Determining the Masses of Stars. Many of the stars in the

Suppose the two lightning bolts shown in Fig. 37.5a are simultaneous to an observer on the train. Show that they are not simultaneous to an observer on the ground. Which lightning strike does the ground observer measure to come first?

The positive muon an unstable particle, lives on average (measured in its own frame of reference) before decaying. (a) If such a particle is moving, with respect to the laboratory, with a speed of what average lifetime is measured in the laboratory? (b) What average distance, measured in the laboratory, does the particle move before decaying?

How fast must a rocket travel relative to the earth so that time in the rocket slows down to half its rate as measured by earth-based observers? Do present-day jet planes approach such speeds?

A spaceship flies past Mars with a speed of relative to the surface of the planet. When the spaceship is directly overhead, a signal light on the Martian surface blinks on and then off. An observer on Mars measures that the signal light was on for (a) Does the observer on Mars or the pilot on the spaceship measure the proper time? (b) What is the duration of the light pulse measured by the pilot of the spaceship?

The negative pion is an unstable particle with an average lifetime of (measured in the rest frame of the pion). (a) If the pion is made to travel at very high speed relative to a laboratory, its average lifetime is measured in the laboratory to be Calculate the speed of the pion expressed as a fraction of (b) What distance, measured in the laboratory, does the pion travel during its average lifetime?

As you pilot your space utility vehicle at a constant speed toward the moon, a race pilot flies past you in her spaceracer at a constant speed of relative to you. At the instant the spaceracer passes you, both of you start timers at zero. (a) At the instant when you measure that the spaceracer has traveled past you, what does the race pilot read on her timer? (b) When the race pilot reads the value calculated in part (a) on her timer, what does she measure to be your distance from her? (c) At the instant when the race pilot reads the value calculated in part (a) on her timer, what do you read on yours?

A spacecraft flies away from the earth with a speed of relative to the earth and then returns at the same speed. The spacecraft carries an atomic clock that has been carefully synchronized with an identical clock that remains at rest on earth. The spacecraft returns to its starting point 365 days (1 year) later, as measured by the clock that remained on earth. What is the difference in the elapsed times on the two clocks, measured in hours? Which clock, the one in the spacecraft or the one on earth, shows the shorter elapsed time?

An alien spacecraft is flying overhead at a great distance as you stand in your backyard. You see its searchlight blink on for The first officer on the spacecraft measures that the searchlight is on for (a) Which of these two measured times is the proper time? (b) What is the speed of the spacecraft relative to the earth expressed as a fraction of the speed of light c?

A spacecraft of the Trade Federation flies past the planet Coruscant at a speed of . A scientist on Coruscant measures the length of the moving spacecraft to be The spacecraft later lands on Coruscant, and the same scientist measures the length of the now stationary spacecraft. What value does she get?

A meter stick moves past you at great speed. Its motion relative to you is parallel to its long axis. If you measure the length of the moving meter stick to be 1.00 ft (1ft = 0.3048 m) - for example, by comparing it to a 1-foot ruler that is at rest relative to you - at what speed is the meter stick moving relative to you?

Why Are We Bombarded by Muons? Muons are unstable subatomic particles that decay to electrons with a mean lifetime of \(2.2 \mu \mathrm{s}\). They are produced when cosmic rays bombard the upper atmosphere about 10 km above the earth's surface, and they travel very close to the speed of light. The problem we want to address is why we see any of them at the earth's surface. (a) What is the greatest distance a muon could travel during its \(2.2-\mu \mathrm{s}\) lifetime? (b) According to your answer in part (a), it would seem that muons could never make it to the ground. But the \(2.2-\mu \mathrm{s}\) lifetime is measured in the frame of the muon, and muons are moving very fast. At a speed of 0.999c, what is the mean lifetime of a muon as measured by an observer at rest on the earth? How far would the muon travel in this time? Does this result explain why we find muons in cosmic rays? (c) From the point of view of the muon, it still lives for only \(2.2 \mu \mathrm{s}\), so how does it make it to the ground? What is the thickness of the 10 km of atmosphere through which the muon must travel, as measured by the muon? Is it now clear how the muon is able to reach the ground?

An unstable particle is created in the upper atmosphere from a cosmic ray and travels straight down toward the surface of the earth with a speed of relative to the earth. A scientist at rest on the earths surface measures that the particle is created at an altitude of (a) As measured by the scientist, how much time does it take the particle to travel the to the surface of the earth? (b) Use the length- contraction formula to calculate the distance from where the particle is created to the surface of the earth as measured in the particles frame. (c) In the particles frame, how much time does it take the particle to travel from where it is created to the surface of the earth? Calculate this time both by the time dilation formula and from the distance calculated in part (b). Do the two results agree?

As measured by an observer on the earth, a spacecraft runway on earth has a length of (a) What is the length of the runway as measured by a pilot of a spacecraft flying past at a speed of relative to the earth? (b) An observer on earth measures the time interval from when the spacecraft is directly over one end of the runway until it is directly over the other end. What result does she get? (c) The pilot of the spacecraft measures the time it takes him to travel from one end of the runway to the other end. What value does he get?

A rocket ship flies past the earth at 85.0% of the speed of light. Inside, an astronaut who is undergoing a physical examination is having his height measured while he is lying down parallel to the direction the rocket ship is moving. (a) If his height is measured to be 2.00 m by his doctor inside the ship, what height would a person watching this from earth measure for his height? (b) If the earth-based person had measured 2.00 m, what would the doctor in the spaceship have measured for the astronauts height? Is this a reasonable height? (c) Suppose the astronaut in part (a) gets up after the examination and stands with his body perpendicular to the direction of motion. What would the doctor in the rocket and the observer on earth measure for his height now?

An observer in frame \(S^{\prime}\) is moving to the right (+x-direction) at speed u = 0.600c away from a stationary observer in frame S. The observer in \(S^{\prime}\) measures the speed \(v^{\prime}\) of a particle moving to the right away from her. What speed v does the observer in S measure for the particle if (a) \(v^{\prime}=0.400 c\); (b) \(v^{\prime}=0.900 c\); (c) \(v^{\prime}=0.990 c\)?

Space pilot Mavis zips past Stanley at a constant speed relative to him of 0.800c. Mavis and Stanley start timers at zero when the front of Mavis’s ship is directly above Stanley. When Mavis reads 5.00 s on her timer, she turns on a bright light under the front of her spaceship. (a) Use the Lorentz coordinate transformation derived in Example 37.6 to calculate x and t as measured by Stanley for the event of turning on the light. (b) Use the time dilation formula, Eq. (37.6), to calculate the time interval between the two events (the front of the spaceship passing overhead and turning on the light) as measured by Stanley. Compare to the value of t you calculated in part (a). (c) Multiply the time interval by Mavis’s speed, both as measured by Stanley, to calculate the distance she has traveled as measured by him when the light turns on. Compare to the value of x you calculated in part (a).

A pursuit spacecraft from the planet Tatooine is attempting to catch up with a Trade Federation cruiser. As measured by an observer on Tatooine, the cruiser is traveling away from the planet with a speed of The pursuit ship is traveling at a speed of relative to Tatooine, in the same direction as the cruiser. (a) For the pursuit ship to catch the cruiser, should the velocity of the cruiser relative to the pursuit ship be directed toward or away from the pursuit ship? (b) What is the speed of the cruiser relative to the pursuit ship?

An extraterrestrial spaceship is moving away from the earth after an unpleasant encounter with its inhabitants. As it departs, the spaceship fires a missile toward the earth. An observer on earth measures that the spaceship is moving away with a speed of 0.600c. An observer in the spaceship measures that the missile is moving away from him at a speed of 0.800c. As measured by an observer on earth, how fast is the missile approaching the earth?

Two particles are created in a high-energy accelerator and move off in opposite directions. The speed of one particle, as measured in the laboratory, is and the speed of each particle relative to the other is What is the speed of the second particle, as measured in the laboratory?

Two particles in a high-energy accelerator experiment are approaching each other head-on, each with a speed of as measured in the laboratory. What is the magnitude of the velocity of one particle relative to the other?

Two particles in a high-energy accelerator experiment approach each other head-on with a relative speed of 0.890c. Both particles travel at the same speed as measured in the laboratory. What is the speed of each particle, as measured in the laboratory?

An enemy spaceship is moving toward your starfighter with a speed, as measured in your frame, of The enemy ship fires a missile toward you at a speed of relative to the enemy ship (Fig. E37.22). (a) What is the speed of the missile relative to you? Express your answer in terms of the speed of light. (b) If you measure that the enemy ship is away from you when the missile is fired, how much time, measured in your frame, will it take the missile to reach you?

An imperial spaceship, moving at high speed relative to the planet Arrakis, fires a rocket toward the planet with a speed of relative to the spaceship. An observer on Arrakis measures that the rocket is approaching with a speed of What is the speed of the spaceship relative to Arrakis? Is the spaceship moving toward or away from Arrakis?

Electromagnetic radiation from a star is observed with an earth-based telescope. The star is moving away from the earth with a speed of 0.600c. If the radiation has a frequency of \(8.64\times10^{14}\mathrm{\ Hz}\) in the rest frame of the star, what is the frequency measured by an observer on earth?

Tell It to the Judge. (a) How fast must you be approaching a red traffic light \((\lambda=675 \mathrm{\ nm})\) for it to appear yellow \((\lambda=575 \mathrm{\ nm})\)? Express your answer in terms of the speed of light. (b) If you used this as a reason not to get a ticket for running a red light, how much of a fine would you get for speeding? Assume that the fine is $1.00 for each kilometer per hour that your speed exceeds the posted limit of 90 km/h.

