Modern Physics

Kenneth S. Krane

Chapter 2

The Special Theory of Relativity - all with Video Answers

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Chapter Questions

Problem 1

You are piloting a small airplane in which you want to reach a destination that is $750 \mathrm{km}$ due north of your starting location. Once you are airborne, you find that (due to a strong but steady wind) to maintain a northerly course you must point the nose of the plane at an angle that is $22^{\circ}$ west of true north. From previous flights on this route in the absence of wind, you know that it takes you 3.14 h to make the journey. With the wind blowing, you find that it takes $4.32 \mathrm{h}$. A fellow pilot calls you to ask about the wind velocity (magnitude and direction). What is your report?

Supratim Pal

Supratim Pal

Numerade Educator

Problem 2

A moving sidewalk $95 \mathrm{m}$ in length carries passengers at a speed of $0.53 \mathrm{m} / \mathrm{s}$. One passenger has a normal walking speed of $1.24 \mathrm{m} / \mathrm{s}$. $(a)$ If the passenger stands on the sidewalk without walking, how long does it take her to travel the length of the sidewalk? $(b)$ If she walks at her normal walking speed on the sidewalk, how long does it take to travel the full length? $(c)$ When she reaches the end of the sidewalk, she suddenly realizes that she left a package at the opposite end. She walks rapidly back along the sidewalk at double her normal walking speed to retrieve the package. How long does it take her to reach the package?

Narayan Hari

Numerade Educator

Problem 3

A shift of one fringe in the Michelson-Morley experiment corresponds to a change in the round-trip travel time along one arm of the interferometer by one period of vibration of light (about $\left.2 \times 10^{-15} \mathrm{s}\right)$ when the apparatus is rotated by $90^{\circ} .$ Based on the results of Example 2.3 what velocity through the ether would be deduced from a shift of one fringe? (Take the length of the interferometer arm to be $11 \mathrm{m} .$ )

Anand Jangid

Numerade Educator

Problem 4

The distance from New York to Los Angeles is about $4000 \mathrm{km}$ and should take about $40 \mathrm{h}$ in a car driving at $100 \mathrm{km} / \mathrm{h} .(a)$ How much shorter than $4000 \mathrm{km}$ is the distance according to the car travelers? (b) How much less than $40 \mathrm{h}$ do they age during the trip?

Narayan Hari

Numerade Educator

Problem 5

How fast must an object move before its length appears to be contracted to one-half its proper length?

Narayan Hari

Numerade Educator

Problem 6

An astronaut must journey to a distant planet, which is 300 light-years from Earth. What speed will be necessary if the astronaut wishes to age only 12 years during the round trip?

Aparna Shakti

Numerade Educator

Problem 7

The proper lifetime of a certain particle is $120.0 \mathrm{ns}$. (a) How long does it live in the laboratory if it moves at $v=0.950 \mathrm{c} ?$ (b) How far does it travel in the laboratory during that time? $(c)$ What is the distance traveled in the laboratory according to an observer moving with the particle?

Narayan Hari

Numerade Educator

Problem 8

High-energy particles are observed in laboratories by photographing the tracks they leave in certain detectors; the length of the track depends on the speed of the particle and its lifetime. A particle moving at $0.993 c$ leaves a track $1.15 \mathrm{mm}$ long. What is the proper lifetime of the particle?

Narayan Hari

Numerade Educator

Problem 9

Carry out the missing steps in the derivation of Equation 2.17

Suzanne W.

Numerade Educator

Problem 10

Two spaceships approach the Earth from opposite directions. According to an observer on the Earth, ship $A$ is moving at a speed of $0.743 c$ and ship $B$ at a speed of $0.831 c .$ What is the velocity of ship $A$ as observed from ship $B ?$ Of ship $B$ as observed from ship $A$ ?

Anand Jangid

Numerade Educator

Problem 11

Rocket $A$ leaves a space station with a speed of 0.811 c. Later, rocket $B$ leaves in the same direction with a speed of $0.665 c .$ What is the velocity of rocket $A$ as observed from rocket $B ?$

Yaqub Khan

Numerade Educator

Problem 12

One of the strongest emission lines observed from distant galaxies comes from hydrogen and has a wavelength of $122 \mathrm{nm}$ (in the ultraviolet region). $(a)$ How fast must a galaxy be moving away from us in order for that line to be observed in the visible region at $366 \mathrm{nm} ?$ ( $b$ ) What would be the wavelength of the line if that galaxy were moving toward us at the same speed?

