Answers to Selected Problems - West Valley College

Problem Sets Physics 4D
(Tipler, P1.46) Two spaceships, each 100 m long when measured at rest, travel toward each other with
speeds of 0.85c relative to Earth.
a) How long is each ship as measured by someone on Earth.
b) How fast is each ship traveling as measured by an observer on the other?
c) How long is one ship when measured by an observer on the other?
d) At time t = 0 on earth, the fronts of the ships are together as they just begin to pass each other. At
what time on earth are their ends together?
e) Sketch accurately scaled diagrams in the frame of one of the ships showing the passing of the other
ship.
*Answer(s)
a) 52.7 m
b) 0.987 c
c) 16.1 m
d) 2.07 x 10 -7 s
e) The “other” ship will appear shorter by a factor of
Melvin J. Vaughn Page 1 of 61 West Valley College

Problem Sets Physics 4D
The Captain of the Federation starship Enterprise observes two photon torpedoes explode at two points
exactly 1500 meters apart. The first explosion occurs at (x1, y1, z1, t1) = (0, 0, 0, 0) and the second at (x2,
y2, z2, t2) = (1500 m, 0, 0, 0).
a) How far apart are these two events according to a Klingon battlecruiser moving along the +x axis at
.800c relative to Enterprise?
b) Are the events simultaneous to the Klingons? If not, which event occurred first and what is their time
difference?
Answer(s)
Melvin J. Vaughn Page 2 of 61 West Valley College

Problem Sets Physics 4D
c) Ann observes two light pulses to be emitted from the same location, but separated in time by 3.00 μs.
Bill sees the emission of the same two pulses separated in time by 9.00 μs.
a) How fast is Bill moving relative to Ann?
b) According to Bill, what is the separation in space of the two pulses?
Answer(s)
Melvin J. Vaughn Page 3 of 61 West Valley College

Problem Sets Physics 4D
Suppose our Sun is about to explode. In an effort to escape, we depart in a spaceship at v = 0.80c and
head toward the star Tau Ceti, 12 light years away (in Earth’s reference frame). When we reach the
midpoint of our journey from Earth, we see our Sun explode and, unfortunately, at the same instant we
see Tau Ceti explode as well.
a) In the spaceship’s frame of reference, should we conclude that the two explosions occurred
simultaneously? If not, which occurred first?
b) In a frame of reference in which the Sun and Tau Ceti are at rest, did they explode simultaneously? If
not, which exploded first?
Answer(s)
a) Simultaneous.
b) The Sun exploded first, 9.6 yrs before Tau Ceti.
Melvin J. Vaughn Page 4 of 61 West Valley College

Problem Sets Physics 4D
(Tipler, P1.57) Two observers agree to test time dilation. They use identical clocks and one observer in
frame S’ moves with speed v = 0.6c relative to the other observer in frame S. When their origins coincide,
they start their clocks. They agree to send a signal when their clocks read 60 min and to send a
confirmation signal when each receives the other’s signal.
a) When does the observer in S receive the first signal from the observer in S’? When does she receive
the confirmation signal?
b) Make a table showing the times in S when toe observer sent the first signal, received the first signal,
and received the confirmation signal. How does this table compare with one constructed by the
observer in S’?
*Answer(s)
a) The total time according to S: treceived = 120 minutes
b) The total time for the confirmation signal: tconfirmation = 240 minutes
c)
S S’
tsent 60 min. 60 min.
treceived 120 min. 120 min.
tconfirmation 240 min. 240 min.
Melvin J. Vaughn Page 5 of 61 West Valley College