A source of electromagnetic radiation is moving in a radial direction relative to you. The frequency you measure is 1.25 times the frequency measured in the rest frame of the source. What is the speed of the source relative to you? Is the source moving toward you or away from you?

A proton has momentum with magnitude \(p_0\) when its speed is 0.400c. In terms of \(p_0\), what is the magnitude of the proton's momentum when its speed is doubled to 0.800c?

When Should You Use Relativity? As you have seen, relativistic calculations usually involve the quantity When is appreciably greater than 1, we must use relativistic formulas instead of Newtonian ones. For what speed (in terms of ) is the value of (a) 1.0% greater than 1; (b) 10% greater than 1; (c) 100% greater than 1?

(a) At what speed is the momentum of a particle twice as great as the result obtained from the nonrelativistic expression Express your answer in terms of the speed of light. (b) A force is applied to a particle along its direction of motion. At what speed is the magnitude of force required to produce a given acceleration twice as great as the force required to produce the same acceleration when the particle is at rest? Express your answer in terms of the speed of light.

As measured in an earth-based frame, a proton is moving in the ?x-direction at a speed of . (a) What force (magnitude and direction) is required to produce an acceleration in the that has magnitude ? (b) What magnitude of acceleration does the force calculated in part (a) give to a proton that is initially at rest?

An electron is acted upon by a force of \(5.00\times10^{-15}\mathrm{\ N}\) due to an electric field. Find the acceleration this force produces in each case: (a) The electrons speed is 1.00 km/s. (b) The electrons speed is \(2.50\times10^8\mathrm{\ m}/\mathrm{s}\) and the force is parallel to the velocity.

Relativistic Baseball. Calculate the magnitude of the force required to give a 0.145-kg baseball an acceleration \(a=1.00\mathrm{\ m}/\mathrm{s}^2\) in the direction of the baseball’s initial velocity when this velocity has a magnitude of (a) 10.0 m/s; (b) 0.900c; (c) 0.990c. (d) Repeat parts (a), (b), and (c) if the force and acceleration are perpendicular to the velocity.

What is the speed of a particle whose kinetic energy is equal to (a) its rest energy and (b) five times its rest energy?

If a muon is traveling at 0.999c, what are its momentum and kinetic energy? (The mass of such a muon at rest in the laboratory is 207 times the electron mass.)

A proton (rest mass \(1.67\times10^{-27}\mathrm{\ kg}\)) has total energy that is 4.00 times its rest energy. What are (a) the kinetic energy of the proton; (b) the magnitude of the momentum of the proton; (c) the speed of the proton?

(a) How much work must be done on a particle with mass to accelerate it (a) from rest to a speed of and (b) from a speed of to a speed of (Express the answers in terms of ) (c) How do your answers in parts (a) and (b) compare?

(a) By what percentage does your rest mass increase when you climb to the top of a ten-story building? Are you aware of this increase? Explain. (b) By how many grams does the mass of a spring with force constant change when you compress it by Does the mass increase or decrease? Would you notice the change in mass if you were holding the spring? Explain.

A 60.0-kg person is standing at rest on level ground. How fast would she have to run to (a) double her total energy and (b) increase her total energy by a factor of 10?

An Antimatter Reactor. When a particle meets its antiparticle, they annihilate each other and their mass is converted to light energy. The United States uses approximately of energy per year. (a) If all this energy came from a futuristic antimatter reactor, how much mass of matter and antimatter fuel would be consumed yearly? (b) If this fuel had the density of iron and were stacked in bricks to form a cubical pile, how high would it be? (Before you get your hopes up, antimatter reactors are a long way in the futureif they ever will be feasible.)

Electrons are accelerated through a potential difference of 750 kV, so that their kinetic energy is (a) What is the ratio of the speed of an electron having this energy to the speed of light, c? (b) What would the speed be if it were computed from the principles of classical mechanics?

A particle has rest mass \(6.64 \times 10^{-27}\) kg and \(2.10 \times 10^{-18} \mathrm{~kg} \cdot \mathrm{m} / \mathrm{s}\) momentum (a) What is the total energy (kinetic plus rest energy) of the particle? (b) What is the kinetic energy of the particle? (c) What is the ratio of the kinetic energy to the rest energy of the particle?

A speck of dust is accelerated from rest to a speed of by a constant N force. (a) If the nonrelativistic mechanics is used, how far does the object travel to reach its final speed? (b) Using the correct relativistic treatment of Section 37.8, how far does the object travel to reach its final speed? (c) Which distance is greater? Why?

Compute the kinetic energy of a proton (mass \(1.67\times10^{-27}\mathrm{\ kg}\)) using both the nonrelativistic and relativistic expressions, and compute the ratio of the two results (relativistic divided by nonrelativistic) for speeds of (a) \(8.00\times10^7\mathrm{\ m}/\mathrm{s}\) and (b) \(2.85\times10^8\mathrm{\ m}/\mathrm{s}\).

What is the kinetic energy of a proton moving at (a) 0.100c; (b) 0.500c; (c) 0.900c? How much work must be done to (d) increase the proton’s speed from 0.100c to 0.500c and (e) increase the proton's speed from 0.500c to 0.900c? (f ) How do the last two results compare to results obtained in the nonrelativistic limit?

(a) Through what potential difference does an electron have to be accelerated, starting from rest, to achieve a speed of 0.980c? (b) What is the kinetic energy of the electron at this speed? Express your answer in joules and in electron volts.

Creating a Particle. Two protons (each with rest mass \(M=1.67 \times 10^{-27} \mathrm{~kg}\)) are initially moving with equal speeds in opposite directions. The protons continue to exist after a collision that also produces an \(\eta^{0}\) particle (see Chapter 44). The rest mass of the \(\eta^{0}\) is \(m=9.75 \times 10^{-28} \mathrm{~kg}\) (a) If the two protons and the \(\eta^{0}\) are all at rest after the collision, find the initial speed of the protons, expressed as a fraction of the speed of light. (b) What is the kinetic energy of each proton? Express your answer in MeV. (c) What is the rest energy of the \(\eta^{0}\) expressed in MeV? (d) Discuss the relationship between the answers to parts (b) and ©

The sun produces energy by nuclear fusion reactions, in which matter is converted into energy. By measuring the amount of energy we receive from the sun, we know that it is producing energy at a rate of \(3.8\times10^{26}\mathrm{\ W}\). (a) How many kilograms of matter does the sun lose each second? Approximately how many tons of matter is this (1 ton = 2000 lbs)? (b) At this rate, how long would it take the sun to use up all its mass?

Inside a spaceship flying past the earth at three-fourths the speed of light, a pendulum is swinging. (a) If each swing takes 1.50 s as measured by an astronaut performing an experiment inside the spaceship, how long will the swing take as measured by a person at mission control on earth who is watching the experiment? (b) If each swing takes 1.50 s as measured by a person at mission control on earth, how long will it take as measured by the astronaut in the spaceship?

After being produced in a collision between elementary particles, a positive pion must travel down a 1.90-km-long tube to reach an experimental area. A particle has an average lifetime (measured in its rest frame) of the we are considering has this lifetime. (a) How fast must the travel if it is not to decay before it reaches the end of the tube? (Since will be very close to write and give your answer in terms of rather than ) (b) The has a rest energy of What is the total energy of the at the speed calculated in part (a)?

A cube of metal with sides of length a sits at rest in a frame S with one edge parallel to the x-axis. Therefore, in S the cube has volume \(a^3\). Frame S’ moves along the x-axis with a speed u. As measured by an observer in frame S’, what is the volume of the metal cube?

The starships of the Solar Federation are marked with the symbol of the federation, a circle, while starships of the Denebian Empire are marked with the empires symbol, an ellipse whose major axis is 1.40 times longer than its minor axis in Fig. P37.51). How fast, relative to an observer, does an empire ship have to travel for its marking to be confused with the marking of a federation ship?

A space probe is sent to the vicinity of the star Capella, which is 42.2 light-years from the earth. (A light-year is the distance light travels in a year.) The probe travels with the speed of 0.9930c. An astronaut recruit on board is 19 years old when the probe leaves the earth. What is her biological age when the probe reaches Capella?

A particle is said to be extremely relativistic when its kinetic energy is much greater than its rest energy. (a) What is the speed of a particle (expressed as a fraction of ) such that the total energy is ten times the rest energy? (b) What is the percentage difference between the left and right sides of Eq. (37.39) if \(\left(m c^{2}\right)^{2}\) is neglected for a particle with the speed calculated in part (a)?

Everyday Time Dilation. Two atomic clocks are carefully synchronized. One remains in New York, and the other is loaded on an airliner that travels at an average speed of 250 m/s and then returns to New York. When the plane returns, the elapsed time on the clock that stayed behind is 4.00 h. By how much will the readings of the two clocks differ, and which clock will show the shorter elapsed time? (Hint: Since \(u \ll c\), you can simplify \(\sqrt{1-u^{2} / c^{2}}\) by a binomial expansion.) Text Transcription: u ll c sqrt 1-u^2/c^2

The Large Hadron Collider (LHC). Physicists and engineers from around the world have come together to build the largest accelerator in the world, the Large Hadron Collider (LHC) at the CERN Laboratory in Geneva, Switzerland. The machine will accelerate protons to kinetic energies of 7 TeV in an underground ring 27 km in circumference. (For the latest news and more information on the LHC, visit www.cern.ch.) (a) What speed will protons reach in the LHC? (Since is very close to write and give your answer in terms of ) (b) Find the relativistic mass, of the accelerated protons in terms of their rest mass.

