Physics For Scientists And Engineers 6E - part 320

Problems

1277

18. Two cards have straight edges. Suppose that the top edge

of one card crosses the bottom edge of another card at a
small angle, as in Figure Q39.18a. A person slides the cards
together at a moderately high speed. In what direction
does the intersection point of the edges move? Show that
it can move at a speed greater than the speed of light.

A small flashlight is suspended in a horizontal plane

and  set  into  rapid  rotation.  Show  that  the  spot  of  light  it
produces on a distant screen can move across the screen at
a speed greater than the speed of light. (If you use a laser
pointer,  as  in  Figure  Q39.18b,  make  sure  the  direct  laser
light cannot enter a person’s eyes.) Argue that these exper-
iments do not invalidate the principle that no material, no
energy,  and  no  information  can  move  faster  than  light
moves in a vacuum.

19. Describe how the results of Example 39.7 would change if,

instead of fast space vehicles, two ordinary cars were
approaching each other at highway speeds.

20. Two objects are identical except that one is hotter than the

other. Compare how they respond to identical forces.

21. With regard to reference frames, how does general relativ-

ity differ from special relativity?

22. Two identical clocks are in the same house, one upstairs in

a bedroom, and the other downstairs in the kitchen.
Which clock runs more slowly? Explain.

23. A thought experiment. Imagine ants living on a merry-

go-round turning at relativistic speed, which is their two-
dimensional world. From measurements on small circles
they  are  thoroughly  familiar  with  the  number  2.  When
they  measure  the  circumference  of  their  world,  and
divide  it  by  the  diameter,  they  expect  to  calculate  the
number  2 ! 3.141 59.  .  .  .  We  see  the  merry-go-round
turning at relativistic speed. From our point of view, the
ants’  measuring  rods  on  the  circumference  are  experi-
encing  length  contraction  in  the  tangential  direction;
hence  the  ants  will  need  some  extra  rods  to  fill  that
entire  distance.  The  rods  measuring  the  diameter,
however,  do  not  contract,  because  their  motion  is
perpendicular to their lengths. As a result, the computed
ratio does not agree with the number 2. If you were an
ant,  you  would  say  that  the  rest  of  the  universe  is
spinning  in  circles,  and  your  disk  is  stationary.  What
possible  explanation  can  you  then  give  for  the  discrep-
ancy, in light of the general theory of relativity?

(b)

(a)

Figure Q39.18

Section 39.1 The Principle of Galilean Relativity

1. A 2 000-kg car moving at 20.0 m/s collides and locks

together with a 1 500-kg car at rest at a stop sign. Show that
momentum is conserved in a reference frame moving at
10.0 m/s in the direction of the moving car.

2. A ball is thrown at 20.0 m/s inside a boxcar moving along

the tracks at 40.0 m/s. What is the speed of the ball

relative to the ground if the ball is thrown (a) forward
(b) backward (c) out the side door?

In  a  laboratory  frame  of  reference,  an  observer  notes
that  Newton’s  second  law  is  valid.  Show  that  it  is  also
valid  for  an  observer  moving  at  a  constant  speed,  small
compared with the speed of light, relative to the labora-
tory frame.

= straightforward, intermediate, challenging

= full solution available in the Student Solutions Manual and Study Guide

= coached solution with hints available at http://www.pse6.com

= computer useful in solving problem

= paired numerical and symbolic problems

P R O B L E M S

1278

C H A P T E R 3 9 • Relativity

Show that Newton’s second law is not valid in a reference
frame moving past the laboratory frame of Problem 3 with
a constant acceleration.

Section 39.2 The Michelson–Morley Experiment
Section 39.3 Einstein’s Principle of Relativity
Section 39.4 Consequences of the Special

Theory of Relativity

5. How fast must a meter stick be moving if its length is

measured to shrink to 0.500 m?

6. At what speed does a clock move if it is measured to run at

a rate that is half the rate of a clock at rest with respect to
an observer?