Narayan Hari

Numerade Educator

Problem 13

A physics professor claims in court that the reason he went through the red light $(\lambda=650 \mathrm{nm})$ was that, due to his motion, the red color was Doppler shifted to green $(\lambda=550 \mathrm{nm}) .$ How fast was he going?

Narayan Hari

Numerade Educator

Problem 14

Three rods are joined to form a 45 - 45-90 triangle, which is placed at rest in the $x^{\prime} y^{\prime}$ plane with its hypotenuse along the $x^{\prime}$ axis. $O^{\prime}$ moves away from $O$ in the $x$ direction at a speed of $0.92 \mathrm{c} .$ What are the angles of the triangle according to $O ?$

Suzanne W.

Numerade Educator

Problem 15

In the Relativistic Heavy Ion Collider (an accelerator at Brookhaven National Laboratory in New York), nuclei of gold atoms are accelerated to a speed of $0.99995 c$ and then forced to undergo head-on collisions. The radius of a gold nucleus is about $7.0 \mathrm{fm}$. Compared to its normal density, what is the density of a gold nucleus traveling at this speed?

Suzanne W.

Numerade Educator

Problem 16

Derive the Lorentz velocity transformations for $v_{x}^{\prime}$ and $v_{z}^{\prime}$

Suzanne W.

Numerade Educator

Problem 17

Observer $O$ fires a light beam in the $y$ direction $\left(v_{y}=c\right)$. Use the Lorentz velocity transformation to find $v_{x}^{\prime}$ and $v_{1}^{\prime}$ and show that $O^{\prime}$ also measures the value $c$ for the speed of light. Assume $O^{\prime}$ moves relative to $O$ with velocity $u$ in the $x$ direction.

Suzanne W.

Numerade Educator

Problem 18

A light bulb at point $x$ in the frame of reference of $O$ blinks on and off at intervals $\Delta t=t_{2}-t_{1} .$ Observer $O^{\prime}$ moving relative to $O$ at speed $u$, measures the interval to be $\Delta t^{\prime}=t_{2}^{\prime}-t_{1}^{\prime} .$ Use the Lorentz transformation expressions to derive the time dilation expression relating $\Delta t$ and $\Delta t^{\prime}$

Narayan Hari

Numerade Educator

Problem 19

A neutral $K$ meson at rest decays into two $\pi$ mesons, which travel in opposite directions along the $x$ axis with speeds of $0.815 \mathrm{c} .$ If instead the $K$ meson were moving in the positive $x$ direction with a velocity of $0.453 c,$ what would be the velocities of the two $\pi$ mesons?

Narayan Hari

Numerade Educator

Problem 20

A rod in the reference frame of observer $O$ makes an angle of $34^{\circ}$ with the $x$ axis. According to observer $O^{\prime},$ who is in motion in the $x$ direction with velocity $u$, the rod makes an angle of $52^{\circ}$ with the $x$ axis. Find the velocity $u$.

Suzanne W.

Numerade Educator

Problem 21

Two events occur at locations separated by a distance of $49.5 \mathrm{m}$ and by a time interval of $0.528 \mu \mathrm{s},$ according to observer $O$. Observer $O^{\prime}$ is in motion away from $O$ with a speed of $0.685 c$ in the $x$ direction. According to $O^{\prime},$ what are the spatial and time separations of the events?

Suzanne W.

Numerade Educator

Problem 22

According to observer $O,$ a blue flash occurs at $x_{b}=$ $10.4 \mathrm{m}$ when $t_{\mathrm{b}}=0.124 \mu \mathrm{s},$ and a red flash occurs at $x_{\mathrm{r}}=23.6 \mathrm{m}$ when $t_{\mathrm{r}}=0.138 \mu \mathrm{s} .$ According to observer
$O^{\prime},$ who is in motion relative to $O$ at velocity $u,$ the two flashes appear to be simultaneous. Find the velocity $u$.

