Physics: Principles with Applications

Douglas C. Giancoli

Chapter 26

THE SPECIAL THEORY OF RELATIVITY - all with Video Answers

Educators

Chapter Questions

Problem 1

(I) A spaceship passes you at a speed of 0.850c. You measure its length to be 44.2 m. How long would it be when at rest?

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Problem 2

(I) A certain type of elementary particle travels at a speed of 2.70 $\times$ 10$^8$ m/s. At this speed, the average lifetime is measured to be 4.76 $\times$ 10$^{-6}$ s. What is the particle's lifetime
at rest?

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Problem 3

(II) You travel to a star 135 light-years from Earth at a speed of 2.90 $\times 10^8$ m/s. What do you measure this distance to be?

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Problem 4

(II) What is the speed of a pion if its average lifetime is measured to be 4.40 $\times 10^{-8}$ s? At rest, its average lifetime is 2.60 $\times 10^{-8}$ s.

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Problem 5

(II) In an Earth reference frame, a star is 49 light-years away. How fast would you have to travel so that to you the distance would be only 35 light-years?

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Problem 6

(II) At what speed v will the length of a 1.00-m stick look 10.0% shorter (90.0 cm)?

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Problem 7

(II) At what speed do the relativistic formulas for ($a$) length and ($b$) time intervals differ from classical values by 1.00%? (This is a reasonable way to estimate when to use relativistic calculations rather than classical.)

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Problem 8

(II) You decide to travel to a star 62 light-years from Earth at a speed that tells you the distance is only 25 lightyears. How many years would it take you to make the trip?

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Problem 9

(II) A friend speeds by you in her spacecraft at a speed of 0.720$c$. It is measured in your frame to be 4.80 m long and 1.35 m high. ($a$) What will be its length and height at rest? ($b$) How many seconds elapsed on your friend's watch when 20.0 s passed on yours? ($c$) How fast did you appear to be traveling according to your friend? (d) How many seconds elapsed on your watch when she saw 20.0 s pass on hers?

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Problem 10

(II) A star is 21.6 light-years from Earth. How long would it take a spacecraft traveling 0.950c to reach that star as measured by observers: ($a$) on Earth, ($b$) on the spacecraft? ($c$) What is the distance traveled according to observers on the spacecraft? ($d$) What will the spacecraft occupants
compute their speed to be from the results of ($b$) and ($c$)?

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Problem 11

(II) A fictional news report stated that starship $Enterprise$ had just returned from a 5-year voyage while traveling at 0.70$c$. ($a$) If the report meant 5.0 years of $Earth time$, how
much time elapsed on the ship? ($b$) If the report meant 5.0 years of $ship time$, how much time passed on Earth?

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Problem 12

(II) A box at rest has the shape of a cube 2.6 m on a side. This box is loaded onto the flat floor of a spaceship and the spaceship then flies past us with a horizontal speed of 0.80$c$. What is the volume of the box as we observe it?

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Problem 13

(III) Escape velocity from the Earth is 11.2 km/s. What would be the percent decrease in length of a 68.2-m-long spacecraft traveling at that speed as seen from Earth?

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Problem 14

(III) An unstable particle produced in an accelerator experiment travels at constant velocity, covering 1.00 m in 3.40 ns in the lab frame before changing ("decaying") into other particles. In the rest frame of the particle, determine ($a$) how long it lived before decaying, ($b$) how far it moved before decaying.

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Problem 15

(III) How fast must a pion be moving on average to travel 32 m before it decays? The average lifetime, at rest, is 2.6 $\times 10^{-8}$ s.

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Problem 16

(I) What is the momentum of a proton traveling at $v = 0.68c$?

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Problem 17

(II) ($a$) A particle travels at $v = 0.15c$. By what percentage will a calculation of its momentum be wrong if you use the classical formula? ($b$) Repeat for $v = 0.75c$

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Problem 18

(II) A particle of mass m travels at a speed $v = 0.22c$. At what speed will its momentum be doubled?

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Problem 19

(II) An unstable particle is at rest and suddenly decays into two fragments. No external forces act on the particle or its fragments. One of the fragments has a speed of 0.60$c$ and a mass of 6.68 $\times 10^{-27}$ kg, while the other has a mass of 6.67 $\times 10^{-27}$ kg, What is the speed of the less massive fragment?

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Problem 20

(II) What is the percent change in momentum of a proton that accelerates from ($a$) $0.45c$ to $0.85c$, ($b$) $0.85c$ to $0.98c$?

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Problem 21

(I) Calculate the rest energy of an electron in joules and in
$M$e$V$ $\big($1 $M$e$V =$ 1.60 $\times 10^{-13} J \big)$.