7. An astronaut is traveling in a space vehicle that has a speed

of 0.500c relative to the Earth. The astronaut measures her
pulse  rate  at  75.0  beats  per  minute.  Signals  generated  by
the astronaut’s pulse are radioed to Earth when the vehicle
is  moving  in  a  direction  perpendicular  to  the  line  that
connects  the  vehicle  with  an  observer  on  the  Earth.
(a) What  pulse  rate  does  the  Earth  observer  measure?
(b) What If? What would be the pulse rate if the speed of
the space vehicle were increased to 0.990c?

An  astronomer  on  Earth  observes  a  meteoroid  in  the
southern sky approaching the Earth at a speed of 0.800c.
At  the  time  of  its  discovery  the  meteoroid  is  20.0 ly  from
the Earth. Calculate (a) the time interval required for the
meteoroid  to  reach  the  Earth  as  measured  by  the  Earth-
bound astronomer, (b) this time interval as measured by a
tourist on the meteoroid, and (c) the distance to the Earth
as measured by the tourist.
An  atomic  clock  moves  at  1 000 km/h  for  1.00 h  as  mea-
sured  by  an  identical  clock  on  the  Earth.  How  many
nanoseconds slow will the moving clock be compared with
the Earth clock, at the end of the 1.00-h interval?

10. A muon formed high in the Earth’s atmosphere travels at

speed v ! 0.990c for a distance of 4.60 km before it
decays into an electron, a neutrino, and an antineutrino
(+

3 &

). (a) How long does the muon live, as

measured in its reference frame? (b) How far does the
Earth travel, as measured in the frame of the muon?

A spacecraft with a proper length of 300 m takes

0.750 +s to pass an Earth observer. Determine the speed of
the spacecraft as measured by the Earth observer.

12.

(a) An object of proper length L

takes a time interval 't

to  pass  an  Earth  observer.  Determine  the  speed  of  the
object as measured by the Earth observer. (b) A column
of tanks, 300 m long, takes 75.0 s to pass a child waiting
at  a  street  corner  on  her  way  to  school.  Determine  the
speed of the armored vehicles. (c) Show that the answer
to part (a) includes the answer to Problem 11 as a special
case,  and  includes  the  answer  to  part  (b)  as  another
special case.

11.

Problem 43 in Chapter 4 can be assigned with this section.

13.

Review problem. In 1963 Mercury astronaut Gordon Cooper
orbited  the  Earth  22  times.  The  press  stated  that  for  each
orbit  he  aged  2  millionths  of  a  second  less  than  he  would
have if he had remained on the Earth. (a) Assuming that he
was  160 km  above  the  Earth  in  a  circular  orbit,  determine
the time difference between someone on the Earth and the
orbiting  astronaut  for  the  22  orbits.  You  will  need  to  use
the approximation

, for small x. (b) Did

the press report accurate information? Explain.

14. For what value of v does * ! 1.010 0? Observe that for

speeds lower than this value, time dilation and length con-
traction are effects amounting to less than 1%.

15. A friend passes by you in a spacecraft traveling at a high

speed. He tells you that his craft is 20.0 m long and that
the identically constructed craft you are sitting in is 19.0 m
long. According to your observations, (a) how long is your
spacecraft, (b) how long is your friend’s craft, and (c) what
is the speed of your friend’s craft?

16.

The  identical  twins  Speedo  and  Goslo  join  a  migration
from the Earth to Planet X. It is 20.0 ly away in a reference
frame in which both planets are at rest. The twins, of the
same age, depart at the same time on different spacecraft.
Speedo’s  craft  travels  steadily  at  0.950c,  and  Goslo’s  at
0.750c.  Calculate  the  age  difference  between  the  twins
after  Goslo’s  spacecraft  lands  on  Planet  X.  Which  twin  is
the older?

17.

