Physics

John D. Cutnell, Kenneth W. Johnson

Chapter 28

Special Relativity - all with Video Answers

Educators

Chapter Questions

Problem 1

A particle known as a pion lives for a short time before breaking
apart into other particles. Suppose that a pion is moving at a speed of
$0.990 c,$ and an observer who is stationary in a laboratory measures the
pion's lifetime to be $3.5 \times 10^{-8} \mathrm{s}$ . (a) What is the lifetime according to a hypothetical person who is riding along with the pion? (b) According
to this hypothetical person, how far does the laboratory move before the
pion breaks apart?

Netra Sharma

Netra Sharma

University of Wisconsin - Milwaukee

Problem 2

A radar antenna is rotating and makes one revolution every 25 $\mathrm{s}$ , as
measured on earth. However, instruments on a spaceship moving with
respect to the earth at a speed $v$ measure that the antenna makes one
revolution every 42 $\mathrm{s}$ . What is the ratio $v / c$ of the speed $v$ to the speed $c$
of light in a vacuum?

Netra Sharma

University of Wisconsin - Milwaukee

Problem 3

Suppose that you are planning a trip in which a spacecraft is to
travel at a constant velocity for exactly six months, as measured by a
clock on board the spacecraft, and then return home at the same speed.
Upon your return, the people on earth will have advanced exactly one hundred years into the future. According to special relativity, how fast
must you travel? Express your answer to five significant figures as a
multiple of $c-$ for example, 0.95585$c .$

Netra Sharma

University of Wisconsin - Milwaukee

Problem 4

Suppose that you are traveling on board a spacecraft that is
moving with respect to the earth at a speed of 0.975$c$ . You are breathing
at a rate of 8.0 breaths per minute. As monitored on earth, what is your
breathing rate?

Netra Sharma

University of Wisconsin - Milwaukee

Problem 5

A $6.00-\mathrm{kg}$ object oscillates back and forth at the end of a spring
whose spring constant is 76.0 $\mathrm{N} / \mathrm{m}$ . An observer is traveling at a speed
of $1.90 \times 10^{8} \mathrm{m} / \mathrm{s}$ relative to the fixed end of the spring. What does this
observer measure for the period of oscillation?

Netra Sharma

University of Wisconsin - Milwaukee

Problem 6

A spaceship travels at a constant speed from earth to a planet orbiting another star. When the spacecraft arrives, 12 years have elapsed
on earth, and 9.2 years have elapsed on board the ship. How far away
(in meters) is the planet, according to observers on earth?

Netra Sharma

University of Wisconsin - Milwaukee

Problem 7

As observed on earth, a certain type of bacterium is known to double in
number every 24.0 hours. Two cultures of these bacteria are prepared,
each consisting initially of one bacterium. One culture is left on earth and the other placed on a rocket that travels at a speed of 0.866$c$ relative to the
earth. At a time when the earthbound culture has grown to 256 bacteria,
how many bacteria are in the culture on the rocket, according to an earth-
based observer?

Narayan Hari

Numerade Educator

Problem 8

Suppose the straight-line distance between New York and San
Francisco is $4.1 \times 10^{6} \mathrm{m}$ (neglecting the curvature of the earth). A UFO
is flying between these two cities at a speed of 0.70 $\mathrm{c}$ relative to the earth.
What do the voyagers aboard the UFO measure for this distance?

Netra Sharma

University of Wisconsin - Milwaukee

Problem 9

How fast must a meter stick be moving if its length is observed
to shrink to one-half of a meter?

Netra Sharma

University of Wisconsin - Milwaukee

Problem 10

The distance from earth to the center of our galaxy is about 23000 ly $\left(1 \mathrm{ly}=1 \text { light-year }=9.47 \times 10^{15} \mathrm{m}\right),$ as measured by an earth-based observer. A spaceship is to make this journey at a speed of
0.9990$c$ . According to a clock on board the spaceship, how long will it take
to make the trip? Express your answer in years $\left(1 \mathrm{yr}=3.16 \times 10^{7} \mathrm{s}\right)$ .

