College Physics: A Strategic Approach

Randall D. Knight, Brian Jones, Stuart Field

Chapter 27

Relativity - all with Video Answers

Educators

Chapter Questions

Problem 1

A sprinter crosses the finish line of a race. The roar of the crowd in front approaches her at a speed of 355 mis. The roar from the crowd behind her approaches at 335 mis. What are the speed of sound and the speed of the sprinter?

Narayan Hari

Narayan Hari

Numerade Educator

Problem 2

A baseball pitcher can throw a ball with a speed of $40 \mathrm{m} / \mathrm{s}$. He is in the back of a pickup truck that is driving away from you. He throws the ball in your direction, and it floats toward you at a lazy $10 \mathrm{m} / \mathrm{s}$. What is the speed of the truck?

Zulfiqar Ali

Numerade Educator

Problem 3

A boy on a skateboard coasts along at $5 \mathrm{m} / \mathrm{s} .$ He has a ball that he can throw at a speed of $10 \mathrm{m} / \mathrm{s}$. What is the ball's speed relative to the ground if he throws the ball (a) forward or (b) backward?

Narayan Hari

Numerade Educator

Problem 4

A boat takes 3.0 hours to travel $30 \mathrm{km}$ down a river, then 5.0 hours to return. How fast is the river flowing?

Lisa Tarman

Numerade Educator

Problem 5

When the moving sidewalk at the airport is broken, as it often seems to be, it takes you 50s to walk from your gate to baggage claim. When it is working and you stand on the moving sidewalk the entire way, without walking, it takes 75s to travel the same distance. How long will it take you to travel from the gate to baggage claim if you walk while riding on the moving sidewalk?

Narayan Hari

Numerade Educator

Problem 6

An assembly line has a staple gun that rolls to the left at $1.0 \mathrm{m} / \mathrm{s}$ while parts to be stapled roll past it to the right at $3.0 \mathrm{m} / \mathrm{s} .$ The staple gun fires 10 staples per second. How far apart are the staples in the finished part?

Lisa Tarman

Numerade Educator

Problem 7

An out-of-control alien spacecraft is diving into a star at a speed of $1.0 \times 10^{8} \mathrm{m} / \mathrm{s} .$ At what speed, relative to the spacecraft, is the starlight approaching?

Narayan Hari

Numerade Educator

Problem 8

A starship blasts past the earth at $2.0 \times 10^{8} \mathrm{m} / \mathrm{s}$. Just after passing the earth, the starship fires a laser beam out its back. With what speed does the laser beam approach the earth?

Guilherme Barros

Guilherme Barros

Numerade Educator

Problem 9

You are flying at $0.99 c$ with respect to Kara. At the exact instant you pass Kara, she fires a very short laser pulse in the same direction you're heading.
a. After 1.0 s has elapsed on Kara's watch, what does Kara say the distance is between you and the laser pulse?
b. After $1.0 \mathrm{s}$ has elapsed on your watch, what do you say the distance is between you and the laser pulse?

Guilherme Barros

Guilherme Barros

Numerade Educator

Problem 10

You are standing at the point $x=300 \mathrm{m}, y=400 \mathrm{m}$ in a reference frame with synchronized clocks. What time does your clock show at the instant you see the clock at the origin showing $3.00 \mu$ s?

Laszlo Zalavari

Laszlo Zalavari

Numerade Educator

Problem 11

Bjorn is standing at $x=600 \mathrm{m}$. Firecracker I explodes at the origin and firecracker 2 explodes at $x=900 \mathrm{m}$. The flashes from both explosions reach Bjorn's eye at $t=3.0 \mu \mathrm{s}$. At what time did each firecracker explode?