An  interstellar  space  probe  is  launched  from  the  Earth.
After  a  brief  period  of  acceleration  it  moves  with  a
constant velocity, with a magnitude of 70.0% of the speed
of light. Its nuclear-powered batteries supply the energy to
keep its data transmitter active continuously. The batteries
have  a  lifetime  of  15.0 yr  as  measured  in  a  rest  frame.
(a) How  long  do  the  batteries  on  the  space  probe  last  as
measured by Mission Control on the Earth? (b) How far is
the  probe  from  the  Earth  when  its  batteries  fail,  as  mea-
sured  by  Mission  Control?  (c)  How  far  is  the  probe  from
the Earth when its batteries fail, as measured by its built-in
trip  odometer?  (d) For  what  total  time  interval  after
launch  are  data  received  from  the  probe  by  Mission
Control? Note that radio waves travel at the speed of light
and fill the space between the probe and the Earth at the
time of battery failure.

18.

Review  problem. An  alien  civilization  occupies  a  brown
dwarf,  nearly  stationary  relative  to  the  Sun,  several
lightyears  away.  The  extraterrestrials  have  come  to  love
original  broadcasts  of  I  Love  Lucy,  on  our  television
channel  2,  at  carrier  frequency  57.0 MHz.  Their  line  of
sight  to  us  is  in  the  plane  of  the  Earth’s  orbit.  Find  the
difference  between  the  highest  and  lowest  frequencies
they  receive  due  to  the  Earth’s  orbital  motion  around
the Sun.

19.

Police  radar  detects  the  speed  of  a  car  (Fig.  P39.19)  as
follows.  Microwaves  of  a  precisely  known  frequency  are
broadcast  toward  the  car.  The  moving  car  reflects  the
microwaves  with  a  Doppler  shift.  The  reflected  waves  are
received and combined with an attenuated version of the
transmitted wave. Beats occur between the two microwave
signals.  The  beat  frequency  is  measured.  (a)  For  an
electromagnetic  wave  reflected  back  to  its  source  from  a
mirror  approaching  at  speed  v,  show  that  the  reflected

√

1 # x

% 1 # x

Problems

1279

wave has frequency

where f

source

is the source frequency. (b) When v is much

less than c, the beat frequency is much smaller than the
transmitted frequency. In this case use the approximation
f & f

source

% 2 f

source

and show that the beat frequency can

be written as f

beat

2v/). (c) What beat frequency is mea-

sured  for  a  car  speed  of  30.0 m/s  if  the  microwaves  have
frequency  10.0 GHz?  (d)  If  the  beat  frequency  measure-
ment  is  accurate  to  % 5 Hz,  how  accurate  is  the  velocity
measurement?

f ! f

source

c & v
c # v

the emission of the same two pulses separated in time by
9.00 +s. (a) How fast is Mark moving relative to Suzanne?
(b) According to Mark, what is the separation in space of
the two pulses?

23.

A moving rod is observed to have a length of 2.00 m and to
be oriented at an angle of 30.0° with respect to the direc-
tion of motion, as shown in Figure P39.23. The rod has a
speed of 0.995c. (a) What is the proper length of the rod?
(b) What is the orientation angle in the proper frame?

Figure P39.19

20.

The red shift. A light source recedes from an observer with a
speed v

source

that is small compared with c. (a) Show that

the fractional shift in the measured wavelength is given by
the approximate expression

This  phenomenon  is  known  as  the  red  shift,  because  the
visible  light  is  shifted  toward  the  red.  (b)  Spectroscopic
measurements  of  light  at  ) ! 397 nm  coming  from  a
galaxy in Ursa Major reveal a red shift of 20.0 nm. What is
the recessional speed of the galaxy?

21. A physicist drives through a stop light. When he is pulled

over, he tells the police officer that the Doppler shift made
the red light of wavelength 650 nm appear green to him,
with a wavelength of 520 nm. The police officer writes out
a traffic citation for speeding. How fast was the physicist
traveling, according to his own testimony?

