College Physics With an Integrated Approach to Forces and Kinematics

Alan Giambattista, Betty McCarthy Richardson , Robert C. Richardson

Chapter 26

Relativity - all with Video Answers

Educators

Chapter Questions

Problem 1

An engineer in a train moving toward the station with a velocity $v=0.60 c$ lights a signal flare as he reaches a marker $1.0 \mathrm{~km}$ from the station (according to a scale laid out on the ground). By how much time, on the stationmaster's clock, does the arrival of the optical signal precede the arrival of the train?

Dading Chen

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Problem 2

The light-second is a unit of distance; 1 light-second is the distance that light travels in 1 second. (a) Find the conversion between light-seconds and meters: 1 lightsecond $=? \mathrm{~m}$. (b) What is the speed of light in units of light-seconds per second?

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Problem 3

A spaceship traveling at speed $0.13 c$ away from Earth sends a radio transmission to Earth. (a) According to Galilean relativity, at what speed would the transmission travel relative to Earth? (b) Using Einstein's postulates, at what speed does the transmission travel relative to Earth?

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Problem 4

Event A happens at the spacetime coordinates $(x, y, z, t)=(2 \mathrm{~m}, 3 \mathrm{~m}, 0,0.1 \mathrm{~s})$ and event $\mathrm{B}$ happens at
the spacetime coordinates $(x, y, z, t)=\left(0.4 \times 10^{8} \mathrm{~m},\right.$, $3 \mathrm{~m}, 0,0.2 \mathrm{~s}) .$ (a) Is it possible that event A caused event B? (b) If event B occurred at $\left(0.2 \times 10^{8} \mathrm{~m}, 3 \mathrm{~m}, 0,0.2 \mathrm{~s}\right)$
instead, would it then be possible that event A caused event B? [Hint: How fast would a signal need to travel to get from event $\mathrm{A}$ to the location of $\mathrm{B}$ before event $\mathrm{B}$ occurred?]

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Problem 5

An astronaut wears a new Rolex watch on a journey at a speed of $2.0 \times 10^{8} \mathrm{~m} / \mathrm{s}$ with respect to Earth. According to mission control in Houston, the trip lasts $12.0 \mathrm{~h}$. How long is the trip as measured on the Rolex?

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Problem 6

An unstable particle called the pion has a mean lifetime of $25 \mathrm{~ns}$ in its own rest frame. A beam of pions travels through the laboratory at a speed of $0.60 c$. (a) What is the mean lifetime of the pions as measured in the laboratory frame? (b) How far does a pion travel (as measured by laboratory observers) during this time?

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Problem 7

Suppose your handheld calculator will show six places beyond the decimal point. How fast (in meters per second) would an object have to be traveling so that the decimal places in the value of $\gamma$ can actually be seen on your calculator display? That is, how fast should an object travel so that $\gamma=1.000001 ?$ [Hint: Use the binomial approximation.]

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Problem 8

A spaceship is traveling away from Earth at $0.87 \mathrm{c}$. The astronauts report home by radio every $12 \mathrm{~h}$ (by their own clocks). At what interval are the reports sent to Earth, according to Earth clocks?

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Problem 9

A spaceship travels at constant velocity from Earth to a point 710 ly away as measured in Earth's rest frame. The ship's speed relative to Earth is $0.9999 \mathrm{c}$. A passenger is 20 yr old when departing from Earth. (a) How old is the passenger when the ship reaches its destination, as measured by the ship's clock? (b) If the spaceship sends a radio signal back to Earth as soon as it reaches its destination, in what year, by Earth's calendar, does the signal reach Earth? The spaceship left Earth in the year 2000 .

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Problem 10

A clock moves at a constant velocity of $8.0 \mathrm{~km} / \mathrm{s}$ with respect to Earth. If the clock ticks at intervals of one second in its rest frame, how much more than a second elapses between ticks of the clock as measured by an observer at rest on Earth? [Hint: Use the binomial approximation $\left.\left(1-v^{2} / c^{2}\right)^{-1 / 2} \approx 1+v^{2} /\left(2 c^{2}\right) .\right]$

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Problem 11

A plane trip lasts $8.0 \mathrm{~h}$; the average speed of the plane during the flight relative to Earth is $220 \mathrm{~m} / \mathrm{s}$. What is the time difference between an atomic clock on board the plane and one on the ground, assuming they were synchronized before the flight? (Ignore general relativistic complications due to gravity and the acceleration of the plane.)