A nuclear bomb containing 12.0 lg of plutonium explodes. The sum of the rest masses of the products of the explosion is less than the original rest mass by one part in \(10^4\). (a) How much energy is released in the explosion? (b) If the explosion takes place in \(4.00\ \mu\mathrm{s}\), what is the average power developed by the bomb? (c) What mass of water could the released energy lift to a height of 1.00 km?

Cerenkov Radiation. The Russian physicist P. A. Cerenkov discovered that a charged particle traveling in a solid with a speed exceeding the speed of light in that material radiates electromagnetic radiation. (This is analogous to the sonic boom produced by an aircraft moving faster than the speed of sound in air; see Section 16.9. Cerenkov shared the 1958 Nobel Prize for this discovery.) What is the minimum kinetic energy (in electron volts) that an electron must have while traveling inside a slab of crown glass (n = 1.52) in order to create this Cerenkov radiation?

A photon with energy is emitted by an atom with mass which recoils in the opposite direction. (a) Assuming that the motion of the atom can be treated nonrelativistically, compute the recoil speed of the atom. (b) From the result of part (a), show that the recoil speed is much less than whenever is much less than the rest energy of the atom.

In an experiment, two protons are shot directly toward each other, each moving at half the speed of light relative to the laboratory. (a) What speed does one proton measure for the other proton? (b) What would be the answer to part (a) if we used only nonrelativistic Newtonian mechanics? (c) What is the kinetic energy of each proton as measured by (i) an observer at rest in the laboratory and (ii) an observer riding along with one of the protons? (d) What would be the answers to part (c) if we used only nonrelativistic Newtonian mechanics?

Two protons are moving away from each other. In the frame of each proton, the other proton has a speed of . What does an observer in the rest frame of the earth measure for the speed of each proton?

Frame \(S^{\prime}\) has an x-component of velocity u relative to frame S, and at \(t=t^{\prime}=0\) the two frames coincide (see Fig. 37.3). A light pulse with a spherical wave front is emitted at the origin of \(S^{\prime}\) at time \(t^{\prime}=0\) Its distance \(x^{\prime}\) from the origin after a time \(t^{\prime}\) is given by \(x^{\prime 2}=c^{2} t^{\prime 2}\). Use the Lorentz coordinate transformation to transform this equation to an equation in x and t, and show that the result is \(x^{2}=c^{2} t^{2}\); that is, the motion appears exactly the same in frame of reference S as it does in \(S^{\prime}\); the wave front is observed to be spherical in both frames.

In certain radioactive beta decay processes, the beta particle (an electron) leaves the atomic nucleus with a speed of 99.95% the speed of light relative to the decaying nucleus. If this nucleus is moving at 75.00% the speed of light in the laboratory reference frame, find the speed of the emitted electron relative to the laboratory reference frame if the electron is emitted (a) in the same direction that the nucleus is moving and (b) in the opposite direction from the nucleuss velocity. (c) In each case in parts (a) and (b), find the kinetic energy of the electron as measured in (i) the laboratory frame and (ii) the reference frame of the decaying nucleus.

CALC A particle with mass accelerated from rest by a constant force will, according to Newtonian mechanics, continue to accelerate without bound; that is, as Show that according to relativistic mechanics, the particles speed approaches as [Note: A useful integral is

Two events are observed in a frame of reference S to occur at the same space point, the second occurring 1.80s after the first. In a second frame S’ moving relative to S, the second event is observed to occur 2.35s after the first. What is the difference between the positions of the two events as measured in S’?

Two events observed in a frame of reference have positions and times given by and respectively. (a) Frame moves along the just fast enough that the two events occur at the same position in Show that in the time interval between the two events is given by where and Hence show that if there is no frame in which the two events occur at the same point. The interval is sometimes called the proper time interval for the events. Is this term appropriate? (b) Show that if there is a different frame of reference in which the two events occur simultaneously. Find the distance between the two events in express your answer in terms of and This distance is sometimes called a proper length. Is this term appropriate? (c) Two events are observed in a frame of reference to occur simultaneously at points separated by a distance of In a second frame moving relative to along the line joining the two points in the two events appear to be separated by What is the time interval between the events as measured in [ S? Hint: Apply the result obtained in part (b).]

Albert in Wonderland. Einstein and Lorentz, being avid tennis players, play a fast-paced game on a court where they stand from each other. Being very skilled players, they play without a net. The tennis ball has mass You can ignore gravity and assume that the ball travels parallel to the ground as it travels between the two players. Unless otherwise specified, all measurements are made by the two men. (a) Lorentz serves the ball at What is the balls kinetic energy? (b) Einstein slams a return at What is the balls kinetic energy? (c) During Einsteins return of the ball in part (a), a white rabbit runs beside the court in the direction from Einstein to Lorentz. The rabbit has a speed of relative to the two men. What is the speed of the rabbit relative to the ball? (d) What does the rabbit measure as the distance from Einstein to Lorentz? (e) How much time does it take for the rabbit to run according to the players? (f ) The white rabbit carries a pocket watch. He uses this watch to measure the time (as he sees it) for the distance from Einstein to Lorentz to pass by under him. What time does he measure?

One of the wavelengths of light emitted by hydrogen atoms under normal laboratory conditions is in the red portion of the electromagnetic spectrum. In the light emitted from a distant galaxy this same spectral line is observed to be Doppler-shifted to in the infrared portion of the spectrum. How fast are the emitting atoms moving relative to the earth? Are they approaching the earth or receding from it?

Measuring Speed by Radar. A baseball coach uses a radar device to measure the speed of an approaching pitched baseball. This device sends out electromagnetic waves with frequency and then measures the shift in frequency of the waves reflected from the moving baseball. If the fractional frequency shift produced by a baseball is what is the baseballs speed in (Hint: Are the waves Doppler-shifted a second time when reflected off the ball?)

Space Travel? Travel to the stars requires hundreds or thousands of years, even at the speed of light. Some people have suggested that we can get around this difficulty by accelerating the rocket (and its astronauts) to very high speeds so that they will age less due to time dilation. The fly in this ointment is that it takes a great deal of energy to do this. Suppose you want to go to the immense red giant Betelgeuse, which is about 500 light-years away. (A light-year is the distance that light travels in a year.) You plan to travel at constant speed in a 1000-kg rocket ship (a little over a ton), which, in reality, is far too small for this purpose. In each case that follows, calculate the time for the trip, as measured by people on earth and by astronauts in the rocket ship, the energy needed in joules, and the energy needed as a percentage of U.S. yearly use (which is ). For comparison, arrange your results in a table showing E (in J), and E (as % of U.S. use). The rocket ships speed is (a) 0.50c; (b) 0.99c; (c) 0.9999c. On the basis of your results, does it seem likely that any government will invest in such high-speed space travel any time soon?

A spaceship moving at constant speed relative to us broadcasts a radio signal at constant frequency As the spaceship approaches us, we receive a higher frequency after it has passed, we receive a lower frequency. (a) As the spaceship passes by, so it is instantaneously moving neither toward nor away from us, show that the frequency we receive is not and derive an expression for the frequency we do receive. Is the frequency we receive higher or lower than (Hint: In this case, successive wave crests move the same distance to the observer and so they have the same transit time. Thus equals Use the time dilation formula to relate the periods in the stationary and moving frames.) (b) A spaceship emits electromagnetic waves of frequency as measured in a frame moving with the ship. The spaceship is moving at a constant speed relative to us. What frequency do we receive when the spaceship is approaching us? When it is moving away? In each case what is the shift in frequency, (c) Use the result of part (a) to calculate the frequency and the frequency shift we receive at the instant that the ship passes by us. How does the shift in frequency calculated here compare to the shifts calculated in part (b)?

In a particle accelerator a proton moves with constant speed 0.750c in a circle of radius 628 m. What is the net force on the proton?

CP The French physicist Armand Fizeau was the first to measure the speed of light accurately. He also found experimentally that the speed, relative to the lab frame, of light traveling in a tank of water that is itself moving at a speed relative to the lab frame is where is the index of refraction of water. Fizeau called the dragging coefficient and obtained an experimental value of What value of do you calculate from relativistic transformations?

Lorentz Transformation for Acceleration. Using a method analogous to the one in the text to find the Lorentz transformation formula for velocity, we can find the Lorentz transformation for acceleration. Let frame \(S^{\prime}\) have a constant x-component of velocity u relative to frame S. An object moves relative to frame S along the x-axis with instantaneous velocity \(v_x\) and instantaneous acceleration \(a_x\). (a) Show that its instantaneous acceleration in frame \(S^{\prime}\) is \(a_{x}^{\prime}=a_{x}\left(1-\frac{u^{2}}{c^{2}}\right)^{3 / 2}\left(1-\frac{u v_{x}}{c^{2}}\right)^{-3}\) [Hint: Express the acceleration in \(S^{\prime}\) as \(a_{x}^{\prime}=d v_{x}^{\prime} / d t^{\prime}\). Then use Eq. (37.21) to express \(d t^{\prime}\) in terms of dt and dx, and use Eq. (37.22) to express \(d v_{x}^{\prime}\) in terms of u and \(dv_x\) The velocity of the object in S is \(v_{x}=d x / d t\).] (b) Show that the acceleration in frame S can be expressed as \(a_{x}=a_{x}^{\prime}\left(1-\frac{u^{2}}{c^{2}}\right)^{3 / 2}\left(1+\frac{u v_{x}^{\prime}}{c^{2}}\right)^{-3}\) where \(v_{x}^{\prime}=d x^{\prime} / d t^{\prime}\) is the velocity of the object in frame \(S^{\prime}\).

