Physics

Robert Coleman Richardson; Betty McCarthy Richardson; Alan Giambattista

Chapter 26

Relativity - all with Video Answers

Educators

Chapter Questions

Problem 1

An engineer in a train moving toward the station with a velocity $v=0.60 \mathrm{c}$ lights a signal flare as he reaches a marker $1.0 \mathrm{km}$ from the station (according to a scale laid out on the ground). By how much time, on the stationmaster's clock, does the arrival of the optical signal precede the arrival of the train?

Dading Chen

Dading Chen

Numerade Educator

Problem 2

The light-second is a unit of distance; 1 light-second is the distance that light travels in 1 second.
(a) Find the conversion between light-seconds and meters: 1 lightsecond $=? \mathrm{m} .$ (b) What is the speed of light in units of light-seconds per second?

Dading Chen

Numerade Educator

Problem 3

A spaceship traveling at speed $0.13 \mathrm{c}$ away from Earth sends a radio transmission to Earth.
(a) According to Galilean relativity, at what speed would the transmission travel relative to Earth?
(b) Using Einstein's postulates, at what speed does the transmission travel relative to Earth?

Dading Chen

Numerade Educator

Problem 4

Event A happens at the spacetime coordinates $(x, y, z, t)=$ $(2 \mathrm{m}, 3 \mathrm{m}, 0,0.1 \mathrm{s})$ and event $\mathrm{B}$ happens at the spacetime coordinates $(x, y, z, t)=\left(0.4 \times 10^{8} \mathrm{m}, 3 \mathrm{m}, 0,0.2 \mathrm{s}\right)$
(a) Is it possible that event A caused event B?
(b) If event $B$ occurred at $\left(0.2 \times 10^{8} \mathrm{m}, 3 \mathrm{m}, 0,0.2 \mathrm{s}\right)$ instead, would it then be possible that event A caused event B? [Hint: How fast would a signal need to travel to get from event $A$ to the location of $B$ before event $B$ occurred?]

Narayan Hari

Numerade Educator

Problem 5

An astronaut wears a new Rolex watch on a journey at a speed of $2.0 \times 10^{8} \mathrm{m} / \mathrm{s}$ with respect to Earth. According to mission control in Houston, the trip lasts 12.0 h. How long is the trip as measured on the Rolex?

Dading Chen

Numerade Educator

Problem 6

An unstable particle called the pion has a mean lifetime of $25 \mathrm{ns}$ in its own rest frame. A beam of pions travels through the laboratory at a speed of $0.60 c .$ (a) What is the mean lifetime of the pions as measured in the laboratory frame?
(b) How far does a pion travel (as measured by laboratory observers) during this time?

Dading Chen

Numerade Educator

Problem 7

Suppose your handheld calculator will show six places beyond the decimal point. At what minimum speed would an object have to be traveling so that gamma can be seen to be different from 1 on your calculator display? That is, how fast should an object travel so that $\gamma=1.000001 ?[$Hint: Use the binomial approximation. $]$

Narayan Hari

Numerade Educator

Problem 8

A spaceship is traveling away from Earth at $0.87 c .$ The astronauts report home by radio every $12 \mathrm{h}$ (by their own clocks). At what interval are the reports sent to Earth, according to Earth clocks?

Dading Chen

Numerade Educator

Problem 9

A spaceship travels at constant velocity from Earth to a point 710 ly away as measured in Earth's rest frame. The ship's speed relative to Earth is $0.9999 \mathrm{c}$. A passenger is 20 yr old when departing from Earth.
(a) How old is the passenger when the ship reaches its destination, as measured by the ship's clock? (b) If the spaceship sends a radio signal back to Earth as soon as it reaches its destination, in what year, by Earth's calendar, does the signal reach Earth? The spaceship left Earth in the year 2000 .

Dading Chen

Numerade Educator

Problem 10

A clock moves at a constant velocity of $8.0 \mathrm{km} / \mathrm{s}$ with respect to Earth. If the clock ticks at intervals of one second in its rest frame, how much more than a second elapses between ticks of the clock as measured by an observer at rest on Earth? [Hint: Use the binomial approximation.

Jack Hou

Numerade Educator

Problem 11

A plane trip lasts $8.0 \mathrm{h} ;$ the average speed of the plane during the flight relative to Earth is $220 \mathrm{m} / \mathrm{s}$. What is the time difference between an atomic clock on board
the plane and one on the ground, assuming they were synchronized before the flight? (Ignore general relativistic complications due to gravity and the acceleration of the plane.)