Narayan Hari

Numerade Educator

Problem 12

Bianca is standing at $x=600 \mathrm{m}$. Firecracker $1,$ at the origin, and firecracker $2,$ at $x=900 \mathrm{m},$ explode simultaneously. The flash from firecracker 1 reaches Bianca's eye at $t=3.0 \mu \mathrm{s} .$ At what time does she see the flash from firecracker $2 ?$

Narayan Hari

Numerade Educator

Problem 13

You are standing at $x=9.0 \mathrm{km}$. Lightning bolt 1 strikes at $x=0 \mathrm{km}$ and lightning bolt 2 strikes at $x=$ $12.0 \mathrm{km} .$ Both flashes reach your eye at the same time. Your assistant isstanding at $x=3.0 \mathrm{km} .$ Does your assistant see the flashes at the same time? If not, which does she see first and what is the time difference between the two?

Narayan Hari

Numerade Educator

Problem 14

A light flashes at position $x=0$ m. One microsecond later, a light flashes at position $x=1000 \mathrm{m}$. In a second reference frame, moving along the $x$-axis at speed $v,$ the two flashes are simultaneous. Is this second frame moving to the right or to the left relative to the original frame?

Laszlo Zalavari

Laszlo Zalavari

Numerade Educator

Problem 15

Jose is looking to the east. Lightning bolt 1 strikes a tree $300 \mathrm{m}$ from him. Lightning bolt 2 strikes a barn 900 m from him in the same direction. Jose sees the tree strike $1.0 \mu \mathrm{s}$ before he sees the barn strike. According to Jose, were the lightning strikes simultaneous? If not, which occurred first and what was the time difference between the two?

Narayan Hari

Numerade Educator

Problem 16

Your $1000-\mathrm{m}-$ long starship has warning lights at each end that, to you, flash simultaneously every minute. You are moving directly away from the planet Zerkon at $0.70 c .$ To a Zerkonian, do the lights flash simultaneously? If not, which flashes first the light at the front of your ship or the trailing one?

Narayan Hari

Numerade Educator

Problem 17

There is a lightbulb exactly halfway between the front and rear of a long hallway in your spaceship. Your ship is traveling at $0.5 \mathrm{c}$ relative to the earth. The bulb is suddenly turned on. Let event 1 be when the light from the bulb first strikes the front end of the hall, and event 2 be when it strikes the rear end. Which event occurs first, or are they simultaneous, in the frame of (a) the spaceship and (b) a person standing on the earth?

Narayan Hari

Numerade Educator

Problem 18

A cosmic ray travels $60 \mathrm{km}$ through the earth's atmosphere in $400 \mu \mathrm{s},$ as measured by experimenters on the ground. How long does the journey take according to the cosmic ray?

Guilherme Barros

Guilherme Barros

Numerade Educator

Problem 19

At what speed relative to a laboratory does a clock tick at half the rate of an identical clock at rest in the laboratory? Give your answer as a fraction of $c$.

Guilherme Barros

Guilherme Barros

Numerade Educator

Problem 20

An astronaut travels to a star system 4.5 ly away at a speed of $0.90 \mathrm{c} .$ Assume that the time needed to accelerate and decelerate is negligible.
a. How long does the journey take according to Mission Control on earth?
b. How long does the journey take according to the astronaut?
c. How much time elapses between the launch and the arrival of the first radio message from the astronaut saying that she has arrived?

Narayan Hari

Numerade Educator

Problem 21

A starship voyages to a distant planet 10 ly away. The explorers stay 1 yr, return at the same speed, and arrive back on earth 26 yr after they left. Assume that the time needed to accelerate and decelerate is negligible.
a. What is the speed of the starship?
b. How much time has elapsed on the astronauts' chronometers?

Prabhu Ramji

Numerade Educator

Problem 22

At what speed, as a fraction of $c,$ will a moving rod have a length $60 \%$ that of an identical rod at rest?

Guilherme Barros

Guilherme Barros

Numerade Educator

Problem 23

Jill claims that her new rocket is $100 \mathrm{m}$ long. As she flies past your house, you measure the rocket's length and find that it is only $80 \mathrm{m}$. Should Jill be cited for exceeding the $0.5 \mathrm{c}$ speed limit?

Zulfiqar Ali

Numerade Educator

Problem 24

A muon travels $60 \mathrm{km}$ through the atmosphere at a speed of $0.9997 \mathrm{c}$. According to the muon, how thick is the atmosphere?

