Name _________________
Solutions to Test 1
September 19, 2014
This test consists of three parts. Please note that in parts II and III, you can skip one
question of those offered.
Possibly useful formulas:


 
F  qE  qu  B
p  qRB
 
u pc

c E
f 
f0
 1  v cos  c 
x    x  vt 
t     t  vx c 2 
y  y,
z  z
1   
E     E  vpx 
px    px  vE c
py  p y
pz  pz
n
2

u x 
ux  v
1  vu x c 2
uy
u y 
 1  vu x c 2 
u z 
uz
 1  vu x c 2 
 1  n  12 n  n  1  2 
Part I: Multiple Choice [20 points]
For each question, choose the best answer (2 points each)
1. The speed of light in vacuum as viewed by some observer depends on
A) The speed of the source (only)
B) The speed of the observer (only)
C) The speed of the source AND the speed of the observer
D) Neither the speed of the source nor the speed of the observer
E) The observer’s IQ divided by the source’s shoe size
2. Under what condition can you absolutely determine which of two events A and B
occurred first; that is, when will all observers agree which came first?
A) When the separation is timelike
B) When the separation is lightlike  This answer was also accepted
C) When the two events occur at the same place
D) You can always tell
E) You can never tell
3. Suppose two observers are examining the same system but one is moving relative to
the other along the x-axis. Which of the following momenta components pi or energy
E will the two agree on?
A) px but not py, pz, nor E
B) py and pz, but not px nor E
C) E, but not any component of momentum
D) E and px, but not py nor pz
E) All three components of the momentum pi, but not the energy E
4. In particle colliders, particles are forced to move in circles by _______ fields
A) Gravitational B) Scalar
C) Electric D) Magnetic E) None of these
5. If tachyons exist, with a negative mass squared, which of the following would
describe their speed?
A) It would always be slower than c
B) It would always be faster than c
C) It would always be equal to c
D) It could be at speed c or slower, but not faster
E) It could have any speed between 0 and infinity
6. The binomial approximation   1  v 2 2c 2 works pretty well for a particle moving
A) slowly
B) quickly C) at c
D) at any constant speed E) never
7. The Lorentz transformation tells you how to write primed coordinates in terms of
unprimed. If we wanted to write x in terms of the primed coordinates, which would
be the correct formula?
A) x    t   vx 
B) x    t   vx 
C) x    x  vt  
D) x    x  vt  
E) None of the above
8. In which of the following ways is time UNLIKE the other three dimensions of space?
A) Observers may disagree on space coordinates, but they agree on time
B) Observers may disagree on time coordinates, but they agree on space
C) Though the pace of time may differ for different observers, they all agree on
whether two events are simultaneous
D) Objects can move through space, but not through time
E) Time enters the distance formula with the opposite sign
9. The effective mass of two objects moving opposite directions, with velocity v each
with mass m, will be
A) 2m
B) 2m/
C) 2m
D) 0
E) None of these
10. Suppose a pole vaulter runs through a barn with a pole the same length as the barn. A
farmer briefly closes both doors at the same time, showing the pole is actually shorter
than the barn. How does the pole vaulter explain the fact that his pole fits in the
barn?
A) He is accelerating, and hence not in an inertial frame, so his perspective is wrong
B) Since he is the one that is truly moving, he actually sees his pole as short
C) He will actually see the doors of the barn hitting his pole, so it doesn’t fit
D) He will see the pole tilted at an angle, so it fits
E) He will claim that farmer didn’t close the doors simultaneously
Part II: Short answer [20 points]
Choose two of the following questions and give a short answer (1-3 sentences)
(10 points each).
11. Suppose we have two events in spacetime, with coordinates P1   x1 , y1 , z1 , t1  and
P2   x2 , y2 , z2 , t2  . Under what circumstances can we conclude that event P2 is in
the absolute future of P1? At least one formula is required.
You can only have one in the absolute past or future of the other if the separation
2
2
2
2
is timely, so that  x2  x1    y2  y1    z2  z1   c 2  t2  t1   0 . However, to make
sure P2 is in the future of P1, we also need the time to be greater for P2, so that t2  t1 .
12. You are running out of time to finish this test, so you decide to work on the test
while running around the room in circles very fast. You reason that since the
room is moving relative to you, time slows down relative to you, and this will
give you extra time to finish the test. Assuming you can run at high speeds,
explain why this is or isn’t a good idea.
Although the time in the room seems to slow down to you, you are constantly
accelerating, and hence your reasoning is flawed. In fact, given a choice between moving
at a constant velocity (sitting still) and a constantly changing velocity (running in
circles), the time is always maximized by moving at constant speed. Hence you are
actually making your situation worse.
13. Explain qualitatively why the equations F = ma and F = dp/dt cannot both be
true in relativity. Also, explain which of these is preferred, and why (a
conservation law should be mentioned).
These equations are equivalent if and only if p = mv, as is easily proven, but in
relativity, this formula is incorrect. Hence the formulas are incompatible in relativity.
The second formula, F = dp/dt, combined with Newton’s third law, results in
conservation of momentum, and hence this formula is preferred.
Part III: Calculation: [60 points]
Choose three of the following four questions and perform the indicated calculations (20
points each)
14. Luke Skywalker is flying his X-wing starfighter (length 12.50 m) past the
Millenium Falcon, flown by Han Solo. He notices as he passes that when his
watch advances 11.0 seconds, while Solo’s only advances 4.00 seconds.
(a) What is the relative speed, as a fraction of c, of the two spacecraft? Can we
say for sure which of the two is really moving, and which one is stationary?
From Luke’s perspective, Solo is moving past at high speed, so he will see Han’s
watch as slowing down. We use the formula t   to relate the two times. Since
Luke’s watch sees t  11 s , whereas Solo’s watch advances   4 s , we have
1
1 v c
2
2
 