Narayan Hari

Numerade Educator

Problem 23

Suppose the speed of light were $1000 \mathrm{mi} / \mathrm{h} .$ You are traveling on a flight from Los Angeles to Boston, a distance of 3000 mi. The plane's speed is a constant $600 \mathrm{mi} / \mathrm{h}$. You leave Los Angeles at 10: 00 am, as indicated by your wristwatch and by a clock in the airport. $(a)$ According to your watch, what time is it when you land in Boston? $(b)$ In the Boston airport is a clock that is synchronized to read exactly the same time as the clock in the Los Angeles airport. What time does that clock read when you land in Boston? $(c)$ The following day when the Boston clock that records Los Angeles time reads 10: 00 am, you leave Boston to return to Los Angeles on the same airplane. When you land in Los Angeles, what are the times read on your watch and on the airport clock?

Suzanne W.

Numerade Educator

Problem 24

Suppose rocket traveler Amelia has a clock made on Earth. Every year on her birthday she sends a light signal to brother Casper on Earth. $(a)$ At what rate does Casper receive the signals during Amelia's outward journey? $(b)$ At what rate does he receive the signals during her return journey? (c) How many of Amelia's birthday signals does Casper receive during the journey that he measures to last 20 years?

Suzanne W.

Numerade Educator

Problem 25

Suppose Amelia traveled at a speed of $0.80 c$ to a star that (according to Casper on Earth) is 8.0 light-years away. Casper ages 20 years during Amelia's round trip. How much younger than Casper is Amelia when she returns to Earth?

Suzanne W.

Numerade Educator

Problem 26

Make a drawing similar to Figure 2.20 showing the worldlines of Casper and Amelia from Casper's frame of reference. Divide the worldline for Amelia's outward journey into 8 equal segments (for the 8 birthdays that Amelia celebrates). For each birthday, draw a line that represents a light signal that Amelia sends to Casper on her birthday. Do the same for Amelia's return journey. (a) According to Casper's time, when does he receive the signal showing Amelia celebrating her 8 th birthday after leaving Earth? (b) How long does it take for Casper to receive the signals showing Amelia celebrating birthdays 9 through $16 ?$

Suzanne W.

Numerade Educator

Problem 27

Two twins make a round-trip journey from Earth to a star that is 12 light-years distant. Alice travels at a speed of $0.6 c .$ Bob departs 10 years after Alice and travels at a speed of $0.8 c .$ (a) Show that the two twins arrive back on Earth at the same time. $(b)$ Which twin is the younger when they return?

Suzanne W.

Numerade Educator

Problem 28

Agnes makes a round trip at a constant speed to a star that is 16 light-years distant from Earth, while twin brother Bert remains on Earth. When Agnes returns to Earth, she reports that she has celebrated 20 birthdays during her journey. (a) What was her speed during her journey? (b) How old is Bert when she returns?

Suzanne W.

Numerade Educator

Problem 29

(a) Using the relativistically correct final velocities for the collision shown in Figure $2.21 a\left(v_{1 f}^{\prime}=-0.585 c, \quad v_{2 f}^{\prime}=\right.$
$+0.294 c$ ), show that relativistic kinetic energy is conserved according to observer $O^{\prime} .$
(b) Using the relativistically correct final velocities for the collision shown in Figure $2.21 b\left(v_{1 f}=-0.051 c, v_{2 f}=+0.727 c\right),$ show that relativistic kinetic energy is conserved according to observer $O$.

Suzanne W.

Numerade Educator

Problem 30

Find the momentum, kinetic energy, and total energy of a proton moving at a speed of 0.835 c.

Suzanne W.

Numerade Educator

Problem 30

Find the momentum, kinetic energy, and total energy of a proton moving at a speed of $0.835 c$

Suzanne W.

Numerade Educator

Problem 31

An electron is moving with a kinetic energy of $0.923 \mathrm{MeV}$. What is its speed?

Anand Jangid

Numerade Educator

Problem 32

The work-energy theorem relates the change in kinetic energy of a particle to the work done on it by an external force: $\Delta K=W=\int F d x .$ Writing Newton's second law as $F=d p / d t,$ show that $W=\int v d p$ and integrate by parts using the relativistic momentum to obtain Equation 2.34

Suzanne W.

Numerade Educator

Problem 33

For what range of velocities of a particle of mass $m$ can we use the classical expression for kinetic energy $\frac{1}{2} m v^{2}$ to within an accuracy of 1 percent?