Dading Chen

Numerade Educator

Problem 12

Fill in the missing algebraic steps in the derivation of the time dilation equation $[\mathrm{Eq} \cdot(26-4)]$

Narayan Hari

Numerade Educator

Problem 13

A spaceship travels toward Earth at a speed of $0.97 c$ The occupants of the ship are standing with their torsos parallel to the direction of travel. According to Earth observers, they are about $0.50 \mathrm{m}$ tall and $0.50 \mathrm{m}$ wide. What are the occupants"
(a) height and
(b) width according to others on the spaceship?

Dading Chen

Numerade Educator

Problem 14

While the spaceship in Problem 13 continues to travel in the same direction, one of the occupants lies on his side, so that now his torso is perpendicular to the direction of travel and his width is parallel to the travel direction. What are the (a) height and (b) width of this occupant according to an Earth observer?

Dading Chen

Numerade Educator

Problem 15

A cosmic ray particle travels directly over an American football field, from one goal line to the other, at a speed of $0.50 c .$ (a) If the length of the field between goal lines in the Earth frame is $91.5 \mathrm{m}(100 \mathrm{yd}),$ what length is measured in the rest frame of the particle? (b) How long does it take the particle to go from one goal line to the other according to Earth observers? (c) How long does it take in the rest frame of the particle?

Narayan Hari

Numerade Educator

Problem 16

A laboratory measurement of the coordinates of the ends of a moving meterstick, taken at the same time in the laboratory, gives the result that one end of the stick is $0.992 \mathrm{m}$ due north of the other end. If the stick is moving due north, what is its speed with respect to the lab?

Narayan Hari

Numerade Educator

Problem 17

Two spaceships are moving directly toward each other with a relative velocity of $0.90 \mathrm{c}$. If an astronaut measures the length of his own spaceship to be $30.0 \mathrm{m},$ how long is the spaceship as measured by an astronaut in the other ship?

Narayan Hari

Numerade Educator

Problem 18

A spaceship is moving at a constant velocity of $0.70 c$ relative to an Earth observer. The Earth observer measures the length of the spaceship to be $40.0 \mathrm{m}$. How long is the spaceship as measured by its pilot?

Dading Chen

Numerade Educator

Problem 19

A spaceship moves at a constant velocity of $0.40 c$ relative to an Earth observer. The pilot of the spaceship is holding a rod, which he measures to be $1.0 \mathrm{m}$ long. (a) The rod is held perpendicular to the direction of motion of the spaceship. How long is the rod according to the Earth observer? (b) After the pilot rotates the rod and holds it parallel to the direction of motion of the spaceship, how long is it according to the Earth observer?

Dading Chen

Numerade Educator

Problem 20

A rectangular plate of glass, measured at rest, has sides $30.0 \mathrm{cm}$ and $60.0 \mathrm{cm}$
(a) As measured in a reference frame moving parallel to the $60.0 \mathrm{cm}$ edge at speed $0.25 c$ with respect to the glass, what are the lengths of the sides? (b) How fast would a reference frame have to move in the same direction so that the plate of glass viewed in that frame is square?

Dading Chen

Numerade Educator

Problem 21

A futuristic train moving in a straight line with a uniform speed of $0.80 \mathrm{c}$ passes a series of communications towers. The spacing between the towers, according to an observer on the ground, is $3.0 \mathrm{km}$. A passenger on the train uses an accurate stopwatch to see how often a tower passes him.
(a) What is the time interval the passenger measures between the passing of one tower and the next? (b) What is the time interval an observer on the ground measures for the train to pass from one tower to the next?

Dading Chen

Numerade Educator

Problem 22

An astronaut in a rocket moving at $0.50 c$ toward the Sun finds himself halfway between Earth and the Sun. According to the astronaut, how far is he from Earth? In the frame of the Sun, the distance from Earth to the Sun is $1.50 \times 10^{11} \mathrm{m}$

Dading Chen

Numerade Educator

Problem 23

The mean (average) lifetime of a muon in its rest frame is $2.2 \mu \mathrm{s}$. A beam of muons is moving through the lab with speed 0.994 c. How far on average does a muon travel through the lab before it decays?

Dading Chen

Numerade Educator

Problem 24

The Tevatron is a particle accelerator at Fermilab that accelerates protons and antiprotons to high energies in an underground ring. Scientists observe the results of collisions between the particles. The protons are accelerated until they have speeds only $100 \mathrm{m} / \mathrm{s}$ slower than the speed of light. The circumference of the ring is $6.3 \mathrm{km} .$ What is the circumference according to an observer moving with the protons? [Hint: Let $v=c-u$ where $v$ is the proton speed and $u=100 \mathrm{m} / \mathrm{s} .1$

Narayan Hari

Numerade Educator

Problem 25

Kurt is measuring the speed of light in an evacuated chamber aboard a spaceship traveling with a constant velocity of $0.60 c$ with respect to Earth. The light is moving in the direction of motion of the spaceship. siu-Ling is on Earth watching the experiment. With what speed does the light in the vacuum chamber travel, according to Siu-Ling's observations?