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Problem 22

(I) When a uranium nucleus at rest breaks apart in the process known as $fission$ in a nuclear reactor, the resulting fragments have a total kinetic energy of about 200 $M$e$V$. How much mass was lost in the process?

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Problem 23

(I) The total annual energy consumption in the United States is about 1 $\times 10^{20}J$. How much mass would have to be converted to energy to fuel this need?

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Problem 24

(I) Calculate the mass of a proton (1.67 $\times 10^{-27}$ kg) in $M$e$V/c^2$.

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Problem 25

(I) A certain chemical reaction requires 4.82 $\times 10^4 J$ of energy input for it to go. What is the increase in mass of the products over the reactants?

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Problem 26

(II) Calculate the kinetic energy and momentum of a proton traveling 2.90 $\times 10^8$ m/s.

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Problem 27

(II) What is the momentum of a 950-$M$e$V$ proton (that is, its kinetic energy is 950 $M$e$V$)?

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Problem 28

(II) What is the speed of an electron whose kinetic energy
is 1.12 $M$e$V$?

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Problem 29

(II) ($a$) How much work is required to accelerate a proton from rest up to a speed of 0.985$c$? ($b$) What would be the momentum of this proton?

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Problem 30

(II) At what speed will an object's kinetic energy be 33% of its rest energy?

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Problem 31

(II) Determine the speed and the momentum of an electron ($m = 9.11 \times 10^{-31}$ kg) whose $KE$ equals its rest energy.

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Problem 32

(II) A proton is traveling in an accelerator with a speed of 1.0 $\times 10^8$ m/s.
By what factor does the proton's kinetic energy increase if its speed is doubled?

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Problem 33

(II) How much energy can be obtained from conversion of 1.0 gram of mass? How much mass could this energy raise to a height of 1.0 km above the Earth's surface?

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Problem 34

(II) To accelerate a particle of mass m from rest to speed 0.90$c$ requires work $W_1$. To accelerate the particle from speed 0.90$c$ to 0.99$c$ requires work $W_2$. Determine the ratio $W_2/W_1$.

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Problem 35

(II) Suppose there was a process by which two photons, each with momentum 0.65 $M$e$V/c$, could collide and make a single particle. What is the maximum mass that the particle could possess?

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Problem 36

(II) What is the speed of a proton accelerated by a potential
difference of 165 $MV$?

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Problem 37

(II) What is the speed of an electron after being accelerated
from rest by 31,000 $V$?

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Problem 38

(II) The kinetic energy of a particle is 45 MeV. If the
momentum 121 $M$e$V/c$, is what is the particle's mass?

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Problem 39

(II) Calculate the speed of a proton ($m = 1.67 \times 10^{-27}$ kg) whose kinetic energy is exactly half ($a$) its total energy, ($b$) its rest energy.

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Problem 40

(II) Calculate the kinetic energy and momentum of a proton traveling ($m = 1.67 \times 10^{-27}$ kg) traveling 8.65 $\times 10^7$ m/s. By what percentages would your calculations have been in
error if you had used classical formulas?

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Problem 41

(II) Suppose a spacecraft of mass 17,000 kg is accelerated to 0.15$c$. ($a$) How much kinetic energy would it have? ($b$) If you used the classical formula for kinetic energy, by what percentage would you be in error?

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Problem 42

(II) A negative muon traveling at 53% the speed of light collides head on with a positive muon traveling at 65% the speed of light. The two muons (each of mass 105.7 $M$e$V/c^2$ ) annihilate, and produce how much electromagnetic energy?

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Problem 43

(II) Two identical particles of mass $m$ approach each other at equal and opposite speeds, $v$. The collision is completely inelastic and results in a single particle at rest. What is the mass of the new particle? How much energy was lost in the collision? How much kinetic energy was lost in this collision?

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Problem 44

(III) The americium nucleus, $^{241}_{95}Am$ decays to a neptunium nucleus, $^{237}_{93}Np$ by emitting an alpha particle of mass 4.00260 u and kinetic energy 5.5 $M$e$V$. Estimate the mass
of the neptunium nucleus, ignoring its recoil, given that the americium mass is 241.05682 $u$.

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Problem 45

(III) Show that the kinetic energy $KE$ of a particle of mass $m$ is related to its momentum $p$ by the equation $$p = \sqrt{KE^2 + 2KE\;mc^2}/c.$$

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Problem 46

(III) What magnetic field $B$ is needed to keep 998-$G$e$V$ protons revolving in a circle of radius 1.0 km? Use the relativistic mass. The proton's "rest mass" is 0.938 $G$e$V/c^2$. $\big(1G$e$V = 10^9$ e$V$.) [$Hint$: In relativity, $m_{rel}v^2/r = qvB$ is still valid in a magnetic field, where $m_{rel} = \gamma m$.]