Problem Sets Physics 4D
A very fast pole vaulter lives in the country. One day, while practicing, he notices a 10 m long barn with
the doors open at both ends. He decides to run through the barn a 0.866 c while carrying his 16 m long
pole. The farmer, who sees him coming, says, “Aha! This guy’s pole is length contracted. There will be
a short interval of time when the pole is entirely inside the barn. If I’m quick, I can simultaneously close
both barn doors while the pole vaulter and his pole are inside.” The pole vaulter, who sees the farmer
beside the barn, thinks to himself, “That farmer is crazy. The barn is length contracted shorter than my
pole. My 16 m long pole cannot fit into that short barn. If the farmer closes both doors just as the tip of
my pole reaches the back door, the front door will break off the end of my pole.”
So, the farmer bets the pole vaulter that he can close both doors while the pole is in the barn. As the
conditions of the bet, the farmer and vaulter agree that:
When the farmer sees the front of the pole reach the back door, the farmer will open the back door.
When the farmer sees the rear of the pole pass through the front door, the farmer will close both
doors.
Can the farmer close the doors without breaking the pole?
Show that, when properly analyzed, the farmer and the pole vaulter agree on the outcome. Your analysis
should contain both quantitative calculations and a written explanation.
Furthermore, this analysis must contain a thorough discussion of each event (the closing of the front door,
the closing of the rear door, and the opening of the rear door). For example: Were the events
simultaneous to each observer? If not, when and where did each event occur? How are these events
represented by a spacetime diagram?
Answer(s)
In the farmer’s reference frame:
1) At t = 0 s, both doors are closed
2) At t = +7.7 ns the back door is opened
In the pole vaulter’s reference frame:
1) At t = -57 ns the back door is cloded
2) At t = 0 42 ns the back door is opened
3) At t = 0 s the front door is closed.
Since, in the pole vaulter’s reference frame, the back door is closed, then opened, followed by the fro
door being opened, there is no paradox because there is no time at which the doors are closed at the
same time.
There is no paradox in the farmer’s reference frame, because there is a time at which the pole completely
fits within the barn, so . . . they BOTH WIN the bet!!
Melvin J. Vaughn Page 6 of 61 West Valley College

Problem Sets Physics 4D
Show the relativity of time by proving the time dilation equation for an observer moving at velocity v
relative to a light clock (as presented during lecture).
Answer(s)
t Melvin J. Vaughn Page 7 of 61 West Valley College
t
1
0
2
v
2
c

Problem Sets Physics 4D
1) If you were traveling with respect to the stars at a speed close to the speed of light, you could detect it
because
a) your mass would increase
b) your heart would slow down
c) you would shrink
d) . . . all of the above
e) you could never tell your speed by changes in you
2) Astronomers on Earth measure a distant star to be eight light years away, and when they decide to
send an astronaut to explore it at nearly the speed of light, the astronaut has only aged two years,
while several years pass on earth. When they discuss the time difference, the astronaut says that it
is because of time dilation, yet the astronomers say it must be due to length contraction. Who is
right?
a) the astronomers
b) the astronaut
c) neither, the reason lies in relativistic momentum
d) they are both correct, it’s the same phenomenon
Melvin J. Vaughn Page 8 of 61 West Valley College

Problem Sets Physics 4D
Sam leaves Venus in a spaceship headed to Mars and passes Sally, who is on Earth, with a relative
speed of 0.5c.
a) Each measures the Venus-Mars voyage time. Who measures a proper time: Sam, Sally, or neither?
b) On the way, Sam sends a pulse of light to mars. Each measures the travel time of the pulse. Who
measures a proper time?
Answer(s)
a) Sam
b) neither
Melvin J. Vaughn Page 9 of 61 West Valley College

Problem Sets Physics 4D
An unstable high-energy particle enters a detector and leaves a track of length 1.05 mm before it decays.
Its speed relative to the detector was 0.992c. What is its proper lifetime? That is, how long would the
particle have lasted before decay had it been at rest with respect to the detector?
Answer(s)
Melvin J. Vaughn Page 10 of 61 West Valley College

Problem Sets Physics 4D
The premise of the Planet of the Apes movies and book is that hibernating astronauts travel far into
Earth’s future, to a time when human civilization has been replaced by an ape civilization. Considering
only special relativity, determine how far into Earth’s future the astronauts would travel if they slept for
120 y while traveling relative to Earth with a speed of 0.9990c, first outward from earth and then back
again.
Answer(s)
Melvin J. Vaughn Page 11 of 61 West Valley College

Problem Sets Physics 4D
The center of our Milky Way galaxy is about 23000 ly away.
a) To eight significant figures, at what fraction of the speed of light would you need to travel exactly
23000 ly (measured in the Galaxy frame) in exactly 30 y (measured in your frame)?
b) Measured in your frame and in light-years, what length of the Galaxy would pass by you during the
trip?
Answer(s)
Melvin J. Vaughn Page 12 of 61 West Valley College

Problem Sets Physics 4D
The proper length of one spaceship is three times that of another. The two spaceships are traveling in
the same direction and, while both are passing overhead, an Earth observer measures the two
spaceships to have the same length. If the slower spaceship is moving with a speed of 0.35c, determine
the speed of the faster spaceship.
Answer(s)
Melvin J. Vaughn Page 13 of 61 West Valley College