Zulfiqar Ali

Numerade Educator

Problem 25

II The Stanford Linear Accelerator (SLAC) accelerates electrons to $v=0.99999997 \mathrm{c}$ in a $3.2-\mathrm{km}$ -long tube. If they travel the length of the tube at full speed (they don't, because they are accelerating), how long is the tube in the electrons' reference frame?

Guilherme Barros

Guilherme Barros

Numerade Educator

Problem 26

Our Milky Way galaxy is 100,000 ly in diameter. A spaceship crossing the galaxy measures the galaxy's diameter to be a mere 1.0 ly.
a. What is the speed of the spaceship relative to the galaxy?
b. How long is the crossing time as measured in the galaxy's reference frame?

Zulfiqar Ali

Numerade Educator

Problem 27

The $X-15$ rocket-powered plane holds the record for the fastest speed ever attained by a manned aircraft, at $2020 \mathrm{m} / \mathrm{s}$. At this speed, by how much is the $15.5-\mathrm{m}$ -long aircraft length contracted?
Hint: Use the binomial approximation.

Guilherme Barros

Guilherme Barros

Numerade Educator

Problem 28

A rocket cruising past earth at $0.800 c$ shoots a bullet out the back door, opposite the rocket's motion, at $0.900 \mathrm{c}$ relative to the rocket. What is the bullet's speed relative to the earth?

Narayan Hari

Numerade Educator

Problem 29

A base on Planet X fires a missile toward an oncoming space fighter. The missile's speed according to the base is $0.85 \mathrm{c}$. The space fighter measures the missile's speed as $0.96 \mathrm{c}$. How fast is the space fighter traveling relative to Planet X?

Narayan Hari

Numerade Educator

Problem 30

A solar flare blowing out from the sun at $0.90 c$ is overtaking a rocket as it flies away from the sun at $0.80 c$ According to the crew on board, with what speed is the flare gaining on the rocket?

Laszlo Zalavari

Laszlo Zalavari

Numerade Educator

Problem 31

A proton is accelerated to $0.999 \mathrm{c}$.
a. What is the proton's momentum?
b. By what factor does the proton's momentum exceed its Newtonian momentum?

Guilherme Barros

Guilherme Barros

Numerade Educator

Problem 32

A 1.0 g particle has momentum $400,000 \mathrm{kg} \cdot \mathrm{m} / \mathrm{s}$. What is the particle's speed?

Zulfiqar Ali

Numerade Educator

Problem 33

At what speed is a particle's momentum twice its Newtonian value?

Zulfiqar Ali

Numerade Educator

Problem 34

What is the speed of a particle whose momentum is $m c ?$

Zulfiqar Ali

Numerade Educator

Problem 35

What are the kinetic energy, the rest energy, and the total energy of a 1.0 g particle with a speed of $0.80 c ?$

Guilherme Barros

Guilherme Barros

Numerade Educator

Problem 36

A quarter-pound hamburger with all the fixings has a mass of $200 \mathrm{g} .$ The food energy of the hamburger $(480$ food calories) is 2 MJ.
a. What is the energy equivalent of the mass of the hamburger?
b. By what factor does the energy equivalent exceed the food energy?

Zulfiqar Ali

Numerade Educator

Problem 37

How fast must an electron move so that its total energy is $10 \%$ more than its rest mass energy?

Zulfiqar Ali

Numerade Educator

Problem 38

At what speed is a particle's kinetic energy twice its rest energy?

Zulfiqar Ali

Numerade Educator

Problem 39

A firecracker explodes at $x=0 \mathrm{m}, t=0 \mu \mathrm{s}$. A second explodes at $x=300 \mathrm{m}, t=2.0 \mu \mathrm{s}$. What is the proper time between these events?