Netra Sharma

University of Wisconsin - Milwaukee

Problem 11

A tourist is walking at a speed of 1.3 $\mathrm{m} / \mathrm{s}$ along a $9.0-\mathrm{km}$ path
that follows an old canal. If the speed of light in a vacuum were 3.0 $\mathrm{m} / \mathrm{s}$ ,
how long would the path be, according to the tourist?

Netra Sharma

University of Wisconsin - Milwaukee

Problem 12

A Martian leaves Mars in a spaceship that is heading to Venus.
On the way, the spaceship passes earth with a speed $v=0.80 c$ relative
to it. Assume that the three planets do not move relative to each other
during the trip. The distance between Mars and Venus is $1.20 \times 10^{11} \mathrm{m}$ as measured by a person on earth. (a) What does the Martian measure
for the distance between Mars and Venus? (b) What is the time of the
trip (in seconds) as measured by the Martian?

Netra Sharma

University of Wisconsin - Milwaukee

Problem 13

Two spaceships $A$ and $B$ are exploring a new planet. Relative to
this planet, spaceship $A$ has a speed of 0.60$c$ , and spaceship $B$ has a
speed of 0.80$c$ . What is the ratio $D_{A} / D_{B}$ of the values for the planet's
diameter that each spaceship measures in a direction that is parallel to
its motion?

Netra Sharma

University of Wisconsin - Milwaukee

Problem 14

An unstable high-energy particle is created in the laboratory, and it
moves at a speed of 0.990$c .$ Relative to a stationary reference frame
fixed to the laboratory, the particle travels a distance of $1.05 \times 10^{-3} \mathrm{m}$
before disintegrating. What are (a) the proper distance and distance measured by a hypothetical person traveling with the
particle? Determine the particle's (c) proper lifetime and $(\mathrm{d})$ its
dilated lifetime.

Netra Sharma

University of Wisconsin - Milwaukee

Problem 15

As the drawing shows, a carpenter on a space station has constructed a $30.0^{\circ}$ ramp. A rocket moves past the space station with a relative speed of 0.730$c$ in a direction parallel to side $x_{0} .$ What does a person aboard the rocket measure for the angle of the ramp?

Netra Sharma

University of Wisconsin - Milwaukee

Problem 16

An object is made of glass and has the shape of a cube 0.11 $\mathrm{m}$ on a
side, according to an observer at rest relative to it. However, an observer
moving at high speed parallel to one of the object's edges and knowing
that the object's mass is 3.2 $\mathrm{kg}$ determines its density to be 7800 $\mathrm{kg} / \mathrm{m}^{3}$
which is much greater than the density of glass. What is the moving
observer's speed (in units of $c )$ relative to the cube?

Netra Sharma

University of Wisconsin - Milwaukee

Problem 17

A rectangle has the dimensions of 3.0 $\mathrm{m} \times 2.0 \mathrm{m}$ when viewed by
someone at rest with respect to it. When you move past the rectangle
along one of its sides, the rectangle looks like a square. What dimensions
do you observe when you move at the same speed along the adjacent side
of the rectangle?

Netra Sharma

University of Wisconsin - Milwaukee

Problem 18

At what speed is the magnitude of the relativistic momentum of a
particle three times the magnitude of the nonrelativistic momentum?

Netra Sharma

University of Wisconsin - Milwaukee

Problem 19

What is the magnitude of the relativistic momentum of a proton with
a relativistic total energy of $2.7 \times 10^{-10} \mathrm{J} ?$

Netra Sharma

University of Wisconsin - Milwaukee

Problem 20

A spacecraft has a nonrelativistic (or classical) momentum
whose magnitude is $1.3 \times 10^{13} \mathrm{kg} \cdot \mathrm{m} / \mathrm{s}$ . The spacecraft moves at such a
speed that the pilot measures the proper time interval between two events
to be one-half the dilated time interval. Find the relativistic momentum
of the spacecraft.

Netra Sharma

University of Wisconsin - Milwaukee

Problem 21

A woman is 1.6 $\mathrm{m}$ tall and has a mass of 55 kg. She moves past
an observer with the direction of the motion parallel to her height.
The observer measures her relativistic momentum to have a magnitude
of $2.0 \times 10^{10} \mathrm{kg} \cdot \mathrm{m} / \mathrm{s}$ s. What does the observer measure for her height?

