Principles of Physics a Calculus Based Text

Raymond A. Serway, John W. Jewett, Jr.

Chapter 9

Relativity - all with Video Answers

Educators

Chapter Questions

Problem 1

In a laboratory frame of reference, an observer notes that Newton's second law is valid. Assume forces and masses are measured to be the same in any reference frame for speeds small compared with the speed of light. (a) Show that Newton's second law is also valid for an observer moving at a constant speed, small compared with the speed of light, relative to the laboratory frame. (b) Show that Newton's second law is not valid in a reference frame moving past the laboratory frame with a constant acceleration.

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Problem 2

A car of mass $2000 \mathrm{kg}$ moving with a speed of $20.0 \mathrm{m} / \mathrm{s}$ collides and locks together with a 1500 -kg car at rest at a stop sign. Show that momentum is conserved in a reference frame moving at $10.0 \mathrm{m} / \mathrm{s}$ in the direction of the moving car.

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Problem 3

How fast must a meterstick be moving if its length is measured to shrink to $0.500 \mathrm{m}$ ?

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Problem 4

An astronaut is traveling in a space vehicle moving at $0.500 c$ relative to the Earth. The astronaut measures her pulse rate at 75.0 beats per minute. Signals generated by the astronaut's pulse are radioed to the Earth when the vehicle is moving in a direction perpendicular to the line that connects the vehicle with an observer on the Earth. (a) What pulse rate does the Earth-based observer measure? (b) What If? What would be the pulse rate if the speed of the space vehicle were increased to $0.990 c$ ?

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Problem 5

At what speed does a clock move if it is measured to run at a rate one-half the rate of a clock at rest with respect to an observer?

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Problem 6

A muon formed high in the Earth's atmosphere is measured by an observer on the Earth's surface to travel at speed $v=$ $0.990 c$ for a distance of $4.60 \mathrm{km}$ before it decays into an electron, a neutrino, and an antineutrino $\left(\mu^{-} \rightarrow \mathrm{e}^{-}+\nu+\bar{\nu}\right)$ (a) For what time interval does the muon live as measured in its reference frame? (b) How far does the Earth travel as measured in the frame of the muon?

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Problem 7

An astronomer on the Earth observes a meteoroid in the southern sky approaching the Earth at a speed of $0.800 c$. At the time of its discovery the meteoroid is 20.0 ly from the Earth. Calculate (a) the time interval required for the meteoroid to reach the Earth as measured by the Earthbound astronomer, (b) this time interval as measured by a tourist on the meteoroid, and (c) the distance to the Earth as measured by the tourist.

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Problem 8

For what value of $v$ does $\gamma=1.0100 ?$ Observe that for speeds lower than this value, time dilation and length contraction are effects amounting to less than $1 \%$.

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Problem 9

An atomic clock moves at $1000 \mathrm{km} / \mathrm{h}$ for $1.00 \mathrm{h}$ as measured by an identical clock on the Earth. At the end of the 1.00 -h interval, how many nanoseconds slow will the moving clock be compared with the Earth-based clock?

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Problem 10

The identical twins Speedo and Goslo join a migration from the Earth to Planet X, 20.0 ly away in a reference frame in which both planets are at rest. The twins, of the same age, depart at the same moment on different spacecraft. Speedo's spacecraft travels steadily at $0.950 c$ and Goslo's at $0.750 c .$ (a) Calculate the age difference between the twins after Goslo's spacecraft lands on Planet X. (b) Which twin is older?

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Problem 11

A spacecraft with a proper length of $300 \mathrm{m}$ passes by an observer on the Earth. According to this observer, it takes $0.750 \mu \mathrm{s}$ for the spacecraft to pass a fixed point. Determine the speed of the spacecraft as measured by the Earth-based observer.

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Problem 12

A spacecraft with a proper length of $L_{p}$ passes by an observer on the Earth. According to this observer, it takes a time interval $\Delta t$ for the spacecraft to pass a fixed point. Determine the speed of the object as measured by the Earth-based observer.

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Problem 13

A friend passes by you in a spacecraft traveling at a high speed. He tells you that his craft is $20.0 \mathrm{m}$ long and that the identically constructed craft you are sitting in is $19.0 \mathrm{m}$ long. According to your observations, (a) how long is your spacecraft, (b) how long is your friend's craft, and (c) what is the speed of your friend's craft?