Problem Sets Physics 4D
Two powerless rockets are on a collision course. Rocket A is moving at 0.800c relative to Ann, an Earth
observer. Rocket B is moving at -0.600 c relative to Ann. The two rockets are initially 2.52 x 10 12 m apart
and both are 50.0 m in length, as measured by Ann.
a) What are their respective proper lengths?
b) What is the length of each rocket as measured by an observer in the other rocket?
c) According to Ann, how long before the rockets collide?
d) According to rocket A, how long before they collide?
e) According to rocket B, how long before they collide?
f) If both crews are capable of total evacuation within 90 minutes (their own time), will there be any
casualties?
*Answer(s)
a) ΔL0A = 83.3 m
b) ΔL0B = 62.5 m
c) ΔtAnn = 6000 s = 100 min.
d) ΔtA = 5.33 x 103 s = 88.8 min.
e) ΔtB = 7.10 x 103 s = 118 min.
f) Since it takes 90 minutes for each crew to abandon its ship, the crew in rocketship A will be able to
escape in time, but the crew in rocketship B will not escape.
Melvin J. Vaughn Page 14 of 61 West Valley College

Problem Sets Physics 4D
Event 1 takes place at the origin of reference frame S, while Event 2 takes place at an x-coordinate of
3.00 ly in reference frame S. What is the minimum separation in time for Event 1 to be causally affected
by Event 2? Analyze by
a) drawing a spacetime diagram
b) evaluating the spacetime interval.
Answer(s)
Melvin J. Vaughn Page 15 of 61 West Valley College

Problem Sets Physics 4D
2) (Tipler, P1.17) Consider a clock at rest at the origin of the laboratory frame.
a) Draw a spacetime diagram that illustrates that this clock ticks slow when observed from the
reference frame of a rocket moving with respect to the laboratory at v = 0.800c.
b) When 10.0 s have elapsed on the rocket clock, how many have ticked by on the lab clock?
Answer(s)
Melvin J. Vaughn Page 16 of 61 West Valley College

Problem Sets Physics 4D
(Tipler, P1.42) In frame S, event B occurs 2.00 μs after event A and at Δx = 1.50 km from event A.
a) How fast must an observer be moving along the +x axis so that events A and B occur simultaneously?
b) Is it possible for event B to precede event A for some observer?
c) Draw a spacetime diagram that illustrates your answers to a) and b).
d) Compute the spacetime interval and proper distance between the events.
*Answer(s)
a) v = 1.20 x 10 8 m/s = 0.400 c
b) Yes, it is possible for event B to precede event A if v > 0.400 c for some observer.
c)
d) Δs = 1.37 x 10 3 m; Δx = 1.37 x 10 3 m
Melvin J. Vaughn Page 17 of 61 West Valley College

Problem Sets Physics 4D
(Tipler, P1.52) An observer in frame S standing at the origin observes two flashes of colored light
separated spatially by Δx = 2400 m. A blue flash occurs first, followed by a red flash 5.00 μs later. An
observer in S’ moving along the x axis at speed v relative to S also observes the flashes 5.00 μs apart
and with a separation of 2400 m, but the red flash is observed first. Find the magnitude and direction of v.
Answer(s)
Melvin J. Vaughn Page 18 of 61 West Valley College

Problem Sets Physics 4D
While onboard a starship, you intercept signals from four shuttle craft that are moving either directly
toward or directly away from you. The signals have the same proper frequency f0. The speed and
direction (both relative to you) of the shuttle craft are
a) 0.3c toward,
b) 0.6c toward,
c) 0.3c away, and
d) 0.6c away.
Rank the shuttle craft according to the frequency you receive, greatest first.
Answer(s)
Melvin J. Vaughn Page 19 of 61 West Valley College

Problem Sets Physics 4D
You wake up at night in your berth on a train to find yourself being “pulled” to one side of the train. You
naturally assume that the train is rounding a curve, but you are puzzled that you don’t hear any sounds of
motion. Offer another possible explanation that involves only gravity, not acceleration of your frame of
reference.
Answer(s)
Melvin J. Vaughn Page 20 of 61 West Valley College

Problem Sets Physics 4D
We readily note the bending of light by reflection and refraction, but why are we not aware of the bending
of light by gravity?
Answer(s)
Melvin J. Vaughn Page 21 of 61 West Valley College

Problem Sets Physics 4D
Why do we say that light travels in straight lines? Is it strictly accurate to say that a laser beam provides a
perfectly straight line for purposes of surveying? Explain.
Answer(s)
Melvin J. Vaughn Page 22 of 61 West Valley College

