Name _________________
Test 1
September 20, 2013
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
E     E  vpx 
t     t  vx c 2 
y  y,
z  z
u pc

c E
f 
x    x  vt 
f0
 1  v cos  c 
1   
px    px  vE c 2 
py  p y
pz  pz
n
 1  n  12 n  n  1  2 
ux 
uy 
uz 
ux  v
1  vux c 2
uy
 1  vux c 2 
uz
 1  vux c 2 
Part I: Multiple Choice [20 points]
For each question, choose the best answer (2 points each)
1. For small velocities v, we can approximate 1  v2 c2 as
A) 1  v c
B) 1  v c
C) 1  iv c
D) 1  v2 2c2
E) 1  v2 2c2
2. If I see a clock going by at sufficient speed, it will look like it is slowed down by a
factor of two. If someone moving with the clock looks at my clock, he will say my
clock
A) Is slowed down by a factor of two
B) Is speeded up by a factor of two
C) Is slowed down, but by a factor that is generally different from two
D) Is speeded up, but by a factor that is generally different from two
E) May be speeded up or slowed down, depending on which of us is actually moving
3. As the velocity of a massive particle approach c, the momentum approaches ___ and
the energy approaches ___
A) mc, 12 mc 2
B) mc, mc2 C) mc, 
D) , mc2
E) , 
4. Suppose a spacecraft is moving relative to us, and is emitting electromagnetic
radiation with frequency f0. In which circumstances will the detected frequency f be
lower than f0?
A) If it is moving directly away from us (only)
B) If it is moving directly towards us (only)
C) If it is moving perpendicular to our line of sight (only)
D) A and C are true, but not B
E) B and C are true, but not A
5. Which of the following equations is still valid in special relativity?
A) p = mv
B) F = ma C) W = Fd
D) E = ½mv2 E) none of these are valid
6. Suppose I try to communicate over long distances instantaneously by making a long
rigid rod and moving one end, with my partner noticing that his end moves in
response. This would fail because
A) No matter how slowly I move it, it will Lorentz contract, counteracting my effects
B) There is no such thing as a rigid rod in special relativity
C) Time dilation will ensure that there is a delay before it reaches the other end
D) The rod will distort spacetime, creating a black hole
E) Actually, this method would work
7. Ultimately, the Galilean transformation has to be replaced by the Lorentz boost
formula because the Galilean transformation does not
A) Include the effects of length contraction
B) Include the effects of time dilation
C) Preserve the four-dimensional distance formula
D) Allow for any kind of Doppler shift
E) Include the effects of the binomial expansion
8. The fourth component of space is time. What is the fourth component of momentum?
A) Force
B) Energy C) Distance D) Mass
E) Angular momentum
9. Which quantity below is not always conserved, according to special relativity?
A) Mass (only)
B) Momentum (only)
C) Energy (only)
D) Mass and momentum
E) Actually, all of these are always conserved
10. For known particles, the quantity E2 – c2p2 is always
A) Positive, but never negative or zero
B) Negative, but never zero or positive
C) Zero, but never negative or positive
D) Negative or zero, but never positive
E) Positive or zero, but never negative
Part II: Short answer [20 points]
Choose two of the following questions and give a short answer (1-3 sentences)
(10 points each).
11. We wish to make a negatively charged electron go in a clockwise circle,
as sketched at right. What sort of field, and which direction should we
have it point, to accomplish this?
12. Suppose I have 1 kg of some substance, and I now add energy to it by one of the
following: (a) heating it, (b) moving it; adding kinetic energy to it, (c) having it
undergo a chemical reaction that adds energy to it, (d) adding nuclear energy to it, (e)
compressing it to add mechanical energy. In which of these cases, if any, does its
mass increase?
13. If you push on an object of mass m with constant force F, it should undergo constant
acceleration, a, and hence it will eventually exceed the speed of light c. Explain
either why this reasoning is incorrect, or why the result does not, in fact, violate
relativity. What does happen if you push on an object with a constant force for a long
time?
Part III: Calculation: [60 points]
Choose three of the following four questions and perform the indicated calculations (20
points each)
14. In the year 2100, a pair of spacecraft are each launched towards the star Sirius, 9 ly
away (ly = cy). Spacecraft A travels directly there at a steady speed of v = 0.25c.
Spacecraft B, due to engine trouble, is stuck at rest for 18 years, but then they get their
engines working and travel thereafter at v = 0.5c, so that both arrive in 2136.
(a) What is the total time experienced by each of the travelers, from 2100–2136?
(b) Which of the two spacecraft experienced less total proper time? Explain
qualitatively how you could have known the answer to this without doing part (a).
(c) As spacecraft A approaches Sirius, how quickly do they see spacecraft B catching
up to them?
15. The Super Duper Hyperloop passenger transportation signal
system consists of cylindrical cars that (when at rest)
u
are 20.0 m long, and 3.0 m in diameter. They are
then accelerated to a speed of u = 1.00×108 m/s before being injected into the
transport tunnel.
(a) What are the dimensions of the cars, as viewed by a stationary observer?
(b) A businesswoman boards the cars when her watch says (correctly) the time is
9:20, and gets off when her watch says the time is 9:53. What time is it really
when she gets off? Ignore time zones.
(c) As the train approaches a signal light, a red light with frequency 407 THz shines
signals the driver that the train needs to stop. At what frequency does the driver
of the train see the light?
e–
e+
16. The BaBar experiment produces the (4s) particle
by colliding electrons with energy E1 = 9.0 GeV and positrons with energy E2 = 3.1
GeV head on, as sketched at right. The electron and positron are essentially massless.
(a) What is the momentum (in GeV/c) of the initial electron and positron? Don’t
forget that momentum is a vector!
(b) What is the momentum and energy of the (4s) particles that are produced?
(c) What is the mass (in GeV/c2) and velocity (as a fraction of c) of the resulting
(4s) particles?
17. A proton with mass m = 1.673×10-27 kg and charge e = +1.602×10-19 C is initially
moving to the left at velocity ui = –2.40×108 m/s. Suddenly an electric field is turned
on pointing to the right of magnitude E = 3.00×108 V/m. After a short time, it is
found that the proton is now moving to the right at uf = +2.40×108 m/s
(a) What is the initial and final momentum of the proton, in kgm/s? What is the
initial and final energy of the proton, in J? How much did each of them change?
(b) What is the force on the proton, in N (note: CV = Nm)?
(c) How long (in s) would it take for this change in direction to occur? What is the
total displacement that the particle has moved during this process?