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Problem 14

An interstellar space probe is launched from the Earth. After a brief period of acceleration it moves with a constant velocity, with a magnitude of $70.0 \%$ of the speed of light. Its nuclear-powered batteries supply the energy to keep its data transmitter active continuously. The batteries have a lifetime of 15.0 yr as measured in a rest frame. (a) How long do the batteries on the space probe last as measured by Mission Control on the Earth? (b) How far is the probe from the Earth when its batteries fail as measured by Mission Control? (c) How far is the probe from the Earth when its batteries fail as measured by its built-in trip odometer? (d) For what total time interval after launch are data received from the probe by Mission Control? Note that radio waves travel at the speed of light and fill the space between the probe and the Earth at the time of battery failure.

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Problem 15

A moving rod is observed to have a length of $\ell=2.00 \mathrm{m}$ and to be oriented at an angle of $\theta=30.0^{\circ}$ with respect to the direction of motion as shown in Figure P9.15. The rod has a speed of $0.995 c .$ (a) What is the proper length of the rod? (b) What is the orientation angle in the proper frame?

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Problem 16

Shannon observes two light pulses to be emitted from the same location, but separated in time by $3.00 \mu$ s. Kimmie observes the emission of the same two pulses to be separated in time by $9.00 \mu$ s. (a) How fast is Kimmie moving relative to Shannon? (b) According to Kimmie, what is the separation in space of the two pulses?

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Problem 17

A red light flashes at position $x_{\mathrm{R}}=3.00 \mathrm{m}$ and time $t_{\mathrm{R}}=1.00 \times 10^{-9} \mathrm{s},$ and a blue light flashes at $x_{\mathrm{B}}=5.00 \mathrm{m}$ and $t_{\mathrm{B}}=9.00 \times 10^{-9} \mathrm{s},$ all measured in the S reference frame. Reference frame $\mathrm{S}^{\prime}$ moves uniformly to the right and has its origin at the same point as $\mathrm{S}$ at $t=t^{\prime}=0 .$ Both flashes are observed to occur at the same place in $\mathrm{S}^{\prime}$. (a) Find the relative speed between $\mathrm{S}$ and $\mathrm{S}^{\prime}$. (b) Find the location of the two flashes in frame $\mathrm{S}^{\prime}$. (c) At what time does the red flash occur in the $\mathrm{S}^{\prime}$ frame?

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Problem 18

Keilah, in reference frame $\mathrm{S},$ measures two events to be simultaneous. Event A occurs at the point $(50.0 \mathrm{m}, 0,0)$ at the instant 9: 00: 00 Universal time on January 15,2010. Event $\mathrm{B}$ occurs at the point $(150 \mathrm{m}, 0,0)$ at the same moment. Torrey, moving past with a velocity of $0.800 c \hat{\mathbf{i}},$ also observes the two events. In her reference frame $S^{\prime},$ which event occurred first and what time interval elapsed between the events?

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Problem 19

An enemy spacecraft moves away from the Earth at a speed of $v=0.800 c \text { (Fig. } \mathrm{P} 9.19) .$ A galactic patrol spacecraft pursues at a speed of $u=0.900 c$ relative to the Earth. Observers on the Earth measure the patrol craft to be overtaking the enemy craft at a relative speed of $0.100 c$. With what speed is the patrol craft overtaking the enemy craft as measured by the patrol craft's crew?

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Problem 20

A spacecraft is launched from the surface of the Earth with a velocity of $0.600 c$ at an angle of $50.0^{\circ}$ above the horizontal positive $x$ axis. Another spacecraft is moving past with a velocity of $0.700 c$ in the negative $x$ direction. Determine the magnitude and direction of the velocity of the first spacecraft as measured by the pilot of the second spacecraft.

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Problem 21

Figure $\mathrm{P} 9.21$ shows a jet of material (at the upper right) being ejected by galaxy $\mathrm{M} 87$ (at the lower left). Such jets are believed to be evidence of supermassive black holes at the center of a galaxy. Suppose two jets of material from the center of a galaxy are ejected in opposite directions. Both jets move at $0.750 c$ relative to the galaxy center. Determine the speed of one jet relative to the other.