1  v2 c2 
1

2
t



11.0 s
 2.75 ,
4.00 s
1
 0.1322 ,
2.752
v 2 c 2  1  0.1322  0.8678,
v c  0.8678  0.9316 .
Since all velocities are relative in relativity, there is no way to tell which of the two is
really moving, as it is arbitrary.
(b) How long does the X-wing appear to Han Solo?
The length given is the proper length, since Luke is seeing the S-wing in his own
frame, so Lp  12.50 m . The length as viewed by Han Solo will be L  L p  , so
L
Lp


12.50 m
 4.545 m .
2.75
(c) To Luke, the Millenium Falcon appears to be the same length as his
starfighter. How long is the Millenium Falcon according to Han Solo?
The Millenium falcon appears to be 12.50 m long, but this is as viewed by Luke,
who is not moving along with it, so in this case, L  12.50 m . The proper length is
therefore L p   L , so
Lp   L  2.75 12.50 m  34.38 m .
15. Atomic hydrogen atoms
us
alien
clou
produce radio waves with a
frequency of 1420 MHz. Suppose a cloud of hydrogen gas is moving directly
away from me, as sketched above, at a velocity of v = 1.80  108 m/s.
(a) What frequency would I observe for this cloud?
Because the object is moving directly away from us, this corresponds to an angle
of    . We therefore have, using the Doppler shift formula
 1.80 108 m/s 
1420
MHz
1





8
f 1  v2 c2
f0
 3.00  10 m/s 
f 
 0

1.80  108 m/s
 1  v cos  c  1  v cos  c
1
cos 
3.00 108 m/s
2
1420 MHz 

1  0.6002 1420 MHz  0.800 

 710 MHz .
1  0.600
1.600
(b) An alien culture is along the line of sight to the cloud, and moving at a speed
of 2.50  108 m/s compared to us. From their point of view, how fast is the
cloud moving?
This just uses the subtraction of velocity formula, namely
1.80 108 m/s    2.50  108 m/s 

ux  v

u x 
8
8
uxv
1  2 1   2.50  10 m/s 1.80  10 m/s 
2
c
 3.00 108 m/s 
0.70 108 m/s

 1.40  108 m/s .
1  0.500
(c) If the alien culture measured the frequency from the cloud, would they see a
higher or lower frequency? You do not need to perform a calculation.
Because the cloud, according to the aliens, is moving towards them, there will be
a blue shift. This means the frequency will be higher.
16. A charged pion + at rest has mass 139.6
MeV/c2. It decays spontaneously to a massless
+

+
neutrino  traveling to the left with energy E =
29.8 MeV, and to a muon+.
(a) What is the momentum of the neutrino in MeV/c? What is the energy and
momentum of the muon?


The neutrino is massless, so we have E  c p , so p  E c  29.8 MeV/c . Since
the particle is moving to the left, we should probably make it negative, or even add a
vector i to make it clear it is moving to the left:

p  29.8i MeV/c
The initial energy, for the pion, is just E  mc 2  139.6 MeV . It has no momentum. We
can then use conservation of energy and momentum to get these quantities for the muon,
so we have
E  E  E  139.6 MeV    29.8 MeV   109.8 MeV ,



p  p  p  0   29.78i MeV/c   29.78i MeV/c .
(b) What is the mass of the muon, in MeV/c2? What is its velocity as a fraction
of c?
We can find its mass using the formula
2
 mc 2   E 2  c 2 p 2  109.8 MeV 2   29.78 MeV 2  ,
mc 2  11170 MeV 2  105.7 MeV .
The velocity is then just
 
u pc 29.78i MeV


 0.271i .
c E
109.8 MeV
(c) A muon is itself unstable. In its own frame, its average lifetime is 2.197 s.
What is the average expected lifetime for this moving muon?
This average lifetime is the proper time, so we use the equation t   , and find
t   

1 u c
2
2

2.197  s
1  0.2712
 2.282  s .
17. The intergalactic launcher accelerates spacecraft of mass 1950 kg from rest by
pushing them along a track that reaches from the Sun to Neptune (d = 4.501012
m) with a force of 2.57  107 N.
(a) What is the initial and final energy of the spacecraft (in J)?
The spacecraft is at rest initially, so E   mc 2  mc 2 , so we have
Ei  mc 2  1950 kg   3.00 108 m/s   1.755 1020 J.
2
We can calculate the final energy by using work, using the formula W = Fd. We
therefore have
W  Fd   2.57 107 N  4.50 1012 m   1.156 1020 J .
The final energy is then
E f  Ei  W  1.755 1020 J   1.156 1020 J   2.9111020 J
(b) What is the initial and final momentum (in kgm/s) and velocity (in m/s) of
the spacecraft?
The spacecraft starts at rest and hence has momentum and velocity zero. The
final momentum is most easily calculated using E 2  p 2 c 2   mc 2  , so we have
2
p 2 c 2  E 2   mc 2    2.911 1020 J   1.755 1020 J   5.397 1040 J 2 ,
2
2
2
pc  5.397  1040 J 2  2.323 1020 J ,
2.323  1020 J
 7.744 1011 kg  m/s .
p
8
3.00  10 m/s
The final velocity can then be calculated in a variety of ways, such as
u pc 2.323 1020 J


 0.798 ,
c E 2.911 1020 J
u  0.798  3.00  108 m/s   2.394  108 m/s .
(c) How long does it take to launch a ship this way?
We use the formula F  dp dt , or for constant force, p  F t , and solve for
t :
p 7.744 1011 kg  m/s
t 

 3.01104 s  8.37 h .
7
F
2.57 10 N