Guilherme Barros

Guilherme Barros

Numerade Educator

Problem 40

You're standing on an asteroid when you see your best friend rocketing by in her new spaceship. As she goes by, you notice that the front and rear of her ship coincide exactly with the $400-\mathrm{m}-$ diameter of another nearby asteroid that is stationary with respect to you. However, you happen to know that your friend's spaceship measured $500 \mathrm{m}$ long in the showroom. What is your friend's speed relative to you?

Narayan Hari

Numerade Educator

Problem 41

A subatomic particle moves through the laboratory at $0.90 \mathrm{c}$. Laboratory experimenters measure its lifetime, from creation to annihilation, to be 2.3 ps $\left(1 \mathrm{ps}=1\right.$ picosecond $\left.=10^{-12} \mathrm{s}\right)$ According to the particle, how long did it live?

Narayan Hari

Numerade Educator

Problem 42

You and Maria each own identical spaceships. As you fly past Maria, you measure her ship to be $90 \mathrm{m}$ long and your own ship to be $100 \mathrm{m}$ long.
a. How long does Maria measure your ship to be?
b. How fast is Maria moving relative to you?

Guilherme Barros

Guilherme Barros

Numerade Educator

Problem 43

A very fast-moving train car passes you, moving to the right at $0.50 c .$ You measure its length to be $12 \mathrm{m} .$ Your friend David flies past you to the right at a speed relative to you of $0.80 \mathrm{c}$. How long does David measure the train car to be?

Narayan Hari

Numerade Educator

Problem 44

Two events in reference frame S occur $10 \mu$ s apart at the same point in space. Frame $\mathrm{S}^{\prime}$ travels at speed $v=0.90 \mathrm{c}$ relative to frame S.
a. What is the time interval between the events in reference frame $\mathrm{S}^{\prime} ?$
b. What is the distance between the events in frame $S^{\prime} ?$

Laszlo Zalavari

Laszlo Zalavari

Numerade Educator

Problem 45

A spaceship heads directly toward an asteroid at a relative speed of $0.75 \mathrm{c} .$ When it is $3.0 \times 10^{8} \mathrm{m}$ from the asteroid, as measured in the asteroid's frame, the rocket fires a laser pulse at the asteroid. According to the rocket captain, what is the time interval between when the pulse hits the asteroid and when the rocket passes the asteroid, barely missing it?

Narayan Hari

Numerade Educator

Problem 46

Because of the earth's rotation, a person living on top of a mountain moves at a faster speed than someone at sea level. The mountain dweller's clocks thus run slowly compared to those at sea level. If the average life span of a hermit is 80 years, on average how much longer would a hermit dwelling on the top of a $3000-\mathrm{m}-$ high mountain live compared to a sea-level hermit?

Narayan Hari

Numerade Educator

Problem 47

Two spaceships approach each other at speeds of $0.80 c$ and $0.90 c$ relative to the sun. When they are $5.0 \times 10^{9} \mathrm{m}$ apart, as measured in the sun's reference frame, they each set their clocks to zero. By how much do their clocks differ at the instant they pass each other?
Hint: First find how long it takes the spaceships to pass in the sun's reference frame.

Narayan Hari

Numerade Educator

Problem 48

You fly $5000 \mathrm{km}$ across the United States on an airliner at $250 \mathrm{m} / \mathrm{s} .$ You return two days later at the same speed.
a. Have you aged more or less than your friends at home?
b. By how much?
Hint: Use the binomial approximation.

Zulfiqar Ali

Numerade Educator

Problem 49

A cube has a density of $2000 \mathrm{kg} / \mathrm{m}^{3}$ while at rest in the laboratory. What is the cube's density as measured by an experimenter in the laboratory as the cube moves through the laboratory at $90 \%$ of the speed of light in a direction perpendicular to one of its faces?

Zulfiqar Ali

Numerade Educator

Problem 50

All railroad car that is $20 \mathrm{m}$ long when at rest passes Rachael, who is standing on the ground. She records that it takes $120 \mathrm{ns}$ to go by. How fast is the car moving, as a fraction of $c ?$

Guilherme Barros

Guilherme Barros

Numerade Educator

Problem 51

A spaceship flies past an experimenter who measures its length to be one-half the length he had measured when the spaceship was at rest. An astronaut aboard the spaceship notes that his clock ticks at 1-second intervals. What is the time between ticks as measured by the experimenter?