CALC A Realistic Version of the Twin Paradox. A rocket ship leaves the earth on January 1, 2100. Stella, one of a pair of twins born in the year 2075, pilots the rocket (reference frame ); the other twin, Terra, stays on the earth (reference frame ). The rocket ship has an acceleration of constant magnitude in its own reference frame (this makes the pilot feel at home, since it simulates the earths gravity). The path of the rocket ship is a straight line in the in frame (a) Using the results of Challenge Problem 37.73, show that in Terras earth frame the rockets acceleration is du dt = ga1 - u2 c2 b 3>2 where is the rockets instantaneous velocity in frame (b) Write the result of part (a) in the form where is a function of and integrate both sides. (Hint: Use the integral given in Problem 37.63.) Show that in Terras frame, the time when Stella attains a velocity is (c) Use the time dilation formula to relate and (infinitesimal time intervals measured in frames and respectively). Combine this result with the result of part (a) and integrate as in part (b) to show the following: When Stella attains a velocity relative to Terra, the time that has elapsed in frame is Here arctanh is the inverse hyperbolic tangent. (Hint: Use the integral given in Challenge Problem 5.124.) (d) Combine the results of parts (b) and (c) to find in terms of and alone. (e) Stella accelerates in a straight-line path for five years (by her clock), slows down at the same rate for five years, turns around, accelerates for five years, slows down for five years, and lands back on the earth. According to Stellas clock, the date is January 1, 2120. What is the date according to Terras clock?

CP Determining the Masses of Stars. Many of the stars in the sky are actually binary stars, in which two stars orbit about their common center of mass. If the orbital speeds of the stars are high enough, the motion of the stars can be detected by the Doppler shifts of the light they emit. Stars for which this is the case are called spectroscopic binary stars. Figure P37.75 shows the simplest case of a spectroscopic binary star: two identical stars, each with mass orbiting their center of mass in a circle of radius The plane of the stars orbits is edge-on to the line of sight of an observer on the earth. (a) The light produced by heated hydrogen gas in a laboratory on the earth has a frequency of In the light received from the stars by a telescope on the earth, hydrogen light is observed to vary in frequency between and Determine whether the binary star system as a whole is moving toward or away from the earth, the speed of this motion, and the orbital speeds of the stars. (Hint: The speeds involved are much less than so you may use the approximate result given in Section 37.6.) (b) The light from each star in the binary system varies from its maximum frequency to its minimum frequency and back again in 11.0 days. Determine the orbital radius and the mass of each star. Give your answer for in kilograms and as a multiple of the mass of the sun, Compare the value of to the distance from the earth to the sun, (This technique is actually used in astronomy to determine the masses of stars. In practice, the problem is more complicated because the two stars in a binary system are usually not identical, the orbits are usually not circular, and the plane of the orbits is usually tilted with respect to the line of sight from the earth.)

CP CALC Relativity and the Wave Equation. (a) Consider the Galilean transformation along the and In frame the wave equation for electromagnetic waves in a vacuum is where represents the electric field in the wave. Show that by using the Galilean transformation the wave equation in frame is found to be This has a different form than the wave equation in Hence the Galilean transformation violates the first relativity postulate that all physical laws have the same form in all inertial reference frames. (Hint: Express the derivatives and in terms of and by use of the chain rule.) (b) Repeat the analysis of part (a), but use the Lorentz coordinate transformations, Eqs. (37.21), and show that in frame the wave equation has the same form as in frame Explain why this shows that the speed of light in vacuum is in both frames and

CP Kaon Production. In high-energy physics, new particles can be created by collisions of fast- moving projectile particles with stationary particles. Some of the kinetic energy of the incident particle is used to create the mass of the new particle. A protonproton collision can result in the creation of a negative kaon and a positive kaon (a) Calculate the minimum kinetic energy of the incident proton that will allow this reaction to occur if the second (target) proton is initially at rest. The rest energy of each kaon is and the rest energy of each proton is (Hint: It is useful here to work in the frame in which the total momentum is zero. But note that the Lorentz transformation must be used to relate the velocities in the laboratory frame to those in the zerototal-momentum frame.) (b) How does this calculated minimum kinetic energy compare with the total rest mass energy of the created kaons? (c) Suppose that instead the two protons are both in motion with velocities of equal magnitude and opposite direction. Find the minimum combined kinetic energy of the two protons that will allow the reaction to occur. How does this calculated minimum kinetic energy compare with the total rest mass energy of the created kaons? (This example shows that when colliding beams of particles are used instead of a stationary target, the energy requirements for producing new particles are reduced substantially.)

Problem 3E How fast must a rocket travel relative to the earth so that time in the rocket “slows down” to half its rate as measured by earth-based observers? Do present-day jet planes approach such speeds?

Problem 4DQ What do you think would be different in everyday life if the speed of light were 10 m/s instead of 3.00 × 108 m/s?

Problem 4E A spaceship flies past Mars with a speed of 0.985c relative to the surface of the planet. When the spaceship is directly overhead, a signal light on the Martian surface blinks on and then off. An observer on Mars measures that the signal light was on for 75.0 ?s. (a) Does the observer on Mars or the pilot on the spaceship measure the proper time? (b) What is the duration of the light pulse measured by the pilot of the spaceship?

Problem 9E A spacecraft of the Trade Federation flies past the planet Coruscant at a speed of 0.600c. A scientist on Coruscant measures the length of the moving spacecraft to be 74.0 m. The spacecraft later lands on Coruscant, and the same scientist measures the length of the now stationary spacecraft. What value does she get?

A student asserts that a material particle must always have a speed slower than that of light, and a massless particle must always move at exactly the speed of light. Is she correct? If so, how do massless particles such as photons and neutrinos acquire this speed? Can’t they start from rest and accelerate? Explain.

Problem 10E A meter stick moves past you at great speed. Its motion relative to you is parallel to its long axis. If you measure the length of the moving meter stick to be 1.00 ft (1 ft = 0.3048 m)—for example, by comparing it to a 1-foot ruler that is at rest relative to you—at what speed is the meter stick moving relative to you?

The speed of light relative to still water is \(2.25 \times 10^{8} \mathrm{~m} / \mathrm{s}\). If the water is moving past us, the speed of light we measure depends on the speed of the water. Do these facts violate Einstein’s second postulate? Explain.

Problem 11E Why Are We Bombarded by Muons? Muons are unstable subatomic particles that decay to electrons with a mean lifetime of 2.2 ?s. They are produced when cosmic rays bombard the upper atmosphere about 10 km above the earth’s surface, and they travel very close to the speed of light. The problem we want to address is why we see any of them at the earth’s surface. (a) What is the greatest distance a muon could travel during its 2.2-?s lifetime? (b) According to your answer in part (a), it would seem that muons could never make it to the ground. But the 2.2-?s lifetime is measured in the frame of the muon, and muons are moving very fast. At a speed of 0.999c, what is the mean lifetime of a muon as measured by an observer at rest on the earth? How far would the muon travel in this time? Does this result explain why we find muons in cosmic rays? (c) From the point of view of the muon, it still lives for only 2.2 ?s, so how does it make it to the ground? What is the thickness of the 10 km of atmosphere through which the muon must travel, as measured by the muon? Is it now clear how the muon is able to reach the ground?

Problem 12DQ When a monochromatic light source moves toward an observer, its wavelength appears to be shorter than the value measured when the source is at rest. Does this

Problem 28E When Should You Use Relativity? As you have seen, relativistic calculations usually involve the quantity ?. When ? is appreciably greater than 1, we must use relativistic formulas instead of Newtonian ones. For what speed v (in terms of c) is the value of ? (a) 1.0% greater than 1; (b) 10% greater than 1; (c) 100% greater than 1?

(a) At what speed is the momentum of a particle twice as great as the result obtained from the nonrelativistic expression mv? Express your answer in terms of the speed of light. (b) A force is applied to a particle along its direction of motion. At what speed is the magnitude of force required to produce a given acceleration twice as great as the force required to produce the same acceleration when the particle is at rest? Express your answer in terms of the speed of light.

As measured in an earth-based frame, a proton is moving in the +x-direction at a speed of 2.30 \(\times\) 108 s m/s. (a) What force (magnitude and direction) is required to produce an acceleration in the ?x-direction that has magnitude 2.30 \(\times\) 108 s m/s\(^{2}\)? (b) What magnitude of acceleration does the force calculated in part (a) give to a proton that is initially at rest?

Problem 40E Electrons are accelerated through a potential difference of 750 kV, so that their kinetic energy is 7.50 × 105 eV. (a) What is the ratio of the speed v of an electron having this energy to the speed of light, c? (b) What would the speed be if it were computed from the principles of classical mechanics?

Problem 41E A particle has rest mass 6.64 × 10-27 kg and momentum 2.10 × 10-18 kg ? m/s. (a) What is the total energy (kinetic plus rest energy) of the particle? (b) What is the kinetic energy of the particle? (c) What is the ratio of the kinetic energy to the rest energy of the particle?

Problem 42E A 0.100-?g speck of dust is accelerated from rest to a speed of 0.900c by a constant 1.00 × 106 N force. (a) If the non-relativistic mechanics is used, how far does the object travel to reach its final speed? (b) Using the correct relativistic treatment of Section 37.8, how far does the object travel to reach its final speed? (c) Which distance is greater? Why?

