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University Physics with Modern Physics

Roger A. Freedman, Hugh D. Young

Chapter 37

Relativity - all with Video Answers

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Chapter Questions

04:58

Problem 1

Suppose the two lightning bolts shown in Fig. $37.5 \mathrm{a}$ are simultaneous to an observer on the train. Show that they are not simultaneous to an observer on the ground. Which lightning strike does the ground observer measure to come first?

Narayan Hari
Narayan Hari
Numerade Educator
02:34

Problem 2

The positive muon $\left(\mu^{+}\right),$ an unstable particle, lives on average $2.20 \times 10^{-6} \mathrm{~s}$ (measured in its own frame of reference) before decaying. (a) If such a particle is moving, with respect to the laboratory, with a speed of $0.900 c$, what average lifetime is measured in the laboratory?
(b) What average distance, measured in the laboratory, does the particle move before decaying?

Ceren Uzun
Ceren Uzun
Texas Tech University
01:24

Problem 3

How fast must a rocket travel relative to the earth so that time in the rocket "slows down" to half its rate as measured by earth-based observers? Do present-day jet planes approach such speeds?

Elan Stopnitzky
Elan Stopnitzky
Numerade Educator
01:11

Problem 4

A spaceship flies past Mars with a speed of $0.985 c$ relative to the surface of the planet. When the spaceship is directly overhead, a signal light on the Martian surface blinks on and then off. An observer on Mars measures that the signal light was on for $75.0 \mu$ s. (a) Does the observer on Mars or the pilot on the spaceship measure the proper time? (b) What is the duration of the light pulse measured by the pilot of the spaceship?

Narayan Hari
Narayan Hari
Numerade Educator
02:32

Problem 5

The negative pion $\left(\pi^{-}\right)$ is an unstable particle with an average lifetime of $2.60 \times 10^{-8} \mathrm{~s}$ (measured in the rest frame of the pion). (a) If the pion is made to travel at very high speed relative to a laboratory, its average lifetime is measured in the laboratory to be $4.20 \times 10^{-7} \mathrm{~s}$. Calculate the speed of the pion expressed as a fraction of $c$. (b) What distance, measured in the laboratory, does the pion travel during its average lifetime?

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
03:41

Problem 6

$\mathrm{As}$ you pilot your space utility vehicle at a constant speed toward the moon, a race pilot flies past you in her spaceracer at a constant speed of $0.800 c$ relative to you. At the instant the spaceracer passes you, both of you start timers at zero. (a) At the instant when you measure that the spaceracer has traveled $1.20 \times 10^{8} \mathrm{~m}$ past you, what does the race pilot read on her timer? (b) When the race pilot reads the value calculated in part (a) on her timer, what does she measure to be your distance from her? (c) At the instant when the race pilot reads the value calculated in part (a) on her timer, what do you read on yours?

Ceren Uzun
Ceren Uzun
Texas Tech University
03:50

Problem 7

An alien spacecraft is flying overhead at a great distance as you stand in your backyard. You see its searchlight blink on for $0.150 \mathrm{~s}$ The first officer on the spacecraft measures that the searchlight is on for $12.0 \mathrm{~ms}$. (a) Which of these two measured times is the proper time?
(b) What is the speed of the spacecraft relative to the earth, expressed as a fraction of the speed of light $c ?$

Ceren Uzun
Ceren Uzun
Texas Tech University
02:58

Problem 8

At $x=x^{\prime}=0$ and $t=t^{\prime}=0$ a clock ticks aboard an extremely fast spaceship moving past us in the $+x$ -direction with a Lorentz factor of 100 so $v \approx c .$ The captain hears the clock tick again $1.00 \mathrm{~s}$ later. Where and when do we measure the second tick to occur?

Mirza Aslam Beig
Mirza Aslam Beig
Numerade Educator
01:18

Problem 9

A spacecraft of the Trade Federation flies past the planet Coruscant at a speed of $0.600 c$. A scientist on Coruscant measures the length of the moving spacecraft to be $74.0 \mathrm{~m}$. The spacecraft later lands on Coruscant, and the same scientist measures the length of the now stationary spacecraft. What value does she get?

Robert Zaballa
Robert Zaballa
Numerade Educator
02:47

Problem 10

A meter stick moves past you at great speed. Its motion relative to you is parallel to its long axis. If you measure the length of the moving meter stick to be $1.00 \mathrm{ft}(1 \mathrm{ft}=0.3048 \mathrm{~m})-$ for example, by comparing it to a 1 foot ruler that is at rest relative to you-at what speed is the meter stick moving relative to you?

Robert Zaballa
Robert Zaballa
Numerade Educator
04:42

Problem 11

Muons are unstable subatomic particles that decay to electrons with a mean lifetime of $2.2 \mu \mathrm{s}$ They are produced when cosmic rays bombard the upper atmosphere about $10 \mathrm{~km}$ above the earth's surface, and they travel very close to the speed of light. The problem we want to address is why we see any of them at the earth's surface. (a) What is the greatest distance a muon could travel during its $2.2 \mu$ s lifetime? (b) According to your answer in part (a), it would seem that muons could never make it to the ground. But the $2.2 \mu \mathrm{s}$ lifetime is measured in the frame of the muon, and muons are moving very fast. At a speed of $0.999 c,$ what is the mean lifetime of a muon as measured by an observer at rest on the earth? How far would the muon travel in this time? Does this result explain why we find muons in cosmic rays? (c) From the point of view of the muon, it still lives for only $2.2 \mu \mathrm{s},$ so how does it make it to the ground? What is the thickness of the $10 \mathrm{~km}$ of atmosphere through which the muon must travel, as measured by the muon? Is it now clear how the muon is able to reach the ground?

Mirza Aslam Beig
Mirza Aslam Beig
Numerade Educator
02:34

Problem 12

A rocket ship flies past the earth at $91.0 \%$ of the speed of light. Inside, an astronaut who is undergoing a physical examination is having his height measured while he is lying down parallel to the direction in which the ship is moving.
(a) If his height is measured to be $2.00 \mathrm{~m}$ by his doctor inside the ship, what height would a person watching this from the earth measure? (b) If the earth-based person had measured $2.00 \mathrm{~m},$ what would the doctor in the spaceship have measured for the astronaut's height? Is this a reasonable height? (c) Suppose the astronaut in part (a) gets up after the examination and stands with his body perpendicular to the direction of motion. What would the doctor in the rocket and the observer on earth measure for his height now?