Dading Chen

Numerade Educator

Problem 26

Particle $A$ is moving with a constant velocity $v_{\mathrm{AE}}=+0.90 \mathrm{c}$ relative to an Earth observer. Particle $B$ moves in the opposite direction with a constant velocity $v_{\mathrm{BE}}=-0.90 \mathrm{c}$ relative to the same Earth observer. What is the velocity of particle $B$ as seen by particle $A ?$

Dading Chen

Numerade Educator

Problem 27

A man on the Moon observes two spaceships coming toward him from opposite directions at speeds of $0.60 c$ and $0.80 \mathrm{c}$. What is the relative speed of the two ships as measured by a passenger on either one of the spaceships?

Dading Chen

Numerade Educator

Problem 28

Rocket ship Able travels at $0.400 \mathrm{c}$ relative to an Earth observer. According to the same observer, rocket ship Able overtakes a slower moving rocket ship Baker that moves in the same direction. The captain of Baker sees Able pass her ship at 0.114 c. Determine the speed of Baker relative to the Earth observer.

Dading Chen

Numerade Educator

Problem 29

Relative to the laboratory, a proton moves to the right with a speed of $\frac{4}{5} c,$ while relative to the proton, an electron moves to the left with a speed of $\frac{5}{7} c .$ What is the speed of the electron relative to the lab?

Dading Chen

Numerade Educator

Problem 30

As observed from Earth, rocket Alpha moves with speed $0.90 \mathrm{c}$ and rocket $B$ bravo travels with a speed of $0.60 \mathrm{c}$. They are moving along the same line toward a head-on collision. What is the speed of rocket Alpha as measured from rocket $B$ bravo?

Narayan Hari

Numerade Educator

Problem 31

Electron A is moving west with speed $\frac{3}{5} c$ relative to the lab. Electron B is also moving west with speed $\frac{4}{3} c$ relative to the lab. What is the speed of electron B in a frame of reference in which electron A is at rest?

Narayan Hari

Numerade Educator

Problem 32

A proton moves at $0.90 c .$ What is its momentum?

Narayan Hari

Numerade Educator

Problem 33

An electron has momentum of magnitude $2.4 \times$ $10^{-22} \mathrm{kg} \cdot \mathrm{m} / \mathrm{s} .$ What is the electron's speed?

Dading Chen

Numerade Educator

Problem 34

By what factor is the momentum of a particle moving at $0.60 c$ greater than the momentum of the same particle moving at $0.30 c ?$

Narayan Hari

Numerade Educator

Problem 35

A particle is initially moving at $0.60 \mathrm{c}$. If its momentum increases by a factor of $2.0,$ what is its speed?

Jack Hou

Numerade Educator

Problem 36

The International Space Station (ISS) has a mass of $4.5 \times 10^{5} \mathrm{kg}$ and orbits Earth at a speed of $7.7 \mathrm{km} / \mathrm{s} .$ By what percentage does the approximate momentum of the ISS calculated nonrelativistically differ from the relativistic momentum? [Hint: Use one of the approximations in Appendix A.9.]

Jack Hou

Numerade Educator

Problem 37

A spaceship of mass $m$ is traveling away from Earth at speed $v$. Its momentum has magnitude $2.5 \mathrm{mv}$. (a) Find $v$.
(b) An astronaut on the spaceship has a watch that ticks once every second. How often does the watch tick as measured by an Earth observer?

Jack Hou

Numerade Educator

Problem 38

How much energy is released by a nuclear reactor if the total mass of the fuel decreases by $1.0 \mathrm{g} ?$

Dading Chen

Numerade Educator

Problem 39

Two lumps of putty are moving in opposite directions, each one having a speed of $30.0 \mathrm{m} / \mathrm{s}$. They collide and stick together. After the collision the combined lumps are at rest. If the mass of each lump was $1.00 \mathrm{kg}$ before the collision, and no energy is lost to the environment, what is the change in mass of the system due to the collision?