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Problem 12

Fill in the missing algebraic steps in the derivation of the time dilation equation [Eq. (26-1)].

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Problem 13

A spaceship travels toward Earth at a speed of $0.97 c$. The occupants of the ship are standing with their torsos parallel to the direction of travel. According to Earth observers, they are about $0.50 \mathrm{~m}$ tall and $0.50 \mathrm{~m}$ wide. What are the occupants' (a) height and (b) width according to others on the spaceship?

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Problem 14

While the spaceship in Problem 13 continues to travel in the same direction, one of the occupants lies on his side, so that now his torso is perpendicular to the direction of travel and his width is parallel to the travel direction. What are the (a) height and (b) width of this occupant according to an Earth observer?

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Problem 15

A cosmic ray particle travels directly over a football field, from one goal line to the other, at a speed of $0.50 c$.
(a) If the length of the field between goal lines in the Earth frame is $91.5 \mathrm{~m}(100 \mathrm{yd})$, what length is measured in the rest frame of the particle? (b) How long does it take the particle to go from one goal line to the other according to Earth observers? (c) How long does it take in the rest frame of the particle?

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Problem 16

A laboratory measurement of the coordinates of the ends of a moving meterstick, taken at the same time in the laboratory, gives the result that one end of the stick is $0.992 \mathrm{~m}$ due north of the other end. If the stick is moving due north, what is its speed with respect to the lab?

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Problem 17

Two spaceships are moving directly toward one another with a relative velocity of $0.90 c$. If an astronaut measures the length of his own spaceship to be $30.0 \mathrm{~m}$, how long is the spaceship as measured by an astronaut in the other ship?

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Problem 18

A spaceship is moving at a constant velocity of $0.70 c$ relative to an Earth observer. The Earth observer measures the length of the spaceship to be $40.0 \mathrm{~m}$. How long is the spaceship as measured by its pilot?

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Problem 19

A spaceship moves at a constant velocity of $0.40 c$ relative to an Earth observer. The pilot of the spaceship is holding a rod, which he measures to be $1.0 \mathrm{~m}$ long.
(a) The rod is held perpendicular to the direction of motion of the spaceship. How long is the rod according to the Earth observer? (b) After the pilot rotates the rod and holds it parallel to the direction of motion of the spaceship, how long is it according to the Earth observer?

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Problem 20

A rectangular plate of glass, measured at rest, has sides $30.0 \mathrm{~cm}$ and $60.0 \mathrm{~cm}$. (a) As measured in a reference frame moving parallel to the $60.0-\mathrm{cm}$ edge at speed $0.25 c$ with respect to the glass, what are the lengths of the sides? (b) How fast would a reference frame have to move in the same direction so that the plate of glass viewed in that frame is square?

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Problem 21

A futuristic train moving in a straight line with a uniform speed of $0.80 c$ passes a series of communications towers. The spacing between the towers, according to an observer on the ground, is $3.0 \mathrm{~km}$. A passenger on the train uses an accurate stopwatch to see how often a tower passes him. (a) What is the time interval the passenger measures between the passing of one tower and the next? (b) What is the time interval an observer on the ground measures for the train to pass from one tower to the next?

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Problem 22

An astronaut in a rocket moving at $0.50 c$ toward the Sun finds himself halfway between Earth and the Sun. According to the astronaut, how far is he from Earth? In the frame of the Sun, the distance from Earth to the Sun is $1.50 \times 10^{11} \mathrm{~m}$.

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Problem 23

The mean (average) lifetime of a muon in its rest frame is $2.2 \mu$ s. A beam of muons is moving through the lab with speed $0.994 c$. How far on average does a muon travel through the lab before it decays?

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Problem 24

The Tevatron is a particle accelerator at Fermilab that accelerates protons and antiprotons to high energies in an underground ring. Scientists observe the results of collisions between the particles. The protons are accelerated until they have speeds only $100 \mathrm{~m} / \mathrm{s}$ slower than the speed of light. The circumference of the ring is $6.3 \mathrm{~km}$. What is the circumference according to an observer moving with the protons? [Hint: Let $v=c-u$ where $v$ is the proton speed and $u=100 \mathrm{~m} / \mathrm{s}$.]