Netra Sharma

University of Wisconsin - Milwaukee

Problem 22

Three particles are listed in the table. The mass and speed of
each particle are given as multiples of the variables $m$ and $v,$ which have
the values $m=1.20 \times 10^{-8} \mathrm{kg}$ and $v=0.200 c .$ The speed of light in a
vacuum is $c=3.00 \times 10^{8} \mathrm{m} / \mathrm{s}$ . Determine the momentum for each
particle according to special relativity.

Netra Sharma

University of Wisconsin - Milwaukee

Problem 23

Starting from rest, two skaters push off against each other on
smooth level ice, where friction is negligible. One is a woman and one is
a man. The woman moves away with a velocity of $+2.5 \mathrm{m} / \mathrm{s}$ relative to
the ice. The mass of the woman is $54 \mathrm{kg},$ and the mass of the man is
88 $\mathrm{kg}$ . Assuming that the speed of light is $3.0 \mathrm{m} / \mathrm{s},$ so that the relativistic
momentum must be used, find the recoil velocity of the man relative to
the ice. (Hint: This problem is similar to Example 6 in Chapter 7.)

Netra Sharma

University of Wisconsin - Milwaukee

Problem 24

Radium is a radioactive element whose nucleus emits an $\alpha$ particle
(a helium nucleus) with a kinetic energy of about $7.8 \times 10^{-13} \mathrm{J}$
$(4.9 \mathrm{MeV})$ . To what amount of mass is this energy equivalent?

Netra Sharma

University of Wisconsin - Milwaukee

Problem 25

How much work must be done on an electron to accelerate it
from rest to a speed of 0.990$c ?$

Netra Sharma

University of Wisconsin - Milwaukee

Problem 26

Review Conceptual Example 9 for background pertinent to this
problem. Suppose that the speed of light in a vacuum were one million
times smaller than its actual value, so that $c=3.00 \times 10^{2} \mathrm{m} / \mathrm{s}$ . The spring
constant of a spring is 850 $\mathrm{N} / \mathrm{m}$ . Determine how far you would have to
compress the spring from its equilibrium length in order to increase its
mass by 0.010 $\mathrm{g}$ .

Netra Sharma

University of Wisconsin - Milwaukee

Problem 27

Suppose that one gallon of gasoline produces $1.1 \times 10^{8} \mathrm{J}$ of energy,
and this energy is sufficient to operate a car twenty miles. An aspirin
tablet has a mass of 325 $\mathrm{mg}$ . If the aspirin could be converted completely
into thermal energy, how many miles could the converted completely

Netra Sharma

University of Wisconsin - Milwaukee

Problem 28

Two kilograms of water are changed (a) from ice at $0^{\circ} \mathrm{C}$ into
liquid water at $0^{\circ} \mathrm{C}$ and $\quad$ (b) from liquid water at $100^{\circ} \mathrm{C}$ into steam at
$100^{\circ} \mathrm{C}$ . For each situation, determine the change in mass of the water.

Netra Sharma

University of Wisconsin - Milwaukee

Problem 29

Determine the ratio of the relativistic kinetic energy to the
nonrelativistic kinetic energy $\left(\frac{1}{2} m v^{2}\right)$ when a particle has a speed of
(a) $1.00 \times 10^{-3} c$ and $(\mathbf{b}) 0.970 c .$

Netra Sharma

University of Wisconsin - Milwaukee

Problem 30

Multiple-Concept Example 6 reviews the principles that play a role
in this problem. A nuclear power reactor generates $3.0 \times 10^{9} \mathrm{W}$ of
power. In one year, what is the change in the mass of the nuclear fuel due
to the energy being taken from the reactor?

Netra Sharma

University of Wisconsin - Milwaukee

Problem 31

Multiple-Concept Example 6 explores the approach taken in problems such as this one. Quasars are believed to be the nuclei of galaxies
in the early stages of their formation. Suppose that a quasar radiates
electromagnetic energy at the rate of $1.0 \times 10^{41} \mathrm{W}$ . At what rate (in $\mathrm{kg} / \mathrm{s} )$
is the quasar losing mass as a result of this radiation?

