Show that linear momentum is conserved in Example2.9 as measured by Mary

Michelson used the motion of the Earth around the sun to try to determine the effects of the ether. Can you think of a more convenient experiment with a higher speed that Michelson might have used in the 1880s? What about today?

If you wanted to set out today to fi nd the effects of the ether, what experimental apparatus would you want to use? Would a laser be included? Why?

For what reasons would Michelson and Morley repeat their experiment on top of a mountain? Why would they perform the experiment in summer and winter?

Does the fact that Maxwells equations do not need to be modifi ed because of the special theory of relativity, whereas Newtons laws of motion do, mean that Maxwells work is somehow greater or more signifi - cant than Newtons? Explain.

The special theory of relativity has what effect on measurements done today? (a) None whatsoever, because any correction would be negligible. (b) We need to consider the effects of relativity when objects move close to the speed of light. (c) We should always make a correction for relativity because Newtons laws are basically wrong. (d) It doesnt matter, because we cant make measurements where relativity would matter.

Why did it take so long to discover the theory of relativity? Why didnt Newton fi gure it out?

Can you think of a way you can make yourself older than those born on your same birthday?

Will metersticks manufactured on Earth work correctly on spaceships moving at high speed? Explain.

Devise a system for you and three colleagues, at rest with you, to synchronize your clocks if your clocks are too large to move and are separated by hundreds of miles.

In the experiment to verify time dilation by fl ying the cesium clocks around the Earth, what is the order of the speed of the four clocks in a system fi xed at the center of the Earth, but not rotating?

Can you think of an experiment to verify length contraction directly? Explain.

Would it be easier to perform the muon decay experiment in the space station orbiting above Earth and then compare with the number of muons on Earth? Explain.

On a spacetime diagram, can events above t 0 but not in the shaded area in Figure 2.25 affect the future? Explain.

Why dont we also include the spatial coordinate z when drawing the light cone?

What would be a suitable name for events connected by s 2 0?

Is the relativistic Doppler effect valid only for light waves? Can you think of another situation in which it might be valid?

In Figure 2.22, why can a real worldline not have a slope less than one?

Explain how in the twin paradox, we might arrange to compare clocks at the beginning and end of Marys journey and not have to worry about acceleration effects.

In each of the following pairs, which is the more massive: a relaxed or compressed spring, a charged or uncharged capacitor, or a piston-cylinder when closed or open?

In the fi ssion of 235U, the masses of the fi nal products are less than the mass of 235U. Does this make sense? What happens to the mass?

In the fusion of deuterium and tritium nuclei to produce a thermonuclear reaction, where does the kinetic energy that is produced come from?

Mary, the astronaut, wants to travel to the star system Alpha Centauri, which is 4.3 lightyears away. She wants to leave on her 30th birthday, travel to Alpha Centauri but not stop, and return in time for her wedding to Vladimir on her 35th birthday. What is most likely to happen? (a) Vladimir is a lucky man, because he will marry Mary after she completes her journey. (b) Mary will have to hustle to get in her wedding gown, and the wedding is likely to be watched by billions of people. (c) It is a certainty that Mary will not reach Alpha Centauri if she wants to marry Vladimir as scheduled. (d) Mary does reach Alpha Centauri before her 35th birthday and sends a radio message to Vladimir from Alpha Centauri that she will be back on time. Vladimir is relieved to receive the message before the wedding date.

A salesman driving a very fast car was arrested for driving through a traffi c light while it was red, according to a policeman parked near the traffi c light. The salesman said that the light was actually green to him, because it was Doppler shifted. Is he likely to be found innocent? Explain.

A spaceship of length 40 m at rest is observed to be 20 m long when in motion. How fast is it moving?

The Concorde traveled 8000 km between two places in North America and Europe at an average speed of 375 m/s. What is the total difference in time between two similar atomic clocks, one on the airplane and one at rest on Earth during a one-way trip? Consider only time dilation and ignore other effects such as Earths rotation.

A mechanism on Earth used to shoot down geosynchronous satellites that house laser-based weapons is fi nally perfected and propels golf balls at 0.94c. (Geosynchronous satellites are placed 3.58  104 km above the surface of the Earth.) (a) What is the distance from the Earth to the satellite, as measured by a detector placed inside the golf ball? (b) How much time will it take the golf ball to make the journey to the satellite in the Earths frame? How much time will it take in the golf balls frame?

Two events occur in an inertial system K at the same time but 4 km apart. What is the time difference measured in a system K moving parallel to these two events when the distance separation of the events is measured to be 5 km in K?