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Problem 25

Kurt is measuring the speed of light in an evacuated chamber aboard a spaceship traveling with a constant velocity of $0.60 c$ with respect to Earth. The light is moving in the direction of motion of the spaceship. SiuLing is on Earth watching the experiment. With what speed does the light in the vacuum chamber travel, according to Siu-Ling's observations?

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Problem 26

The rogue starship Galaxa is being chased by the battlecruiser Millenia. The Millenia is catching up to the Galaxa at a rate of $0.55 c$ when the captain of the Millenia decides it is time to fire a missile. First the captain shines a laser range finder to determine the distance to the Galaxa and then he fires a missile that is moving at a speed of $0.45 c$ with respect to the Millenia. What speed does the Galaxa measure for (a) the laser beam and (b) the missile as they both approach the starship?

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Problem 27

Particle $A$ is moving with a constant velocity $v_{\mathrm{AE}}=+0.90 c$ relative to an Earth observer. Particle $B$ moves in the opposite direction with a constant velocity $v_{\mathrm{BE}}=-0.90 c$ relative to the same Earth observer. What is the velocity of particle $B$ as seen by particle $A$ ?

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Problem 28

A man on the Moon observes two spaceships coming toward him from opposite directions at speeds of $0.60 c$ and $0.80 c$. What is the relative speed of the two ships as measured by a passenger on either one of the spaceships?

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Problem 29

Rocket ship Able travels at $0.400 c$ relative to an Earth observer. According to the same observer, rocket ship Able overtakes a slower moving rocket ship Baker that moves in the same direction. The captain of Baker sees Able pass her ship at $0.114 c$. Determine the speed of Baker relative to the Earth observer.

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Problem 30

Relative to the laboratory, a proton moves to the right with a speed of $\frac{4}{5} c$, while relative to the proton, an electron moves to the left with a speed of $\frac{5}{7} c .$ What is the speed of the electron relative to the lab?

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Problem 31

As observed from Earth, rocket Alpha moves with speed $0.90 c$ and rocket Bravo travels with a speed of $0.60 c$. They are moving along the same line toward a head-on collision. What is the speed of rocket Alpha as measured from rocket Bravo?

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Problem 32

Electron A is moving west with speed $\frac{3}{5} c$ relative to the lab. Electron $\mathrm{B}$ is also moving west with speed $\frac{4}{5} c$ relative to the lab. What is the speed of electron $\mathrm{B}$ in a frame of reference in which electron A is at rest?

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Problem 33

A proton moves at $0.90 c$. What is its momentum in SI units?

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Problem 34

An electron has momentum of magnitude $2.4 \times 10^{-22} \mathrm{~kg} \cdot \mathrm{m} / \mathrm{s}$. What is the electron's speed?

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Problem 35

The fastest vehicle flown by humans on a regular basis is the Space Shuttle which has a mass of about $1 \times 10^{5} \mathrm{~kg}$ and travels at a speed of about $8 \times 10^{3} \mathrm{~m} / \mathrm{s}$. Find the momentum of the Space Shuttle using relativistic formulae and then again by using nonrelativistic formulae. By what percent do the relativistic and nonrelativistic momenta of the Space Shuttle differ? [Hint: You might want to use one of the approximations in Appendix A.5.]

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Problem 36

A body has a mass of $12.6 \mathrm{~kg}$ and a speed of $0.87 c$.
(a) What is the magnitude of its momentum? (b) If a constant force of $424.6 \mathrm{~N}$ acts in the direction opposite to the body's motion, how long must the force act to bring the body to rest?

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Problem 37

A constant force, acting for $3.6 \times 10^{4} \mathrm{~s}(10 \mathrm{~h})$, brings a spaceship of mass $2200 \mathrm{~kg}$ from rest to speed $0.70 c$.
(a) What is the magnitude of the force? (b) What is the initial acceleration of the spaceship? Comment on the magnitude of the answer.

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Problem 38

How much energy is released by a nuclear reactor if the total mass of the fuel decreases by $1.0 \mathrm{~g}$ ?