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Problem 22

The nonrelativistic expression for the momentum of a particle, $p=m u,$ agrees with experiment if $u \ll c .$ For what speed does the use of this equation give an error in the measured momentum of (a) $1.00 \%$ and (b) $10.0 \%$ ?

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Problem 23

Calculate the momentum of an electron moving with a speed of (a) $0.0100 c,$ (b) $0.500 c$, and (c) $0.900 c$.

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Problem 24

Show that the speed of an object having momentum of magnitude $p$ and mass $m$ is $$u=\frac{c}{\sqrt{1+(m c / p)^{2}}}$$

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Problem 25

A golf ball travels with a speed of $90.0 \mathrm{m} / \mathrm{s}$. By what fraction does its relativistic momentum magnitude $p$ differ from its classical value $m u$ ? That is, find the ratio $(p-m u) / m u$.

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Problem 26

The speed limit on a certain roadway is $90.0 \mathrm{km} / \mathrm{h}$. Suppose speeding fines are made proportional to the amount by which a vehicle's momentum exceeds the momentum it would have when traveling at the speed limit. The fine for driving at $190 \mathrm{km} / \mathrm{h}$ (that is, $100 \mathrm{km} / \mathrm{h}$ over the speed limit) is 80.0 dollars. What, then, is the fine for traveling (a) at $1090 \mathrm{km} / \mathrm{h}$ ? (b) At $1000000090 \mathrm{km} / \mathrm{h} ?$

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Problem 27

An unstable particle at rest spontaneously breaks into two fragments of unequal mass. The mass of the first fragment is $2.50 \times 10^{-28} \mathrm{kg},$ and that of the other is $1.67 \times 10^{-27} \mathrm{kg} .$ If
the lighter fragment has a speed of $0.893 c$ after the breakup, what is the speed of the heavier fragment?

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Problem 28

Show that for any object moving at less than one-tenth the speed of light, the relativistic kinetic energy agrees with the result of the classical equation $K=\frac{1}{2} m u^{2}$ to within less than $1\%$. Therefore, for most purposes, the classical equation is sufficient to describe these objects.

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Problem 29

Determine the energy required to accelerate an electron from (a) $0.500 c$ to $0.900 c$ and (b) $0.900 c$ to $0.990 c$.

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Problem 30

(a) Find the kinetic energy of a 78.0 -kg spacecraft launched out of the solar system with speed $106 \mathrm{km} / \mathrm{s}$ by using the classical equation $K=\frac{1}{2} m u^{2}$. (b) What If? Calculate its kinetic energy using the relativistic equation. (c) Explain the result of comparing the answers of parts (a) and (b).

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Problem 31

An electron has a kinetic energy five times greater than its rest energy. Find (a) its total energy and (b) its speed.

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Problem 32

A cube of steel has a volume of $1.00 \mathrm{cm}^{3}$ and a mass of $8.00 \mathrm{g}$ when at rest on the Earth. If this cube is now given a speed $u=0.900 c,$ what is its density as measured by a stationary observer? Note that relativistic density is defined as $E_{R} / c^{2} V$.

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Problem 33

The rest energy of an electron is $0.511 \mathrm{MeV}$. The rest energy of a proton is $938 \mathrm{MeV}$. Assume both particles have kinetic energies of $2.00 \mathrm{MeV}$. Find the speed of (a) the electron and (b) the proton. (c) By what factor does the speed of the electron exceed that of the proton? (d) Repeat the calculations in parts (a) through (c) assuming both particles have kinetic energies of $2000 \mathrm{MeV}$.

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Problem 34

An unstable particle with mass $m=3.34 \times 10^{-27} \mathrm{kg}$ is initially at rest. The particle decays into two fragments that fly off along the $x$ axis with velocity components $u_{1}=0.987 c$ and $u_{2}=-0.868 c .$ From this information, we wish to determine the masses of fragments 1 and 2. (a) Is the initial system of the unstable particle, which becomes the system of the two fragments, isolated or nonisolated? (b) Based on your answer to part (a), what two analysis models are appropriate for this situation? (c) Find the values of $\gamma$ for the two fragments after the decay. (d) Using one of the analysis models in part (b), find a relationship between the masses $m_{1}$ and $m_{2}$ of the fragments. (e) Using the second analysis model in part (b), find a second relationship between the masses $m_{1}$ and $m_{2}$. (f) Solve the relationships in parts (d) and (e) simultaneously for the masses $m_{1}$ and $m_{2}$.