Guilherme Barros

Guilherme Barros

Numerade Educator

Problem 52

Marissa's spaceship approaches Joseph's at a speed of 0.99 c. As Marissa passes Joseph, they synchronize their clocks to both read $t=0$ s. When Marissa's clock reads 100 s, she sends a light signal back to Joseph. According to his clock, when does he receive this signal?

Narayan Hari

Numerade Educator

Problem 53

At a speed of $0.90 c,$ a spaceship travels to a star that is $9.0 \mathrm{ly}$ distant.
a. According to a scientist on earth, how long does the trip take?
b. According to a scientist on the spaceship, how long does the trip take?
c. According to the scientist on the spaceship, what is the distance traveled during the trip?
d. At what speed do observers on the spaceship see the star approaching them?

Laszlo Zalavari

Laszlo Zalavari

Numerade Educator

Problem 54

In an attempt to reduce the extraordinarily long travel times for voyaging to distant stars, some people have suggested traveling at close to the speed of light. Suppose you wish to visit the red giant star Betelgeuse, which is 430 ly away, and that you want your 20,000 kg rocket to move so fast that you age only 20 years during the round trip.
a. How fast must the rocket travel relative to earth?
b. How much energy is needed to accelerate the rocket to this speed?
c. How many times larger is this energy than the total energy used by the United States in the year $2012,$ which was roughly $1.0 \times 10^{20} \mathrm{J} ?$

Laszlo Zalavari

Laszlo Zalavari

Numerade Educator

Problem 55

A rocket traveling at $0.500 c$ sets out for the nearest star, Alpha Centauri, which is 4.25 ly away from earth. It will return to earth immediately after reaching Alpha Centauri. What distance will the rocket travel and how long will the journey last according to (a) stay-at-home earthlings and (b) the rocket crew? (c) Which answers are the correct ones, those in part a or those in part b?

Narayan Hari

Numerade Educator

Problem 56

A distant quasar is found to be moving away from the earth at $0.80 c .$ A galaxy closer to the earth and along the same line of sight is moving away from us at $0.20 \mathrm{c} .$ What is the recessional speed of the quasar as measured by astronomers in the other galaxy?

Guilherme Barros

Guilherme Barros

Numerade Educator

Problem 57

Two rockets approach each other. Each is traveling at $0.75 c$ in the earth's reference frame. What is the speed of one rocket relative to the other?

Guilherme Barros

Guilherme Barros

Numerade Educator

Problem 58

A railcar has a velocity of $0.5 c$ relative to the ground. To another train traveling on the same track, the railcar's velocity is $-0.2 c .$ What is the velocity of the train relative to the ground?

Narayan Hari

Numerade Educator

Problem 58

A railcar has a velocity of $0.5 c$ relative to the ground. To another train traveling on the same track, the railcar's velocity is $-0.2 c .$ What is the velocity of the train relative to the ground?

Laszlo Zalavari

Laszlo Zalavari

Numerade Educator

Problem 59

A military jet traveling at $1500 \mathrm{m} / \mathrm{s}$ has engine trouble and the pilot must bail out. Her ejection seat shoots her forward at $300 \mathrm{m} / \mathrm{s}$ relative to the jet. According to the Lorentz velocity transformation, by how much is her velocity relative to the ground less than the $1800 \mathrm{m} / \mathrm{s}$ predicted by Galilean relativity?
Hint: Use the binomial approximation.

Narayan Hari

Numerade Educator

Problem 60

James, Daniella, and Tara all possess identical clocks. As Daniella passes James in her rocket, James observes that her clock runs at $80 \%$ the rate of his clock. As Tara passes in her rocket, in the same direction as Daniella, James observes that her clock runs at $70 \%$ the rate of his clock. At what rate, relative to her clock, does Daniella observe Tara's clock to run?