Imagine that in another universe the speed of light is only 100 m/s. (a) A person traveling along an interstate highway at 120 km/h ages at what fraction of the rate of a person at rest? (b) This traveler passes by a meterstick at rest on the highway. How long does the meterstick appear?

In another universe where the speed of light is only 100 m/s, an airplane that is 40 m long at rest and fl ies at 300 km/h will appear to be how long to an observer at rest?

Two systems K and K synchronize their clocks at t t 0 when their origins are aligned as system K passes by system K along the x axis at relative speed 0.8c. At time t 3 ns, Frank (in system K) shoots a proton gun having proton speeds of 0.98c along his x axis. The protons leave the gun at x 1 m and arrive at a target 120 m away. Determine the event coordinates (x, t) of the gun fi ring and of the protons arriving as measured by observers in both systems K and K.

A spaceship is moving at a speed of 0.84c away from an observer at rest. A boy in the spaceship shoots a pro ton gun with protons having a speed of 0.62c. What is the speed of the protons measured by the observer at rest when the gun is shot (a) away from the observer and (b) toward the observer?

A proton and an antiproton are moving toward each other in a head-on collision. If each has a speed of 0.8c with respect to the collision point, how fast are they moving with respect to each other?

Imagine the speed of light in another universe to be only 100 m/s. Two cars are traveling along an interstate highway in opposite directions. Person 1 is traveling 110 km/h, and person 2 is traveling 140 km/h. How fast does person 1 measure person 2 to be traveling? How fast does person 2 measure person 1 to be traveling?

In the Fizeau experiment described in Example 2.5, suppose that the water is flowing at a speed of 5 m/s. Find the difference in the speeds of two beams of light, one traveling in the same direction as the water and the other in the opposite direction. Use n 1.33 for water.

Three galaxies are aligned along an axis in the order A, B, C. An observer in galaxy B is in the middle and observes that galaxies A and C are moving in opposite directions away from him, both with speeds 0.60c. What is the speed of galaxies B and C as observed by someone in galaxy A?

Consider the gedanken experiment discussed in Section 2.6 in which a giant fl oodlight stationed 400 km above the Earths surface shines its light across the moons surface. How fast does the light fl ash across the moon?

A group of scientists decide to repeat the muon decay experiment at the Mauna Kea telescope site in Hawaii, which is 4205 m above sea level. They count 104 muons during a certain time period. Repeat the calculation of Section 2.7 and fi nd the classical and relativistic number of muons expected at sea level. Why did they decide to count as many as 104 muons instead of only 103?

Consider a reference system placed at the U.S. Naval Observatory in Washington, D.C. Two planes take off from Washington Dulles Airport, one going eastward and one going westward, both carrying a cesium atomic clock. The distance around the Earth at 39 latitude (Washington, D.C.) is 31,000 km, and Washington rotates about the Earths axis at a speed of 360 m/s. Calculate the predicted differences between the clock left at the observatory and the two clocks in the airplanes (each traveling at 300 m/s) when the airplanes return to Washington. Include the rotation of the Earth but no general relativistic effects. Compare with the predictions given in the text.

Derive the results in Table 2.1 for the frequencies f and f  . During what time period do Frank and Mary receive these frequencies?

Derive the results in Table 2.1 for the time of the total trip and the total number of signals sent in the frame of both twins. Show your work.

Use the Lorentz transformation to prove that s 2 s 2 .

Prove that for a timelike interval, two events can never be considered to occur simultaneously.

Prove that for a spacelike interval, two events cannot occur at the same place in space.

Given two events, (x1, t1) and (x2, t2), use a spacetime diagram to fi nd the speed of a frame of reference in which the two events occur simultaneously. What values may s2 have in this case?

(a) Draw on a spacetime diagram in the fi xed system a line expressing all the events in the moving system that occur at t 0 if the clocks are synchronized at t t 0. (b) What is the slope of this line? (c) Draw lines expressing events occurring for the four times t4, t3, t2, and t1 where t4  t3  0  t2  t1. (d) How are these four lines related geometrically?

Consider a fi xed and a moving system with their clocks synchronized and their origins aligned at t t 0. (a) Draw on a spacetime diagram in the fi xed system a line expressing all the events occurring at t 0. (b) Draw on this diagram a line expressing all the events occurring at x 0. (c) Draw all the worldlines for light that pass through t t 0. (d) Are the x and ct axes perpendicular? Explain.

Use the results of the two previous problems to show that events simultaneous in one system are not simultaneous in another system moving with respect to the fi rst. Consider a spacetime diagram with x, ct and x, ct axes drawn such that the origins coincide and the clocks were synchronized at t t 0. Then consider events 1 and 2 that occur simultaneously in the fi xed system. Are they simultaneous in the moving system?