Narayan Hari

Numerade Educator

Problem 40

A white dwarf is a star that has exhausted its nuclear fuel and lost its outer mass so that it consists only of its dense, hot inner core. It will cool unless it gains mass from some nearby star. It may form a binary system with such a star and gradually gain mass up to the limit of 1.4 times the mass of the Sun. If the white dwarf were to start to exceed the limit, it would explode into a supernova. How much energy is released by the explosion of a white dwarf at its limiting mass if $80.0 \%$ of its mass is

Narayan Hari

Numerade Educator

Problem 41

A lambda hyperon $\Lambda^{0}$ (mass $=1115 \mathrm{MeV} / \mathrm{c}^{2}$ ) at rest decays into a neutron $n$ (mass $=940 \mathrm{MeV} / \mathrm{c}^{2}$ ) and a pion $\pi^{0}\left(\mathrm{mass}=135 \mathrm{MeV} / \mathrm{c}^{2}\right):$ $$
\Lambda^{0} \rightarrow \mathrm{n}+\pi^{0}
$$

Narayan Hari

Numerade Educator

Problem 42

Radon decays as follows: ${ }^{222} \mathrm{Rn} \rightarrow{ }^{218} \mathrm{Po}+\alpha .$ The mass of the radon-222 nucleus is $221.97039 \mathrm{u},$ the mass of the polonium- 218 nucleus is $217.96289 \mathrm{u},$ and the mass of the alpha particle is 4.00151 u. How much energy is released in the decay? $\left(1 \mathrm{u}=931.494 \mathrm{MeV} / \mathrm{c}^{2}\right.$.)

Dading Chen

Numerade Educator

Problem 43

The energy to accelerate a starship comes from combining matter and antimatter. When this is done, the total rest energy of the matter and antimatter is converted to other forms of energy. Suppose a star-ship with a mass of $2.0 \times 10^{5} \mathrm{kg}$ accelerates to $0.3500 \mathrm{c}$ from rest. How much matter and antimatter must be converted to kinetic
energy for this to occur?

Bettina Hanlon

Numerade Educator

Problem 44

A laboratory observer measures an electron's energy to be $1.02 \times 10^{-13} \mathrm{J} .$ What is the electron's speed?

Dading Chen

Numerade Educator

Problem 45

A muon with rest energy 106 MeV is created at an altitude of $4500 \mathrm{m}$ and travels downward toward Earth's surface. An observer on Earth measures its speed as $0.980 c .$ (a) What is the muon's total energy in the Earth observer's frame of reference?
(b) What is the muon's total energy in the muon's frame of reference?

Dading Chen

Numerade Educator

Problem 46

An object of mass $0.12 \mathrm{kg}$ is moving at $1.80 \times 10^{8} \mathrm{m} / \mathrm{s}$ What is its kinetic energy in joules?

Dading Chen

Numerade Educator

Problem 47

When an electron travels at $0.60 c,$ what is its total energy in mega-electron-volts?

Dading Chen

Numerade Educator

Problem 48

An observer in the laboratory finds that an electron's total energy is $5.0 \mathrm{mc}^{2} .$ What is the magnitude of the electron's momentum (as a multiple of $m c$ ) as observed in the laboratory?

Dading Chen

Numerade Educator

Problem 49

The rest energy of an electron is 0.511 MeV. What momentum (in MeV/c) must an electron have in order that its total energy be 3.00 times its rest energy?

Dading Chen

Numerade Educator

Problem 50

An electron has a total energy of 6.5 MeV. What is its momentum (in $\mathrm{MeV} / \mathrm{c}$ )?

Dading Chen

Numerade Educator

Problem 51

An electron accelerator used in a hospital for cancer treatment produces a beam of electrons with kinetic energy $25 \mathrm{MeV}$.
(a) What is the speed of the electrons produced by this accelerator? (b) If the end of the electron accelerator is placed $15 \mathrm{cm}$ from the patient, how long, in the reference frame of the electrons, do they take to travel this distance?

Bettina Hanlon

Numerade Educator

Problem 52

A typical hospital accelerator built for proton beam therapy accelerates protons from rest by passing them through an electric potential difference of magnitude 75 MV. Find the speed of these protons.

Narayan Hari

Numerade Educator

Problem 53

PET scans involve the use of positron-emitting isotopes like carbon-11 and fluorine-18. These isotopes can be produced at hospital-based accelerators that first accelerate deuterons (hydrogen-2 nuclei) and then direct the deuterons onto a solid or gaseous target. Suppose a deuteron (rest energy 1875.6 MeV ) is accelerated to a kinetic energy of 2.50 MeV. What is its speed in meters per second?

Narayan Hari

Numerade Educator

Problem 54

In a medical treatment known as fast-neutron therapy, neutrons of kinetic energy 25 MeV are directed toward a patient"s tumor. Neutrons are known to decay, when at rest, with an average lifetime of 886 s. What is the lifetime, as measured in the laboratory, of 25 MeV neutrons?

