we have
fZ (z)
Answer
First note that E[Xin ] =
1
n+1 .
Then
100
.
3
2
E[Xi2 ] = 100E[X1 ] =
E[Y ] =
i
Also note, that as the Xi s are independent
Xi2
Var(Y ) = Var
1 1
5 9
2
= 100Var(X1 ) = 100
i
=
80
9
Then we not
Problem 3: Algebra
Our task is to show that the interval I = (ct)2 (x)2 (y)2
(z)2 between two spacetime events is invariant under Lorentz transformations. For brevity, we will drop the symbols (which
Problem 2: Mr. Tompkins and the word when
(a) Distance as measured in town frame is d = 5 km. So the distance
as seen by the moving traveler is contracted and is d which is given by
d = d/ = d
1
= 4
Homework 2 Solutions
Problem 1: The Train-Tunnel Paradox
In order to resolve this apparent paradox, we will examine the
three key events, both in the rest frame of the tunnel and in the rest
frame of
Problem 4: Distorted Meter Stick
(a) With = 4/5, = 5/3. In the frame S0 , we know the length
l0 = 100 cm and the angle 0 = 60 , so we can calculate x0 = 50 cm
and y0 = 86.6 cm. In the frame S, x is co
Physics 128, Modern Physics
Final, December ll, 2012 . Name
Question 1 (15 points- 5 points each)
Consider the following Lorentz transformation between coordinates in O and 0. Consider O an
(c) Missile A gets launched at Cape Canaveral at time t = 3 AU/c
with an initial speed c. The missile deccelerates as it travels the xdirection through space (one dimensional), and at the position x 2