Problem 43E Compute the kinetic energy of a proton (mass 1.67 × 10-27 kg) using both the nonrelativistic and relativistic expressions, and compute the ratio of the two results (relativistic divided by nonrelativistic) for speeds of (a) 8.00 × 107 m/s and (b) 2.85 × 108 m/s.

What is the kinetic energy of a proton moving at (a) 0.100c; (b) 0.500c; (c) 0.900c? How much work must be done to (d) increase the proton’s speed from 0.100c to 0.500c and (e) increase the proton’s speed from 0.500c to 0.900c? (f) How do the last two results compare to results obtained in the nonrelativistic limit?

Problem 45E (a) Through what potential difference does an electron have to be accelerated, starting from rest, to achieve a speed of 0.980c? (b) What is the kinetic energy of the electron at this speed? Express your answer in joules and in electron volts.

Problem 64P Two events are observed in a frame of reference S to occur at the same space point, the second occurring 1.80 s after the first. In a second frame S' moving relative to S, the second event is observed to occur 2.35 s after the first. What is the difference between the positions of the two events as measured in S'.

Two events observed in a frame of reference have positions and times given by \(x_{1}\), \(t_{1}\) and \(x_{2}\), \(t_{2}\) respectively. (a) Frame \(S^{\prime}\) moves along the -axis just fast enough that the two events occur at the same position in \(S^{\prime}\). Show that in \(S^{\prime}\) the time interval \(\Delta t^{\prime}\) between the two events is given by \(\Delta t^{\prime}=\sqrt{(\Delta t)^{2}-\left(\frac{\Delta x}{c}\right)^{2}}\) where \(\Delta x=x_{2}-x_{1}\) and \(\Delta t=t_{2}-t_{1}\). Hence show that if \(\Delta x>c\) \(\Delta t\), there is no frame \(S^{\prime}\) in which the two events occur at the same point. The interval \(\Delta t^{\prime}\) is sometimes called the proper time interval for the events. Is this term appropriate? (b) Show that if \(\Delta x>c\) \(\Delta t\),there is a different frame of reference in which the two events occur simultaneously. Find the distance between the two events in \(S^{\prime}\); express your answer in terms of \(\Delta x\), \(\Delta t\), and This distance is sometimes called a proper length. Is this term appropriate? (c) Two events are observed in a frame of reference \(S^{\prime}\) to occur simultaneously at points separated by a distance of 2.50 m. In a second frame \(S^{\prime}\) moving relative to \(S^{\prime}\) along the line joining the two points in \(S^{\prime}\) the two events appear to be separated by 5.00 m. What is the time interval between the events as measured in \(S^{\prime}\)? [Hint: Apply the result obtained in part (b) Equation Transcription: Text Transcription: x_1 t_1 x_2 t_2 S' S' S' Deltat' Deltat'=sqrt (deltat)^2-(deltax over c)2 Deltax=x_2-x_1 Deltat=t_2-t_1 Delta x>c Delta t S' Delta t’ Delta x>c Deltat S' S' Delta x Delta t S' S' S'

Problem 66P Albert in Wonderland. Einstein and Lorentz, being avid tennis players, play a fast-paced game on a court where they stand 20.0 m from each other. Being very skilled players, they play without a net. The tennis ball has mass 0.0580 kg. You can ignore gravity and assume that the ball travels parallel to the ground as it travels between the two players. Unless otherwise specified, all measurements are made by the two men. (a) Lorentz serves the ball at 80.0 m/s. What is the ball’s kinetic energy? (b) Einstein slams a return at 1.80 × 108 m/s. What is the ball’s kinetic energy? (c) During Einstein’s return of the ball in part (a), a white rabbit runs beside the court in the direction from Einstein to Lorentz. The rabbit has a speed of 2.20 × 108m/s relative to the two men. What is the speed of the rabbit relative to the ball? (d) What does the rabbit measure as the distance from Einstein to Lorentz? (e) How much time does it take for the rabbit to run 20.0 m, according to the players? (f) The white rabbit carries a pocket watch. He uses this watch to measure the time (as he sees it) for the distance from Einstein to Lorentz to pass by under him. What time does he measure?

Kaon Production. In high-energy physics, new particles can be created by collisions of fast-moving projectile particles with stationary particles. Some of the kinetic energy of the incident particle is used to create the mass of the new particle. A proton–proton collision can result in the creation of a negative kaon \(\left(K^{-}\right)\) and a positive kaon \(\left(K^{+}\right)\). \(p+p \rightarrow p+p+K^{-}+K^{+}\) (a) Calculate the minimum kinetic energy of the incident proton that will allow this reaction to occur if the second (target) proton is initially at rest. The rest energy of each kaon is 493.7 MeV and the rest energy of each proton is 938.3 MeV. (Hint: It is useful here to work in the frame in which the total momentum is zero. But note that the Lorentz transformation must be used to relate the velocities in the laboratory frame to those in the zero-total-momentum frame.) (b) How does this calculated minimum kinetic energy compare with the total rest mass energy of the created kaons? (c) Suppose that instead the two protons are both in motion with velocities of equal magnitude and opposite direction. Find the minimum combined kinetic energy of the two protons that will allow the reaction to occur. How does this calculated minimum kinetic energy compare with the total rest mass energy of the created kaons? (This example shows that when colliding beams of particles are used instead of a stationary target, the energy requirements for producing new particles are reduced substantially.) .Equation Transcription: Text Transcription: K- K+ p+p rarrow p+p+K^-+K^+

Problem 1DQ You are standing on a train platform watching a high-speed train pass by. A light inside one of the train cars is turned on and then a little later it is turned off. (a) Who can measure the proper time interval for the duration of the light: you or a passenger on the train? (b) Who can measure the proper length of the train car: you or a passenger on the train? (c) Who can measure the proper length of a sign attached to a post on the train platform: you or a passenger on the train? In each case explain your answer.

If simultaneity is not an absolute concept, does that mean that we must discard the concept of causality? If event A is to cause event B, A must occur first. Is it possible that in some frames Appears to be the cause of B, and in others B appears to be the cause of A? Explain.

Suppose the two lightning bolts shown in Fig. 37.5a are simultaneous to an observer on the train. Show that they are not simultaneous to an observer on the ground. Which lightning strike does the ground observer measure to come first?

Problem 2E The positive muon (?+), an unstable particle, lives on average 2.20 × 10-6 s (measured in its own frame of reference) before decaying. (a) If such a particle is moving, with respect to the laboratory, with a speed of 0.900c, what average lifetime is measured in the laboratory? (b) What average distance, measured in the laboratory, does the particle move before decaying?

rocket is moving to the right at \(\frac{1}{2}\) the speed of light relative to the earth. A light bulb in the center of a room inside the rocket suddenly turns on. Call the light hitting the front end of the room event A and the light hitting the back of the room Event B (Fig. Q37.3). Which event occurs first, A or B, or are they simultaneous, as viewed by (a) an astronaut riding in the rocket and (b) a person at rest on the earth? Equation transcription: Text transcription: frac{1}{2}

The average life span in the United States is about 70 years. Does this mean that it is impossible for an average person to travel a distance greater than 70 light-years away from the earth? (A light-year is the distance light travels in a year.) Explain.

Problem 5E The negative pion (?-) is an unstable particle with an average lifetime of 2.60 × 10-8 s (measured in the rest frame of the pion). (a) If the pion is made to travel at very high speed relative to a laboratory, its average lifetime is measured in the laboratory to be 4.20 × 10-7 s. Calculate the speed of the pion expressed as a fraction of c. (b) What distance, measured in the laboratory, does the pion travel during its average lifetime?

Problem 6DQ You are holding an elliptical serving platter. How would you need to travel for the serving platter to appear round to another observer?

Problem 6E As you pilot your space utility vehicle at a constant speed toward the moon, a race pilot flies past you in her space racer at a constant speed of 0.800c relative to you. At the instant the space racer passes you, both of you start timers at zero. (a) At the instant when you measure that the space racer has traveled 1.20 × 108 m past you, what does the race pilot read on her timer? (b) When the race pilot reads the value calculated in part (a) on her timer, what does she measure to be your distance from her? (c) At the instant when the race pilot reads the value calculated in part (a) on her timer, what do you read on yours?

Problem 7DQ Two events occur at the same space point in a particular inertial frame of reference and are simultaneous in that frame. Is it possible that they may not be simultaneous in a different inertial frame? Explain.

Problem 7E A spacecraft flies away from the earth with a speed of 4.80 × 106 m/s relative to the earth and then returns at the same speed. The spacecraft carries an atomic clock that has been carefully synchronized with an identical clock that remains at rest on earth. The spacecraft returns to its starting point 365 days (1 year) later, as measured by the clock that remained on earth. What is the difference in the elapsed times on the two clocks, measured in hours? Which clock, the one in the spacecraft or the one on earth, shows the shorter elapsed time?

A high-speed train passes a train platform. Larry is a passenger on the train, Adam is standing on the train platform, and David is riding a bicycle toward the platform in the same direction as the train is traveling. Compare the length of a train car as measured by Larry, Adam, and David.

Problem 8E An alien spacecraft is flying overhead at a great distance as you stand in your backyard. You see its searchlight blink on for 0.150 s. The first officer on the spacecraft measures that the searchlight is on for 12.0 ms. (a) Which of these two measured times is the proper time? (b) What is the speed of the spacecraft relative to the earth, expressed as a fraction of the speed of light c?

Problem 9DQ The theory of relativity sets an upper limit on the speed that a particle can have. Are there also limits on the energy and momentum of a particle? Explain.