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Problem 35

A proton moves at $0.950 c .$ Calculate its (a) rest energy, (b) total energy, and (c) kinetic energy.

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Problem 36

Show that the energy-momentum relationship in Equation $9.22, E^{2}=p^{2} c^{2}+\left(m c^{2}\right)^{2},$ follows from the expressions $E=\gamma m c^{2}$ and $p=\gamma m u$.

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Problem 37

A proton in a high-energy accelerator moves with a speed of $c / 2 .$ Use the work-kinetic energy theorem to find the work required to increase its speed to (a) $0.750 c$ and (b) $0.995 c$.

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Problem 38

An object having mass $900 \mathrm{kg}$ and traveling at speed $0.850 c$ collides with a stationary object having mass 1400 kg. The two objects stick together. Find (a) the speed and (b) the mass of the composite object.

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Problem 39

A pion at rest $\left(m_{\pi}=273 m_{e}\right)$ decays to a muon $\left(m_{\mu}=\right.$$207 m_{e})$ and an antineutrino $\left(m_{\bar{\nu}} \approx 0\right) .$ The reaction is written $\pi^{-} \rightarrow \mu^{-}+\bar{\nu} .$ Find (a) the kinetic energy of the muon and (b) the energy of the antineutrino in electron volts.

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Problem 40

In a nuclear power plant, the fuel rods last 3 yr before they are replaced. The plant can transform energy at a maximum possible rate of $1.00 \mathrm{GW}$. Supposing it operates at $80.0 \%$ capacity for 3.00 yr, what is the loss of mass of the fuel?

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Problem 41

The power output of the Sun is $3.85 \times 10^{26} \mathrm{W}$. By how much does the mass of the Sun decrease each second?

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Problem 42

When $1.00 \mathrm{g}$ of hydrogen combines with $8.00 \mathrm{g}$ of oxygen, $9.00 \mathrm{g}$ of water is formed. During this chemical reaction, $2.86 \times 10^{5} \mathrm{J}$ of energy is released. (a) Is the mass of the water larger or smaller than the mass of the reactants? (b) What is the difference in mass? (c) Explain whether the change in mass is likely to be detectable.

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Problem 43

A global positioning system (GPS) satellite moves in a circular orbit with period 11 h 58 min. (a) Determine the radius of its orbit. (b) Determine its speed. (c) The nonmilitary GPS signal is broadcast at a frequency of $1575.42 \mathrm{MHz}$ in the reference frame of the satellite. When it is received on the Earth's surface by a GPS receiver (Fig. P9.43), what is the fractional change in this frequency due to time dilation as described by special relativity? (d) The gravitational "blueshift" of the frequency according to general relativity is a separate effect. It is called a blueshift to indicate a change to a higher frequency. The magnitude of that fractional change is given by $$\frac{\Delta f}{f}=\frac{\Delta U_{g}}{m c^{2}}$$ where $U_{g}$ is the change in gravitational potential energy of an object-Earth system when the object of mass $m$ is moved between the two points where the signal is observed. Calculate this fractional change in frequency due to the change in position of the satellite from the Earth's surface to its orbital position. (e) What is the overall fractional change in frequency due to both time dilation and gravitational blueshift?

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Problem 44

In $1963,$ astronaut Gordon Cooper orbited the Earth 22 times. The press stated that for each orbit, he aged two-millionths of a second less than he would have had he remained on the Earth. (a) Assuming Cooper was $160 \mathrm{km}$ above the Earth in a circular orbit, determine the difference in elapsed time between someone on the Earth and the orbiting astronaut for the 22 orbits. You may use the approximation $$\frac{1}{\sqrt{1-x}} \approx 1+\frac{x}{2}$$ for small $x$. (b) Did the press report accurate information? Explain.

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Problem 45

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 the galaxy is $2.00 \times 10^{6}$ ly away and 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 1000-metric-ton spacecraft? (c) What is the cost of this energy if it is purchased at a typical consumer price for electric energy of $0.110 / \mathrm{kWh}$ ?