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Problem 47

(I) A person on a rocket traveling at 0.40$c$ (with respect to
the Earth) observes a meteor come from behind and pass
her at a speed she measures as 0.40$c$. How fast is the
meteor moving with respect to the Earth?

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Problem 48

(II) Two spaceships leave Earth in opposite directions, each with a speed of 0.60$c$ with respect to Earth. ($a$) What is the velocity of spaceship 1 relative to spaceship 2? ($b$) What is the velocity of spaceship 2 relative to spaceship 1?

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Problem 49

(II) A spaceship leaves Earth traveling at 0.65$c$. A second spaceship leaves the first at a speed of 0.82$c$ with respect to the first. Calculate the speed of the second ship with respect to Earth if it is fired ($a$) in the same direction the first spaceship is already moving, ($b$) directly backward
toward Earth.

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Problem 50

(II) An observer on Earth sees an alien vessel approach at a speed of 0.60$c$. The fictional starship $Enterprise$ comes to the rescue (Fig. 26$-$13), overtaking the aliens while moving directly toward Earth at a speed of 0.90$c$ relative to Earth. What is the relative speed of one vessel as seen
by the other?

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Problem 51

(II) A spaceship in distress sends out two escape pods in opposite directions. One travels at a speed $v_1 = +0.70c$ in one direction, and the other travels at a speed $v_2 = -0.80c$
in the other direction, as observed from the spaceship. What speed does the first escape pod measure for the second escape pod?

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Problem 52

(II) Rocket $A$ passes Earth at a speed of 0.65$c$. At the same time, rocket $B$ passes Earth moving 0.95$c$ relative to Earth in the same direction as $A$. How fast is $B$ moving relative to $A$ when it passes $A$?

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Problem 53

(II) Your spaceship, traveling at 0.90$c$, needs to launch a probe out the forward hatch so that its speed relative to the planet that you are approaching is 0.95$c$. With what speed must it leave your ship?

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Problem 54

What is the speed of a particle when its kinetic energy equals its rest energy? Does the mass of the particle affect the result?

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Problem 55

The nearest star to Earth is Proxima Centauri, 4.3 lightyears away. ($a$) At what constant velocity must a spacecraft travel from Earth if it is to reach the star in 4.9 years, as measured by travelers on the spacecraft? ($b$) How long does the trip take according to Earth observers?

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Problem 56

According to the special theory of relativity, the factor $\gamma$ that determines the length contraction and the time dilation is given by $\gamma = 1/\sqrt{1 - v^2/c^2}$. Determine the numerical values of $\gamma$ for an object moving at speed $v = 0.01c$, 0.05c, 0.10c, 0.20c, 0.30c, 0.40c, 0.50c, 0.60c, 0.70c, 0.80c, 0.90c, 0.95c, and 0.99c. Make a graph of $\gamma$ versus $v$.

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Problem 57

A healthy astronaut's heart rate is Flight doctors on Earth can monitor an astronaut's vital signs
remotely while in flight. How fast would an astronaut be flying away from Earth if the doctor measured her having a heart rate of 25 beats / min?

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Problem 58

($a$) What is the speed $v$ of an electron whose kinetic energy is 14,000 times its rest energy? You can state the answer as the difference $c - v$. Such speeds are reached in the Stanford Linear Accelerator, SLAC. ($b$) If the electrons travel in the lab through a tube 3.0 km long (as at SLAC), how long is this tube in the electrons' reference frame? [$Hint$: Use the binomial expansion.]

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Problem 59

What minimum amount of electromagnetic energy is needed to produce an electron and a positron together? A positron is a particle with the same mass as an electron, but has the opposite charge. (Note that electric charge is conserved in this process. See Section 27$-$6.)

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Problem 60

How many grams of matter would have to be totally destroyed to run a 75-$W$ lightbulb for 1.0 year?

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Problem 61

A free neutron can decay into a proton, an electron, and a neutrino. Assume the neutrino's mass is zero; the other masses can be found in the Table inside the front cover. Determine the total kinetic energy shared among the three particles when a neutron decays at rest.

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Problem 62

An electron ($m = 9.11 \times 10^{-31}$kg) is accelerated from rest to speed $v$ by a conservative force. In this process, its potential energy decreases by 6.20 $\times 10^{-14} J$. Determine
the electron's speed, $v$.

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Problem 63

The Sun radiates energy at a rate of about 4 $\times 10^{26}W$. ($a$) At what rate is the Sun's mass decreasing? ($b$) How long does it take for the Sun to lose a mass equal to that of Earth? ($c$) Estimate how long the Sun could last if it radiated constantly at this rate.

