CHAPTER 28 SPECIAL RELATIVITY
CONCEPTUAL QUESTIONS
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1.
REASONING AND SOLUTION The speed of light postulate states that the speed of light
in a vacuum, measured in any inertial reference frame, always has the same value of c, no
matter how fast the source of the light and the observer are moving relative to each other.
The speed of light in water is c/n, where n = 1.33 is the refractive index of water. Thus,
the speed of light in water is less than c. This does not violate the speed of light postulate,
because the postulate refers to the speed of light in a vacuum, not in a physical medium.
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2.
REASONING AND SOLUTION A baseball player at home plate hits a pop fly straight up
(the beginning event) that is caught by the catcher at home plate (the ending event).
a. A spectator in the stands is at rest relative to these events; therefore, this spectator would
record the proper time interval between them t0.
b. A spectator sitting on the couch and watching the game on TV is at rest relative to these
events; therefore, this spectator would also record the proper time interval between them t0.
c. The third baseman running in to cover the play is moving relative to the events; therefore,
the third baseman will not record the proper time interval between them. He will record a
dilated time interval t given by the time-dilation equation (Equation 28.1):
t  t0
1  v 2 / c 2 , where v is the relative speed between the observer who measures t0
and the observer who measures t.
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3.
REASONING AND SOLUTION The earth spins on its axis once each day. To a person
viewing the earth from an inertial reference frame in space, a clock at the equator would run
more slowly than a clock at the north (magnetic) pole. The observer in the inertial reference
frame is not in the rest frame of either clock, so he will measure a dilated time for both
clocks. According to the time dilation equation (Equation 28.1): t  t0
1  v2 / c2 ,
where v is the relative speed between the observer who measures t0 and the observer who
measures t. Both clocks move around the earth's rotation axis with the same angular speed
 as that of the earth. The linear speed of each clock is given by v  r , where r is the
distance from the clock to the rotation axis. The clock at the equator has the greater value of
r, and, therefore, the greater linear speed v. According to Equation 28.1, the observer in the
inertial reference frame will see the clock with the larger linear speed v register the longer
time interval t. Therefore, the clock at the equator will appear to run slower when viewed
by the inertial observer.
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Chapter 28 Conceptual Questions
4.
177
REASONING AND SOLUTION
a. You are standing at a railroad crossing watching a train go by. Both you and a passenger
in the train are looking at a clock on the train. The passenger on the train is at rest relative to
the clock; therefore, the passenger on the train measures the proper time interval.
b. A passenger in the train car is at rest relative to the car; therefore, the passenger on the
train measures the proper length of the train car.
c. Since you are standing at the railroad crossing, you are at rest relative to the track;
therefore, you measure the proper distance between the railroad ties under the track.
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5.
REASONING AND SOLUTION There are tables that list data for the various particles of
matter that physicists have discovered. Often, such tables list the masses of the particles in
units of energy, such as MeV, rather than kilograms. This is possible because of the
equivalence between mass and energy. The total energy E of a moving object is related to its
mass m and its speed v by Equation 28.4: E  mc 2 1  v 2 / c 2 . The mass of a particle
listed in the table is actually the total energy of the particle when it is at rest (v = 0 m/s); the
rest energy of an object of mass m is given by Equation 28.5: E0  mc 2 .
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6.
REASONING AND SOLUTION When an object is accelerated from rest to a speed v, the
object acquires kinetic energy in addition to its rest energy. The total energy E of the object
is the sum of its rest energy E0 and its kinetic energy KE. According to Equation 28.6, the
kinetic energy of an object of mass m moving at speed v is
KE  E  E0  mc 2  1 1  v 2 / c 2  1 . The work required to accelerate an object from


rest to speed v is, from the work-energy theorem (Chapter 6), equal to the change in the
kinetic energy of the object. Since the object begins at rest, its change in kinetic energy is
equal to its final kinetic energy and is given by Equation 28.6.
According to Equation 28.6, it is easier to accelerate an electron to a speed that is close to
the speed of light, compared to accelerating a proton to the same speed. This is because the
mass of the electron is much less than that of the proton. Hence, the final kinetic energy of
the proton, will be greater than that of the electron at the same speed v. Therefore, the
proton will require more work to get it to the same speed as the electron.


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7.
8
REASONING AND SOLUTION Light travels in water at a speed of 2.26 10 m/s . It is
possible for a particle that has mass to travel through water at a faster speed than this. But
objects that have mass cannot reach the speed of light in a vacuum, that is, they cannot
attain a speed of c  3.00 10 8 m / s . It is the speed of light in a vacuum that is the ultimate
speed, not the speed of light in a material like water.
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178
8.
SPECIAL RELATIVITY
REASONING AND SOLUTION Two positive, electric charges separated by a finite
distance have more mass than when they are infinitely far apart. The electric potential
energy of the two-particle system is zero when the particles are infinitely far apart. Work is
required to bring two positive charges in from infinity, so that there is a finite separation
between them; therefore, the electric potential energy and consequently the total energy of
the two-particle system increases as they are brought in from infinity. Consistent with the
theory of special relativity, any increase in the total energy of the system, including a change
in the electric potential energy, is equivalent to an increase in the mass of the system.