Bettina Hanlon

Numerade Educator

Problem 55

An experimental form of cancer therapy involves the use of a beam of highly ionized carbon atoms with a charge of $+6 e$ (all six electrons have been removed). The mass of the ions is $11.172 \mathrm{GeV} / \mathrm{c}^{2}$. If the accelerator is $7.50 \mathrm{m}$ long and the ions are accelerated through a 125 MV potential difference, what are
(a) the ion's kinetic energy, (b) the speed of the ions as measured in the lab frame, and (c) the length of the accelerator in the reference frame of the ions?

Sanat Mukherjee

Sanat Mukherjee

Numerade Educator

Problem 56

For a nonrelativistic particle of mass $m,$ show that $K=p^{2} /(2 m) \cdot[$Hint: Start with the nonrelativistic expressions for kinetic energy $K$ and momentum $p .1$

Narayan Hari

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Problem 57

Find the conversion between the momentum unit MeV/c and the SI unit of momentum.

Dading Chen

Numerade Educator

Problem 58

Find the conversion between the mass unit MeV/ $c^{2}$ and the SI unit of mass.

Dading Chen

Numerade Educator

Problem 59

In a beam of electrons used in a diffraction experiment, each electron is accelerated to a kinetic energy of 150 keV. (a) Are the electrons relativistic? Explain.
(b) How fast are the electrons moving?

Dading Chen

Numerade Educator

Problem 60

Derive the energy-momentum relation
$$
E^{2}=E_{0}^{2}+(p c)^{2}
$$
Start by squaring the definition of total energy $\left(E=K+E_{0}\right)$ and then use the relativistic expressions for momentum and kinetic energy [Eqs. ( $26-15$ ) and ( $26-18$ )].

Jack Hou

Numerade Educator

Problem 61

Starting with the energy-momentum relation $E^{2}=E_{0}^{2}+(p c)^{2}$ and the definition of total energy show that $(p c)^{2}=K^{2}+2 K E_{0}[\mathrm{Eq} \cdot(26-23)]$

Narayan Hari

Numerade Educator

Problem 62

Show that Eq. $(26-23)$ reduces to the nonrelativistic relationship between momentum and kinetic energy, $K \approx p^{2} /(2 m),$ for $K \ll E_{0}$

Narayan Hari

Numerade Educator

Problem 63

Show that each of these statements implies that $v \ll c,$ which means that $v$ can be considered a nonrelativistic speed:
$\begin{array}{llll}\text { (a) } \gamma-1 \ll 1 & \text { [Eq. } & (26-26)] ; & \text { ; }\end{array}$
(b) $K \ll m c^{2}[\mathrm{Eq} \cdot(26-27)] ;(\mathrm{c}) p \ll m c[\mathrm{Eq} \cdot(26-28)]$
(d) $K \approx p^{2} /(2 m)$

Sanat Mukherjee

Sanat Mukherjee

Numerade Educator

Problem 64

The rogue starship Galaxa is being chased by the battle cruiser Millenia. The Millenia is catching up to the Galaxa at a rate of 0.55 $c$ when the captain of the Millenia decides it is time to fire a missile. First the captain shines a laser range finder to determine the distance to the Galaxa and then he fires a missile that is moving at a speed of $0.45 c$ with respect to the Millenia. What speed does the Galaxa measure for (a) the laser beam and
(b) the missile as they both approach the starship?

Bettina Hanlon

Numerade Educator

Problem 65

Refer to Example $26.2 .$ One million muons are moving toward the ground at speed $0.9950 \mathrm{c}$ from an altitude of $4500 \mathrm{m}$. In the frame of reference of an observer on the ground, what are (a) the distance traveled by the muons; (b) the time of flight of the muons;
(c) the time interval during which half of the muons decay; and (d) the number of muons that survive to reach sea level? [Hint: The answers to (a) to (c) are not the same as the corresponding quantities in the muons' reference frame. Is the answer to (d) the same?]

Dading Chen

Numerade Educator

Problem 66

Two atomic clocks are synchronized. One is put aboard a spaceship that leaves Earth at $t=0$ at a speed of $0.750 \mathrm{c}$. (a) When the spaceship has been traveling for $48.0 \mathrm{h}$ (according to the atomic clock on board), it sends a radio signal back to Earth. When would the signal be received on Earth, according to the atomic clock on Earth? (b) When the Earth clock says that the spaceship has been gone for $48.0 \mathrm{h}$, it sends a radio signal to the spaceship. At what time (according to the spaceship"s clock) does the spaceship receive the signal?