An unstable particle is created in the upper atmosphere from a cosmic ray and travels straight down toward the surface of the earth with a speed of 0.99540c relative to the earth. A scientist at rest on the earth’s surface measures that the particle is created at an altitude of 45.0 km. (a) As measured by the scientist, how much time does it take the particle to travel the 45.0 km to the surface of the earth? (b) Use the length-contraction formula to calculate the distance from where the particle is created to the surface of the earth as measured in the particle’s frame. (c) In the particle’s frame, how much time does it take the particle to travel from where it is created to the surface of the earth? Calculate this time both by the time dilation formula and from the distance calculated in part (b). Do the two results agree?

Problem 13DQ In principle, does a hot gas have more mass than the same gas when it is cold? Explain. In practice, would this be a measurable effect? Explain.

Problem 13E As measured by an observer on the earth, a spacecraft runway on earth has a length of 3600 m. (a) What is the length of the runway as measured by a pilot of a spacecraft flying past at a speed of 4.00 × 107 m/s relative to the earth? (b) An observer on earth measures the time interval from when the spacecraft is directly over one end of the runway until it is directly over the other end. What result does she get? (c) The pilot of the spacecraft measures the time it takes him to travel from one end of the runway to the other end. What value does he get?

Problem 14DQ Why do you think the development of Newtonian mechanics preceded the more refined relativistic mechanics by so many years?

A rocket ship flies past the earth at 91.0% of the speed of light. Inside, an astronaut who is undergoing a physical examination is having his height measured while he is lying down parallel to the direction in which the ship is moving. (a) If his height is measured to be 2.00 m by his doctor inside the ship, what height would a person watching this from the earth measure? (b) If the earth-based person had measured 2.00 m, what would the doctor in the spaceship have measured for the astronaut’s height? Is this a reasonable height? (c) Suppose the astronaut in part (a) gets up after the examination and stands with his body perpendicular to the direction of motion. What would the doctor in the rocket and the observer on earth measure for his height now?

Problem 15E An observer in frame S’ is moving to the right (+x-direction) at speed u = 0.600c away from a stationary observer in frame S. The observer in S’ measures the speed v’ of a particle moving to the right away from her. What speed v does the observer in S measure for the particle if (a)v’ = 0.400c; (b) v’ = 0.900c; (c) v’ = 0.990c?

37.16 - Space pilot Mavis zips past Stanley at a constant speed relative to him of Mavis and Stanley start timers at zero when the front of Mavis's ship is directly above Stanley. When Mavis reads on her timer, she turns on a bright light under the front of her spaceship. (a) Use the Lorentz coordinate transformation derived in Example to calculate and as measured by Stanley for the event of turning on the light. (b) Use the time dilation formula, Eq. (37.6), to calculate the time interval between the two events (the front of the spaceship passing overhead and turning on the light) as measured by Stanley. Compare to the value of you calculated in part (a). (c) Multiply the time interval by Mavis's speed, both as measured by Stanley, to calculate the distance she has traveled as measured by him when the light turns on. Compare to the value of you calculated in part (a

A pursuit spacecraft from the planet Tatooine is attempting to catch up with a Trade Federation cruiser. As measured by an observer on Tatooine, the cruiser is traveling away from the planet with a speed of 0.600c. The pursuit ship is traveling at a speed of 0.800c relative to Tatooine, in the same direction as the cruiser. (a) For the pursuit ship to catch the cruiser, should the velocity of the cruiser relative to the pursuit ship be directed toward or away from the pursuit ship? (b) What is the speed of the cruiser relative to the pursuit ship?

Problem 18E An extraterrestrial spaceship is moving away from the earth after an unpleasant encounter with its inhabitants. As it departs, the spaceship fires a missile toward the earth. An observer on earth measures that the spaceship is moving away with a speed of 0.600c. An observer in the spaceship measures that the missile is moving away from him at a speed of 0.800c. As measured by an observer on earth, how fast is the missile approaching the earth?

Problem 19E Two particles are created in a high-energy accelerator and move off in opposite directions. The speed of one particle, as measured in the laboratory, is 0.650c, and the speed of each particle relative to the other is 0.950c. What is the speed of the second particle, as measured in the laboratory?

Problem 20E Two particles in a high-energy accelerator experiment are approaching each other head-on, each with a speed of 0.9520c as measured in the laboratory. What is the magnitude of the velocity of one particle relative to the other?

Two particles in a high-energy accelerator experiment approach each other head-on with a relative speed of 0.890c. Both particles travel at the same speed as measured in the laboratory. What is the speed of each particle, as measured in the laboratory?

An enemy spaceship is moving toward your starfighter with a speed, as measured in your frame, of The enemy ship fires a missile toward you at a speed of relative to the enemy ship (Fig. E37.22). (a) What is the speed of the missile relative to you? Express your answer in terms of the speed of light. (b) If you measure that the enemy ship is \(8.00 \times 10^{6} \mathrm{~km}\) away from you when the missile is fired, how much time, measured in your frame, will it take the missile to reach you? Equation transcription: Text transcription: 8.00 \times 10^{6}{~km}

Problem 23E An imperial spaceship, moving at high speed relative to the planet Arrakis, fires a rocket toward the planet with a speed of 0.920c relative to the spaceship. An observer on Arrakis measures that the rocket is approaching with a speed of 0.360c. What is the speed of the spaceship relative to Arrakis? Is the spaceship moving toward or away from Arrakis?

Problem 24E Electromagnetic radiation from a star is observed with an earth based telescope. The star is moving away from the earth with a speed of 0.600c. If the radiation has a frequency of 8.64 × 1014 Hz in the rest frame of the star, what is the frequency measured by an observer on earth?

Problem 25E Tell It to the Judge. (a) How fast must you be approaching a red traffic light (? = 675 nm) for it to appear yellow (? = 575 nm)? Express your answer in terms of the speed of light. (b) If you used this as a reason not to get a ticket for running a red light, how much of a fine would you get for speeding? Assume that the fine is $1.00 for each kilometer per hour that your speed exceeds the posted limit of 90 km/h.

Problem 26E A source of electromagnetic radiation is moving in a radial direction relative to you. The frequency you measure is 1.25 times the frequency measured in the rest frame of the source. What is the speed of the source relative to you? Is the source moving toward you or away from you?

Problem 27E A proton has momentum with magnitude p0 when its speed is 0.400c. In terms of p0, what is the magnitude of the proton’s momentum when its speed is doubled to 0.800c?

Problem 31E An electron is acted upon by a force of 5.00 × 10-15 N due to an electric field. Find the acceleration this force produces in each case: (a) The electron’s speed is 1.00 km/s. (b) The electron’s speed is 2.50 × 108 m/s and the force is parallel to the velocity.

Problem 32E Relativistic Baseball. Calculate the magnitude of the force required to give a 0.145-kg baseball an acceleration a = 1.00 m/s2 in the direction of the baseball’s initial velocity when this velocity has a magnitude of (a) 10.0 m/s; (b) 0.900c; (c) 0.990c. (d) Repeat parts (a), (b), and (c) if the force and acceleration are perpendicular to the velocity.

Problem 33E What is the speed of a particle whose kinetic energy is equal to (a) its rest energy and (b) five times its rest energy?

Problem 34E If a muon is traveling at 0.999c, what are its momentum and kinetic energy? (The mass of such a muon at rest in the laboratory is 207 times the electron mass.)

A proton (rest mass 1.67 \(\times\) 10-27 kg) has total energy that is 4.00 times its rest energy. What are (a) the kinetic energy of the proton; (b) the magnitude of the momentum of the proton; (c) the speed of the proton?

(a) How much work must be done on a particle with mass m to accelerate it (a) from rest to a speed of 0.090c and (b) from a speed of 0.900c to a speed of 0.990c? (Express the answers in terms of mc2.) (c) How do your answers in parts (a) and (b) compare?

Problem 37E (a) By what percentage does your rest mass increase when you climb 30 m to the top of a ten-story building? Are you aware of this increase? Explain. (b) By how many grams does the mass of a 12.0-g spring with force constant 200 N/cm change when you compress it by 6.0 cm? Does the mass increase or decrease? Would you notice the change in mass if you were holding the spring? Explain.

A 60.0-kg person is standing at rest on level ground. How fast would she have to run to (a) double her total energy and (b) increase her total energy by a factor of 10?

Problem 39E An Antimatter Reactor. When a particle meets its antiparticle, they annihilate each other and their mass is converted to light energy. The United States uses approximately 1.0 × 1020 J of energy per year. (a) If all this energy came from a futuristic antimatter reactor, how much mass of matter and antimatter fuel would be consumed yearly? (b) If this fuel had the density of iron (7.86 g/cm3) and were stacked in bricks to form a cubical pile, how high would it be? (Before you get your hopes up, antimatter reactors are a long way in the future—if they ever will be feasible.)

Problem 46E Creating a Particle. Two protons (each with rest mass M = 1.67 × 10-27 kg) are initially moving with equal speeds in opposite directions. The protons continue to exist after a collision that also produces an ?0 particle (see Chapter 44). The rest mass of the ?0 is m = 9.75 × 10-28 kg. (a) If the two protons and the ?0 are all at rest after the collision, find the initial speed of the protons, expressed as a fraction of the speed of light. (b) What is the kinetic energy of each proton? Express your answer in MeV. (c) What is the rest energy of the ?0, expressed in MeV? (d) Discuss the relationship between the answers to parts (b) and (c).