Mirza Aslam Beig
Mirza Aslam Beig
Numerade Educator
03:41

Problem 13

As measured by an observer on the earth, a spacecraft runway on earth has a length of $3600 \mathrm{~m}$. (a) What is the length of the runway as measured by a pilot of a spacecraft flying past at a speed of $4.00 \times 10^{7} \mathrm{~m} / \mathrm{s}$ relative to the earth? (b) An observer on earth measures the time interval from when the spacecraft is directly over one end of the runway until it is directly over the other end. What result does she get?
(c) The pilot of the spacecraft measures the time it takes him to travel from one end of the runway to the other end. What value does he get?

Keshav Singh
Keshav Singh
Numerade Educator
03:16

Problem 14

The Lorentz coordinate transformation assumes that $t=t^{\prime}$ at $x=x^{\prime}=0 .$ At what other values of $x$ and $x^{\prime}$ does $t=t^{\prime} ?$

Alan Gavel
Alan Gavel
Numerade Educator
04:45

Problem 15

An observer in frame $S^{\prime}$ is moving to the right $(+x$ -direction $)$ at speed $u=0.600 c$ away from a stationary observer in frame $S .$ The observer in $S^{\prime}$ measures the speed $v^{\prime}$ of a particle moving to the right away from her. What speed $v$ does the observer in $S$ measure for the particle if (a) $v^{\prime}=0.400 c ;$ (b) $v^{\prime}=0.900 c ;$ (c) $v^{\prime}=0.990 c ?$

Keshav Singh
Keshav Singh
Numerade Educator
04:56

Problem 16

Space pilot Mavis zips past Stanley at a constant speed relative to him of $0.800 \mathrm{c}$. Mavis and Stanley start timers at zero when the front of Mavis's ship is directly above Stanley. When Mavis reads 5.00 s on her timer, she turns on a bright light under the front of her spaceship.
(a) Use the Lorentz coordinate transformation derived in Example 37.6 to calculate $x$ and $t$ as measured by Stanley for the event of turning on the light. (b) Use the time dilation formula, Eq. (37.6), to calculate the time interval between the two events (the front of the spaceship passing overhead and turning on the light) as measured by Stanley. Compare to the value of $t$ you calculated in part (a). (c) Multiply the time interval by Mavis's speed, both as measured by Stanley, to calculate the distance she has traveled as measured by him when the light turns on. Compare to the value of $x$ you calculated in part (a).

Ceren Uzun
Ceren Uzun
Texas Tech University
02:35

Problem 17

A pursuit spacecraft from the planet Tatooine is attempting to catch up with a Trade Federation cruiser. As measured by an observer on Tatooine, the cruiser is traveling away from the planet with a speed of $0.600 c$. The pursuit ship is traveling at a speed of $0.800 c$ relative to Tatooine, in the same direction as the cruiser. (a) For the pursuit ship to catch the cruiser, should the velocity of the cruiser relative to the pursuit ship be directed toward or away from the pursuit ship? (b) What is the speed of the cruiser relative to the pursuit ship?

Keshav Singh
Keshav Singh
Numerade Educator
04:49

Problem 18

An enemy spaceship is moving toward your starfighter with a speed, as measured in your frame, of $0.400 c$. The enemy ship fires a missile toward you at a speed of $0.700 c$ relative to the enemy ship (Fig. E37.18). (a) What is the speed of the missile relative to you? Express your answer in terms of the speed of light. (b) If you measure that the enemy ship is $8.00 \times 10^{6} \mathrm{~km}$ away from you when the missile is fired, how much time, measured in your frame, will it take the missile to reach you?

Linda Winkler
Linda Winkler
Numerade Educator
02:03

Problem 19

Two particles are created in a high-energy accelerator and move off in opposite directions. The speed of one particle, as measured in the laboratory, is $0.650 c,$ and the speed of each particle relative to the other is $0.950 c$. What is the speed of the second particle, as measured in the laboratory?

Elan Stopnitzky
Elan Stopnitzky
Numerade Educator
02:10

Problem 20

Two particles in a high-energy accelerator experiment are approaching each other head-on, each with a speed of $0.9380 c$ as measured in the laboratory. What is the magnitude of the velocity of one particle relative to the other?

Ceren Uzun
Ceren Uzun
Texas Tech University
04:00

Problem 21

Two particles in a high-energy accelerator experiment approach each other head-on with a relative speed of $0.890 c .$ Both particles travel at the same speed as measured in the laboratory. What is the speed of each particle, as measured in the laboratory?

Linda Winkler
Linda Winkler
Numerade Educator
01:55

Problem 22

Electromagnetic radiation from a star is observed with an earth-based telescope. The star is moving away from the earth at a speed of $0.520 c$. If the radiation has a frequency of $8.64 \times 10^{14} \mathrm{~Hz}$ in the rest frame of the star, what is the frequency measured by an observer on earth?

Ceren Uzun
Ceren Uzun
Texas Tech University
05:38

Problem 23

Tell It to the Judge. (a) How fast must you be approaching a red traffic light $(\lambda=675 \mathrm{nm})$ for it to appear yellow $(\lambda=575 \mathrm{nm}) ?$ Express your answer in terms of the speed of light. (b) If you used this as a reason not to get a ticket for running a red light, how much of a fine would you get for speeding? Assume that the fine is $\$ 1.00$ for each kilometer per hour that your speed exceeds the posted limit of $90 \mathrm{~km} / \mathrm{h}$.

Keshav Singh
Keshav Singh
Numerade Educator
02:42

Problem 24

A source of electromagnetic radiation is moving in a radial direction relative to you. The frequency you measure is 1.25 times the frequency measured in the rest frame of the source. What is the speed of the source relative to you? Is the source moving toward you or away from you?