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Problem 39

Two lumps of putty are moving in opposite directions, each one having a speed of $30.0 \mathrm{~m} / \mathrm{s}$. They collide and stick together. After the collision the combined lumps are at rest. If the mass of each lump was $1.00 \mathrm{~kg}$ before the collision, and no energy is lost to the environment, what is the change in mass of the system due to the collision? ( $\mathbf{h}$ tutorial: colliding bullets)

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Problem 40

The solar energy arriving at the outer edge of Earth's atmosphere from the Sun has intensity $1.4 \mathrm{~kW} / \mathrm{m}^{2}$.
(a) How much mass does the Sun lose per day? (b) What percent of the Sun's mass is this?

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Problem 41

A white dwarf is a star that has exhausted its nuclear fuel and lost its outer mass so that it consists only of its dense, hot inner core. It will cool unless it gains mass from some nearby star. It can form a binary system with such a star and gradually gain mass up to the limit of $1.4$ times the mass of the Sun. If the white dwarf were to start to exceed the limit, it would explode into a supernova. How much energy is released by the explosion of a white dwarf at its limiting mass if $80.0 \%$ of its mass is converted to energy?

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Problem 42

A lambda hyperon $\Lambda^{0}\left(\right.$ mass $\left.=1115 \mathrm{MeV} / c^{2}\right)$ at rest decays into a neutron $\mathrm{n}\left(\mathrm{mass}=940 \mathrm{MeV} / c^{2}\right)$ and a pion $\left(\right.$ mass $\left.=135 \mathrm{MeV} / c^{2}\right)$
$$
\Lambda^{0} \rightarrow \mathrm{n}+\pi^{0}
$$
What is the total kinetic energy of the neutron and pion?

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Problem 43

Radon decays as follows: ${ }^{222} \mathrm{Rn} \rightarrow{ }^{218} \mathrm{Po}+\alpha$. The mass of the radon-222 nucleus is $221.97039 \mathrm{u}$, the mass of the polonium- 218 nucleus is $217.96289 \mathrm{u}$, and the mass of the alpha particle is $4.00151 \mathrm{u}$. How much energy is released in the decay? $\left(1 \mathrm{u}=931.494 \mathrm{MeV} / \mathrm{c}^{2} .\right)$

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Problem 44

The energy to accelerate a starship comes from combining matter and antimatter. When this is done the total rest energy of the matter and antimatter is converted to other forms of energy. Suppose a starship with a mass of $2.0 \times 10^{5} \mathrm{~kg}$ accelerates to $0.3500 c$ from rest. How much matter and antimatter must be converted to kinetic energy for this to occur?

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Problem 45

A laboratory observer measures an electron's energy to be $1.02 \times 10^{-13} \mathrm{~J}$. What is the electron's speed?

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Problem 46

A muon with rest energy $106 \mathrm{MeV}$ is created at an altitude of $4500 \mathrm{~m}$ and travels downward toward Earth's surface. An observer on Earth measures its speed as $0.980 \mathrm{c}$. (a) What is the muon's total energy in the Earth observer's frame of reference? (b) What is the muon's total energy in the muon's frame of reference?

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Problem 47

An object of mass $0.12 \mathrm{~kg}$ is moving at $1.80 \times 10^{8} \mathrm{~m} / \mathrm{s}$. What is its kinetic energy in joules?

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Problem 48

When an electron travels at $0.60 c$, what is its total energy in mega-electron-volts?

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Problem 49

An observer in the laboratory finds that an electron's total energy is $5.0 \mathrm{mc}^{2}$. What is the magnitude of the electron's momentum (as a multiple of $m c$ ), as observed in the laboratory?

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Problem 50

The rest energy of an electron is $0.511 \mathrm{MeV}$. What momentum (in $\mathrm{MeV} / \mathrm{c}$ ) must an electron have in order that its total energy be $3.00$ times its rest energy?

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Problem 51

An electron has a total energy of $6.5 \mathrm{MeV}$. What is its momentum (in $\mathrm{MeV} / c)$ ?

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Problem 52

For a nonrelativistic particle of mass $m$, show that $K=p^{2} /(2 m) .$ [Hint: Start with the nonrelativistic expressions for kinetic energy $K$ and momentum $p .]$

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Problem 53

Find the conversion between the momentum unit MeV/c and the SI unit of momentum.

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Problem 54

Find the conversion between the mass unit $\mathrm{MeV} / \mathrm{c}^{2}$ and the SI unit of mass.

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Problem 55

In a beam of electrons used in a diffraction experiment, each electron is accelerated to a kinetic energy of $150 \mathrm{keV}$.
(a) Are the electrons relativistic? Explain.
(b) How fast are the electrons moving?