Section 39.5 The Lorentz Transformation

Equations

22.

Suzanne observes two light pulses to be emitted from the
same location, but separated in time by 3.00 +s. Mark sees

)

source

Direction of motion

30.0

2.00 m

Trent Steffler/David R. Frazier Photo Library

Figure P39.23

Figure P39.26

24.

An observer in reference frame S sees two events as simul-
taneous. Event A occurs at the point (50.0 m, 0, 0) at the
instant 9:00:00 Universal time, 15 January 2004. Event B
occurs at the point (150 m, 0, 0) at the same moment. A
second observer, moving past with a velocity of 0.800c

iˆ,

also observes the two events. In her reference frame S",
which event occurred first and what time interval elapsed
between the events?

25.

A red light flashes at position x

3.00 m and time t

1.00 $ 10

s, and a blue light flashes at x

5.00 m and

9.00 $ 10

s, all measured in the S reference frame.

Reference frame S" has its origin at the same point as S at
t ! t" ! 0;  frame  S" moves  uniformly  to  the  right.  Both
flashes  are  observed  to  occur  at  the  same  place  in  S".
(a) Find the relative speed between S and S". (b) Find the
location  of  the  two  flashes  in  frame  S".  (c)  At  what  time
does the red flash occur in the S" frame?

Section 39.6 The Lorentz Velocity Transformation

Equations

26. A Klingon spacecraft moves away from the Earth at a speed

of 0.800c (Fig. P39.26). The starship Enterprise pursues at a
speed of 0.900c relative to the Earth. Observers on the Earth
see the Enterprise overtaking the Klingon craft at a relative
speed of 0.100c. With what speed is the Enterprise overtaking
the Klingon craft as seen by the crew of the Enterprise?

u = 0.900c

′

v = 0.800c

Two jets of material from the center of a radio galaxy

are ejected in opposite directions. Both jets move at 0.750c

27.

1280

C H A P T E R 3 9 • Relativity

stationary observer? Note that relativistic density is defined
as E

41.

An unstable particle with a mass of 3.34 $ 10

kg is

initially at rest. The particle decays into two fragments that
fly off along the x axis with velocity components 0.987c and
#

0.868c. Find the masses of the fragments. (Suggestion:

Conserve both energy and momentum.)

42.

An object having mass 900 kg and traveling at speed 0.850c
collides with a stationary object having mass 1 400 kg. The
two objects stick together. Find (a) the speed and (b) the
mass of the composite object.
Show that the energy–momentum relationship E

(mc

)

follows from the expressions E ! *mc

and p ! *mu.

44. In a typical color television picture tube, the electrons are

accelerated through a potential difference of 25 000 V.
(a) What speed do the electrons have when they strike the
screen? (b) What is their kinetic energy in joules?

45. Consider electrons accelerated to an energy of 20.0 GeV in

the 3.00-km-long Stanford Linear Accelerator. (a) What is
the * factor for the electrons? (b) What is their speed?
(c) How long does the accelerator appear to them?

46. Compact high-power lasers can produce a 2.00-J light

pulse of duration 100 fs, focused to a spot 1 +m in diame-
ter. (See Mourou and Umstader, “Extreme Light,” Scientific
American,  May  2002,  page  81.)  The  electric  field  in  the
light  accelerates  electrons  in  the  target  material  to  near
the  speed  of  light.  (a)  What  is  the  average  power  of  the
laser  during  the  pulse?  (b)  How  many  electrons  can  be
accelerated  to  0.999 9c if  0.010 0%  of  the  pulse  energy  is
converted into energy of electron motion?
A  pion  at  rest  (m

273m

) decays to a muon (m

207m

) and an antineutrino

. The reaction is

written 2

: +

. Find the kinetic energy of the muon

and the energy of the antineutrino in electron volts. (Sug-
gestion: Conserve both energy and momentum.)