Problem Sets Physics 4D
Would we notice the slowing down or speeding up of a clock if we carried it to the bottom of a very deep
well?
Answer(s)
We wouldn’t notice it at all. However, an observer at the surface would observe that the clock carried to
the bottom of the well runs slightly faster, since the gravitational field is slightly weaker there.
Melvin J. Vaughn Page 23 of 61 West Valley College

Problem Sets Physics 4D
The celebrated equation E = mc 2 or m = E/c 2 (C is the speed of light) tells us how much mass loss, m, must
be suffered by a nuclear reactor in order to generate a given amount of energy, E. Which of the following
statements is correct? Why or why not?
a) The same equation E = mc 2 or m = E/c 2 , also tells us how much mass loss, m, must be suffered by a
flashlight battery when the flashlight puts out a given amount of energy, E.
b) The equation E = mc 2 applies to nuclear energy in a reactor, but not to chemical energy in a battery.
Answer(s)
Melvin J. Vaughn Page 24 of 61 West Valley College

Problem Sets Physics 4D
Armed with highly sensitive detection equipment, you are in the front of a railroad car that is accelerating
forward. Your friend at the rear of the car shines green light toward you.
a) Do you find the light to be red-shifted (lower frequency and energy), blue-shifted (higher frequency
and energy), or neither?
b) Explain by using the principle of equivalence.
*Answer(s)
a) Since the train is accelerating away from the green light, the light will be red-shifted, indicating that it
possesses lower frequency and energy relative to the observer in the front of the train.
b) The Principle of Equivalence states that there is no distinction between a uniform gravitational field
and a uniformly accelerating frame of reference. Therefore, the uniform acceleration of the train is
indistinguishable of gravity, therefore, causing the green light to “lose energy” as it follows the
“curvature” in the train’s spacetime.
Melvin J. Vaughn Page 25 of 61 West Valley College

Problem Sets Physics 4D
A spaceship, moving away from Earth at a speed of 0.900c, reports back by transmitting at a frequency
(measured in the spaceship frame) of 100 MHz. To what frequency must Earth receivers be tuned to
receive the report?
*Answer(s)
22.9 MHz
Melvin J. Vaughn Page 26 of 61 West Valley College

Problem Sets Physics 4D
A spaceship is moving away from Earth at speed 0.20c. A source on the rear of the ship emits light at
wavelength 450 nm according to someone on the ship. What
a) wavelength and
b) color (blue, green, yellow, or red) are detected by someone on Earth watching the ship?
*Answer(s)
a) 550 nm;
b) yellow
Melvin J. Vaughn Page 27 of 61 West Valley College

Problem Sets Physics 4D
The mass of an electron is 9.10938188 x 10 -31 kg. To six significant figures, find the speed for an electron
with kinetic energy K = 100.000 MeV.
Answer(s)
0.99987c
Melvin J. Vaughn Page 28 of 61 West Valley College

Problem Sets Physics 4D
A 5.00 grain aspirin tablet has a mass of 320. mg. For how many kilometers would the energy equivalent
of this mass power an automobile? Assume 12.75 km/L and a heat of combustion of 3.65 J/L for the
gasoline used in the automobile.
Answer(s)
1.01 x 10 14 km
Melvin J. Vaughn Page 29 of 61 West Valley College

Problem Sets Physics 4D
Answer the following:
a) How much energy is released in the explosion of a fission bomb containing 3.0 kg of fissionable
material? Assume that 0.10% of the mass is converted to energy.
b) What mass of TNT would have to explode to provide this same energy release? Assume that each
mole of TNT liberates 3.4 x 10 6 J of energy on exploding. The molecular mass of TNT is 0.227
kg/mol.
c) For the same mass of explosive, what is the ratio of the energy released in a nuclear explosion to that
released in a TNT explosion?
Answer(s)
Melvin J. Vaughn Page 30 of 61 West Valley College

Problem Sets Physics 4D
How much work must be done to increase the speed of an electron
a) from 0.18c to 0.19c?
b) 0.98c to 0.99c?
Note: The speed increase is 0.01c in both cases.
Answer(s)
Melvin J. Vaughn Page 31 of 61 West Valley College

Problem Sets Physics 4D
What must be the momentum of a particle of rest mass m0 so that the total energy of the particle is 3.00
times its rest energy?
Answer(s)
Melvin J. Vaughn Page 32 of 61 West Valley College

