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Physics 264L First Midterm Exam
Professor Greenside
Wednesday, September 30, 2015
This exam is closed book and no electronic devices are allowed. The test will last the entire time from
10:05 am to 11:20 am. Please note the following:
1. Unless indicated otherwise, please answer all questions on extra blank pages so that you will have
plenty of space to write clearly.
2. Please write your name and the problem number at the top of each extra page. Before handing in
your exam, please sort your pages in order of increasing problem number and then staple all the pages
together.
3. When writing on the extra blank pages, please write clearly and logically. You will lose points if the
grader and I can not easily read and understand what you write.
4. Unless otherwise stated, you need to justify your answers, at least briefly with short phrases or sketches.
In particular, algebra without any verbal explanation or description will be given little or no credit,
even if the reasoning and answer are correct.
5. Do not hesitate to ask for help during the exam if you need clarification of a problem.
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Shorter Questions
For the questions in this section, please circle “true” or “false” on these pages for the true-false questions
or circle the single best answer to the multiple choice questions. You can write an optional brief explanation
of your reasoning in the space just below the question, which can get you some partial credit in case
you circle the wrong answer.
1. (4 points) True or false: according to special relativity, a massive particle can have an arbitrarily
large momentum magnitude p even though the speed of the particle can never exceed the speed of
light.
2. (4 points) True or false: according to the theory of special relativity, Newton’s second law ma = F
is wrong.
3. (4 points) True or false: a Duke basketball will have a spherical shape in all inertial frames of
reference.
4. (4 points) True or false: if an event A precedes an event B at the same spatial point in one inertial
frame of reference, then event A will precede event B in all other inertial frames of reference.
5. (4 points) A nuclear physicist studies particles emitted by a beam of radioactive nuclei in order to
test a theory that the average rates at which particles are emitted parallel to the beam in the forward
and backward directions are equal. True or false: if over a 10-hour period, the physicist measures
9,996 particles in the backwards direction and 10,103 particles in the forward direction, the theory
must be wrong.
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6. (4 points) True or false: an initially horizontal beam of neutrons and an initially horizontal beam
of photons (both in vacuum) fall at the same rate in Earth’s gravitational field.
7. (4 points) True or false: when you look at the center of the Sun with a high precision telescope on
Earth, the center of the Sun is actually in a slightly different location.
8. (4 points) True or false: for a rocket undergoing constant acceleration parallel to its axis, time
passes more slowly at the front of the rocket compared to the back of the rocket.
9. (4 points) Consider a long straight train that is moving with constant velocity v = vx̂ in the positive
x direction of some inertial frame S. A stationary observer on the train sees a person on the train
running towards the back of the train with the constant speed v (so the velocity of the runner is −vx̂
with respect to the train). True or false: there is no time dilation of the runner’s watch in the
frame S.
10. (4 points) True or false: for a triangle defined by three non-collinear points on a plastic disk (say
a CD) that is rotating with constant angular speed ω about an axis through the disk’s center and
perpendicular to the plane of the disk, the sum of the interior angles of the triangle is exactly 180◦ .
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11. (4 points) A rod moves with constant speed u ≪ c in the positive y direction of an inertial frame of
reference S such that it always remains parallel to the x axis of S as shown here:
S y
S'
v
u
x
Then in an inertial frame S ′ that moves with large constant speed v ≫ u in the positive x direction, a
stationary observer in S’ will measure the rod
(a) to be parallel to the x axis.
(b) to be rotated clockwise (rod has negative slope).
(c) to be rotate counterclockwise (rod has positive slope).
Longer Questions
Unless stated otherwise, please write all of your answers on the extra blank pages.
12. Two neutrons approach each other on the x-axis of some coordinate system, and each neutron has the
same speed v = βc according to a certain stationary observer.
(a) (8 points) In terms of β and the rest mass m of a neutron, what is the total energy E of one
neutron as measured by someone in the rest frame of the other neutron?
(b) (2 points)
mechanics.
For β ≪ 1, show that your answer reduces to the expected result for Newtonian
13. (10 points) In an inertial frame S with Cartesian coordinates x and y, a rocket Ry moves with
constant velocity uŷ along the positive y-axis, while rocket Rx moves with constant velocity vx̂ along
the positive x direction, with both rockets starting at the origin x = y = 0 at time t = 0. Rocket Ry
has a light beacon that continuously emits monochromatic light in all directions with frequency f in
its rest frame.
(a) (2 points) Is the frequency f ′ of this light as measured by someone in rocket Rx larger, the
same, or smaller than f ? Justify your answer briefly and qualitatively.
(b) (8 points) Derive a quantitative expression for the frequency f ′ of this light as measured by
someone in rocket Rx ? (You can treat the rockets as point objects for this problem.)
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14. (10 points) Two rockets O and O’ with equal proper lengths approach each other on nearby parallel
lines with equal and opposite velocities close to the speed of light (as measured by some observer) as
shown in panel (a) of this figure:
(a)
(b)
(c)
The rocket O on the right has a gun in its tail at point b that fires bullets perpendicular to the direction
of motion, and this rocket has been programmed to fire its gun when its tip at point a passes the tail a’
of the other rocket, as shown in panel (b). According to an observer in rocket O, rocket O’ on the left
is Lorentz contracted so rocket O expects the bullet to miss the other rocket, as indicated in panel (b).
But according to an observer in rocket O’, it is rocket O that is Lorentz contracted and therefore
rocket O’ should be struck by a bullet, as shown in panel (c).
Discuss whether the bullet actually hits or misses rocket O’. As part of your answer, explain why there
is an error in panel (b) or in panel (c), and explain what is that error.
15. (12 points) Two observers decide to test time dilation using identical atomic clocks. The first
observer is at rest at the origin x = 0 of an inertial frame S while the second observer is at rest at the
origin x′ = 0 of an inertial frame S ′ that moves with constant speed v = 0.6c with respect to S along
the x-axis of S. When the origins of S and S’ coincide, both observers start their clocks with times t = 0
and t′ = 0. When 60 minutes have passed according to their own clocks, each observer sends a light
signal to the other observer, and then, upon receiving that light signal, each observer instantly sends
a confirmation light signal back to the other observer.
In units of minutes, at what times t1 and t2 does the observer in S receive the two light signals, and
at what times t′1 and t′2 does the observer in S’ receive the two light signals? (Your answers will all be
positive integers, so don’t leave your answer in terms of unevaluated parameters like γ.)
16. A particle with rest mass m1 and that is moving with speed v1 collides with and sticks to a stationary
particle of mass m2 .
(a) (8 points) In terms of m1 , v1 , and m2 , what is the final speed of the composite particle according
to special relativity?
(b) (4 points) For v ≪ c, show that your answer reduces to the expected formula for a non-relativistic
inelastic collision.
(c) (4 points) Simplify your formula in the ultra-relativistic regime γ ≫ 1, and discuss briefly
whether your simplified formula makes physical sense.
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Points lost:
Total score:
Grade:
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• Problem 2:
• Problem 3:
• Problem 4:
• Problem 5:
• Problem 6:
• Problem 7:
• Problem 8:
• Problem 9:
• Problem 10:
• Problem 11:
• Problem 12:
• Problem 13:
• Problem 14:
• Problem 15:
• Problem 16:
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