Dading Chen

Numerade Educator

Problem 67

A spaceship passes over an observation station on Earth. Just as the nose of the ship passes the station, a light in the nose of the ship flashes. As the tail of the ship passes the station, a light flashes in the ship's tail. According to an Earth observer, 50.0 ns elapses between the two events. In the astronaut's reference frame, the length of the ship is $12.0 \mathrm{m}$. (a) How fast is the ship traveling according to an Earth observer?
(b) What is the elapsed time between light flashes in the astronaut"s frame of reference?

Dading Chen

Numerade Educator

Problem 68

Octavio, traveling at a speed of $0.60 c,$ passes Tracy and her barn. Tracy, who is at rest with respect to her barn, says that the barn is $16 \mathrm{m}$ long in the direction in which Octavio is traveling, $4.5 \mathrm{m}$ high, and $12 \mathrm{m}$ deep.
(a) What does Tracy say is the volume of her barn?
(b) What volume does Octavio measure?

Dading Chen

Numerade Educator

Problem 69

A spaceship resting on Earth has a length of $35.2 \mathrm{m}$. As it departs on a trip to another planet, it has a length of $30.5 \mathrm{m}$ as measured by the Earthbound observers. The Earthbound observers also notice that one of the
astronauts on the spaceship exercises for 22.2 min. How long would the astronaut herself say that she exercises?

Dading Chen

Numerade Educator

Problem 70

At the $10.0 \mathrm{km}$ long Stanford Linear Accelerator, electrons with rest energy of 0.511 MeV have been accelerated to a total energy of 46 GeV. How long is the accelerator as measured in the reference frame of the
electrons?

Dading Chen

Numerade Educator

Problem 71

Consider the following decay process: $\pi^{+} \rightarrow \mu^{+}+v$ The mass of the pion $\left(\pi^{+}\right)$ is $139.6 \mathrm{MeV} / \mathrm{c}^{2},$ the mass of the muon $\left(\mu^{+}\right)$ is 105.7 MeV/c $^{2}$, and the mass of the neutrino ( $\nu$ ) is negligible. If the pion is initially at rest, what is the total kinetic energy of the decay products?

Narayan Hari

Numerade Educator

Problem 72

A neutron (mass 939.565 MeV $/ c^{2}$ ) disintegrates into a proton (mass 938.272 MeV/c $^{2}$ ), an electron (mass. $0.5110 \mathrm{MeV} / \mathrm{c}^{2}$, and an antineutrino (mass negligibly small). What is the sum of the kinetic energies of the particles produced, if the neutron was at rest?

Bettina Hanlon

Numerade Educator

Problem 73

A starship takes 3.0 days to travel between two distant space stations according to its own clocks. Instruments on one of the space stations indicate that the trip took 4.0 days. How fast did the starship travel relative to that space station?

Dading Chen

Numerade Educator

Problem 74

Two spaceships are observed from Earth to be approaching each other along a straight line. Ship A moves at $0.40 \mathrm{c}$ relative to the Earth observer, and ship $\mathrm{B}$ moves at $0.50 c$ relative to the same observer. What speed does the captain of ship A report for the speed of ship B?

Dading Chen

Numerade Educator

Problem 75

A neutron, with rest energy $939.6 \mathrm{MeV}$, has momentum 935 MeV/c downward. What is its total energy?

Dading Chen

Numerade Educator

Problem 76

Suppose that as you travel away from Earth in a spaceship, you observe another ship pass you heading in the same direction and measure its speed to be $0.50 \mathrm{c}$. As you look back at Earth, you measure Earth's speed relative to you to be $0.90 \mathrm{c}$. What is the speed of the ship that passed you according to Earth observers?

Dading Chen

Numerade Educator

Problem 77

(a) If you measure the ship that passes you in Problem 76 to be $24 \mathrm{m}$ long, how long will the observers on Earth measure that ship to be?
(b) If there is a rod on your spaceship that you measure to be $24 \mathrm{m}$ long, how long will the observers on Earth measure your rod to be?
(c) How long do the observers on the passing ship measure your rod to be?

Sanat Mukherjee

Sanat Mukherjee

Numerade Educator

Problem 78

Verify that the collision between the proton and the nitrogen nucleus in Example 26.4 is elastic.