Keshav Singh
Keshav Singh
Numerade Educator
06:03

Problem 25

A particle zips by us with a Lorentz factor of $1.12 .$ Then another particle zips by us moving at twice the speed of the first particle. (a) What is the Lorentz factor of the second particle? (b) If the particles were moving with a speed much less than $c,$ the magnitude of the momentum of the second particle would be twice that of the first. However, what is the ratio of the magnitudes of momentum for these relativistic particles?

Linda Winkler
Linda Winkler
Numerade Educator
13:57

Problem 26

Calculate the magnitude of the force required to give a $0.145 \mathrm{~kg}$ baseball an acceleration $a=1.00 \mathrm{~m} / \mathrm{s}^{2}$ in the direction of the baseball's initial velocity when this velocity has a magnitude of (a) $10.0 \mathrm{~m} / \mathrm{s} ;$ (b) $0.900 c ;$ (c) $0.990 c$
(d) Repeat parts (a),
(b), and (c) if the force and acceleration are perpendicular to the velocity.

Declan Nell
Declan Nell
Numerade Educator
04:10

Problem 27

A proton has momentum with magnitude $p_{0}$ when its speed is $0.400 c .$ In terms of $p_{0},$ what is the magnitude of the proton's momentum when its speed is doubled to $0.800 c ?$

Linda Winkler
Linda Winkler
Numerade Educator
06:52

Problem 28

A spaceship has length $120 \mathrm{~m},$ diameter $25 \mathrm{~m},$ and mass $4.0 \times 10^{3} \mathrm{~kg}$ as measured by its crew. As the spaceship moves parallel to its cylindrical axis and passes us, we measure its length to be $90 \mathrm{~m}$.
(a) What do we measure its diameter to be? (b) What do we measure the magnitude of its momentum to be?

Declan Nell
Declan Nell
Numerade Educator
03:20

Problem 29

(a) At what speed is the momentum of a particle twice as great as the result obtained from the nonrelativistic expression $m v ?$ Express your answer in terms of the speed of light. (b) A force is applied to a particle along its direction of motion. At what speed is the magnitude of force required to produce a given acceleration twice as great as the force required to produce the same acceleration when the particle is at rest? Express your answer in terms of the speed of light.

Keshav Singh
Keshav Singh
Numerade Educator
03:09

Problem 30

The sun produces energy by nuclear fusion reactions, in which matter is converted into energy. By measuring the amount of energy we receive from the sun, we know that it is producing energy at a rate of $3.8 \times 10^{26} \mathrm{~W}$. (a) How many kilograms of matter does the sun lose each second? Approximately how many tons of matter is this $(1$ ton $=2000 \mathrm{lb}) ?$
(b) At this rate, how long would it take the sun to use up all its mass?

Declan Nell
Declan Nell
Numerade Educator
01:13

Problem 31

What is the speed of a particle whose kinetic energy is equal to (a) its rest energy and (b) five times its rest energy?

Salamat Ali
Salamat Ali
Numerade Educator
03:40

Problem 32

If a muon is traveling at $0.999 c,$ what are its momentum and kinetic energy? (The mass of such a muon at rest in the laboratory is 207 times the electron mass.)

Keshav Singh
Keshav Singh
Numerade Educator
06:06

Problem 33

A proton (rest mass $1.67 \times 10^{-27} \mathrm{~kg}$ ) has total energy that is 4.00 times its rest energy. What are (a) the kinetic energy of the proton;
(b) the magnitude of the momentum of the proton; (c) the proton's speed?

Linda Winkler
Linda Winkler
Numerade Educator
06:20

Problem 34

(a) How much work must be done on a particle with mass $m$ to accelerate it (a) from rest to a speed of $0.090 c$ and (b) from a speed of $0.900 c$ to a speed of $0.990 c ?$ (Express the answers in terms of $\left.m c^{2} .\right)$
(c) How do your answers in parts (a) and (b) compare?

Declan Nell
Declan Nell
Numerade Educator
04:08

Problem 35

A particle has rest mass $6.64 \times 10^{-27} \mathrm{~kg}$ and momentum $2.10 \times 10^{-18} \mathrm{~kg} \cdot \mathrm{m} / \mathrm{s} .$ (a) What is the total energy (kinetic plus rest energy) of the particle? (b) What is the kinetic energy of the particle? (c) What is the ratio of the kinetic energy to the rest energy of the particle?

Keshav Singh
Keshav Singh
Numerade Educator
01:56

Problem 36

Electrons are accelerated through a potential difference of $750 \mathrm{kV},$ so that their kinetic energy is $7.50 \times 10^{5} \mathrm{eV}$. (a) What is the ratio of the speed $v$ of an electron having this energy to the speed of light, $c ?$ (b) What would the speed be if it were computed from the principles of classical mechanics?

Elan Stopnitzky
Elan Stopnitzky
Numerade Educator
04:59

Problem 37

Compute the kinetic energy of a proton (mass $\left.1.67 \times 10^{-27} \mathrm{~kg}\right)$ using both the nonrelativistic and relativistic expressions, and compute the ratio of the two results (relativistic divided by nonrelativistic) for speeds of (a) $8.00 \times 10^{7} \mathrm{~m} / \mathrm{s}$ and
(b) $2.85 \times 10^{8} \mathrm{~m} / \mathrm{s}$

Linda Winkler
Linda Winkler
Numerade Educator
06:31

Problem 38

Two protons (each with rest mass $M=1.67 \times 10^{-27} \mathrm{~kg}$ ) are initially moving with equal speeds in opposite directions. The protons continue to exist after a collision that also produces an $\eta^{0}$ particle (see Chapter 44 ). The rest mass of the $\eta^{0}$ is $m=9.75 \times 10^{-28} \mathrm{~kg} .$ (a) If the two protons and the $\eta^{0}$ are all at rest after the collision, find the initial speed of the protons, expressed as a fraction of the speed of light. (b) What is the kinetic energy of each proton? Express your answer in $\mathrm{MeV}$. (c) What is the rest energy of the $\eta^{0}$, expressed in MeV? (d) Discuss the relationship between the answers to parts (b) and (c).

Robert Zaballa
Robert Zaballa
Numerade Educator
01:37

Problem 39

(a) Through what potential difference does an electron have to be accelerated, starting from rest, to achieve a speed of $0.980 c ?$
(b) What is the kinetic energy of the electron at this speed? Express your answer in joules and in electron volts.