Problem Sets Physics 4D
Left over from the big-bang at the beginning of our universe, tiny black holes might still wander through
the universe. If one with a mass of 1.00 x 10 11 kg (and a radius of only 1.00 x 10 -16 m) reached Earth,
a) at what distance above your head would its gravitational pull on you match that of Earth’s (Use
Newton’s Universal law of gravitation)?
b) At what distance would your head be sucked into the black hole, with no possibility of escape?
Answer(s)
Melvin J. Vaughn Page 33 of 61 West Valley College

Problem Sets Physics 4D
In a Millikan experiment the distance of rise or fall of a droplet is 0.600 cm and the average time of fall
when E = 0 is 21.0 s. The observed successive rise times when the electric field is on are 46.0 s, 15.5 s,
28.1 s, 12.9 s, 45.3 s, and 20.0 s.
Using the derivation done in class:
q
mg v v
T E
E vT
a) Prove that charge is quantized. (Hint: charge is quantized if ratios of charge are ratios of small whole
numbers: e.g. q1/q2 = 2/5, q2/q3 = 5/7, etc.)
b) Calculate the successive charges on each drop, and, from these results, determine the fundamental
electric charge. Assume a plate separation of 1.60 cm, a potential difference of 4550 V for the
parallel-plate capacitor, and the mass for an oil drop of 1.67 x 10 -14 kg.
Hint: In Millikan’s time, e was known to be between 1.50 x 10 -19 C and 2.00 x 10 -19 C. Dividing q1 through
q6 by each of these values gives the range of integral charges on each drop. For example, if q1 = 8.39 x
10 -19 C, then the range for q1 is 8.39 / 1.5 = 5.6 and 8.39 / 2.0 = 4.2 multiples of electric charge e. Since 5
is an integer between 4.2 and 5.6. we can conclude that q1 = 5e, or e = q1/5. In like manner, determining e
for each charge, then averaging them, will generate the value of the electronic charge as determined in
Millikan’s day to be e = 1.69 x 10 -19 C.
Answer(s): (see Excel File, “p4d_charge_quantization_computation.xls” for calculations)
a)
q1/q2 = 5/8; q1/q3 = 5/6; q1/q4 = 5/9; q1/q5 = 1; q1/q6 = 5/7
q2/q3 = 4/3; q2/q4 = 8/9; q2/q5 = 8/5; q2/q6 = 8/7
q3/q4 = 2/3; q3/q5 = 6/5; q3/q6 = 6/7
q4/q5 = 9/5; q4/q6 = 9/7
q5/q6 = 5/7
n1 = 5
n2 = 8
n3 = 6
n4 = 9
n5 = 5
n6 = 7
b)
q1 = n1∙e = 8.38 x 10 -19 C
q2 = n2∙e =1.36 x 10 -18 C
q3 = n3∙e =1.01 x 10 -18 C
q4 = n4∙e =1.51 x 10 -18 C
q5 = n5∙e =8.42 x 10 -19 C
q6 = n6∙e =1.18 x 10 -18 C
eavg = 1.68 x 10 -19 C
Melvin J. Vaughn Page 34 of 61 West Valley College

Problem Sets Physics 4D
1) In a photoelectric effect experiment, the intensity of the light is increased. As a result:
a) There are more photoelectrons.
b) The photoelectrons are faster.
c) Both a and b.
d) Neither a nor b.
Explain your choice.
Intensity of light is directly proportional to the number of photons. A more intense light delivers a
larger number of photons to the surface, thereby ejecting a larger number of electrons, increasing the
current.
2) In a photoelectric effect experiment, the frequency of the light is increased. As a result:
a) There are more photoelectrons.
b) The photoelectrons are faster.
c) Both a and b.
d) Neither a nor b.
Explain your choice.
The maximum kinetic energy of electrons is directly proportional to the frequency of light, so, the
higher the frequency the faster the photoelectrons.
Melvin J. Vaughn Page 35 of 61 West Valley College

Problem Sets Physics 4D
Metal 1 has a larger work function than metal 2. Both are illuminated with the same short wavelength
ultraviolet light. Do photoelectrons from metal 1 have a higher speed, a lower speed, or the same speed
as photoelectrons from metal 2? Explain.
Answer(s)
Since electrons in metal 1 has a larger work function than in metal 2, they require more of the photon’s
energy to be liberated from the material. Therefore, they will have a lower speed than the photoelectrons
emitted from metal 2.
Melvin J. Vaughn Page 36 of 61 West Valley College