Netra Sharma

University of Wisconsin - Milwaukee

Problem 32

An electron is accelerated from rest through a potential difference that has a magnitude of $2.40 \times 10^{7} \mathrm{V}$ . The mass of the electron is
$9.11 \times 10^{-31} \mathrm{kg}$ , and the negative charge of the electron has a magnitude
of $1.60 \times 10^{-19} \mathrm{C} .$ (a) What is the relativistic kinetic energy (in joules)
of the electron? (b) What is the speed of the electron? Express your
answer as a multiple of $c,$ the speed of light in a vacuum.

Netra Sharma

University of Wisconsin - Milwaukee

Problem 33

An object has a total energy of $5.0 \times 10^{15} \mathrm{J}$ and a kinetic
energy of $2.0 \times 10^{15} \mathrm{J}$ . What is the magnitude of the object's relativistic
momentum?

Netra Sharma

University of Wisconsin - Milwaukee

Problem 34

You are driving down a two-lane country road, and a truck in the
opposite lane is traveling toward you. Suppose that the speed of light in
a vacuum is $c=65 \mathrm{m} / \mathrm{s}$ . Determine the speed of the truck relative to
you when $(\mathrm{a})$ your speed is 25 $\mathrm{m} / \mathrm{s}$ and the truck's speed is 35 $\mathrm{m} / \mathrm{s}$ and
(b) your speed is 5.0 $\mathrm{m} / \mathrm{s}$ and the truck's speed is 55 $\mathrm{m} / \mathrm{s}$ . The speeds
given in parts (a) and (b) are relative to the ground.

Netra Sharma

University of Wisconsin - Milwaukee

Problem 35

A spacecraft approaching the earth launches an exploration
vehicle. After the launch, an observer on earth sees the spacecraft
approaching at a speed of 0.50$c$ and the exploration vehicle approaching
at a speed of 0.70$c$ . What is the speed of the exploration vehicle relative
to the spaceship?

Netra Sharma

University of Wisconsin - Milwaukee

Problem 36

Spaceships of the future may be powered by ion-propulsion
engines in which ions are ejected from the back of the ship to drive it
forward. In one such engine the ions are to be ejected with a speed of
0.80$c$ relative to the spaceship. The spaceship is traveling away from the
earth at a speed of 0.70$c$ relative to the earth. What is the velocity of the ions relative to the earth? Assume that the direction in which the spaceship
is traveling is the positive direction, and be sure to assign the correct plus
or minus signs to the velocities.

Netra Sharma

University of Wisconsin - Milwaukee

Problem 37

The spaceship Enterprise $I$ is moving directly away from earth at a
velocity that an earth-based observer measures to be $+0.65 c$ . A sister
ship, Enterprise $2,$ is ahead of Enterprise $I$ and is also moving directly
away from earth along the same line. The velocity of Enterprise 2 relative
to Enterprise I is $+0.31 c$ . What is the velocity of Enterprise $2,$ as measured
by the earth-based observer?

Netra Sharma

University of Wisconsin - Milwaukee

Problem 38

A person on earth notices a rocket approaching from the right
at a speed of 0.75$c$ and another rocket approaching from the left at 0.65$c .$
What is the relative speed between the two rockets, as measured by a
passenger on one of them?

Netra Sharma

University of Wisconsin - Milwaukee

Problem 39

Refer to Conceptual Example 11 as an aid in solving this
problem. An intergalactic cruiser has two types of guns: a photon cannon
that fires a beam of laser light and an ion gun that shoots ions at a velocity
of 0.950$c$ relative to the cruiser. The cruiser closes in on an alien space-craft at a velocity of 0.800$c$ relative to this spacecraft. The captain fires
both types of guns. At what velocity do the aliens see $(a)$ the laser light
and $(b)$ the ions approach them? At what velocity do the aliens see
(c) the laser light and (d) the ions move away from the cruiser?