Dading Chen

Numerade Educator

Problem 79

Muons are created by cosmic-ray collisions at an elevation $h$ (as measured in Earth's frame of reference) above Earth's surface and travel downward with a constant speed of $0.990 \mathrm{c}$. During any time interval of $1.5 \mu \mathrm{s}$ in the rest frame of the muons, half of the muons present at the beginning of the interval decay. If one fourth of the original muons reach Earth before decaying, about how big is the height $h ?$

Dading Chen

Numerade Educator

Problem 80

Refer to Example $26.1 .$ Ashlin travels at speed $0.800 c$ to a star 30.0 ly from Earth.
(a) Find the distance between Earth and the star in the astronaut's frame of reference. (b) How long (as measured by the astronaut) does it take to travel this distance at a speed of $0.800 c ?$ Compare your answer to the result of Example 26.1 and explain any discrepancy.

Dading Chen

Numerade Educator

Problem 81

A starship is traveling at a speed of $0.78 c$ toward Earth when it experiences a major malfunction and the crew is forced to evacuate. An escape pod that is $12.0 \mathrm{m}$ long with respect to its passengers is ejected from the starship and sent toward Earth at a speed of $0.63 \mathrm{c}$ with respect to the starship. How long is the escape pod as measured by people on Earth?

Bettina Hanlon

Numerade Educator

Problem 82

According to the special theory of relativity, no object that has mass can travel faster than the speed of light. Yoo Jin says she knows something that moves faster than the speed of light. She tells you to consider a rotating beacon on Earth with a powerful laser that can send a beam to the Moon. (a) If the beacon rotates with a period of $6.00 \mathrm{s}$, how fast will light from the laser travel across the Moon's surface?
(b) How do you explain to Yoo Jin that this does not violate the results of the theory of special relativity?

Narayan Hari

Numerade Educator

Problem 83

Harvey claims that he annihilated a 1.00 lb bag of chocolate-chip cookies after playing basketball for $3 \mathrm{h}$
(a) If Harvey had truly annihilated the mass in the cookies, how much energy would be produced? (b) How many kilowatt-hours of electric energy is this?

Narayan Hari

Numerade Educator

Problem 84

A laboratory observer measures an electron's kinetic energy to be $1.02 \times 10^{-13} \mathrm{J} .$ What is the electron's speed?

Narayan Hari

Numerade Educator

Problem 85

A spaceship is moving away from Earth with a constant velocity of $0.80 \mathrm{c}$ with respect to Earth. The spaceship and an Earth station synchronize their clocks, setting both to zero, at an instant when the ship is near Earth. By prearrangement, when the clock on Earth reaches a reading of $1.0 \times 10^{4}$ s, the Earth station sends out a light signal to the spaceship.
(a) In the frame of reference of the Earth station, how far must the signal travel to reach the spaceship? (b) According to an Earth observer, what is the reading of the clock on Earth when the signal is received?

Dading Chen

Numerade Educator

Problem 86

A charged particle is observed to have a total energy of 0.638 MeV when it is moving at $0.600 c .$ If this particle enters a linear accelerator and its speed is increased to $0.980 c,$ what is the new value of the particle's total energy?

Dading Chen

Numerade Educator

Problem 87

A particle decays in flight into two pions, each having a rest energy of $140.0 \mathrm{MeV}$. The pions travel at right angles to each other with equal speeds of $0.900 \mathrm{c}$. Find
(a) the momentum magnitude of the original particle,
(b) its kinetic energy, and (c) its mass in units of MeV/c".

Sanat Mukherjee

Sanat Mukherjee

Numerade Educator

Problem 88

A spaceship is traveling away from Earth at $0.70 \mathrm{c}$ The astronauts report home by radio every $4.0 \mathrm{h}$ (by their own clocks). (a) At what interval are the reports sent to Earth, according to Earth clocks?
(b) At what interval are the reports received by Earth observers, according to their own clocks?

Sanat Mukherjee

Sanat Mukherjee

Numerade Educator

Problem 89

A cosmic-ray proton entering the atmosphere from space has a kinetic energy of $2.0 \times 10^{20} \mathrm{eV} .$ (a) What is its kinetic energy in joules? (b) If all of the kinetic energy of the proton could be harnessed to lift an object of mass $1.0 \mathrm{kg}$ near Earth's surface, how far could the object be lifted?
(c) What is the speed of the proton? $\left[\right.$Hint: Note that $\left.K \gg E_{0 \cdot}\right]$

Sanat Mukherjee

Sanat Mukherjee

Numerade Educator

Problem 90

An astronaut has spent a long time in the International Space Station (ISS) traveling at $7.66 \mathrm{km} / \mathrm{s}$. When he returns to Earth, he is $50 \mathrm{ms}$ younger than his twin brother. How long was he on the ISS? [Hint: Use an approximation from Appendix A.9]