Laszlo Zalavari

Laszlo Zalavari

Numerade Educator

Problem 61

A space beacon on Planet Karma emits a pulse of light every second. Spaceman Trevor flies directly toward the beacon at a speed of $0.95 c .$ According to Trevor, what is the time between one pulse reaching his ship and the next?
Hint: First find the time between pulses reaching the ship in Planet Karma's reference frame.

Laszlo Zalavari

Laszlo Zalavari

Numerade Educator

Problem 62

Two rockets, A and B, approach the earth from opposite directions at speed $0.800 \mathrm{c} .$ The length of each rocket measured in its rest frame is $100 \mathrm{m}$. What is the length of rocket $\mathrm{A}$ as measured by the crew of rocket B?

Guilherme Barros

Guilherme Barros

Numerade Educator

Problem 63

The highest-energy cosmic ray ever detected had an energy of about $3.0 \times 10^{20} \mathrm{eV}$. Assume that this cosmic ray was a proton.
a. What was the proton's speed as a fraction of $c ?$
b. If this proton started at the same time and place as a photon traveling at the speed of light, how far behind the photon would it be after traveling for 1 ly?

Guilherme Barros

Guilherme Barros

Numerade Educator

Problem 64

What is the speed of an electron after being accelerated from rest through a $20 \times 10^{6} \mathrm{V}$ potential difference?

Guilherme Barros

Guilherme Barros

Numerade Educator

Problem 65

What is the speed of a proton after being accelerated from rest through a $50 \times 10^{6} \mathrm{V}$ potential difference?

Zulfiqar Ali

Numerade Educator

Problem 66

The half-life of a muon at rest is $1.5 \mu$ s. Muons that have been accelerated to a very high speed and are then held in a circular storage ring have a half-life of $7.5 \mu$ s.
a. What is the speed of the muons in the storage ring?
b. What is the total energy of a muon in the storage ring? The mass of a muon is 207 times the mass of an electron.

Zulfiqar Ali

Numerade Educator

Problem 67

What is the momentum of a particle with speed $0.95 c$ and total energy $2.0 \times 10^{-10} \mathrm{J} ?$

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 68

What is the momentum of a particle whose total energy is four times its rest energy? Give your answer as a multiple of $m c.$

Zulfiqar Ali

Numerade Educator

Problem 69

What is the total energy, in $\mathrm{MeV}$, of
a. A proton traveling at $99.0 \%$ of the speed of light?
b. An electron traveling at $99.0 \%$ of the speed of light?

Guilherme Barros

Guilherme Barros

Numerade Educator

Problem 70

What is the velocity, as a fraction of $c$, of
a. A proton with $500 \mathrm{GeV}$ total energy?
b. An electron with $2.0 \mathrm{GeV}$ total energy?

Guilherme Barros

Guilherme Barros

Numerade Educator

Problem 71

At what speed is the kinetic energy of a particle twice its Newtonian value?

Zulfiqar Ali

Numerade Educator

Problem 72

The factor $\gamma$ appears in many relativistic expressions. A value $\gamma=1.01$ implies that relativity changes the Newtonian values by approximately $1 \%$ and that relativistic effects can no longer be ignored. At what kinetic energy, in MeV, is $\gamma=1.01$ for (a) an electron, and (b) a proton?

Guilherme Barros

Guilherme Barros

Numerade Educator

Problem 73

The chemical energy of gasoline is $46 \mathrm{MJ} / \mathrm{kg} .$ If gasoline's mass could be completely converted into energy, what mass of gasoline would be needed to equal the chemical energy content of $1.0 \mathrm{kg}$ of gasoline?

Guilherme Barros

Guilherme Barros

Numerade Educator

Problem 74

A standard nuclear power plant generates $3.0 \mathrm{GW}$ of thermal power from the fission of ${ }^{235}$ U. Experiments show that, on average, 0.19 u of mass is lost in each fission of $a^{235}$ U nucleus. How many kilograms of ${ }^{235}$ U undergo fission each year in this power plant?