An astronaut is said to have tried to get out of a traffi c violation for running a red light ( 650 nm) by telling the judge that the light appeared green ( 540 nm) to her as she passed by in her high-powered transport. If this is true, how fast was the astronaut going?

Derive Equation (2.32) for the case where the source is fi xed but the receiver approaches it with velocity v.

Do the complete derivation for Equation (2.33) when the source and receiver are receding with relative velocity v.

A spacecraft traveling out of the solar system at a speed of 0.95c sends back information at a rate of 1400 kHz. At what rate do we receive the information?

Three radio-equipped plumbing vans are broadcasting on the same frequency f 0. Van 1 is moving east of van 2 with speed v, van 2 is fi xed, and van 3 is moving west of van 2 with speed v. What is the frequency of each van as received by the others?

Three radio-equipped plumbing vans are broadcasting on the same frequency f 0. Van 1 is moving north of van 2 with speed v, van 2 is fi xed, and van 3 is moving west of van 2 with speed v. What frequency does van 3 hear from van 2; from van 1?

A spaceship moves radially away from Earth with acceleration 29.4 m/s2 (about 3g). How much time does it take for the sodium streetlamps ( 589 nm) on Earth to be invisible (with a powerful telescope) to the human eye of the astronauts? The range of visible wavelengths is about 400 to 700 nm.

Newtons second law is given by F dp/dt. If the force is always perpendicular to the velocity, show that F ma, where a is the acceleration.

Use the result of the previous problem to show that the radius of a particles circular path having charge q traveling with speed v in a magnetic fi eld perpendicular to the particles path is r p/qB. What happens to the radius as the speed increases as in a cyclotron?

Newtons second law is given by F dp/dt. If the force is always parallel to the velocity, show that F 3ma.

Find the force necessary to give a proton an acceleration of 1019 m/s2 when the proton has a velocity (along the same direction as the force) of (a) 0.01c, (b) 0.1c, (c) 0.9c, and (d) 0.99c.

A particle having a speed of 0.92c has a momentum of 1016 kg # m/s. What is its mass?

A particle initially has a speed of 0.5c. At what speed does its momentum increase by (a) 1%, (b) 10%, (c) 100%?

The Bevatron accelerator at the Lawrence Berkeley Laboratory accelerated protons to a kinetic energy of 6.3 GeV. What magnetic fi eld was necessary to keep the protons traveling in a circle of 15.2 m? (See Problem 56.)

Show that linear momentum is conserved in Example 2.9 as measured by Mary.

Show that 1 2mv2 does not give the correct kinetic energy.

How much ice must melt at 0C in order to gain 2 g of mass? Where does this mass come from? The heat of fusion for water is 334 J/g.

Physicists at the Stanford Linear Accelerator Center (SLAC) bombarded 9-GeV electrons head-on with 3.1-GeV positrons to create B mesons and anti-B mesons. What speeds did the electron and positron have when they collided?

The Tevatron accelerator at the Fermi National Accelerator Laboratory (Fermilab) outside Chicago boosts protons to 1 TeV (1000 GeV) in fi ve stages (the numbers given in parentheses represent the total kinetic energy at the end of each stage): CockcroftWalton (750 keV), Linac (400 MeV), Booster (8 GeV), Main ring or injector (150 GeV), and fi nally the Tevatron itself (1 TeV). What is the speed of the proton at the end of each stage?

Calculate the momentum, kinetic energy, and total energy of an electron traveling at a speed of (a) 0.020c, (b) 0.20c, and (c) 0.90c.

The total energy of a body is found to be twice its rest energy. How fast is it moving with respect to the observer?

A system is devised to exert a constant force of 8 N on an 80-kg body of mass initially at rest. The force pushes the mass horizontally on a frictionless table. How far does the body have to be pushed to increase its mass-energy by 25%?

What is the speed of a proton when its kinetic energy is equal to twice its rest energy?

What is the speed of an electron when its kinetic energy is (a) 10% of its rest energy, (b) equal to the rest energy, and (c) 10 times the rest energy?

Derive the following equation: b v c A1  a E0 E0  K b 2

Prove that  pc/E. This is a useful relation to fi nd the velocity of a highly energetic particle.

A good rule of thumb is to use relativistic equations whenever the kinetic energies determined classically and relativistically differ by more than 1%. Find the speeds when this occurs for (a) electrons and (b) protons.

How much mass-energy (in joules) is contained in a peanut weighing 0.1 ounce? How much mass-energy do you gain by eating 10 ounces of peanuts? Compare this with the food energy content of peanuts, about 100 kcal per ounce.