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Problem 46

An object disintegrates into two fragments. One fragment has mass $1.00 \mathrm{MeV} / \mathrm{c}^{2}$ and momentum $1.75 \mathrm{MeV} / \mathrm{c}$ in the positive $x$ direction, and the other has mass $1.50 \mathrm{MeV} / c^{2}$ and momentum $2.00 \mathrm{MeV} / c$ in the positive $y$ direction. Find (a) the mass and (b) the speed of the original object.

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Problem 47

The net nuclear fusion reaction inside the Sun can be written as $4^{1} \mathrm{H} \rightarrow^{4} \mathrm{He}+E .$ The rest energy of each hydrogen atom is $938.78 \mathrm{MeV}$, and the rest energy of the helium- 4 atom is $3728.4 \mathrm{MeV}$. Calculate the percentage of the starting mass that is transformed to other forms of energy.

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Problem 48

Why is the following situation impossible? On their 40th birthday, twins Speedo and Goslo say good-bye as Speedo takes off for a planet that is 50 ly away. He travels at a constant speed of $0.85 c$ and immediately turns around and comes back to the Earth after arriving at the planet. Upon arriving back at the Earth, Speedo has a joyous reunion with Goslo.

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Problem 49

Massive stars ending their lives in supernova explosions produce the nuclei of all the atoms in the bottom half of the periodic table by fusion of smaller nuclei. This problem roughly models that process. A particle of mass $m=1.99 \times 10^{-26} \mathrm{kg}$ moving with a velocity $\overrightarrow{\mathbf{u}}=0.500 c$ i collides head-on and sticks to a particle of mass $m^{\prime}=m / 3$ moving with the velocity $\overrightarrow{\mathbf{u}}=-0.500 c \mathbf{i} .$ What is the mass of the resulting particle?

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Problem 50

Massive stars ending their lives in supernova explosions produce the nuclei of all the atoms in the bottom half of the periodic table by fusion of smaller nuclei. This problem roughly models that process. A particle of mass $m$ moving along the $x$ axis with a velocity component $+u$ collides headon and sticks to a particle of mass $m / 3$ moving along the $x$ axis with the velocity component $-u .$ (a) What is the mass $M$ of the resulting particle? (b) Evaluate the expression from part (a) in the limit $u \rightarrow 0 .$ (c) Explain whether the result agrees with what you should expect from nonrelativistic physics.

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Problem 51

The cosmic rays of highest energy are protons that have kinetic energy on the order of $10^{13} \mathrm{MeV}$. (a) As measured in the proton's frame, what time interval would a proton of this energy require to travel across the Milky Way galaxy, which has a proper diameter $\sim 10^{5}$ ly? (b) From the point of view of the proton, how many kilometers across is the galaxy?

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Problem 52

An electron has a speed of $0.750 c .$ (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.

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Problem 53

An alien spaceship traveling at $0.600 c$ toward the Earth launches a landing craft. The landing craft travels in the same direction with a speed of $0.800 c$ relative to the mother ship. As measured on the Earth, the spaceship is 0.200 ly from the Earth when the landing craft is launched. (a) What speed do the Earth-based observers measure for the approaching landing craft? (b) What is the distance to the Earth at the moment of the landing craft's launch as measured by the aliens? (c) What travel time is required for the landing craft to reach the Earth as measured by the aliens on the mother ship? (d) If the landing craft has a mass of $4.00 \times 10^{5} \mathrm{kg}$, what is its kinetic energy as measured in the Earth reference frame?

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Problem 54

(a) 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. (b) At what speed does the classical kinetic energy underestimate the experimental value by $1 \%$ ? (c) By $5 \%$ ? (d) By $50 \%$ ?

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Problem 55

A supertrain with a proper length of $100 \mathrm{m}$ travels at a speed of $0.950 c$ as it passes through a tunnel having a proper length of $50.0 \mathrm{m}$. As seen by a trackside observer, is the train ever completely within the tunnel? If so, by how much do the train's ends clear the ends of the tunnel?