Salamat Ali
Salamat Ali
Numerade Educator
01:53

Problem 40

Inside a spaceship flying past the earth at three-fourths the speed of light, a pendulum is swinging. (a) If each swing takes $1.80 \mathrm{~s}$ as measured by an astronaut performing an experiment inside the spaceship, how long will the swing take as measured by a person at mission control (on earth) who is watching the experiment? (b) If each swing takes $1.80 \mathrm{~s}$ as measured by a person at mission control, how long will it take as measured by the astronaut in the spaceship?

Ceren Uzun
Ceren Uzun
Texas Tech University
01:09

Problem 41

The starships of the Solar Federation are marked with the symbol of the federation, a circle, while starships of the Denebian Empire are marked with the empire's symbol, an ellipse whose major axis is 1.40 times longer than its minor axis $(a=1.40 b$ in Fig. P37.41). How fast, relative to an observer, does an empire ship have to travel for its marking to be confused with the marking of a federation ship?

Salamat Ali
Salamat Ali
Numerade Educator
03:04

Problem 42

A cube of metal with sides of length $a$ sits at rest in a frame $S$ with one edge parallel to the $x$ -axis. Therefore, in $S$ the cube has volume $a^{3}$. Frame $S^{\prime}$ moves along the $x$ -axis with a speed $u .$ As measured by an observer in frame $S^{\prime},$ what is the volume of the metal cube?

Laszlo Zalavari
Laszlo Zalavari
Numerade Educator
01:17

Problem 43

A space probe is sent to the vicinity of the star Capella, which is 42.2 light-years from the earth. (A light-year is the distance light travels in a year.) The probe travels with a speed of $0.9930 c$. An astronaut recruit on board is 19 years old when the probe leaves the earth. What is her biological age when the probe reaches Capella?

Salamat Ali
Salamat Ali
Numerade Educator
03:47

Problem 44

$\mathrm{A} \Sigma+$ particle has a mean lifetime of $80.2 \mathrm{ps} .$ A physicist measures that mean lifetime to be 403 ps as the particle moves in his lab. The rest mass of the particle is $2.12 \times 10^{-27} \mathrm{~kg} .$ (a) How fast is the particle moving? (b) How far does it travel, as measured in the lab frame, over one mean lifetime? (c) What are its rest, kinetic, and total energies in the lab frame of reference? (d) What are its rest, kinetic, and total energies in the particle's frame?

Alan Gavel
Alan Gavel
Numerade Educator
07:15

Problem 45

Spaceship $A$ moves past the earth at $0.80 c$ to the west. Spaceship $B$ approaches $A,$ moving to the east. Both spaceship crews measure their relative speed of approach to be $0.98 c .$ What mass would the crews of both spaceships measure for the standard kilogram, kept at rest on the earth, (a) according to classical physics and (b) according to the special theory of relativity?

Declan Nell
Declan Nell
Numerade Educator
06:51

Problem 46

One way to strictly enforce a speed limit would be to alter the laws of nature. Suppose the speed of light were $65 \mathrm{mph}$ and your workplace was 30 miles from your home. Assume you travel to work at a typical driving speed of 60 mph. (a) If you drove at that speed for the round trip to and from work, light, how much would your wristwatch lag your kitchen clock each day? (b) Estimate the length of your car. (c) If you were driving at your estimated driving speed, how long would your car be when viewed from the roadside?
(d) What would be the speed relative to you of similar cars traveling toward you in the opposite lane with the same ground speed as you?
(e) How long would you measure those cars to be? (f) If the total mass of you and your car was $2000 \mathrm{~kg}$, how much work would be required to get you up to speed? (Note: Your rest mass energy in this world is $m c^{2}$, where $c=65$ mph. ) (g) How much work would be required in the real world, where the speed of light is $3.0 \times 10^{8} \mathrm{~m} / \mathrm{s},$ to get you up to speed?

Keshav Singh
Keshav Singh
Numerade Educator
03:43

Problem 47

Physicists and engineers from around the world came together to build the largest accelerator in the world, the Large Hadron Collider (LHC) at the CERN Laboratory in Geneva, Switzerland. The machine accelerates protons to high kinetic energies in an underground ring $27 \mathrm{~km}$ in circumference. (a) What is the speed $v$ of a proton in the $\mathrm{LHC}$ if the proton's kinetic energy is $7.0 \mathrm{TeV} ?$ (Because $v$ is very close to $c,$ write $v=(1-\Delta) c$ and give your answer in terms of $\Delta .$ ) (b) Find the relativistic mass, $m_{\text {rel }}$, of the accelerated proton in terms of its rest mass.

Salamat Ali
Salamat Ali
Numerade Educator
01:35

Problem 48

The net force $\vec{F}$ on a particle of mass $m$ is directed at $30.0^{\circ}$ counterclockwise from the $+x$ -axis. At one instant of time, the particle is traveling in the $+x$ -direction with a speed (measured relative to the earth) of $0.700 c .$ At this instant, what is the direction of the particle's acceleration?

Alan Gavel
Alan Gavel
Numerade Educator
03:01

Problem 49

Two atomic clocks are carefully synchronized. One remains in New York, and the other is loaded on an airliner that travels at an average speed of $250 \mathrm{~m} / \mathrm{s}$ and then returns to New York. When the plane returns, the elapsed time on the clock that stayed behind is $4.00 \mathrm{~h}$. By how much will the readings of the two clocks differ, and which clock will show the shorter elapsed time? (Hint:
since $u \ll c,$ you can simplify $\sqrt{1-u^{2} / c^{2}}$ by a binomial expansion. $)$

Robert Zaballa
Robert Zaballa
Numerade Educator
08:53

Problem 50

The distance to a particular star, as measured in the earth's frame of reference, is 7.11 light-years ( 1 light-year is the distance that light travels in $1 \mathrm{y}$ ). A spaceship leaves the earth and takes $3.35 \mathrm{y}$ to arrive at the star, as measured by passengers on the ship. (a) How long does the trip take, according to observers on earth?
(b) What distance for the trip do passengers on the spacecraft measure?