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Problem 56

Derive the energy-momentum relation
$$
E^{2}=E_{0}^{2}+(p c)^{2}
$$
Start by squaring the definition of total energy $\left(E=K+E_{0}\right)$ and then use the relativistic expressions for momentum and kinetic energy [Eqs. (26-6) and (26-8)].

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Problem 57

Starting with the energy-momentum relation $E^{2}=E_{0}^{2}+$ $(p c)^{2}$ and the definition of total energy, show that $(p c)^{2}=K^{2}+2 K E_{0}[$ Eq. $(26-11)] .$

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Problem 58

Show that Eq. (26-11) reduces to the nonrelativistic relationship between momentum and kinetic energy, $K \approx p^{2} /(2 m)$, for $K \ll E_{0}$

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Problem 59

Show that each of these statements implies that $v \ll c$, which means that $v$ can be considered a nonrelativistic speed: (a) $\gamma-1 \ll 1$ [Eq. (26-14)]; (b) $K \ll m c^{2}$ [Eq. $(26-15)] ;$ (c) $p \ll m c$ [Eq. (26-16)]; (d) $K \approx p^{2} /(2 m)$.

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Problem 60

Octavio, traveling at a speed of $0.60 c$, passes Tracy and her barn. Tracy, who is at rest with respect to her barn, says that the barn is $16 \mathrm{~m}$ long in the direction in which Octavio is traveling, $4.5 \mathrm{~m}$ high, and $12 \mathrm{~m}$ deep.
(a) What does Tracy say is the volume of her barn?
(b) What volume does Octavio measure?

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Problem 61

A spaceship resting on Earth has a length of $35.2 \mathrm{~m}$. As it departs on a trip to another planet, it has a length of $30.5 \mathrm{~m}$ as measured by the Earthbound observers. The Earthbound observers also notice that one of the astronauts on the spaceship exercises for $22.2$ min. How long would the astronaut herself say that she exercises?

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Problem 62

At the $10.0$ -km-long Stanford Linear Accelerator, electrons with rest energy of $0.511 \mathrm{MeV}$ have been accelerated to a total energy of $46 \mathrm{GeV}$. How long is the accelerator as measured in the reference frame of the
electrons?

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Problem 63

Consider the following decay process: $\pi^{+} \rightarrow \mu^{+}+v .$ The mass of the pion $\left(\pi^{+}\right)$ is $139.6 \mathrm{MeV} / c^{2}$, the mass of the muon $\left(\mu^{+}\right)$ is $105.7 \mathrm{MeV} / c^{2}$, and the mass of the neutrino $(v)$ is negligible. If the pion is initially at rest, what is the total kinetic energy of the decay products?

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Problem 64

A neutron (mass $1.00866 \mathrm{u}$ ) disintegrates into a proton (mass $1.00728 \mathrm{u})$, an electron (mass $0.00055 \mathrm{u}$ ), and an antineutrino (mass 0). What is the sum of the kinetic energies of the particles produced, if the neutron is at rest? $\left(1 \mathrm{u}=931.5 \mathrm{MeV} / c^{2} .\right)$

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Problem 65

A starship takes $3.0$ days to travel between two distant space stations according to its own clocks. Instruments on one of the space stations indicate that the trip took $4.0$ days. How fast did the starship travel, relative to that space station?

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Problem 66

Two spaceships are observed from Earth to be approaching each other along a straight line. Ship A moves at $0.40 c$ relative to the Earth observer, and ship B moves at $0.50 c$ relative to the same observer. What speed does the captain of ship A report for the speed of ship B?

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Problem 67

A neutron, with rest energy $939.6 \mathrm{MeV}$, has momentum $935 \mathrm{MeV} / \mathrm{c}$ downward. What is its total energy?

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Problem 68

Suppose that as you travel away from Earth in a spaceship, you observe another ship pass you heading in the same direction and measure its speed to be $0.50 c .$ As you look back at Earth, you measure Earth's speed relative to you to be $0.90 c$. What is the speed of the ship that passed you according to Earth observers?

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Problem 69

(a) If you measure the ship that passes you in Problem 68 to be $24 \mathrm{~m}$ long, how long will the observers on Earth measure that ship to be? (b) If there is a rod on your spaceship that you measure to be $24 \mathrm{~m}$ long, how long will the observers on Earth measure your rod to be? (c) How long do the observers on the passing ship measure your rod to be?