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Problem 64

How much energy would be required to break a helium nucleus into its constituents, two protons and two neutrons? The masses of a proton (including an electron), a neutron, and neutral helium are, respectively, 1.00783 u, 1.00867 u, and 4.00260 u. (This energy difference is called the $total$ $binding$ $energy$ of the $^4_2He$ nucleus.)

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Problem 65

Show analytically that a particle with momentum $p$ and energy $E$ has a speed given by $$v = {pc^2 \over E} = {pc \over \sqrt{m^2c^2 + p^2}}$$
.

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Problem 66

Two protons, each having a speed of 0.990$c$ in the laboratory, are moving toward each other. Determine ($a$) the momentum of each proton in the laboratory, ($b$) the total momentum of the two protons in the laboratory, and ($c$) the momentum of one proton as seen by the other proton.

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Problem 67

When two moles of hydrogen molecules ($H_2$) and one mole of oxygen molecules ($O_2$) react to form two moles of water ($H_2O$). the energy released is 484 kJ. How much does the mass decrease in this reaction? What % of the total original mass is this?

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Problem 68

The fictional starship $Enterprise$ obtains its power by combining matter and antimatter, achieving complete conversion of mass into energy. If the mass of the $Enterprise$ is approximately 6 $\times 10^9$ kg. how much mass must be converted into kinetic energy to accelerate it from rest to one-tenth the speed of light?

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Problem 69

Make a graph of the kinetic energy versus momentum for ($a$) a particle of nonzero mass, and ($b$) a particle with zero mass.

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Problem 70

A spaceship and its occupants have a total mass of 160,000 kg. The occupants would like to travel to a star that is 35 light-years away at a speed of 0.70$c$. To accelerate, the engine of the spaceship changes mass directly to energy. ($a$) Estimate how much mass will be converted to energy
to accelerate the spaceship to this speed. ($b$) Assuming the acceleration is rapid, so the speed for the entire trip can be taken to be 0.70$c$, determine how long the trip will take according to the astronauts on board.

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Problem 71

In a nuclear reaction two identical particles are created, traveling in opposite directions. If the speed of each particle is 0.82$c$, relative to the laboratory frame of reference, what is one particle's speed relative to the other particle?

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Problem 72

A 36,000-kg spaceship is to travel to the vicinity of a star 6.6 light-years from Earth. Passengers on the ship want the (one-way) trip to take no more than 1.0 year. How much work must be done on the spaceship to bring it to the speed necessary for this trip?

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Problem 73

Suppose a 14,500-kg spaceship left Earth at a speed of 0.90c. What is the spaceship's kinetic energy? Compare with the total U.S. annual energy consumption (about $10^{20} J$ ).

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Problem 74

A pi meson of mass $m_\pi$ decays at rest into a muon (mass $m_\mu$ ) and a neutrino of negligible or zero mass. Show that the kinetic energy of the muon is $KE_\mu = (m_\mu - m_\mu)^2c^2/(2m_\pi$).

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Problem 75

An astronaut on a spaceship traveling at 0.75$c$ relative to Earth measures his ship to be 23 m long. On the ship, he eats his lunch in 28 min. ($a$) What length is the spaceship according to observers on Earth? ($b$) How long does the astronaut's lunch take to eat according to observers on Earth?

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Problem 76

Astronomers measure the distance to a particular star to be 6.0 light-years (1ly = distance light travels in 1 year). A spaceship travels from Earth to the vicinity of this star at steady speed, arriving in 3.50 years as measured by clocks on the spaceship. ($a$) How long does the trip take as measured by clocks in Earth's reference frame? ($b$) What distance does the spaceship travel as measured in its own reference frame?

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Problem 77

An electron is accelerated so that its kinetic energy is greater than its rest energy $mc^2$ by a factor of ($a$) 5.00, ($b$) 999. What is the speed of the electron in each case?

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Problem 78

You are traveling in a spaceship at a speed of 0.70$c$ away from Earth. You send a laser beam toward the Earth traveling at velocity c relative to you. What do observers on the Earth measure for the speed of the laser beam?

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Problem 79

A farm boy studying physics believes that he can fit a 13.0-m-long pole into a 10.0-m-long barn if he runs fast enough, carrying the pole. Can he do it? Explain in detail. How does this fit with the idea that when he is running the barn looks even shorter than 10.0 m?

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Problem 80

An atomic clock is taken to the North Pole, while another stays at the Equator. How far will they be out of synchronization after 2.0 years has elapsed? [$Hint$: Use the binomial expansion, Appendix $A$.]

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Problem 81

An airplane travels 1300 km/h around the Earth in a circle of radius essentially equal to that of the Earth, returning to the same place. Using special relativity, estimate the difference in time to make the trip as seen by Earth and by airplane observers. [$Hint$: Use the binomial expansion, Appendix $A$.]

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