Suzanne W.

Numerade Educator

Problem 34

For what range of velocities of a particle of mass $m$ can we use the extreme relativistic approximation $E=p c$ to within an accuracy of 1 percent?

Narayan Hari

Numerade Educator

Problem 35

Use Equations 2.32 and 2.36 to derive Equation 2.39 .

Narayan Hari

Numerade Educator

Problem 36

By carrying the binomial expansion one term farther, find the next term after $\frac{1}{2} m v^{2}$ in the classical approximation of the relativistic kinetic energy. For what value of the speed does this term throw the classical value off by $0.1 \% ?$

Suzanne W.

Numerade Educator

Problem 37

(a) According to observer $O,$ a certain particle has a momentum of $1256 \mathrm{MeV} / \mathrm{c}$ and a total relativistic energy of 1351 MeV. What is the rest energy of this particle?
(b) An observer $O^{\prime}$ in a different frame of reference measures the momentum of this particle to be $857 \mathrm{MeV} / \mathrm{c}$. What does $O^{\prime}$ measure for the total relativistic energy of the particle?

Narayan Hari

Numerade Educator

Problem 38

An electron is moving at a speed of $0.85 c .$ By how much must its kinetic energy increase to raise its speed to $0.91 \mathrm{c} ?$

Suzanne W.

Numerade Educator

Problem 39

What is the change in mass when $1 \mathrm{g}$ of copper is heated from 0 to $100^{\circ} \mathrm{C}$ ? The specific heat capacity of copper is $0.40 \mathrm{J} / \mathrm{g} \cdot \mathrm{K}$

Narayan Hari

Numerade Educator

Problem 40

Find the kinetic energy of an electron moving at a speed of $(a) v=1.00 \times 10^{-4} c ;(b) v=1.00 \times 10^{-2} c ;(c) v=0.300 c$ $(d) v=0.999 c$

Suzanne W.

Numerade Educator

Problem 41

An electron and a proton are each accelerated starting from rest through a potential difference of 12.0 million volts. Find the momentum (in $\mathrm{MeV} / \mathrm{c}$ ) and the kinetic energy (in MeV) of each, and compare with the results of using the classical formulas.

Susan Hallstrom

Susan Hallstrom

Numerade Educator

Problem 42

In a nuclear reactor, each atom of uranium (of atomic mass $235 \mathrm{u}$ ) releases about $210 \mathrm{MeV}$ when it fissions. What is the change in mass when $1.50 \mathrm{kg}$ of uranium- 235 is fissioned?

Narayan Hari

Numerade Educator

Problem 43

(a) Find the energy released as gamma rays when ${ }^{2} \mathrm{H}$ captures a neutron to form ${ }^{3} \mathrm{H}$. Assume all kinetic energies are negligibly small. Atomic masses can be found in Appendix D. (b) Repeat for capture of a neutron by ${ }^{3} \mathrm{He}$ to form ${ }^{4} \mathrm{He}$.

Ren Jie Tuieng

Numerade Educator

Problem 44

A $\pi$ meson of rest energy 139.6 MeV moving at a speed of $0.921 c$ collides with and sticks to a proton of rest energy $938.3 \mathrm{MeV}$ that is at rest. $(a)$ Find the total relativistic energy of the resulting composite particle. (b) Find the total linear momentum of the composite particle. (c) Using the results of $(a)$ and $(b),$ find the rest energy of the composite particle.

Suzanne W.

Numerade Educator

Problem 45

An electron and a positron (an antielectron) make a head-on collision, each moving at $v=0.99999 \mathrm{c} .$ In the collision, the electrons disappear and are replaced by two muons $\left(m c^{2}=105.7 \mathrm{MeV}\right),$ which move off in opposite directions. What is the kinetic energy of each of the muons?

Anand Jangid

Numerade Educator

Problem 46

It is desired to create a particle of mass 9460 MeV/c $^{2}$ in a head-on collision between a proton and an antiproton (each having a mass of 938.3 MeV/c $^{2}$ ) traveling at the same speed. What speed is necessary for this to occur?