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Problem 70

Verify that the collision between the proton and the nitrogen nucleus in Example $26.4$ is elastic.

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Problem 71

Muons are created by cosmic-ray collisions at an elevation $h$ (as measured in Earth's frame of reference) above Earth's surface and travel downward with a constant speed of $0.990 c$. During a time interval of $1.5 \mu \mathrm{s}$ in the rest frame of the muons, half of the muons present at the beginning of the interval decay. If one fourth of the original muons reach Earth before decaying, about how big is the height $h$ ?

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Problem 72

Refer to Example 26.1. Ashlin travels at speed $0.800 c$ to a star $30.0$ ly from Earth.
(a) Find the distance between Earth and the star in the astronaut's frame of reference.
(b) How long (as measured by the astronaut) does it take to travel this distance at a speed of $0.800 c$ ? Compare your answer to the result of Example $26.1$ and explain any discrepancy.

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Problem 73

An extremely relativistic particle is one whose kinetic energy is much larger than its rest energy. Show that for an extremely relativistic particle $E \approx p c$.

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Problem 74

A spaceship is moving away from Earth with a constant velocity of $0.80 c$ with respect to Earth. The spaceship and an Earth station synchronize their clocks, setting both to zero, at an instant when the ship is near Earth. By prearrangement, when the clock on Earth reaches a reading of $1.0 \times 10^{4} \mathrm{~s}$, the Earth station sends out a light signal to the spaceship. (a) In the frame of reference of the Earth station, how far must the signal travel to reach the spaceship? (b) According to an Earth observer, what is the reading of the clock on Earth when the signal is received?

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Problem 75

Refer to Example 26.2. One million muons are moving toward the ground at speed $0.9950 c$ from an altitude of $4500 \mathrm{~m}$. In the frame of reference of an observer on the ground, what are (a) the distance traveled by the muons;
(b) the time of flight of the muons; (c) the time interval during which half of the muons decay; and (d) the number of muons that survive to reach sea level. [Hint: The answers to (a) to (c) are not the same as the corresponding quantities in the muons' reference frame. Is the answer to (d) the same?]

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Problem 76

A particle decays in flight into two pions, each having a rest energy of $140.0 \mathrm{MeV}$. The pions travel at right angles to each other with equal speeds of $0.900 c$. Find (a) the momentum magnitude of the original particle, (b) its kinetic energy, and (c) its mass in units of $\mathrm{MeV} / c^{2}$.

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Problem 77

A charged particle is observed to have a total energy of $0.638 \mathrm{MeV}$ when it is moving at $0.600 c$. If this particle enters a linear accelerator and its speed is increased to $0.980 c$, what is the new value of the particle's total energy?

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Problem 78

A cosmic-ray proton entering the atmosphere from space has a kinetic energy of $2.0 \times 10^{20} \mathrm{eV}$. (a) What is its kinetic energy in joules? (b) If all of the kinetic energy of the proton could be harnessed to lift an object of mass $1.0 \mathrm{~kg}$ near Earth's surface, how far could the object be lifted? (c) What is the speed of the proton? Hint: $v$ is so close to $c$ that most calculators do not keep enough significant figures to do the required calculation, so use the binomial approximation:
$$
\text { if } \gamma \gg 1, \quad \sqrt{1-\frac{1}{\gamma^{2}}} \approx 1-\frac{1}{2 \gamma^{2}}
$$

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Problem 79

A spaceship is traveling away from Earth at $0.87 c$. The astronauts report home by radio every $12 \mathrm{~h}$ (by their own clocks). (a) At what interval are the reports sent to Earth, according to Earth clocks? (b) At what interval are the reports received by Earth observers, according to their own clocks?

