Physics For Scientists And Engineers 6E

Problems

1281

Additional Problems

56.

An astronaut wishes to visit the Andromeda galaxy, making
a one-way trip that will take 30.0 yr in the spacecraft’s
frame of reference. Assume that the galaxy is 2.00 $ 10

away  and  that  the  astronaut’s  speed  is  constant.  (a)  How
fast must he travel relative to the Earth? (b) What will be
the  kinetic  energy  of  his  1 000-metric-ton  spacecraft?
(c) What  is  the  cost  of  this  energy  if  it  is  purchased  at  a
typical consumer price for electric energy: $0.130/kWh?

The cosmic rays of highest energy are protons that

have kinetic energy on the order of 10

MeV. (a) How

long would it take a proton of this energy to travel across
the Milky Way galaxy, having a diameter

* 10

ly, as mea-

sured in the proton’s frame? (b) From the point of view of
the proton, how many kilometers across is the galaxy?

58. An electron has a speed of 0.750c. (a) Find the speed of a

proton that has the same kinetic energy as the electron.
(b) What If? Find the speed of a proton that has the same
momentum as the electron.

59.

Ted  and  Mary  are  playing  a  game  of  catch  in  frame  S",
which  is  moving  at  0.600c with  respect  to  frame  S,  while
Jim, at rest in frame S, watches the action (Fig. P39.59).
Ted throws the ball to Mary at 0.800c (according to Ted)
and  their  separation  (measured  in  S")  is  1.80 $ 10

(a) According to Mary, how fast is the ball moving?
(b) According to Mary, how long does it take the ball to
reach her? (c) According to Jim, how far apart are Ted
and Mary, and how fast is the ball moving? (d) According
to Jim, how long does it take the ball to reach Mary?

57.

operates at 80.0% capacity for 3.00 yr, what is the loss of
mass of the fuel?

52.

Review problem. The total volume of water in the oceans
is approximately 1.40 $ 10

. The density of sea

water is 1 030 kg/m

, and the specific heat of the water is

4 186 J/(kg 0 °C). Find the increase in mass of the oceans
produced by an increase in temperature of 10.0°C.
The power output of the Sun is 3.77 $ 10

W. How much

mass is converted to energy in the Sun each second?

54. A gamma ray (a high-energy photon) can produce an

electron (e

) and a positron (e

) when it enters the

electric field of a heavy nucleus: * : e

. What

minimum gamma-ray energy is required to accomplish this
task? (Note: The masses of the electron and the positron
are equal.)

Section 39.10 The General Theory of Relativity

55.

An  Earth  satellite  used  in  the  global  positioning
system moves in a circular orbit with period 11 h 58 min.
(a) Determine  the  radius  of  its  orbit.  (b)  Determine  its
speed.  (c)  The  satellite  contains  an  oscillator  producing
the  principal  nonmilitary  GPS  signal.  Its  frequency  is
1 575.42 MHz in the reference frame of the satellite. When
it is received on the Earth’s surface, what is the fractional
change in this frequency due to time dilation, as described
by special relativity? (d) The gravitational blue shift of the
frequency  according  to  general  relativity  is  a  separate
effect. The magnitude of that fractional change is given by

where 'U

is the change in gravitational potential energy

of  an  object–Earth  system  when  the  object  of  mass  m is
moved  between  the  two  points  at  which  the  signal  is
observed.  Calculate  this  fractional  change  in  frequency.
(e)  What  is  the  overall  fractional  change  in  frequency?
Superposed  on  both  of  these  relativistic  effects  is  a
Doppler shift that is generally much larger. It can be a red
shift or a blue shift, depending on the motion of a particu-
lar satellite relative to a GPS receiver (Fig. P39.55).

53.

Figure P39.55 This global positioning system (GPS) receiver

incorporates relativistically corrected time calculations in its

analysis of signals it receives from orbiting satellites. This allows

the unit to determine its position on the Earth’s surface to within

a few meters. If these corrections were not made, the location

error would be about 1 km.

′

v = 0.600c

′

Ted

1.80

× 10

Mary

Jim

0.800c

Figure P39.59

Photo courtesy of Garmin Ltd.