Calculate the energy needed to accelerate a spaceship of mass 10,000 kg to a speed of 0.3c for intergalactic space exploration. Compare this with a projected annual energy usage on Earth of 1021 J.

Derive Equation (2.58) for the relativistic kinetic energy and show all the steps, especially the integration by parts.

A test automobile of mass 1000 kg moving at high speed crashes into a wall. The average temperature of the car is measured to rise by 0.5C after the wreck. What is the change in mass of the car? Where does this change in mass come from? (Assume the average specifi c heat of the automobile is close to that of steel, 0.11 cal # g1 # C1.)

A helium nucleus has a mass of 4.001505 u. What is its binding energy?

A free neutron is an unstable particle and beta decays into a proton with the emission of an electron. How much kinetic energy is available in the decay?

The Large Hadron Collider at Europes CERN facility is designed to produce 7.0 TeV (that is, 7.0  1012 eV) protons. Calculate the speed, momentum, and total energy of the protons.

What is the kinetic energy of (a) an electron having a momentum of 40 GeV/c? (b) a proton having a momentum of 40 GeV/c?

A muon has a mass of 106 MeV/c 2. Calculate the speed, momentum, and total energy of a 200-MeV muon.

The reaction 2H  2H S n  3He (where n is a neutron) is one of the reactions useful for producing energy through nuclear fusion. (a) Assume the deuterium nuclei (2H) are at rest and use the atomic mass units of the masses in Appendix 8 to calculate the mass-energy imbalance in this reaction. (Note: You can use atomic masses for this calculation, because the electron masses cancel out.) This amount of energy is given up when this nuclear reaction occurs. (b) What percentage of the initial rest energy is given up?

The reaction 2H  3H S n  4He is one of the reactions useful for producing energy through nuclear fusion. (a) Assume the deuterium (2H) and tritium (3H) nuclei are at rest and use the atomic mass units of the masses in Appendix 8 to calculate the massenergy imbalance in this reaction. This amount of energy is given up when this nuclear reaction occurs. (b) What percentage of the initial rest energy is given up?

Instead of one positive charge outside a conducting wire, as was discussed in Section 2.14 and shown in Figure 2.34, consider a second conducting wire parallel to the fi rst one. Both wires have positive and negative charges, and the wires are electrically neutral. Assume that in both wires the positive charges travel to the right and negative charges to the left. (a) Consider an inertial frame moving with the negative charges of wire 1. Show that the second wire is attracted to the fi rst wire in this frame. (b) Now consider an inertial frame moving with the positive charges of the second wire. Show that the fi rst wire is attracted to the second. (c) Use this argument to show that electrical and magnetic forces are relative?

An  particle has rest energy 1672 MeV and mean lifetime 8.2  1011 s. It is created and decays in a particle track detector and leaves a track 24 mm long. What is the total energy of the  particle?

Show that the following form of Newtons second law satisfi es the Lorentz transformation. Assume the force is parallel to the velocity. F m dv dt 1 31  1v 2 /c 2 2 4 3/2

Use the results listed in Table 2.1 to fi nd (a) the number of signals Frank receives at the rate f and the time at which Frank detects Marys turnaround, and (b) the number of signals Mary receives at the rate f and her clock reading when she turns around. (c) From Franks perspective, fi nd the time for the remainder of the trip (after he detects Marys turnaround), the number of signals he receives at the rate f , the total number of signals he receives, and Marys age, based on that total number of signals. (d) From Marys perspective, fi nd the time for the remainder of the trip (after her turnaround), the number of signals she receives at the rate f , the total number of signals she receives, and Franks age, based on that total number of signals.

For the twins Frank and Mary described in Section 2.8, consider Marys one-way trip at a speed of 0.8c to the star system 8 lightyears from Earth. Compute the spacetime interval s in the fi xed frame and s in the moving frame, and compare the results.

Frank and Mary are twins. Mary jumps on a spaceship and goes to the star system Alpha Centauri (4.30 lightyears away) and returns. She travels at a speed of 0.8c with respect to Earth and emits a radio signal every week. Frank also sends out a radio signal to Mary once a week. (a) How many signals does Mary receive from Frank before she turns around? (b) At what time does the frequency of signals Frank receives suddenly change? How many signals has he received at this time? (c) How many signals do Frank and Mary receive for the entire trip? (d) How much time does the trip take according to Frank and to Mary? (e) How much time does each twin say the other twin will measure for the trip? Do the answers agree with those for (d)?

A police radar gun operates at a frequency of 10.5 GHz. The offi cer, sitting in a patrol car at rest by the highway, directs the radar beam toward a speeding car traveling 80 mph directly away from the patrol car. What is the frequency shift of the refl ected beam, relative to the original radar beam?