Problem 47E The sun produces energy by nuclear fusion reactions, in which matter is converted into energy. By measuring the amount of energy we receive from the sun, we know that it is producing energy at a rate of 3.8 × 1026 W. (a) How many kilograms of matter does the sun lose each second? Approximately how many tons of matter is this (1 ton = 2000 lb)? (b) At this rate, how long would it take the sun to use up all its mass?

Problem 48P Inside a spaceship flying past the earth at three-fourths the speed of light, a pendulum is swinging. (a) If each swing takes 1.80 s as measured by an astronaut performing an experiment inside the spaceship, how long will the swing take as measured by a person at mission control (on earth) who is watching the experiment? (b) If each swing takes 1.80 s as measured by a person at mission control, how long will it take as measured by the astronaut in the spaceship?

After being produced in a collision between elementary particles, a positive pion \(\left(\pi^{+}\right)\) must travel down a 1.90-km-long tube to reach an experimental area. A \(\pi^{+}\) particle has an average lifetime (measured in its rest frame) of \(2.60 \times 10^{-8} \mathrm{~s}\); the \(\pi^{+}\) we are considering has this lifetime. (a) How fast must the \(\pi^{+}\) travel if it is not to decay before it reaches the end of the tube? (Since u will be very close to c, write \(u=(1-\Delta) c\) and give your answer in terms of \(\Delta\) rather than u.) (b) The \(\pi^{+}\) has a rest energy of What is the total energy of the \(\pi^{+}\) at the speed calculated in part (a)?

Problem 50P A cube of metal with sides of length a sits at rest in a frame S with one edge parallel to the x-axis. Therefore, in S the cube has volume a3. Frame S’ moves along the x-axis with a speed u. As measured by an observer in frame S’, what is the volume of the metal cube?

The starships of the Solar Federation are marked with the symbol of the federation, a circle, while starships of the Denebian Empire are marked with the empire’s symbol, an ellipse whose major axis is 1.40 times longer than its minor axis . How fast, relative to an observer, does an empire ship have to travel for its marking to be confused with the marking of a federation ship?

A space probe is sent to the vicinity of the star Capella, which is 42.2 light-years from the earth. (A light-year is the distance light travels in a year.) The probe travels with a speed of 0.9930c. An astronaut recruit on board is 19 years old when the probe leaves the earth. What is her biological age when the probe reaches Capella?

Problem 53P A particle is said to be extremely relativistic when its kinetic energy is much greater than its rest energy. (a) What is the speed of a particle (expressed as a fraction of c) such that the total energy is ten times the rest energy? (b) What is the percentage difference between the left and right sides of Eq (37.39) if (mc2)2 is neglected for a particle with the speed calculated in part (a)?

Everyday Time Dilation. Two atomic clocks are carefully synchronized. One remains in New York, and the other is loaded on an airliner that travels at an average speed of 250 m/s and then returns to New York. When the plane returns, the elapsed time on the clock that stayed behind is 4.00 h. By how much will the readings of the two clocks differ, and which clock will show the shorter elapsed time? (Hint: Since \(u\ \ll\ c\) you can simplify by \(\sqrt{1-u^{2} / c^{2}}\) a binomial expansion.) Text Transcription: u ll c sqrt 1 - u^2/c^2

Problem 55P The Large Hadron Collider (LHC). Physicists and engineers from around the world came together to build the largest accelerator in the world, the Large Hadron Collider (LHC) at the CERN Laboratory in Geneva, Switzerland. The machine accelerates protons to high kinetic energies in an underground ring 27 km in circumference. (a) What is the speed v of a proton in the LHC if the proton’s kinetic energy is 7.0 TeV? (Because v is very close to c, write v = (1 - ?)c and give your answer in terms of ?.) (b) Find the relativistic mass, mrel, of the accelerated proton in terms of its rest mass.

CP A nuclear bomb containing of plutonium explodes. The sum of the rest masses of the products of the explosion is less than the original rest mass by one part in \(10^{4}\). (a) How much energy is released in the explosion? (b) If the explosion takes place in \(4.00 \mu\), what is the average power developed by the bomb? (c) What mass of water could the released energy lift to a height of ? Equation transcription: Text transcription: 10^{4} 4.00 \mu

CP ?erenkov Radiation. The Russian physicist P. A. ?erenkov discovered that a charged particle traveling in a solid with a speed exceeding the speed of light in that material radiates electromagnetic radiation. (This is analogous to the sonic boom produced by an aircraft moving faster than the speed of sound in air; see Section 16.9. ?erenkov shared the 1958 Nobel Prize for this discovery.) What is the minimum kinetic energy (in electron volts) that an electron must have while traveling inside a slab of crown Glass in order to create this ?erenkov radiation?

A photon with energy E is emitted by an atom with mass m, which recoils in the opposite direction (a) Assuming that the motion of the atom can be treated non relativistically, compute the recoil speed of the atom. (b) From the result of part (a), show that the recoil speed is much less than c whenever E is much less than the rest energy \(m c^{2}\) of the atom.

Problem 59P In an experiment, two protons are shot directly toward each other, each moving at half the speed of light relative to the laboratory (a) What speed does one proton measure for the other proton? (b) What would be the answer to part (a) if we used only nonrelativistic Newtonian mechanics? (c) What is the kinetic energy of each proton as measured by (i) an observer al rest in the laboratory and (ii) an observer riding along with one of the protons? (d) What would be the answers to part (e) if we used only nonrelativistic Newtonian mechanics?

Problem 60P Two protons are moving away from each other. In the frame of each proton, the other proton has a speed of 0.600c. What does an observer in the rest frame of the earth measure for the speed of each proton?

Frame \(s^{i}\) has an -component of velocity relative to frame , and at \(t=t^{i}=0\) the two frames coincide (see Fig. 37.3). A light pulse with a spherical wave front is emitted at the origin of \(s^{i}\) at time \(t^{i}=0\). Its distance \(x^{i}\) from the origin after a time is given by \(x^{2}=C^{2} t^{2}\). Use the Lorentz coordinate transformation to transform this equation to an equation in and , and show that the result is \(x^{2}=C^{2} t^{2}\) that is, the motion appears exactly the same in frame of reference as it does in the wave front is observed to be spherical in both frames. Equation transcription: Text transcription: s^{i} t=t^{i}=0 t^{i}=0 x^{i} x^{2}=C^{2} t^{2} x^{2}=C^{2} t^{2}

Problem 62P In certain radioactive beta decay processes, the beta particle (an electron) leaves the atomic nucleus with a speed of 99.95% the speed of light relative to the decaying nucleus. If this nucleus is moving at 75.00% the speed of light in the laboratory reference frame, find the speed of the emitted electron relative to the laboratory reference frame if the electron is emitted (a) in the same direction that the nucleus is moving and (b) in the opposite direction from the nucleus’s velocity. (c) In each case in parts (a) and (b), find the kinetic energy of the electron as measured in (i) the laboratory frame and (ii) the reference frame of the decaying nucleus.

One of the wavelengths of light emitted by hydrogen atoms under normal laboratory conditions is \(\lambda = 656.3 nm\), in the red portion of the electromagnetic spectrum. In the light emitted from a distant galaxy this same spectral line is observed to be Doppler-shifted to \(\lambda = 953.4 nm\), in the infrared portion of the spectrum. How fast are the emitting atoms moving relative to the earth? Are they approaching the earth or receding from it?

Problem 67P One of the wavelengths of light emitted by hydrogen atoms under normal laboratory conditions is ? = 656.3 nm, in the red portion of the electromagnetic spectrum. In the light emitted from a distant galaxy this same spectral line is observed to be Doppler-shifted to ? = 953.4 nm, in the infrared portion of the spectrum. How fast are the emitting atoms moving relative to the earth? Are they approaching the earth or receding from it?

Problem 68P Measuring Speed by Radar. A baseball coach uses a radar device to measure the speed of an approaching pitched baseball. This device sends out electromagnetic waves with frequency f0 and then measures the shift in frequency ?f of the waves reflected from the moving baseball. If the fractional frequency shift produced by a baseball is ?f/f0 = 2.86 × 10-7, what is the baseball’s speed in km/h? (Hint: Are the waves Doppler-shifted a second time when reflected off the ball?)

Problem 69P Space Travel? Travel to the stars requires hundreds or thousands of years, even at the speed of light. Some people have suggested that we can get around this difficulty by accelerating the rocket (and its astronauts) to very high speeds so that they will age less due to time dilation. The fly in this ointment is that it takes a great deal of energy to do this. Suppose you want to go to the immense red giant Betelgeuse, which is about 500 light-years away. (A light year is the distance that light travels in a year.) You plan to travel at constant speed in a 1000-kg rocket ship (a little over a ton), which, in reality, is far too small for this purpose. In each case that follows, calculate the time for the trip, as measured by people on earth and by astronauts in the rocket ship, the energy needed in joules, and the energy needed as a percentage of U.S. yearly use (which is 1.0 × 1020J). For comparison, arrange your results in a table showingvrocket, tearth, trocket (in J), and E (as % of U.S. use). The rocket ship’s speed is (a) 0.50c: (b) 0.99c; (c) 0.9999c. On the basis of your results, does it seem likely that any government will invest in such high-speed space travel any time soon?