60. A rechargeable AA battery with a mass of 25.0 g can supply

a power of 1.20 W for 50.0 min. (a) What is the difference
in mass between a charged and an uncharged battery?
(b) What fraction of the total mass is this mass difference?
The net nuclear fusion reaction inside the Sun can be
written as 4

H :

He & 'E. The rest energy of each hydro-

gen atom is 938.78 MeV and the rest energy of the helium-4
atom is 3 728.4 MeV. Calculate the percentage of the starting
mass that is transformed to other forms of energy.

62.

An object disintegrates into two fragments. One of
the fragments has mass 1.00 MeV/c

and momentum

1.75 MeV/c in the positive x direction. The other fragment
has mass 1.50 MeV/c

and momentum 2.00 MeV/c in the

positive y direction. Find (a) the mass and (b) the speed of
the original object.

61.

1282

C H A P T E R 3 9 • Relativity

63.

An  alien  spaceship  traveling  at  0.600c toward  the  Earth
launches a landing craft with an advance guard of purchas-
ing agents and physics teachers. The lander travels in the
same  direction  with  a  speed  of  0.800c relative  to  the
mother  ship.  As  observed  on  the  Earth,  the  spaceship  is
0.200 ly  from  the  Earth  when  the  lander  is  launched.
(a) What  speed  do  the  Earth  observers  measure  for  the
approaching lander? (b) What is the distance to the Earth
at  the  time  of  lander  launch,  as  observed  by  the  aliens?
(c) How long does it take the lander to reach the Earth as
observed  by  the  aliens  on  the  mother  ship?  (d)  If  the
lander  has  a  mass  of  4.00 $ 10

kg, what is its kinetic

energy as observed in the Earth reference frame?

64.

A  physics  professor  on  the  Earth  gives  an  exam  to  her
students,  who  are  in  a  spacecraft  traveling  at  speed  v
relative  to  the  Earth.  The  moment  the  craft  passes  the
professor, she signals the start of the exam. She wishes her
students  to  have  a  time  interval  T

(spacecraft time) to

complete the exam. Show that she should wait a time inter-
val (Earth time) of

before sending a light signal telling them to stop. (Sugges-
tion: Remember  that  it  takes  some  time  for  the  second
light signal to travel from the professor to the students.)
Spacecraft  I,  containing  students  taking  a  physics  exam,
approaches  the  Earth  with  a  speed  of  0.600c  (relative  to
the  Earth),  while  spacecraft  II,  containing  professors
proctoring  the  exam,  moves  at  0.280c (relative  to  the
Earth) directly toward the students. If the professors stop
the  exam  after  50.0 min  have  passed  on  their  clock,  how
long  does  the  exam  last  as  measured  by  (a)  the  students
(b) an observer on the Earth?

66. Energy reaches the upper atmosphere of the Earth from

the Sun at the rate of 1.79 $ 10

W. If all of this energy

were absorbed by the Earth and not re-emitted, how much
would the mass of the Earth increase in 1.00 yr?
A supertrain (proper length 100 m) travels at a speed of
0.950c as it passes through a tunnel (proper length
50.0 m). As seen by a trackside observer, is the train ever
completely within the tunnel? If so, with how much space
to spare?

68.

Imagine that the entire Sun collapses to a sphere of radius
R

such that the work required to remove a small mass m

from the surface would be equal to its rest energy mc

. This

radius is called the gravitational radius for the Sun. Find R

(It is believed that the ultimate fate of very massive stars is to
collapse beyond their gravitational radii into black holes.)
A particle with electric charge q moves along a straight line
in a uniform electric field

E with a speed of u. The electric

force exerted on the charge is q

E. The motion and the

electric field are both in the x direction. (a) Show that the
acceleration of the particle in the x direction is given by

(b) Discuss the significance of the dependence of the
acceleration on the speed. (c) What If? If the particle

a !

1 #

3/2

69.

67.

65.

T ! T

√

1 # v/c

1 & v/c

starts from rest at x ! 0 at t ! 0, how would you proceed
to find the speed of the particle and its position at time t?

70.

An  observer  in  a  coasting  spacecraft  moves  toward  a
mirror  at  speed  v  relative  to  the  reference  frame  labeled
by S in Figure P39.70. The mirror is stationary with respect
to S. A light pulse emitted by the spacecraft travels toward
the mirror and is reflected back to the craft. The front of
the craft is a distance d from the mirror (as measured by
observers  in  S)  at  the  moment  the  light  pulse  leaves  the
craft. What is the total travel time of the pulse as measured
by  observers  in  (a)  the  S  frame  and  (b)  the  front  of  the
spacecraft?