Problem Sets Physics 4D
The figure shows a typical current vs. potential difference graph for a photoelectric effect experiment.
Directly on the figure, draw and label graphs for the following three situations (assuming that no other
parameters of the experiment are changed):
a) The light intensity is increased.
b) The light frequency is increased.
c) The cathode work function is increased.
Answer(s)
a) Maximum current increases;
b) –Vstop decreases;
c) –Vstop increases.
Melvin J. Vaughn Page 37 of 61 West Valley College
ΔV

Problem Sets Physics 4D
The figure shows a typical current vs. frequency
graph for a photoelectric effect experiment. On the
figure, draw and label graphs for the following three
situations (assuming that no other parameters of
the experiment are changed):
a) The light intensity is increased.
b) The anode-cathode potential difference is
increased.
c) The cathode work function is increased.
Answer(s)
a) Current increases, f0 remains the same;
b) Maximum current increases, f0 remains the same; c) f0 increases
Melvin J. Vaughn Page 38 of 61 West Valley College

Problem Sets Physics 4D
The figure shows a typical stopping potential vs.
frequency graph for a photoelectric effect
experiment. On the figure, draw and label graphs
for the following two situations (assuming that no
other parameters of the experiment are changed):
a) The light intensity is increased.
b) The cathode work function is increased.
Answer(s)
a) remains the same;
b) f0 increases, graph shifts to the right.
Melvin J. Vaughn Page 39 of 61 West Valley College

Problem Sets Physics 4D
When Cesium metal is illuminated with light of wavelength 300 nm, the photoelectrons emitted have a
maximum kinetic energy of 2.23 eV. Find
a) the work function of cesium, and
b) the stopping potential if the incident light has a wavelength of 400 nm.
Answer(s)
Melvin J. Vaughn Page 40 of 61 West Valley College

Problem Sets Physics 4D
An FM radio station of frequency 107.7 MHz puts out a signal of 50,000 W. How many photons/s are
emitted?
Answer(s)
Melvin J. Vaughn Page 41 of 61 West Valley College

Problem Sets Physics 4D
How many photons/s are contained in a beam of electromagnetic radiation of total power 150W if the
source is an
a) AM radio station of 1100 kHz,
b) 8.0 nm x-rays, and
c) 4.0 MeV gamma rays?
Answer(s)
a) 2.06 x 10 29
b) 6.05 x 10 18
c) 2.34 x 10 14
Melvin J. Vaughn Page 42 of 61 West Valley College

Problem Sets Physics 4D
What is the maximum wavelength of incident light that can produce photoelectrons from silver (φ = 4.64
eV)? What will be the maximum kinetic energy of the photoelectrons if the wavelength is halved?
Answer(s)
a) λ = 267.2 nm;
b) K = 4.64 eV
Melvin J. Vaughn Page 43 of 61 West Valley College

Problem Sets Physics 4D
In a photoelectric experiment it is found that a stopping potential of 1.00 V is needed to stop all the
electrons when incident light of wavelength 260 nm is used and 2.30 V is needed for light of wavelength
207 nm. From these data determine Planck’s constant and the work function of the metal.
Answer(s)
h = 4.40 x 10 -15 eV∙s ; φ = 4.1 eV
Melvin J. Vaughn Page 44 of 61 West Valley College

Problem Sets Physics 4D
A photon having 40 keV scatters from a free electron at rest. What is the maximum energy that the
electron can obtain?
Answer(s)
5.41 eV
Melvin J. Vaughn Page 45 of 61 West Valley College

Problem Sets Physics 4D
A photon of wavelength 2.0 nm Compton-scatters from an electron at an angle of 90º. What is the
modified wavelength and the percentage change Δλ/λ?
Answer(s)
2.00243 nm; Δλ/λ 0.122%
Melvin J. Vaughn Page 46 of 61 West Valley College

Problem Sets Physics 4D
(P3.15: Thornton and Rex, 3 rd Ed.) The Spitzer Space Telescope was launched in 2003 to detect infrared
radiation. Suppose that a particle detector on the telescope is sensitive over part of the near infrared
region of wavelengths 980 - 1920 nm. Astronomers want to detect the radiation being emitted from a red
giant star and decide to concentrate on wavelengths from the Paschen series of the line spectra of the
hydrogen atom.
a) What are the known wavelengths in this region? Calculate these values!
b) The detector measures wavelengths of 1334.5, 1138.9, and 1046.1 nm believed to be from the
Paschen series. Why are these wavelengths different than those found in part a)?
c) How fast is the star moving with respect to us? (Hint: Doppler shift)
Answer(s)
a)
k = 4, λ = 1875.63 nm;
k = 5, λ = 1282.17 nm;
k = 6, λ = 1094.12 nm;
k = 7, λ = 1005.22 nm;
k = 8, λ = 954.86 nm
b) The observed spectral lines have been Doppler redshifted.
c) 1.20 x 10 7 m/s
Melvin J. Vaughn Page 47 of 61 West Valley College