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Problem 80

Derivation of the Doppler formula for light. A source and receiver of EM waves move relative to one another at velocity $v$. Let $v$ be positive if the receiver and source are moving apart from one another. The source emits an EM wave at frequency $f_{\mathrm{s}}$ (in the source frame). The time between wavefronts as measured by the source is $T_{\mathrm{s}}=1 / f_{\mathrm{s}}$.
(a) In the receiver's frame, how much time elapses between the emission of wavefronts by the source? Call this $T_{\mathrm{r}}^{\prime}$. (b) $T_{\mathrm{r}}^{\prime}$ is not the time that the receiver measures between the arrival of successive wavefronts because the wavefronts travel different distances. Say that, according to the receiver, one wavefront is emitted at $t=0$ and the next at $t=T_{\mathrm{o}}^{\prime}$. When the first wavefront is emitted, the distance between source and receiver is $d_{\mathrm{r}}$ When the second wavefront is emitted, the distance between source and receiver is $d_{\mathrm{r}}+v T_{\mathrm{r}}^{\prime}$. Each wavefront travels at speed $c$. Calculate the time $T_{\mathrm{r}}$ between the arrival of these two wavefronts as measured by the receiver. (c) The frequency detected by the receiver is $f_{\mathrm{r}}=1 / T_{\mathrm{r}} .$ Show that $f_{\mathrm{r}}$ is given by
$$
f_{\mathrm{r}}=f_{\mathrm{s}} \sqrt{\frac{1-v / \mathrm{c}}{1+v / \mathrm{c}}}
$$

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Problem 81

You are trying to communicate with a rocket ship that is traveling at $1.2 \times 10^{8} \mathrm{~m} / \mathrm{s}$ away from Earth on its way into space. If you send a message at a frequency of $55 \mathrm{kHz}$, to what frequency should the astronauts tune to receive your message? [Hint: Use the result derived in Problem 80.]

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Numerade Educator

Problem 82

An astronaut has spent a long time in the Space Shuttle traveling at $7.860 \mathrm{~km} / \mathrm{s}$. When he returns to Earth, he is $1.0$ s younger than his twin brother. How long was he on the shuttle? [Hint: Use an approximation from Appendix
A.5 and beware of calculator round off errors.]

Dading Chen

Numerade Educator

Problem 83

Two atomic clocks are synchronized. One is put aboard a spaceship that leaves Earth at $t=0$ at a speed of $0.750 \mathrm{c}$. (a) When the spaceship has been traveling for $48.0 \mathrm{~h}$ (according to the atomic clock on board), it sends a radio signal back to Earth. When would the signal be received on Earth, according to the atomic clock on Earth? (b) When the Earth clock says that the spaceship has been gone for $48.0 \mathrm{~h}$, it sends a radio signal to the spaceship. At what time (according to the spaceship's clock) does the spaceship receive the signal?

Dading Chen

Numerade Educator

Problem 84

A spaceship passes over an observation station on Earth. Just as the nose of the ship passes the station, a light in the nose of the ship flashes. As the tail of the ship passes the station, a light flashes in the ship's tail. According to an Earth observer, $50.0 \mathrm{~ns}$ elapses between the two events. In the astronaut's reference frame, the length of the ship is $12.0 \mathrm{~m}$. (a) How fast is the ship traveling according to an Earth observer? (b) What is the elapsed time between light flashes in the astronaut's frame of reference?

Dading Chen

Numerade Educator

Problem 85

A lambda hyperon $\Lambda^{0}\left(\mathrm{mass}=1115.7 \mathrm{MeV} / c^{2}\right)$ at rest in the lab frame decays into a neutron $\mathrm{n}$ (mass $=$ $\left.939.6 \mathrm{MeV} / c^{2}\right)$ and a pion (mass $\left.=135.0 \mathrm{MeV} / c^{2}\right)$ :
$$
\Lambda^{0} \rightarrow \mathrm{n}+\pi^{0}
$$
What are the kinetic energies (in the lab frame) of the neutron and pion after the decay? [Hint: Use Eq. (26-11) to find momentum.]

David Morabito

Numerade Educator

Problem 86

Radon decays as ${ }^{222} \mathrm{Rn} \rightarrow{ }^{218} \mathrm{Po}+\alpha$. The mass of the radon-222 nucleus is $221.97039 \mathrm{u}$, the mass of the polonium- 218 nucleus is $217.96289 \mathrm{u}$, and the mass of the alpha particle is $4.00151 \mathrm{u} .\left(1 \mathrm{u}=931.5 \mathrm{MeV} / \mathrm{c}^{2} .\right)$ If
the radon nucleus is initially at rest in the lab frame, at what speeds (in the lab frame) do the (a) polonium-2 18 nucleus and (b) alpha particle move?

David Morabito

Numerade Educator