Ceren Uzun
Ceren Uzun
Texas Tech University
01:45

Problem 51

The Russian physicist P. A. Cerenkov discovered that a charged particle traveling in a solid with a speed exceeding the speed of light in that material radiates electromagnetic radiation. (This is analogous to the sonic boom produced by an aircraft moving faster than the speed of sound in air; see Section 16.9. Cerenkov shared the 1958 Nobel Prize for this discovery.) What is the minimum kinetic energy (in electron volts) that an electron must have while traveling inside a slab of crown glass $(n=1.52)$ in order to create this Cerenkov radiation?

Salamat Ali
Salamat Ali
Numerade Educator
07:02

Problem 52

Quarks and gluons are fundamental particles that will be discussed in Chapter $44 .$ A proton, which is a bound state of two up quarks and a down quark, has a rest mass of $m_{\mathrm{p}}=1.67 \times 10^{-27} \mathrm{~kg}$. This is significantly greater than the sum of the rest mass of the up quarks, which is $m_{\mathrm{u}}=4.12 \times 10^{-30} \mathrm{~kg}$ each, and the rest mass of the down quark, which is $m_{\mathrm{d}}=8.59 \times 10^{-30} \mathrm{~kg} .$ Suppose we (incorrectly) model the rest energy of the proton $m_{\mathrm{p}} c^{2}$ as derived from the kinetic energy of the three quarks, and we split that energy equally among them.
(a) Estimate the Lorentz factor $\gamma=\left(1-v^{2} / c^{2}\right)^{-1 / 2}$ for each of the up quarks using Eq. $(37.36) .$ (b) Similarly estimate the Lorentz factor $\gamma$ for the down quark.
(c) Are the corresponding speeds $v_{\mathrm{u}}$ and $v_{\mathrm{d}}$ greater than $99 \%$ of the speed of light? (d) More realistically, the quarks are held together by massless gluons, which mediate the strong nuclear interaction. Suppose we model the proton as the three quarks, each with a speed of $0.90 c,$ with the remainder of the proton rest energy supplied by gluons. In this case, estimate the percentage of the proton rest energy associated with gluons.
(e) Model a quark as oscillating with an average speed of $0.90 c$ across the diameter of a proton, $1.7 \times 10^{-15} \mathrm{~m}$. Estimate the frequency of that motion.

Keshav Singh
Keshav Singh
Numerade Educator
01:53

Problem 53

A nuclear bomb containing $12.0 \mathrm{~kg}$ of plutonium explodes. The sum of the rest masses of the products of the explosion is less than the original rest mass by one part in $10^{4}$. (a) How much energy is released in the explosion? (b) If the explosion takes place in $4.00 \mu \mathrm{s}$ what is the average power developed by the bomb? (c) What mass of water could the released energy lift to a height of $1.00 \mathrm{~km} ?$

Salamat Ali
Salamat Ali
Numerade Educator
02:08

Problem 54

In the earth's rest frame, two protons are moving away from each other at equal speed. In the frame of each proton, the other proton has a speed of $0.700 c$. What does an observer in the rest frame of the earth measure for the speed of each proton?

Anand Jangid
Anand Jangid
Numerade Educator
03:59

Problem 55

In a laboratory, a rectangular loop of wire surrounds the origin in the $x z$ -plane, with extent $H$ in the $z$ -direction and extent $L$ in the $x$ direction (Fig. $\mathrm{P} 37.55$ ). The loop carries current $I$ in the counterclockwise direction as viewed from the positive $y$ -axis. A magnetic field $\vec{B}=B \hat{k}$ is present. (a) What is the magnetic dipole moment $\overrightarrow{\boldsymbol{\mu}}$ as seen in the frame $S$ of the laboratory?
(b) What is the torque $\vec{\tau}$ felt by the current loop?
(c) The same loop is viewed from a passing alien spaceship moving with velocity $\overrightarrow{\boldsymbol{v}}=\boldsymbol{v} \hat{\imath}$ as seen from the laboratory. From the point of view of the aliens, the loop is length contracted in the direction of this motion. What is the magnetic dipole moment $\overrightarrow{\boldsymbol{\mu}}^{\prime}$ according to the aliens if their coordinate axes are aligned with those of the humans? (d) The torque on the loop is the same when viewed from the laboratory frame and from the spaceship frame. Accordingly, the magnetic field must be framedependent. What is the magnetic field $\overrightarrow{\boldsymbol{B}}^{\prime}$ in the spaceship frame $S^{\prime} ?$
(e) If the electromagnetic field is $(\overrightarrow{\boldsymbol{E}}, \overrightarrow{\boldsymbol{B}})=(0, \overrightarrow{\boldsymbol{B}})$ in an inertial frame $S,$ then in another frame $S^{\prime}$ moving at velocity $\overrightarrow{\boldsymbol{v}}$ relative to $S,$ what is the component of the magnetic field $\overrightarrow{\boldsymbol{B}}_{\perp}$ perpendicular to $\overrightarrow{\boldsymbol{v}} ?$

Keshav Singh
Keshav Singh
Numerade Educator
03:52

Problem 56

A small sphere with charge $Q$ is resting motionless on an insulated pedestal in a laboratory. In the frame of reference $S$ of the laboratory, the z-axis points upward, there is a magnetic field $\overrightarrow{\boldsymbol{B}}=-B \hat{\jmath},$ and there is no electric field. (a) What is the net force on the sphere?
(b) The same sphere is viewed from a passing spacecraft moving with velocity $\vec{v}=v \hat{\imath}$ as seen from the laboratory. Using the result of Problem $37.55,$ what is the magnetic field $\overrightarrow{\boldsymbol{B}}^{\prime}$ seen by observers in the spacecraft if their coordinate axes are aligned with those of the laboratory? (c) From the perspective of the spacecraft, the sphere has a nonzero velocity and therefore feels a nonzero magnetic force. What is that magnetic force? (d) The net electromagnetic force on the sphere is necessarily the same when viewed from the laboratory frame and from the spaceship frame. This can be true only if there is a nonzero electric field $\overrightarrow{\boldsymbol{E}}^{\prime}$ in the spaceship frame. Determine $\overrightarrow{\boldsymbol{E}}^{\prime} .$ (e) Generalize your conclusion: If the electromagnetic field is $(\overrightarrow{\boldsymbol{E}}, \overrightarrow{\boldsymbol{B}})=(0, \overrightarrow{\boldsymbol{B}})$ in inertial frame $S,$ then in another frame $S^{\prime}$ moving at velocity $\overrightarrow{\boldsymbol{v}}$ relative to $S,$ what is the component of the electric field $E_{\perp}$ perpendicular to $\overrightarrow{\boldsymbol{v}} ?$