Problem Sets Physics 4D
A particle is known to be in a certain interval on the x-axis. A function dependent upon the particle’s
position is given as:
0,
x 0
f( x)
1,
0 x
a) Plot this function.
b) Find the Fourier Series of this function on the given interval.
c) Graph the sum of the first seven terms of this function on the same graph used for part a).
Answer(s)
b)
n
1 1 1 1
f ( x)
sin nx
2 n1
n
Melvin J. Vaughn Page 48 of 61 West Valley College

Problem Sets Physics 4D
Determine the value of the coefficient A if every probable state of a particle is described by the following
wave function:
*Answer(s)
A
2 / 14
1 3
14 14
x sin x Asin 2 x sin 5 x
Melvin J. Vaughn Page 49 of 61 West Valley College

Problem Sets Physics 4D
The Wave Equation for Poker.
You decide to play your friends poker with five cards being dealt per hand. The following table gives the
probability of being dealt a specific hand before the draw using 53-card deck - including Joker. Construct
a wave function that represents your hand before you take a look. Assume each hand is mutually
exclusive (orthogonal). For example, you can hold either a three-of-a-kind or one-pair, not both. (See
next page)
The probability of being dealt...
(before the draw)
Expressed in
percent (%) is...
The odds against it
are...
Number of
possible
combos
5 ACES 0.00* 2,869,684 to 1 1
STRAIGHT-FLUSH
Royal Flush
Other Straight Flush
0.01
0.00**
0.01
14,066 to 1
119,569 to 1
15,942 to 1
FOUR-OF-A-KIND 0.03 3,465 to 1 828
FULL HOUSE 0.15 656 to 1 4,368
FLUSH 0.27 367 to 1 7,804
STRAIGHT 0.71 139 to 1 20,532
THREE-OF-A-KIND
3 Aces
Three-of-a-kind
(Large: K's - 8's)
Three-of-a-kind
(Small: 7's - 2's)
TWO-PAIR
Aces-up
K-up, Q-up, J-up
Small Two-Pair
(Tens-up and under)
2.21
0.37
0.92
Melvin J. Vaughn Page 50 of 61 West Valley College
0.92
4.83
1.10
1.69
2.03
44.3 to 1
271 to 1
108 to 1
108 to 1
19.7 to 1
89.6 to 1
58.0 to 1
48.2 to 1
204
24
180
63,360
10,560
26,400
26,400
138,600
31,680
48,600
58,320
ONE PAIR 42.34 1.36 to 1 1,215,024
PAIR OF A,K,Q,J (Openers)
Aces
Kings
Queens
Jacks
SHORTS
Shorts
(Large: 10's - 7's)
Shorts
(Small: 6's - 2's)
14.19
4.81
3.13
3.13
3.13
28.15
12.51
15.64
6.05 to 1
19.8 to 1
31.0 to 1
31.0 to 1
31.0 to 1
2.55 to 1
6.99 to 1
5.39 to 1
407,184
137,904
89,760
89,760
89,760
807,840
359,040
448,800

Problem Sets Physics 4D
Answer(s)
0.0000* 0.0001 0.0003 0.0015
hand 5acesstraightflush 4ofakind
fullhouse
0.0027 0.0071 0.0221 0.0483
flush straight 3ofakind 2pair
0.4234 0.1419
0.2815
1pair
pairAKQJ shorts
Melvin J. Vaughn Page 51 of 61 West Valley College

Problem Sets Physics 4D
The purpose of the following exercise is to demonstrate that a particle’s wave group velocity is equal to
the particle’s velocity.
A particle’s phase velocity does not have to equal to its particle velocity. In fact, show that a particle’s
deBroglie wave velocity is equal to half its particle velocity:
1
v v
2
phase particle
a) Show that the group velocity of a particle’s wave packet is equal to the derivative of its energy with
respect to its relativistic momentum:
v
group
dE
dp
b) Show that the group velocity of a particle’s wave packet is equal to its particle velocity:
Answer(s)
vgroup vparticle
Melvin J. Vaughn Page 52 of 61 West Valley College