Mirror

v = 0.800c

Figure P39.70

71.

The creation and study of new elementary particles is an
important part of contemporary physics. Especially inter-
esting is the discovery of a very massive particle. To create
a particle of mass M requires an energy Mc

. With enough

energy, an exotic particle can be created by allowing a fast
moving  particle  of  ordinary  matter,  such  as  a  proton,  to
collide  with  a  similar  target  particle.  Let  us  consider  a
perfectly  inelastic  collision  between  two  protons:  an
incident  proton  with  mass  m

, kinetic energy K, and

momentum magnitude p joins with an originally stationary
target proton to form a single product particle of mass M.
You  might  think  that  the  creation  of  a  new  product
particle,  nine  times  more  massive  than  in  a  previous
experiment, would require just nine times more energy for
the  incident  proton.  Unfortunately  not  all  of  the  kinetic
energy  of  the  incoming  proton  is  available  to  create  the
product  particle,  since  conservation  of  momentum
requires that after the collision the system as a whole still
must  have  some  kinetic  energy.  Only  a  fraction  of  the
energy of the incident particle is thus available to create a
new particle. You will determine how the energy available
for particle creation depends on the energy of the moving
proton. Show that the energy available to create a product
particle is given by

From this result, when the kinetic energy K of the incident
proton is large compared to its rest energy m

, we see

that M approaches (2m

1/2

/c. Thus if the energy of the

incoming proton is increased by a factor of nine, the mass
you  can  create  increases  only  by  a  factor  of  three.  This
disappointing result is the main reason that most modern
accelerators, such as those at CERN (in Europe), at Fermi-
lab  (near  Chicago),  at  SLAC  (at  Stanford),  and  at  DESY
(in Germany), use colliding beams. Here the total momen-
tum  of  a  pair  of  interacting  particles  can  be  zero.  The

√

1 &

Answers to Quick Quizzes

1283

center of mass can be at rest after the collision, so in
principle all of the initial kinetic energy can be used for
particle creation, according to

where K is the total kinetic energy of two identical collid-
ing particles. Here if K ,, mc

, we have M directly propor-

tional to K, as we would desire. These machines are
difficult to build and to operate, but they open new vistas
in physics.

72.

A particle of mass m moving along the x axis with a velocity
component & u collides head-on and sticks to a particle of
mass m/3 moving along the x axis with the velocity compo-
nent # u. What is the mass M of the resulting particle?

73.

A rod of length L

moving with a speed v along the hori-

zontal direction makes an angle 4

with respect to the x"

axis. (a) Show that the length of the rod as measured by
a stationary observer is L ! L

[1 # (v

) cos

]

1/2

(b) Show that the angle that the rod makes with the x axis
is given by tan 4 ! * tan 4

. These results show that the

rod is both contracted and rotated. (Take the lower end of
the rod to be at the origin of the primed coordinate
system.)

74.

Suppose  our  Sun  is  about  to  explode.  In  an  effort  to
escape,  we  depart  in  a  spacecraft  at  v ! 0.800c and  head
toward the star Tau Ceti, 12.0 ly away. When we reach the
midpoint  of  our  journey  from  the  Earth,  we  see  our  Sun
explode and, unfortunately, at the same instant we see Tau
Ceti  explode  as  well.  (a)  In  the  spacecraft’s  frame  of
reference,  should  we  conclude  that  the  two  explosions
occurred  simultaneously?  If  not,  which  occurred  first?
(b) What If? In a frame of reference in which the Sun and
Tau  Ceti  are  at  rest,  did  they  explode  simultaneously?  If
not, which exploded first?

75.

Fe nucleus at rest emits a 14.0-keV photon. Use con-

servation of energy and momentum to deduce the
kinetic energy of the recoiling nucleus in electron volts.
(Use Mc

8.60 $ 10

J for the final state of the

nucleus.)

76.

Prepare a graph of the relativistic kinetic energy and

the classical kinetic energy, both as a function of speed, for
an object with a mass of your choice. At what speed does
the classical kinetic energy underestimate the experimen-
tal value by 1%? by 5%? by 50%?