Guilherme Barros

Guilherme Barros

Numerade Educator

Problem 75

The sun radiates energy at the rate $3.8 \times 10^{26} \mathrm{W} .$ The source of this energy is fusion, a nuclear reaction in which mass is transformed into energy. The mass of the sun is $2.0 \times 10^{30} \mathrm{kg} .$
a. How much mass does the sun lose each year?
b. What percentage is this of the sun's total mass?
c. Estimate the lifetime of the sun.

Guilherme Barros

Guilherme Barros

Numerade Educator

Problem 76

The radioactive element radium (Ra) decays by a process known as alpha decay, in which the nucleus emits a helium nucleus. (These high-speed helium nuclei were named alpha particles when radioactivity was first discovered, long before the identity of the particles was established.) The reaction is ${ }^{226} \mathrm{Ra} \rightarrow{ }^{222} \mathrm{Rn}+{ }^{4} \mathrm{He},$ where $\mathrm{Rn}$ is the element radon. The accurately measured atomic masses of the three atoms are $226.025,222.017,$ and $4.003 .$ How much energy is released in each decay? (The energy released in radioactive decay is what makes nuclear waste "hot.")

Narayan Hari

Numerade Educator

Problem 77

The nuclear reaction that powers the sun is the fusion of four protons into a helium nucleus. The process involves several steps, but the net reaction is simply $4 p \rightarrow{ }^{4} \mathrm{He}+$ energy. The mass of a helium nucleus is known to be $6.64 \times 10^{-27} \mathrm{kg}.$
a. How much energy is released in each fusion?
b. What fraction of the initial rest mass energy is this energy?

Zulfiqar Ali

Numerade Educator

Problem 78

When antimatter (which we'll learn more about in Chapter 30 ) interacts with an equal mass of ordinary matter, both matter and antimatter are converted completely into energy in the form of photons. In an antimatter-fueled spaceship, a staple of science fiction, the newly created photons are shot from the back of the ship, propelling it forward. Suppose such a ship has a mass of $2.0 \times 10^{6} \mathrm{kg},$ and carries a mass of fuel equal to $1 \%$ of its mass, or $1.0 \times 10^{4} \mathrm{kg}$ of $\mathrm{matter}$ and an equal mass of antimatter.
a. What is the final speed of the ship, assuming it starts from rest, if all energy released in the matter-antimatter annihilation is transformed into the kinetic energy of the ship?
b. Not only do photons have energy, as you learned in Chapter $25,$ they also have momentum. Explain why, when energy and momentum conservation are both considered, the final speed of the ship will be less than you calculated in part a.

Laszlo Zalavari

Laszlo Zalavari

Numerade Educator

Problem 79

What is the half-life of a pion in the reference frame of the patient undergoing pion therapy?
A. $1.8 \times 10^{-10} \mathrm{s}$
B. $1.8 \times 10^{-8} \mathrm{s}$
C. $1.8 \times 10^{-7} \mathrm{s}$
D. $1.8 \times 10^{-6} \mathrm{s}$

Guilherme Barros

Guilherme Barros

Numerade Educator

Problem 80

According to the pion, what is the distance it travels from the accelerator to the medical bay?
A. $0.30 \mathrm{m}$
B. $3.0 \mathrm{m}$
C. $30 \mathrm{m}$
D. $3000 \mathrm{m}$

Guilherme Barros

Guilherme Barros

Numerade Educator

Problem 81

The proton collision that creates the pion also creates a gamma-ray photon traveling in the same direction as the pion. The photon will get to the medical bay first because it is moving faster. What is the speed of the photon in the pion's reference frame?
A. $0.00005 c$
B. $0.5 c$
C. $0.99995 c$
D. $c$

Guilherme Barros

Guilherme Barros

Numerade Educator

Problem 82

If the pion slows down to $0.99990 c,$ about what percentage of its kinetic energy is lost?
A. $0.03 \%$
?. $0.3 \%$
C. $3 \%$
D. $30 \%$

Narayan Hari

Numerade Educator