Problem Sets Physics 4D
Two waves of equal amplitude are moving down a Slinky at different speeds are given by
( x, t) Asin( k x
t)
1 1 1
a) Write an expression for the resulting superimposed wave.
( x, t) Asin( k x
t)
and 1 2 2
Suppose A = 0.0030 m, k1 = 6.0 m -1 , k2 = 7.0 m -1 , ω1 = 300 s -1 , and ω2 = 250 s -1
b) Graph the superimposed wave at time t = 0.
c) What are the phase and group velocities of this wave?
d) What is Δx between two adjacent zeros of Ψ? (Note: Δx tells us how spread out this wave packet is –
it is the wave envelope.)
e) What is ΔkΔx?
Answer(s)
Melvin J. Vaughn Page 53 of 61 West Valley College

Problem Sets Physics 4D
A particle is known to be in a certain interval on the x-axis. A function dependent upon the particle’s
position is given as:
0,
x 0
f( x)
1,
0 x
a) Plot this function.
b) Find the Fourier Series of this function on the given interval.
c) Graph the sum of the first seven terms of this function on the same graph used for part a).
Answer(s)
b)
n
1 1 1 1
f ( x)
sin nx
2 n1
n
Melvin J. Vaughn Page 54 of 61 West Valley College

Problem Sets Physics 4D
A thin solid barrier in the xy-plane has a 10 μm diameter circular hole. An electron traveling in the zdirection
with vx = 0 m/s passes through the hole. Afterward, is vx still zero? If not, within what range is vx
likely to be?
Answer(s)
Melvin J. Vaughn Page 55 of 61 West Valley College

Problem Sets Physics 4D
Aliens visiting Earth are fascinated by baseball. They are so advanced that they have learned how to
vary to make sure that a pitcher cannot throw a strike with any confidence. Assume the width of the
strike zone is 0.38 m, the speed of the baseball is 35 m/s, the mass of the baseball is 145 g, and the ball
travels a distance of 18 m. What value of is required? (Hint: there are two uncertainties here: the width
of the strike zone and the transverse momentum of the pitched ball.)
*Answer(s)
≥ 0.081 J∙s
Melvin J. Vaughn Page 56 of 61 West Valley College

Problem Sets Physics 4D
What is the wavelength of
a) a photon with energy 1.00 eV,
b) an electron with energy of 1.00 eV,
c) a photon of energy 1.00 GeV, and
d) an electron with energy 1.00 GeV?
Answer(s)
a) 1.24 µm;
b) 1.22 nm;
c) 1.24 fm;
d) 1.24 fm
Melvin J. Vaughn Page 57 of 61 West Valley College

Problem Sets Physics 4D
A 5000 kg elephant is to move through a 1.00 m radius circular opening.
a) How fast must it move in order to diffract through the opening?
b) How fast must an electron move to diffract through the opening?
If the opening is 1.00 nm in radius,
c) how fast must the elephant move to diffract through it?
d) how fast must the electron move to diffract through it?
Answer(s)
a) 6.63 x 10 -38 m/s;
b) 3.64 x 10 -4 m/s;
c) 6.63 x 10 -27 m/s;
d) 3.64 x 10 5 m/s
Melvin J. Vaughn Page 58 of 61 West Valley College

Problem Sets Physics 4D
The highest achievable resolving power of a microscope is limited only by the wavelength used; that is,
the smallest item that can be distinguished has dimensions about equal to the wavelength. Suppose one
wishes to “see” inside an atom. Assuming the atom to have a diameter of 100 pm, this means that one
must be able to resolve a width of, say, 10 pm.
a) If an electron microscope is used, what minimum electron energy is required?
b) If a light microscope is used, what minimum photon energy is required?
c) Which microscope seems more practical? Why?
Answer(s)
a) 15 keV;
b) 120 keV
Melvin J. Vaughn Page 59 of 61 West Valley College

Problem Sets Physics 4D
What is the position uncertainty, in nm, of an electron whose velocity is known to be between 3.48 x 10 5
m/s and 3.58 x 10 5 m/s?
Answer(s)
5.79 nm
Melvin J. Vaughn Page 60 of 61 West Valley College

Problem Sets Physics 4D
Andrea, whose mass is 50 kg, think she’s sitting at rest in her 5.0 m long dorm room as she does her
physics homework. Can Andrea be sure she’s at rest? If not, within what range is her velocity likely to
be?
Answer(s)
vx = vy = vz = ±1.05 x 10 -37 m/s, assuming the dorm room is a box.
Melvin J. Vaughn Page 61 of 61 West Valley College