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Problem 56

A particle with electric charge $q$ moves along a straight line in a uniform electric field $\overrightarrow{\mathbf{E}}$ with speed $u$. The electric force exerted on the charge is $q \overrightarrow{\mathbf{E}}$. The velocity of the particle 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 $$a=\frac{d u}{d t}=\frac{q E}{m}\left(1-\frac{u^{2}}{c^{2}}\right)^{3 / 2}$$ (b) Discuss the significance of the dependence of the acceleration on the speed. (c) What If? If the particle 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 $t ?$

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Problem 57

An observer in a coasting spacecraft moves toward a mirror at speed $v=0.650 c$ relative to the reference frame labeled $\mathrm{S}$ in Figure $\mathrm{P} 9.57 .$ The mirror is stationary with respect to $\mathrm{S}$. A light pulse emitted by the spacecraft travels toward the mirror and is reflected back to the spacecraft. The spacecraft is a distance $d=5.66 \times 10^{10} \mathrm{m}$ from the mirror (as measured by observers in $\mathrm{S}$ ) at the moment the light pulse leaves the spacecraft. What is the total travel time of the pulse as measured by observers in (a) the S frame and (b) the spacecraft?

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Problem 58

An observer in a coasting spacecraft moves toward a mirror at speed $v$ relative to the reference frame labeled $S$ in Figure $\mathrm{P} 9.57 .$ The mirror is stationary with respect to $\mathrm{S}$. A light pulse emitted by the spacecraft travels toward the mirror and is reflected back to the spacecraft. The spacecraft is a distance $d$ from the mirror (as measured by observers in S) at the moment the light pulse leaves the spacecraft. What is the total travel time of the pulse as measured by observers in (a) the S frame and (b) the spacecraft?

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Problem 59

Spacecraft I, containing students taking a physics exam, approaches the Earth with a speed of $0.600 c$ (relative to the Earth), while spacecraft II, containing professors proctoring the exam, moves at $0.280 c$ (relative to the Earth) directly toward the students. If the professors stop the exam after 50.0 min have passed on their clock, for what time interval does the exam last as measured by (a) the students and (b) an observer on the Earth?

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Problem 60

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 $T_{0}$ (spacecraft time) to complete the exam. Show that she should wait a time interval (Earth time) of $$T=T_{0} \sqrt{\frac{1-v / c}{1+v / c}}$$ before sending a light signal telling them to stop. ( Suggestion: Remember that it takes some time for the second light signal to travel from the professor to the students.)

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Problem 61

A gamma ray (a high-energy photon) can produce an electron $\left(\mathrm{e}^{-}\right)$ and a positron $\left(\mathrm{e}^{+}\right)$ of equal mass when it enters the electric field of a heavy nucleus: $\gamma \rightarrow \mathrm{e}^{+}+\mathrm{e}^{-}$. What minimum gamma-ray energy is required to accomplish this task?

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Problem 62

Imagine that the entire Sun, of mass $M_{\mathrm{S}},$ collapses to a sphere of radius $R_{g}$ such that the work required to remove a small mass $m$ from the surface would be equal to its rest energy $m c^{2}$. This radius is called the gravitational radius for the Sun. (a) Use this approach to show that $R_{g}=G M_{S} / c^{2}$. (b) Find a numerical value for $R_{g}$.

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Problem 63

Owen and Dina are at rest in frame $\mathrm{S}^{\prime},$ which is moving at $0.600 c$ with respect to frame S. Theyplayagame of catch while Ed, at rest in frame S, watches the action (Fig. P9.63). Owen throws the ball to Dina at $0.800 c$ (according to Owen), and their separation (measured in $\mathrm{S}^{\prime}$ ) is equal to $1.80 \times 10^{12} \mathrm{m}$. (a) According to Dina, how fast is the ball moving? (b) According to Dina, what time interval is required for the ball to reach her? According to Ed, (c) how far apart are Owen and Dina, (d) how fast is the ball moving, and (e) what time interval is required for the ball to reach Dina?

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Problem 64

A rod of length $L_{0}$ moving with a speed $v$ along the horizontal direction makes an angle $\theta_{0}$ with respect to the $x^{\prime}$ axis. (a) Show that the length of the rod as measured by a stationary observer is $L=L_{0}\left[1-\left(v^{2} / c^{2}\right) \cos ^{2} \theta_{0}\right]^{1 / 2}$ (b) Show that the angle that the rod makes with the $x$ axis is given by $\tan \theta=\gamma \tan \theta_{0} .$ 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.)

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Problem 65

Suppose our Sun is about to explode. In an effort to escape, we depart in a spacecraft at $v=0.800 c$ 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?

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