Therefore, the two-particle system has more mass when the particles have a finite distance
between them, than when they are infinitely far apart.
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9.
REASONING AND SOLUTION One system consists of two stationary electrons separated
by a distance r. Another consists of a positron and an electron, both stationary and separated
by the same distance r. A positron has the same mass as an electron, but it has a positive
electric charge e . Since the charges of unlike sign attract each other, and the charges of
like sign repel each other, more work is required to bring the charges from infinity to
assemble the system that consists of the two stationary electrons. Since more work is
required to assemble the system of two stationary electrons, that system has the greater
electric potential energy. Consistent with mass-energy equivalence predicted by the theory
of special relativity, the system with the larger total energy will have the greater mass.
Therefore, the system that consists of two stationary electrons has the greater mass.
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10. REASONING AND SOLUTION A parallel plate capacitor is initially uncharged. The
capacitor is charged up by removing electrons from one plate and placing them on the other
plate. Work and, therefore, energy is required to charge the capacitor. It is stored in the
electric field between the plates of the charged capacitor in the form of electric potential
energy. Since the charged capacitor has more electric potential energy than the uncharged
capacitor, its also has more total energy than the uncharged capacitor. Therefore, from the
equivalence of mass and energy, we can conclude that the charged capacitor has more mass.
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11. REASONING AND SOLUTION The speed limit on many interstate highways is 65 miles
per hour. If the speed of light were 65 miles per hour, you would never be able to drive at
the speed limit. When the speed, v, of an object is equal to the speed of light, c, the term
1 1  v 2 / c 2 becomes infinitely large. Therefore, according to Equation 28.6, the kinetic
energy of the object would be infinitely large. According to the work-energy theorem, an
infinite amount of work would be required to increase the speed of the object to v = c. Since
an infinite amount of work is not available, it would not be possible for the car to reach the
speed limit.
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Chapter 28 Conceptual Questions
179
12. REASONING AND SOLUTION A person is approaching you in a truck that is traveling
very close to the speed of light. This person throws a baseball toward you. Relative to the
truck, the ball is thrown with a speed nearly equal to the speed of light, so the person on the
truck sees the baseball move away from the truck at a very high speed. Yet you see the
baseball move away from the truck very slowly.
The relative velocities in this problem are:
vBY = velocity of the Baseball relative to You
vBT = velocity of the Baseball relative to the Truck
vTY = velocity of the Truck relative to You
These velocities are related by the velocity-addition formula (Equation 28.8),
vBY 
vBT  vTY
v v
1 + BT 2TY
c
If both vBT and vTY are close to the speed of light, vBY  (c  c) /[1  c 2 / c 2 ]  2c / 2  c ,
and we can conclude that vBY is very close to c. Since both vBY and vTY are measured
relative to the same inertial reference frame, namely your reference frame, you see the
baseball leaving the truck at a relative speed that is the difference between the two speeds,
that is, vBY – vTY. Since vBY and vTY are unequal, but close to the speed of light, the
difference vBY – vTY will be very small. Therefore, you see the baseball move away from
the truck very slowly.
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13. REASONING AND SOLUTION The quantities listed are (a) the time interval between two
events, (b) the length of an object, (c) the speed of light in a vacuum, and (d) the relative
speed between the observers.
a. The time interval between two events depends on the reference frame in which the time
measurements are made. An observer in the rest frame of the two events will measure the
proper time interval. In any other inertial reference frame in motion relative to the rest frame
of the events, the time interval will be longer or dilated. Therefore, two observers in relative
motion will not, in general, agree on the measured value of the time interval between two
events.
b. The length of an object depends on the reference frame in which the length measurement
is made. An observer in the rest frame of the object will measure the proper length of the
object. In any other inertial reference frame in motion relative to the rest frame of the object,
the length will be shorter or contracted in the direction of motion. Therefore, two observers
in relative motion will not, in general, agree on the measured value of the length of the
object.
180
SPECIAL RELATIVITY
c. According to the speed of light postulate, the speed of light in a vacuum will be the same
in all inertial reference frames that are in motion relative to each other. Therefore, two
observers will always measure the speed of light to be the same, regardless of the relative
velocity between them.
d. Since there is no preferred inertial reference frame, two observers in inertial reference
frames that are in motion relative to each other will always measure the same relative speed
between them, regardless of the relative velocity.
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14. REASONING AND SOLUTION If the speed of light were infinitely large, instead of
3 108 m/s , the effects of time dilation and length contraction would not be observable.
According to the time dilation equation (Equation 28.1), t  t0
1  v 2 / c 2 , where v is
the relative speed between the observer who measures t0 and the observer who measures
t. According to the length contraction equation (Equation 28.2), L  L0 1  v 2 / c 2 , where
v is the relative speed between the observer who measures L0 and the observer who
measures L. If c were infinite, then the factor v 2 / c 2 would be equal to zero, 1  v 2 / c 2
would be equal to unity, and it would follow that t  t 0 and L  L0 . In other words, the
time interval for an event would be the same in all inertial reference frames, regardless of the
relative speed between them. Similarly, the length of an object would be the same in all
inertial reference frames, regardless of the relative speed between them. Therefore, if c were
infinitely large, time dilation and length contraction would not be observable.
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