Answers to Quick Quizzes

39.1 (c). While the observers’ measurements differ, both are

correct.

39.2 (d). The Galilean velocity transformation gives us

v ! 110 mi/h & 90 mi/h ! 200 mi/h.

39.3 (d). The two events (the pulse leaving the flashlight and

the pulse hitting the far wall) take place at different loca-
tions for both observers, so neither measures the proper
time interval.

39.4 (a). The two events are the beginning and the end of the

movie, both of which take place at rest with respect to
the spacecraft crew. Thus, the crew measures the proper

2mc

K ! 2mc

1 &

2mc

time interval of 2 h. Any observer in motion with respect
to the spacecraft, which includes the observer on Earth,
will measure a longer time interval due to time dilation.

39.5 (a). If their on-duty time is based on clocks that remain

on the Earth, they will have larger paychecks. A shorter
time interval will have passed for the astronauts in their
frame of reference than for their employer back on the
Earth.

39.6 (c). Both your body and your sleeping cabin are at rest

in your reference frame; thus, they will have their proper
length according to you. There will be no change in
measured lengths of objects, including yourself, within
your spacecraft.

39.7 (d). Time dilation and length contraction depend only

on the relative speed of one observer relative to another,
not on whether the observers are receding or approach-
ing each other.

39.8 (c). Because of your motion toward the source of the

light,  the  light  beam  has  a  horizontal  component  of
velocity  as  measured  by  you.  The  magnitude  of  the
vector  sum  of  the  horizontal  and  vertical  component
vectors  must  be  equal  to  c,  so  the  magnitude  of  the
vertical component must be smaller than c.

39.9 (a). In this case, there is only a horizontal component of

the velocity of the light, and you must measure a speed
of c.

39.10(a) m

; the rest energy of particle 3 is 2E,

while it is E for particles 1 and 2. (b) K

; the

kinetic energy is the difference between the total energy
and the rest energy. The kinetic energy is 4E # 2E ! 2E
for particle 3, 3E # E ! 2E for particle 2, and 2E # E ! E
for particle 1. (c) u

; from Equation 39.26,

E ! *E

. Solving this for the square of the particle speed

u, we find u

(1 # (E

/E )

). Thus, the particle with

the smallest ratio of rest energy to total energy will have
the largest speed. Particles 1 and 3 have the same ratio as
each other, and the ratio of particle 2 is smaller.

2003 by Sidney Harris

A.1

Appendix A • Tables

Mass

slug

1 kilogram

6.852 ! 10

6.024 ! 10

1 gram

6.852 ! 10

6.024 ! 10

1 slug

14.59

1.459 ! 10

8.789 ! 10

1 atomic mass unit

1.660 ! 10

1.137 ! 10

Note: 1 metric ton # 1 000 kg.

Time

min

day

1 second

1.667 ! 10

2.778 ! 10

1.157 ! 10

3.169 ! 10

1 minute

1.667 ! 10

6.994 ! 10

1.901 ! 10

1 hour

3 600

4.167 ! 10

1.141 ! 10

1 day

8.640 ! 10

1 440

2.738 ! 10

1 year

3.156 ! 10

5.259 ! 10

8.766 ! 10

365.2

Speed

m/s

cm/s

ft/s

mi/h

1 meter per second

3.281

2.237

1 centimeter per second

3.281 ! 10

2.237 ! 10

1 foot per second

0.304 8

30.48

0.681 8

1 mile per hour

0.447 0

44.70

1.467

Note: 1 mi/min # 60 mi/h # 88 ft/s.

Force

1 newton

0.224 8

1 pound

4.448

Conversion Factors

Table A.1

Length

in.

1 meter

39.37

3.281

6.214 ! 10

1 centimeter

0.393 7

3.281 ! 10

6.214 ! 10

1 kilometer

3.937 ! 10

3.281 ! 10

0.621 4

1 inch

2.540 ! 10

2.540

2.540 ! 10

8.333 ! 10

1.578 ! 10

1 foot

0.304 8

30.48

3.048 ! 10

1.894 ! 10

1 mile

1 609

1.609 ! 10

1.609

6.336 ! 10

5 280

continued

Physics For Scientists And Engineers 6E - part 321