Narayan Hari

Numerade Educator

Problem 47

A particle of rest energy $m c^{2}$ is moving with speed $v$ in the positive $x$ direction. The particle decays into two particles, each of rest energy $140 \mathrm{MeV}$. One particle, with kinetic energy $282 \mathrm{MeV},$ moves in the positive $x$ direction, and the other particle, with kinetic energy $25 \mathrm{MeV}$, moves in the negative $x$ direction. Find the rest energy of the original particle and its speed.

Suzanne W.

Numerade Educator

Problem 48

Let's consider a different approach to Example $2.21,$ in which two protons collide to form an antiproton. Suppose the two protons collide head-on with equal speeds. (a) In this frame of reference, what is the threshold energy? ( $b$ ) Find the velocity corresponding to the threshold energy. $(c)$ Now transform to a frame of reference in which one of the initial protons is at rest, and find the speed and the energy of the other proton.

Suzanne W.

Numerade Educator

Problem 49

In the muon decay experiment discussed in Section 2.9 as a verification of time dilation, the muons move in the laboratory with a momentum of $3094 \mathrm{MeV} / \mathrm{c}$. Find the dilated lifetime in the laboratory frame. (The proper lifetime is $2.198 \mu \mathrm{s}$.

Suzanne W.

Numerade Educator

Problem 50

Derive the relativistic expression $p^{2} / 2 K=m+K / 2 c^{2}$ which is plotted in Figure 2.28 a.

Suzanne W.

Numerade Educator

Problem 51

Suppose we want to send an astronaut on a round trip to visit a star that is 200 light- years distant and at rest with respect to Earth. The life support systems on the spacecraft enable the astronaut to survive at most 20 years. (a) At what speed must the astronaut travel to make the round trip in 20 years of spacecraft time? $(b)$ How much time passes on Earth during the round trip?

Suzanne W.

Numerade Educator

Problem 52

A "cause" occurs at point $1\left(x_{1}, t_{1}\right)$ and its "effect" occurs at point $2\left(x_{2}, t_{2}\right) .$ Use the Lorentz transformation to find $t_{2}^{\prime}-t_{1}^{\prime},$ and show that $t_{2}^{\prime}-t_{1}^{\prime}>0 ;$ that is, $O^{\prime}$ can never see the "effect" coming before its "cause."

Suzanne W.

Numerade Educator

Problem 53

Observer $O$ sees a red flash of light at the origin at $t=0$ and a blue flash of light at $x=3.65 \mathrm{km}$ at a time $t=8.24 \mu \mathrm{s} .$ What are the distance and the time interval between the flashes according to observer $O^{\prime},$ who moves relative to $O$ in the direction of increasing $x$ with a speed of $0.534 c ?$ Assume that the origins of the two coordinate systems line up at $t=t^{\prime}=0$

Suzanne W.

Numerade Educator

Problem 54

Several spacecraft leave a space station at the same time. Relative to an observer on the station, $A$ travels at $0.65 c$ in the $x$ direction, $B$ at $0.50 c$ in the $y$ direction, $C$ at $0.50 c$ in the negative $x$ direction, and $D$ at $0.50 c$ at $45^{\circ}$ between the $y$ and negative $x$ directions. Find the velocity components, directions, and speeds of $B, C,$ and $D$ as observed from $A$

Susan Hallstrom

Susan Hallstrom

Numerade Educator

Problem 55

Observer $O$ sees a light turn on at $x=524 \mathrm{m}$ when $t=$ $1.52 \mu \mathrm{s}$. Observer $O^{\prime}$ is in motion at a speed of $0.563 c$ in the positive $x$ direction. The two frames of reference are synchronized so that their origins match up $\left(x=x^{\prime}=0\right)$ at $t=t^{\prime}=0 .(a)$ At what time does the light turn on according to $O^{\prime} ?(b)$ At what location does the light turn on in the reference frame of $O^{\prime} ?$

Suzanne W.

Numerade Educator

Problem 56

Suppose an observer $O$ measures a particle of mass $m$ moving in the $x$ direction to have speed $v,$ energy $E,$ and momentum $p$. Observer $O^{\prime}$, moving at speed $u$ in the $x$ direction, measures $v^{\prime}, E^{\prime},$ and $p^{\prime}$ for the same object.
(a) Use the Lorentz velocity transformation to find $E^{\prime}$ and $p^{\prime}$ in terms of $m, u,$ and $v$
(b) Reduce $E^{\prime 2}-\left(p^{\prime} c\right)^{2}$ to its simplest form and interpret the result.