48.

According  to  observer  A,  two  objects  of  equal  mass  and
moving  along  the  x axis  collide  head  on  and  stick  to
each other.  Before  the  collision,  this  observer  measures
that  object  1  moves  to  the  right  with  a  speed  of  3c/4,
while object  2  moves  to  the  left  with  the  same  speed.
According  to  observer  B,  however,  object  1  is  initially  at
rest.  (a)  Determine  the  speed  of  object  2  as  seen  by
observer  B.  (b) Compare  the  total  initial  energy  of  the
system in the two frames of reference.

Section 39.9 Mass and Energy

49. Make an order-of-magnitude estimate of the ratio of mass

increase to the original mass of a flag, as you run it up a
flagpole. In your solution explain what quantities you take
as data and the values you estimate or measure for them.

50. When 1.00 g of hydrogen combines with 8.00 g of oxygen,

9.00 g of water is formed. During this chemical reaction,
2.86 $ 10

J of energy is released. How much mass do the

constituents of this reaction lose? Is the loss of mass likely
to be detectable?

51. In a nuclear power plant the fuel rods last 3 yr before they

are replaced. If a plant with rated thermal power 1.00 GW

% 0)

47.

43.

relative to the galaxy. Determine the speed of one jet
relative to the other.

28.

A spacecraft is launched from the surface of the Earth with a
velocity of 0.600c at an angle of 50.0° above the horizontal
positive x axis. Another spacecraft is moving past, with a
velocity of 0.700c in the negative x direction. Determine the
magnitude and direction of the velocity of the first spacecraft
as measured by the pilot of the second spacecraft.

Section 39.7 Relativistic Linear Momentum

and the Relativistic Form

of Newton’s Laws

29. Calculate the momentum of an electron moving with a

speed of (a) 0.010 0c, (b) 0.500c, and (c) 0.900c.

30.

The nonrelativistic expression for the momentum of a
particle, p ! mu, agrees with experiment if u (( c. For
what speed does the use of this equation give an error in
the momentum of (a) 1.00% and (b) 10.0%?

31.

A golf ball travels with a speed of 90.0 m/s. By what fraction
does its relativistic momentum magnitude p differ from its
classical value mu? That is, find the ratio (p # mu)/mu.

32. Show that the speed of an object having momentum of

magnitude p and mass m is

An unstable particle at rest breaks into two fragments

of unequal mass. The mass of the first fragment is
2.50 $ 10

kg, and that of the other is 1.67 $ 10

kg.

If the lighter fragment has a speed of 0.893c after the
breakup, what is the speed of the heavier fragment?

Section 39.8 Relativistic Energy

34. Determine the energy required to accelerate an electron

from (a) 0.500c to 0.900c and (b) 0.900c to 0.990c.
A proton in a high-energy accelerator moves with a speed
of c/2. Use the work–kinetic energy theorem to find the
work required to increase its speed to (a) 0.750c and
(b) 0.995c.

36. Show that, for any object moving at less than one-tenth the

speed of light, the relativistic kinetic energy agrees with
the result of the classical equation

to within less

than 1%. Thus for most purposes, the classical equation is
good enough to describe these objects, whose motion we
call nonrelativistic.
Find the momentum of a proton in MeV/c units assuming
its total energy is twice its rest energy.

38. Find the kinetic energy of a 78.0-kg spacecraft launched

out of the solar system with speed 106 km/s by using
(a) the classical equation

(b) What If?

Calculate its kinetic energy using the relativistic equation.

A proton moves at 0.950c. Calculate its (a) rest

energy, (b) total energy, and (c) kinetic energy.

40.

A cube of steel has a volume of 1.00 cm

and a mass of

8.00 g when at rest on the Earth. If this cube is now given a
speed u ! 0.900c, what is its density as measured by a

39.

K !

37.

K !

35.

33.

u !

√

1 & (mc/p)

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