A spaceship moving 0.80c direction away from Earth fi res a missile that the spaceship measures to be moving at 0.80c perpendicular to the ships direction of travel. Find the velocity components and speed of the missile as measured by Earth.

An electron has a total energy that is 250 times its rest energy. Determine its (a) kinetic energy, (b) speed, and (c) momentum.

A proton moves with a speed of 0.90c. Find the speed of an electron that has (a) the same momentum as the proton, and (b) the same kinetic energy.

A high-speed K0 meson is traveling at a speed of 0.90c when it decays into a  and a  meson. What are the greatest and least speeds that the mesons may have?

Frank and Mary are twins, and Mary wants to travel to our nearest star system, Alpha Centauri (4.30 lightyears away). Mary leaves on her 30th birthday and intends to return to Earth on her 52nd birthday. (a) Assuming her spaceship returns from Alpha Centauri without stopping, how fast must her spaceship travel? (b) How old will Frank be when she returns?

The International Space Federation constructs a new spaceship that can travel at a speed of 0.995c. Mary, the astronaut, boards the spaceship to travel to Barnards star, which is the second nearest star to our solar system after Alpha Centauri and is 5.98 lightyears away. After reaching Barnards star, the spaceship travels slowly around the star system for three years doing research before returning back to Earth. (a) How much time does her journey take? (b) How much older is her twin Frank than Mary when she returns?

A powerful laser on Earth rotates its laser beam in a circle at a frequency of 0.030 Hz. (a) How fast does the spot that the laser makes on the moon move across the moons landscape? (b) With what rotation frequency should the laser rotate if the laser spot moves across the moons landscape at speed c?

The Lockheed SR-71 Blackbird may be the fastest nonresearch airplane ever built; it traveled at 2200 miles/ hour (983 m/s) and was in operation from 1966 to 1990. Its length is 32.74 m. (a) By what percentage would it appear to be length contracted while in fl ight? (b) How much time difference would occur on an atomic clock in the plane compared to a similar clock on Earth during a fl ight of the Blackbird over its range of 3200 km?

A spaceship is coming directly toward you while you are in the International Space Station. You are told that the spaceship is shining sodium light (with an intense yellow doublet of wavelengths 588.9950 and 589.5924 nm). You have an apparatus that can resolve two closely spaced wavelengths if the difference is   0.55 nm. If you fi nd that you can just resolve the doublet, how fast is the spaceship traveling with respect to you?

Quasars are among the most distant objects in the universe and are moving away from us at very high speeds, as discussed in Chapter 16. Astrophysicists use the redshift parameter z to determine the redshift of such rapidly moving objects. The parameter z is determined by observing a wavelength of a known spectral line of wavelength source on Earth; z /source (  source)/source. Find the speed of two quasars having z values of 1.9 and 4.9.

One possible decay mode of the neutral kaon is K0 S 0  0. The rest energies of the K0 and 0 are 498 MeV and 135 MeV, respectively. The kaon is initially at rest when it decays. (a) How much energy is released in the decay? (b) What are the momentum and relative directions of the two neutral pions (0)?

The sun radiates energy at a rate of 3.9  1026 W. (a) At what rate is the sun losing mass? (b) At that rate, how much time would it take to exhaust the suns fuel supply? The suns mass is 2.0  1030 kg, and you may assume that the reaction producing the energy is about 0.7% effi cient. Compare your answer with the suns expected remaining lifetime, about 5 Gy.

One way astrophysicists have identifi ed extrasolar planets orbiting distant stars is by observing redshifts or blueshifts in the stars spectrum due to the fact that the star and planet each revolve around their common center of mass. (See Scientifi c American, August 2010, p. 41.) Consider a star the size of our sun (mass 1.99  1030 kg), with a planet the size of Jupiter (1.90  1027 kg) in a circular orbit of radius 7.79  1011 m and a period of 11.9 years. (a) Find the speed of the star revolving around the systems center of mass. (b) Assume that Earth is in the planets orbital plane, so that at one point in its orbit the star is moving directly toward Earth, and at the opposite point it moves directly away from Earth. How much is 550-nm light redshifted and blueshifted at those two extreme points?

Small differences in the wavelengths in the suns spectrum are detected when measurements are taken from different parts of the suns disk. Specifi cally, measurements of the 656-nm line in hydrogen taken from opposite sides on the suns equatorone side approaching Earth and the other recedingdiffer from each other by 0.0090 nm. Use this information to fi nd the rotational period of the suns equator. Express your answer in days. (The suns equatorial radius is 6.96  108 m.)