Suzanne W.

Numerade Educator

Problem 57

Repeat Problem 56 for the mass moving in the $y$ direction according to $O$. The velocity $u$ of $O^{\prime}$ is still along the $x$ direction.

Suzanne W.

Numerade Educator

Problem 58

Consider again the situation described in Section $2.6 .$ Amelia's friend Bernice leaves Earth at the same time as Amelia and travels in the same direction at the same speed, but Bernice continues in the original direction when Amelia reaches the planet and turns her ship around. (a) From Bernice's frame of reference, Casper is moving at a velocity of $-0.60 \mathrm{c}$. Draw Casper's worldline in Bernice's frame of reference. (b) Casper celebrates 20 birthdays during Amelia's journey. In Bernice's frame of reference, how long does it take for Casper to celebrate 20 birthdays? $(c)$ In Bernice's frame of reference, draw a worldline representing Amelia's outbound journey to the planet. $(d)$ Calculate Amelia's velocity during her return journey as observed from Bernice's frame of reference, and draw a worldline showing Amelia's return journey. Amelia's and Casper's worldlines should intersect when Amelia return to Earth. $(e)$ Divide Casper's worldline into 20 segments, representing his birthdays. He sends a light signal to Amelia on each birthday. Amelia receives a light signal from Casper just as she arrives at the planet. On which birthday did Casper send this signal? ( $f$ ) Amelia sends Casper a light signal on her 8 th birthday. Draw a line on your diagram representing this light signal. When does Casper receive this signal?

Stanley Enemuo

Numerade Educator

Problem 59

Electrons are accelerated to high speeds by a two-stage machine. The first stage accelerates the electrons from rest to $v=0.99 \mathrm{c} .$ The second stage accelerates the electrons from $0.99 c$ to $0.999 c .$ (a) How much energy does the first stage add to the electrons?
(b) How much energy does the second stage add in increasing the velocity by only 0.9 percent?

Suzanne W.

Numerade Educator

Problem 60

A beam of $2.14 \times 10^{11}$ electrons/s moving at a speed of $0.813 c$ strikes a block of copper that is used as a beam stop. The copper block is a cube measuring $2.54 \mathrm{cm}$ on edge. What is the temperature increase of the block after one hour?

Suzanne W.

Numerade Educator

Problem 61

An electron moving at a speed of $v_{\mathrm{i}}=0.960 \mathrm{c}$ in the positive $x$ direction collides with another electron at rest. After the collision, one electron is observed to move with a speed of $v_{1 f}=0.956 c$ at an angle of $\theta_{1}=9.7^{\circ}$ with the $x$ axis. $(a)$ Use conservation of momentum to find the velocity (magnitude and direction) of the second electron. (b) Based only on the original data given in the problem, use conservation of energy to find the speed of the second electron.

Suzanne W.

Numerade Educator

Problem 62

A pion has a rest energy of 135 MeV. It decays into two gamma-ray photons, bursts of electromagnetic radiation that travel at the speed of light. A pion moving through the laboratory at $v=0.98 c$ decays into two gamma-ray photons of equal energies, making equal angles $\theta$ with the original direction of motion. Find the angle $\theta$ and the energies of the two gamma ray photons.

Suzanne W.

Numerade Educator

Problem 63

Consider the decay of the K meson from Example 2.19 transformed to a frame of reference in which the K meson is at rest. $(a)$ What is the speed of the $\mathrm{K}$ meson in the original reference frame? (b) Using the speed from part $(a)$ to transform to the reference frame in which the $\mathrm{K}$ meson is at rest, find the speed, momentum, and energy of the pi meson in this frame. (c) Find the mass of the unknown particle produced in the decay.

Kai Chen

Princeton University

Problem 64

An electron and a positron (antielectron) are traveling toward each other, the electron with velocity $+0.834 c$ and the positron with velocity $-0.428 c .$ They collide and stick together to form a new composite particle. $(a)$ Find the momentum and energy of the new particle. $(b)$ What is the mass of the new particle? (c) Find the change in kinetic energy in the collision. How is the change in kinetic energy related to the mass of the new particle? (d) Which of the above answers would be different according to an observer in a different reference frame?

Zulfiqar Ali

Numerade Educator