A spaceship moving at constant speed u relative to us broadcasts a radio signal at constant frequency \(f_{0}\). As the spaceship approaches us, we receive a higher frequency f; after it has passed, we receive a lower frequency. (a) As the spaceship passes by, so it is instantaneously moving neither toward nor away from us, show that the frequency we receive is not \(f_{0}\) and derive an expression for the frequency we do receive. Is the frequency we receive higher or lower than \(f_{0}\)? (Hint: In this case, successive wave crests move the same distance to the observer and so they have the same transit time. Thus f equals \(1 / T\). Use the time dilation formula to relate the periods in the stationary and moving frames.) (b) A spaceship emits electromagnetic waves of frequency \(f_0=345\ \mathrm{MHz}\) as measured in a frame moving with the ship. The spaceship is moving at a constant speed 0.758c relative to us. What frequency f do we receive when the spaceship is approaching us? When it is moving away? In each case what is the shift in frequency, \(\begin{array}{ll}f & f_{0}\end{array}\)? (c) Use the result of part (a) to calculate the frequency and the frequency shift \(\left(f-f_{0}\right)\) we receive at the instant that the ship passes by us. How does the shift in frequency calculated here compare to the shifts calculated in part (b)?

Problem 71P CP In a particle accelerator a proton moves with constant speed 0.750c in a circle of radius 628 m. What is the net force on the proton?

CP The French physicist Armand Fizeau was the first to measure the speed of light accurately. He also found experimentally that the speed, relative to the lab frame, of light traveling in a tank of water that is itself moving at a speed V relative to the lab frame is \(v=\frac{c}{n}+K V\) Where n = 1.333 is the index of refraction of water. Fizeau called k the dragging coefficient and obtained an experimental value of k = 0.44. What value of k do you calculate from relativistic transformations?

Using a method analogous to the one in the text to find the Lorentz transformation formula for velocity, we can find the Lorentz transformation for acceleration. Let frame have a constant -component of velocity relative to frame An object moves relative to frame along the -axis with instantaneous velocity and instantaneous acceleration (a) Show that its instantaneous acceleration in frame is \(a_{x}^{1}=a_{x}\left(1-\frac{u^{2}}{c^{2}}\right)^{3 / 2}\left(1+\frac{u v_{x}}{c^{2}}\right)^{-3}\) [Hint: Express the acceleration in as \(a_{x}^{1}=a v_{x}^{1} / a t^{1}\) Then use Eq. (37.21) to express in terms of and , and use Eq. (37.22) to express in terms of and . The velocity of the object in is \(\left.v_{x}=a x / a t .\right]\) (b) Show that the acceleration in frame can be expressed as \(a_{x}=a_{x}^{1}\left(1-\frac{u^{2}}{c^{2}}\right)^{3 / 2}\left(1+\frac{u v_{x}^{1}}{c^{2}}\right)^{-3}\) where \(v_{x}^{1}=a x^{1} / a t^{1}\) is the velocity of the object in frame . Equation transcription: Text transcription: a_{x}^{1}=a_{x}\left(1-\frac{u^{2}}{c^{2}}\right)^{3 / 2}\left(1+\frac{u v_{x}}{c^{2}}\right)^{-3} a_{x}^{1}=a v_{x}^{1} / a t^{1} \left.v_{x}=a x / a t .\right] a_{x}=a_{x}^{1}\left(1-\frac{u^{2}}{c^{2}}\right)^{3 / 2}\left(1+\frac{u v_{x}^{1}}{c^{2}}\right)^{-3} v_{x}^{1}=a x^{1} / a t^{1}

CALC A Realistic Version of the Twin Paradox. rocket ship leaves the earth on January 1, 2100 . Stella, one of a pair of twins born in the year 2075, pilots the rocket (reference frame \(S prime\) ); the other twin, Terra, stays on the earth (reference frame S ). The rocket ship has an acceleration of constant magnitude g in its own reference frame (this makes the pilot feel at home, since it simulates the earth's gravity). The path of the rocket ship is a straight line in the + x -direction in frame S. (a) Using the results of Challenge Problem 37.73, show that in Terra's earth frame S, the rocket's acceleration is \(du/dt = g(1-u^2/c^2)^3/2\) where u is the rocket's instantaneous velocity in frame S. (b) Write the result of part (a) in the form \(d t=f(u) d u\), where is a function of u, and integrate both sides. (Hint: Use the integral given in Problem 37.63.) Show that in Terra's frame, the time when Stella attains a velocity \(v_{1 x}\) is \(t_{1}=\frac{v_{1 x}}{g \sqrt{1-v_{1}^{2} / c^{2}}}\) (c) Use the time dilation formula to relate dt and (infinitesimal time intervals measured in frames S and , respectively). Combine this result with the result of part (a) and integrate as in part (b) to show the following: When Stella attains a velocity \(v_{1 x}\) relative to Terra, the time \(t_{1}^{\prime}\) that has elapsed in frame is \(t_{1}^{\prime}=\frac{c}{g} \operatorname{arctanh}\left(\frac{v_{1 x}}{c}\right)\) Here arctanh is the inverse hyperbolic tangent. (Hint: Use the integral given in Challenge Problem 5.124.) (d) Combine the results of parts (b) and (c) to find \(t_{1}^{\prime}\) in terms of , g, and alone. (e) Stella accelerates in a straight-line path for five years (by her clock), slows down at the same rate for five years, turns around, accelerates for five years, slows down for five years, and lands back on the earth. According to Stella's clock, the date is January 1, 2120 . What is the date according to Terra's clock? Equation Transcription: Text Transcription: S prime du/dt = g(1-u^2/c^2)^3/2 d t=f(u) d u v_1x t_1 prime t_1 prime = c/g arctanh (v_1x/c) T_1 t_{1}^{\prime}=frac{c}{g} operatorname{arctanh}(frac{v_{1 x}}{c}) t_{1}=\frac{v_{1 x}}{g \sqrt{1-v_{1}^{2} / c^{2}}}

CP Determining the Masses of Stars. Many of the stars in the sky are actually binary stars, in which two stars orbit about their common center of mass. If the orbital speeds of the stars are high enough, the motion of the stars can be detected by the Doppler shifts of the light they emit. Stars for which this is the case are called spectroscopic binary stars. Figure shows the simplest case of a spectroscopic binary star: two identical stars, each with mass , orbiting their center of mass in a circle of radius . The plane of the stars' orbits is edge-on to the line of sight of an observer on the earth. (a) The light produced by heated hydrogen gas in a laboratory on the earth has a frequency of \(4.568110 \times 10^{14} \mathrm{~Hz}\). In the light received from the stars by a telescope on the earth, hydrogen light is observed to vary in frequency between \(4.567719 \times 10^{14} \mathrm{~Hz}\) and \(4.568910^{14} \mathrm{~Hz}\). Determine whether the binary star system as a whole is moving toward or away from the earth, the speed of this motion, and the orbital speeds of the stars. (Hint: The speeds involved are much less than , so you may use the approximate result \(\Delta f / f=u / c\) given in Section ) (b) The light from each star in the binary system varies from its maximum frequency to its minimum frequency and back again in days. Determine the orbital radius and the mass of each star. Give your answer for in kilograms and as a multiple of the mass of the sun, \(1.99 \times 10^{30} \mathrm{~kg}\) Compare the value of to the distance from the earth to the sun, \(1.50 \times 10^{11} \mathrm{~m}\). (This technique is actually used in astronomy to determine the masses of stars. In practice, the problem is more complicated because the two stars in a binary system are usually not identical, the orbits are usually not circular, and the plane of the orbits is usually tilted with respect to the line of sight from the earth.) Equation transcription: Text transcription: 4.568110 times 10^{14}{~Hz} 4.567719 times 10^{14}{~Hz} 4.568910^{14}{~Hz} Delta f / f=u / c 1.99 \times 10^{30}{~kg} 1.50 \times 10^{11}{~m}

CP CALC Relativity and the Wave Equation. Consider the Galilean transformation along the x-direction: \(x^{\prime}=x-v t\) and \(t^{t}=t\). In frame S the wave equation for electromagnetic waves in a vacuum is \(\frac{\partial^{2} E(x, t)}{\partial x^{2}}-\frac{1}{c^{2}} \frac{\partial^{2} E(x, t)}{\partial t^{2}}=0\) where E represents the electric field in the wave. Show that by using the Galilean transformation the wave equation in frame \(S^\prime\) is found to be \(\left(1-\frac{v^{2}}{c^{2}}\right) \frac{\partial^{2} E\left(x^{\prime}, t^{\prime}\right)}{\partial x^{\prime 2}}+\frac{2 v}{c^{2}} \frac{\partial^{2} E\left(x^{\prime}, t^{\prime}\right)}{\partial x^{\prime} \partial t^{\prime}}-\frac{1}{c^{2}} \frac{\partial^{2} E\left(x^{\prime}, t^{\prime}\right)}{\partial t^{\prime 2}}=0\) This has a different form than the wave equation in S. Hence the Galilean transformation violates the first relativity postulate that all physical laws have the same form in all inertial reference frames. (Hint: Express the derivatives \(\partial / \partial x\) and \(\partial / \partial x\) in terms of \(\partial / \partial x^{\prime}\) and \(\partial / \partial t^{\prime}\) by use of the chain rule.) (b) Repeat the analysis of part (a), but use the Lorentz coordinate transformations, Eqs. (37.21), and show that in frame \(S^\prime\) the wave equation has the same form as in frame S: \(\frac{\partial^{2} E\left(x^{\prime}, t^{\prime}\right)}{\partial x^{2}}-\frac{1}{c^{2}} \frac{\partial^{2} E\left(x^{\prime}, t^{\prime}\right)}{\partial t^{\prime 2}}=0\) Explain why this shows that the speed of light in vacuum is c in both frames S and \(S^\prime\).