Keshav Singh
Keshav Singh
Numerade Educator
02:12

Problem 57

One of the wavelengths of light emitted by hydrogen atoms under normal laboratory conditions is $\lambda=656.3 \mathrm{nm},$ in the red portion of the electromagnetic spectrum. In the light emitted from a distant galaxy this same spectral line is observed to be Doppler-shifted to $\lambda=953.4 \mathrm{nm},$ in the infrared portion of the spectrum. How fast are the emitting atoms moving relative to the earth? Are they approaching the earth or receding from it?

Salamat Ali
Salamat Ali
Numerade Educator
05:36

Problem 58

Two events are observed in a frame of reference $S$ to occur
at the same space point, the second occurring $1.80 \mathrm{~s}$ after the first. In a frame $S^{\prime}$ moving relative to $S$, the second event is observed to occur $2.15 \mathrm{~s}$ after the first. What is the difference between the positions of the two events as measured in $S^{\prime} ?$

Declan Nell
Declan Nell
Numerade Educator
06:36

Problem 59

A baseball coach uses a radar device to measure the speed of an approaching pitched baseball. This device sends out electromagnetic waves with frequency $f_{0}$ and then measures the shift in frequency $\Delta f$ of the waves reflected from the moving baseball. If the fractional frequency shift produced by a baseball is $\Delta f / f_{0}=2.86 \times 10^{-7},$ what is the baseball's speed in $\mathrm{km} / \mathrm{h} ?$ (Hint: Are the waves Doppler-shifted a second time when reflected off the ball?)

Robert Zaballa
Robert Zaballa
Numerade Educator
03:54

Problem 60

Einstein and Lorentz, avid tennis players, play a fast-paced game on a court where they stand $20.0 \mathrm{~m}$ from each other. They play without a net. The tennis ball has mass $0.0580 \mathrm{~kg} .$ Ignore gravity and assume that the ball travels parallel to the ground as it travels between the two players. Unless otherwise specified, all measurements are made by the two men. (a) Lorentz serves the ball at $80.0 \mathrm{~m} / \mathrm{s}$. What is the ball's kinetic energy?
(b) Einstein slams a return at $1.80 \times 10^{8} \mathrm{~m} / \mathrm{s} .$ What is the ball's kinetic energy? (c) During Einstein's return of the ball in part (a), a white rabbit runs beside the court in the direction from Einstein to Lorentz. The rabbit has a speed of $2.20 \times 10^{8} \mathrm{~m} / \mathrm{s}$ relative to the two men. What is the speed of the rabbit relative to the ball? (d) What does the rabbit measure as the distance from Einstein to Lorentz? (e) How much time does it take for the rabbit to run $20.0 \mathrm{~m},$ according to the players? (f) The white rabbit uses his pocket watch to measure the time (as he sees it) for the distance from Einstein to Lorentz to pass by under him. What time does he measure?

Keshav Singh
Keshav Singh
Numerade Educator
01:55

Problem 61

In a particle accelerator a proton moves at constant speed $0.750 c$ in a circle of radius $628 \mathrm{~m} .$ What is the net force on the proton?

Salamat Ali
Salamat Ali
Numerade Educator
04:41

Problem 62

A spaceship moving at constant speed $u$ relative to us broad. casts a radio signal at constant frequency $f_{0}$. As the spaceship approaches us, we receive a higher frequency $f$; after it has passed, we receive a lower frequency. (a) As the spaceship passes by, so it is instantaneously moving neither toward nor away from us, show that the frequency we receive is not $f_{0},$ and derive an expression for the frequency we do receive. Is the frequency we receive higher or lower than $f_{0} ?$ (Hint: In this case, successive wave crests move the same distance to the observer and so they have the same transit time. Thus $f$ equals $1 / T .$ Use the dilation formula to relate the periods in the stationary and moving frames.) (b) A spaceship emits electromagnetic waves of frequency $f_{0}=345 \mathrm{MHz}$ as measured in a frame moving with the ship. The spaceship is moving at a constant speed $0.758 c$ relative to us. What frequency $f$ do we receive when the spaceship is approaching us? When it is moving away? In each case what is the shift in frequency, $f-f_{0} ?$ (c) Use the result of part (a) to calculate the frequency $f$ and the frequency shift $\left(f-f_{0}\right)$ we receive at the instant that the ship passes by us. How does the shift in frequency calculated here compare to the shifts calculated in part (b)?

Keshav Singh
Keshav Singh
Numerade Educator
03:41

Problem 63

As a research scientist at a linear accelerator, you are studying an unstable particle. You measure its mean lifetime $\Delta t$ as a function of the particle's speed relative to your laboratory equipment. You record the speed of the particle $u$ as a fraction of the speed of light in vacuum $c .$ The table gives the results of your measurements.
$$
\begin{array}{l|lllllll}
u / c & 0.70 & 0.80 & 0.85 & 0.88 & 0.90 & 0.92 & 0.94 \\
\hline \Delta t\left(10^{-8} \mathrm{~s}\right) & 3.57 & 4.41 & 5.02 & 5.47 & 6.05 & 6.58 & 7.62
\end{array}
$$
(a) Your team leader suggests that if you plot your data as $(\Delta t)^{2}$ versus $\left(1-u^{2} / c^{2}\right)^{-1},$ the data points will be fit well by a straight line. Construct this graph and verify the team leader's prediction. Use the best-fit straight line to your data to calculate the mean lifetime of the particle in its rest frame. (b) What is the speed of the particle relative to your lab equipment (expressed as $u / c$ ) if the lifetime that you measure is four times its rest-frame lifetime?