Netra Sharma

University of Wisconsin - Milwaukee

Problem 40

Two identical spaceships are under construction. The constructed
length of each spaceship is 1.50 $\mathrm{km}$ . After being launched, spaceship A
moves away from earth at a constant velocity (speed is 0.850 $\mathrm{c} )$ with
respect to the earth. Spaceship $\mathrm{B}$ follows in the same direction at a different
constant velocity (speed is 0.500 $\mathrm{c}$ ) with respect to the earth. Determine
the length that a passenger on one spaceship measures for the other
spaceship.

Suzanne W.

Numerade Educator

Problem 41

Two atomic particles approach each other in a head-on collision.
Each particle has a mass of $2.16 \times 10^{-25} \mathrm{kg}$ . The speed of each particle is
$2.10 \times 10^{8} \mathrm{m} / \mathrm{s}$ when measured by an observer standing in the laboratory.
(a) What is the speed of one particle as seen by the other particle?
(b) Determine the magnitude of the relativistic momentum of one particle,
as it would be observed by the other.

Netra Sharma

University of Wisconsin - Milwaukee

Problem 42

An electron and a positron have masses of $9.11 \times 10^{-31} \mathrm{kg}$ . They
collide and both vanish, with only electromagnetic radiation appearing
after the collision. If each particle is moving at a speed of 0.20 $\mathrm{c}$ relative
to the laboratory before the collision, determine the energy of the
electromagnetic radiation.

Netra Sharma

University of Wisconsin - Milwaukee

Problem 43

A space traveler moving at a speed of 0.70$c$ with respect
to the earth makes a trip to a distant star that is stationary relative to the
earth. He measures the length of this trip to be 6.5 light-years. What
would be the length of this same trip (in light-years) as measured by a
traveler moving at a speed of 0.90$c$ with respect to the earth?

Netra Sharma

University of Wisconsin - Milwaukee

Problem 44

The speed of an ion in a particle accelerator is doubled from 0.460$c$ to 0.920$c .$ The initial relativistic momentum of the ion is
$5.08 \times 10^{-17} \mathrm{kg} \cdot \mathrm{m} / \mathrm{s}$ . Determine $\quad$ (a) the mass and $(\mathrm{b})$ the magnitude
of the final relativistic momentum of the ion.

Netra Sharma

University of Wisconsin - Milwaukee

Problem 45

A Klingon spacecraft has a speed of 0.75$c$ with respect to the
earth. The Klingons measure 37.0 $\mathrm{h}$ for the interval between two
events on the earth. What value for the time interval would they measure
if their ship had a speed of 0.94$c$ with respect to the earth?

Netra Sharma

University of Wisconsin - Milwaukee

Problem 46

An unstable particle is at rest and suddenly breaks up into two fragments. No extemal forces act on the particle or its fragments. One of the
fragments has a velocity of $+0.800 c$ and a mass of $1.67 \times 10^{-27} \mathrm{kg}$ , and the
other has a mass of $5.01 \times 10^{-27} \mathrm{kg}$ . What is the velocity of the more
massive fragment? (Hint: This problem is similar to Example 6 in Chapter 7.)

Netra Sharma

University of Wisconsin - Milwaukee

Problem 47

The crew of a rocket that is moving away from the earth
launches an escape pod, which they measure to be 45 $\mathrm{m}$ long. The pod
is launched toward the earth with a speed of 0.55$c$ relative to
the rocket. After the launch, the rocket's speed relative to the earth is
0.75$c .$ What is the length of the escape pod as determined by an
observer on earth?

Netra Sharma

University of Wisconsin - Milwaukee

Problem 48

The table gives the total energy and the rest energy for three
objects in terms of an energy increment $\epsilon$ . For each object, determine the
speed as a multiple of the speed $c$ of light in a vacuum.

Netra Sharma

University of Wisconsin - Milwaukee

Problem 49

Twins who are 19.0 years of age leave the earth and travel to a
distant planet 12.0 light-years away. Assume that the planet and earth are
at rest with respect to each other. The twins depart at the same time on
different spaceships. One twin travels at a speed of $0.900 c,$ and the other
twin travels at 0.500$c .$ (a) According to the theory of special relativity,
what is the difference between their ages when meet again at the
earliest possible time? (b) Which twin is older?

Suzanne W.

Numerade Educator