Sanat Mukherjee

Sanat Mukherjee

Numerade Educator

Problem 91

Radon decays as ${ }^{222} \mathrm{Rn} \rightarrow{ }^{218} \mathrm{Po}+\alpha .$ The mass of the radon-222 nucleus is $221.97039 \mathrm{u},$ the mass of the polonium-218 nucleus is $217.96289 \mathrm{u}$, and the mass of the alpha particle is $4.00151 \mathrm{u} .\left(1 \mathrm{u}=931.494 \mathrm{MeV} / \mathrm{c}^{2}\right)$
If the radon nucleus is initially at rest in the lab frame, at what speeds (in the lab frame) do the (a) polonium-218 nucleus and
(b) alpha particle move? Assume that the speeds are nonrelativistic. After you calculate the speeds, verify that this assumption is valid.

Dominador Tan

Numerade Educator

Problem 92

A lambda hyperon $\Lambda^{\circ}\left(\right.$ mass $\left.=1115.7 \mathrm{MeV} / \mathrm{c}^{2}\right)$ at rest in the lab frame decays into a neutron $n$ (mass $=$ $939.6 \mathrm{MeV} / \mathrm{c}^{2}$ ) and a pion $\pi^{0}\left(\mathrm{mass}=135.0 \mathrm{MeV} / \mathrm{c}^{2}\right):$
$$
\Lambda^{0} \rightarrow n+\pi^{0}
$$

Sanat Mukherjee

Sanat Mukherjee

Numerade Educator

Problem 93

A constant force, acting for $3.6 \times 10^{4}$ s $(10 \mathrm{h})$ brings a spaceship of mass $2200 \mathrm{kg}$ from rest to speed $0.70 c$
(a) What is the magnitude of the force? [Hint:
Use the impulse-momentum theorem.
(b) What is the initial acceleration of the spaceship? Comment on the magnitude of the answer.

Narayan Hari

Numerade Educator

Problem 94

An object has a mass of $12.6 \mathrm{kg}$ and a speed of $0.87 \mathrm{c}$
(a) What is the magnitude of its momentum? (b) If a constant force of $424.6 \mathrm{N}$ acts in the direction opposite to the object's motion, how long must the force act to bring the object to rest? [Hint: Use the impulsemomentum theorem.]

Bettina Hanlon

Numerade Educator

Problem 95

The solar energy arriving at the top of Earth's atmosphere from the Sun has intensity $1.4 \mathrm{kW} / \mathrm{m}^{2}$.
(a) How much mass does the Sun lose per day? (b) What percent of the Sun's mass is this?

Narayan Hari

Numerade Educator

Problem 96

Derivation of the Doppler formula for light. A source and observer of EM waves move relative to each other at
velocity $v .$ Let $v$ be positive if the observer and source are moving apart from each other. The source emits an EM wave at frequency $f_{\mathrm{s}}$ (measured in the source frame). The time between wavefronts as measured by the source is $T_{\mathrm{s}}=1 / f_{\mathrm{s}}$. (a) In the observer's frame, how much time elapses between the emission of wavefronts by the source? Call this $T_{\mathrm{s}}^{\prime}$
(b) $T_{s}^{\prime}$ is not the time that
the observer measures between the arrival of successive Wavefronts because the wavefronts travel different distances. Say that, according to the observer, one wavefront is emitted at $t=0$ and the next at $t=T_{\mathrm{s}}^{\prime} .$ When the first wavefront is emitted, the distance between source and observer is $d$. When the second wavefront is emitted, the distance between source and observer is $d+v T_{s}$ Each wavefront travels at speed $c$. Calculate the time $T_{\mathrm{o}}$ between the arrival of these two wavefronts as measured
by the observer. (c) The frequency detected by the observer is $f_{0}=1 / T_{\mathrm{o}^{-}}$ Show that $f_{\mathrm{o}}$ is given by $\mathrm{Eq} \cdot(22-24) \mathrm{z}$
$$
f_{\mathrm{o}}=f_{\mathrm{s}} \sqrt{\frac{1-v / c}{1+v / c}}
$$

Dominador Tan

Numerade Educator

Problem 97

An electron is accelerated through a potential difference of $25.00 \mathrm{MV}$. (a) What would you calculate for the speed of the electron if relativistic equations were not used? (b) What is the actual speed of the electron in this case?

Narayan Hari

Numerade Educator

Problem 98

A particle with charge +e has a total energy of $0.638 \mathrm{MeV}$ when it is moving at $0.600 \mathrm{c}$. If this particle then enters a linear accelerator, what is its speed after it has been accelerated through a 2.6 MV potential difference?

Narayan Hari

Numerade Educator