Keshav Singh
Keshav Singh
Numerade Educator
03:08

Problem 64

The French physicist Armand Fizeau was the first to measure the speed of light accurately. He also found experimentally that the speed, relative to the lab frame, of light traveling in a tank of water that is itself moving at a speed $V$ relative to the lab frame is
$$
v=\frac{c}{n}+k V
$$
where $n=1.333$ is the index of refraction of water. Fizeau called $k$ the dragging coefficient and obtained an experimental value of $k=0.44$. What value of $k$ do you calculate from relativistic transformations?

Robert Zaballa
Robert Zaballa
Numerade Educator
04:22

Problem 65

You are a scientist studying small aerosol particles that are contained in a vacuum chamber. The particles carry a net charge, and you use a uniform electric field to exert a constant force of $8.00 \times 10^{-14} \mathrm{~N}$ on one of them. That particle moves in the direction of the exerted force. Your instruments measure the acceleration of the particle as a function of its speed $v .$ The table gives the results of your measurements for this particular particle.
$$
\begin{array}{l|cccccc}
\boldsymbol{v} / \boldsymbol{c} & 0.60 & 0.65 & 0.70 & 0.75 & 0.80 & 0.85 \\
\hline \boldsymbol{a}\left(\mathbf{1 0}^{\mathbf{3}} \mathbf{m} / \mathbf{s}^{\mathbf{2}}\right) & 20.3 & 17.9 & 14.8 & 11.2 & 8.5 & 5.9
\end{array}
$$
(a) Graph your data so that the data points are well fit by a straight line. Use the slope of this line to calculate the mass $m$ of the particle.
(b) What magnitude of acceleration does the exerted force produce if the speed of the particle is $100 \mathrm{~m} / \mathrm{s} ?$

Keshav Singh
Keshav Singh
Numerade Educator
05:59

Problem 66

You are an astronomer investigating four astronomical sources of infrared radiation. You have identified the nature of each source, so you know the frequency $f_{0}$ of each when it is at rest relative to you. Your detector, which is at rest relative to the earth, measures the frequency $f$ of the moving source. Your results are given in the table.
$$
\begin{array}{l|cccc}
\text { Source } & A & B & C & D \\
\hline f(\mathrm{THz}) & 7.1 & 5.4 & 6.1 & 8.1 \\
f_{0}(\mathrm{THz}) & 9.2 & 8.6 & 7.9 & 8.9
\end{array}
$$
(a) Which source is moving at the highest speed relative to your detector? What is its speed? Is that source moving toward or away from the detector? (b) Which source is moving at the lowest speed relative to your detector? What is its speed? Is that source moving toward or away from the detector? (c) For source $B$, what frequency would your detector measure if the source were moving at the same speed relative to the detector but toward it rather than away from it?

Keshav Singh
Keshav Singh
Numerade Educator
07:57

Problem 67

A fluorescent tube with length $L$ is fixed horizontally above a ticket window in a train station. At time $t=0$ the tube lights up, changing color from white to bright yellow. After a duration $T$ the tube turns off, reverting to its white color. The tube lies along the $x$ -axis of frame $S$ fixed with respect to the station, with its right end at the origin. We can represent the history of the tube using a "spacetime" diagram, as shown in Fig. $\mathrm{P} 37.67 .$ Events 1 and 2 correspond, respectively, to the right and left ends of the tube at time $t=0,$ while events 3 and 4 correspond to the right and left ends of the tube at time $t=T .$ The totality of events corresponding to positions on the tube when it is lit correspond to the yellow region on the diagram. The tube is observed from the window of a rocket train passing the station from left to right at speed $v$. The frame $S^{\prime}$ is fixed with respect to the train such that the $x^{\prime}$ -axis coincides with the $x$ -axis of frame $S^{\prime},$ and such that event 1 occurs at the origin $\left(x^{\prime}, t^{\prime}\right)=(0,0) .$ (a) Use the Lorentz transformations to determine the spacetime coordinates $\left(x^{\prime}, t^{\prime}\right)$ for events $1,2,3,$ and 4 in the train frame $S^{\prime} .$ (b) Construct a spacetime diagram analogous to Figure $\mathrm{P} 37.67$ showing the tube in the train frame $S .$ Carefully label relevant points, using the economical definition $\gamma=\left(1-v^{2} / c^{2}\right)^{-1 / 2}$ when needed. (c) We use the product $c t$ rather than $t$ on the vertical axis of our figure so that both axes have the dimension of distance. Thus the yellow region has a well-defined area. What is the area of the yellow region in the $S$ frame plot? (d) Compute the area of the yellow region in the $S^{\prime}$ frame plot you constructed in part (b). (Hint: The area of a parallelogram is given by the magnitude of the vector product of two vectors describing adjacent edges.) (e) Is the area of the yellow region the same when we change the frame of reference from $S$ to $S^{\prime \prime} ?$

Keshav Singh
Keshav Singh
Numerade Educator
07:49

Problem 68

Many of the stars in the sky are actually binary stars, in which two stars orbit about their common center of mass. If the orbital speeds of the stars are high enough, the motion of the stars can be detected by the Doppler shifts of the light they emit. Stars for which this is the case are called spectroscopic binary stars. Figure $\mathbf{P 3 7 . 6 8}$ shows the simplest case of a spectroscopic binary star:
two identical stars, each with mass $m,$ orbiting their center of mass in a circle of radius $R .$ The plane of the stars' orbits is edge-on to the line of sight of an observer on the earth. (a) The light produced by heated hydrogen gas in a laboratory on the earth has a frequency of $4.568110 \times 10^{14} \mathrm{~Hz}$ In the light received from the stars by a telescope on the earth, hydrogen light is observed to vary in frequency between $4.567710 \times 10^{14} \mathrm{~Hz}$ and $4.568910 \times 10^{14} \mathrm{~Hz}$. Determine whether the binary star system as a whole is moving toward or away from the earth, the speed of this motion, and the orbital speeds of the stars. (Hint: The speeds involved are much less than $c,$ so you may use the approximate result $\Delta f / f=u / c$ given in Section $37.6 .$ ) (b) The light from each star in the binary system varies from its maximum frequency to its minimum frequency and back again in 11.0 days. Determine the orbital radius $R$ and the mass $m$ of each star. Give your answer for $m$ in kilograms and as a multiple of the mass of the sun, $1.99 \times 10^{30} \mathrm{~kg} .$ Compare the value of $R$ to the distance from the earth to the sun, $1.50 \times 10^{11} \mathrm{~m}$. (This technique is actually used in astronomy to determine the masses of stars. In practice, the problem is more complicated because the two stars in a binary system are usually not identical, the orbits are usually not circular, and the plane of the orbits is usually tilted with respect to the line of sight from the earth.)

Keshav Singh
Keshav Singh
Numerade Educator
05:30

Problem 69

In high-energy physics, new particles can be created by collisions of fast-moving projectile particles with stationary particles. Some of the kinetic energy of the incident particle is used to create the mass of the new particle. A proton-proton collision can result in the creation of a negative kaon $\left(\mathrm{K}^{-}\right)$ and a positive $\operatorname{kaon}\left(\mathrm{K}^{+}\right)$
$$
p+p \rightarrow p+p+\mathrm{K}^{-}+\mathrm{K}^{+}
$$
(a) Calculate the minimum kinetic energy of the incident proton that will allow this reaction to occur if the second (target) proton is initially at rest. The rest energy of each kaon is $493.7 \mathrm{MeV},$ and the rest energy of each proton is $938.3 \mathrm{MeV}$. (Hint: It is useful here to work in the frame in which the total momentum is zero. But note that the Lorentz transformation must be used to relate the velocities in the laboratory frame to those in the zero-total-momentum frame.)
(b) How does this calculated minimum kinetic energy compare with the total rest mass energy of the created kaons? (c) Suppose that instead the two protons are both in motion with velocities of equal magnitude and opposite direction. Find the minimum combined kinetic energy of the two protons that will allow the reaction to occur. How does this calculated minimum kinetic energy compare with the total rest mass energy of the created kaons? (This example shows that when colliding beams of particles are used instead of a stationary target, the energy requirements for producing new particles are reduced substantially.)

Keshav Singh
Keshav Singh
Numerade Educator
20:10

Problem 70

(a) Consider the Galilean transformation along the $x$ -direction: $x^{\prime}=x-v t$ and $t^{\prime}=t .$ In frame $S$ the wave equation for electromagnetic waves in a vacuum is
$$
\frac{\partial^{2} E(x, t)}{\partial x^{2}}-\frac{1}{c^{2}} \frac{\partial^{2} E(x, t)}{\partial t^{2}}=0
$$
where $E$ represents the electric field in the wave. Show that by using the Galilean transformation the wave equation in frame $S^{\prime}$ is found to be
$$
\left(1-\frac{v^{2}}{c^{2}}\right) \frac{\partial^{2} E\left(x^{\prime}, t^{\prime}\right)}{\partial x^{\prime 2}}+\frac{2 v}{c^{2}} \frac{\partial^{2} E\left(x^{\prime}, t^{\prime}\right)}{\partial x^{\prime} \partial t^{\prime}}-\frac{1}{c^{2}} \frac{\partial^{2} E\left(x^{\prime}, t^{\prime}\right)}{\partial t^{\prime 2}}=0
$$
This has a different form than the wave equation in $S .$ Hence the Galilean transformation violates the first relativity postulate that all physical laws have the same form in all inertial reference frames. (Hint:
Express the derivatives $\partial / \partial x$ and $\partial / \partial t$ in terms of $\partial / \partial x^{\prime}$ and $\partial / \partial t^{\prime}$ by use of the chain rule.) (b) Repeat the analysis of part (a), but use the Lorentz coordinate transformations, Eqs. (37.21), and show that in frame $S^{\prime}$ the wave equation has the same form as in frame $S:$
$$
\frac{\partial^{2} E\left(x^{\prime}, t^{\prime}\right)}{\partial x^{\prime 2}}-\frac{1}{c^{2}} \frac{\partial^{2} E\left(x^{\prime}, t^{\prime}\right)}{\partial t^{\prime 2}}=0
$$
Explain why this shows that the speed of light in vacuum is $c$ in both frames $S$ and $S^{\prime}$

Declan Nell
Declan Nell
Numerade Educator
01:32

Problem 71

An airplane has a length of $60 \mathrm{~m}$ when measured at rest. When the airplane is moving at $180 \mathrm{~m} / \mathrm{s}(400 \mathrm{mph})$ in the alternate universe, how long would the plane appear to be to a stationary observer?
(a) $24 \mathrm{~m} ;$ (b) $36 \mathrm{~m} ;$ (c) $48 \mathrm{~m} ;$ (d) $60 \mathrm{~m} ;$ (e) $75 \mathrm{~m}$.

Shoukat Ali
Shoukat Ali
Other Schools
01:49

Problem 72

If the airplane of Passage Problem 37.71 has a rest mass of $20,000 \mathrm{~kg},$ what is its relativistic mass when the plane is moving at $180 \mathrm{~m} / \mathrm{s} ?$ (a) $8000 \mathrm{~kg} ;$
(b) $12,000 \mathrm{~kg}$
(c) $16,000 \mathrm{~kg}$
(d) $25,000 \mathrm{~kg}$
(e) $33,300 \mathrm{~kg}$.

Ceren Uzun
Ceren Uzun
Texas Tech University
02:30

Problem 73

In our universe, the rest energy of an electron is approximately $8.2 \times 10^{-14} \mathrm{~J} .$ What would it be in the alternate universe?
(a) $8.2 \times 10^{-8} \mathrm{~J} ;$ (b) $8.2 \times 10^{-26} \mathrm{~J} ;$ (c) $8.2 \times 10^{-2} \mathrm{~J} ;$ (d) $0.82 \mathrm{~J}$.

Nathan Silvano
Nathan Silvano
Numerade Educator
03:00

Problem 74

In the alternate universe, how fast must an object be moving for it to have a kinetic energy equal to its rest mass? (a) $225 \mathrm{~m} / \mathrm{s} ;$ (b) $260 \mathrm{~m} / \mathrm{s}$
(c) $300 \mathrm{~m} / \mathrm{s} ;$ (d) The kinetic energy could not be equal to the rest mass.

Ceren Uzun
Ceren Uzun
Texas Tech University