a. Two balls move as shown in Figure Q27.3. What are the

Problem 1CQ You are in an airplane cruising smoothly at 600 mph. What experiment, if any, could you do that would demonstrate that you are moving, while those on the ground are at rest?

Problem 2CQ Frame S? moves relative to frame S as shown in Figure Q27.2. a. A ball is at rest in frame S?. What are the speed and direction of the ball in frame S? b. A ball is at rest in frame S. What are the speed and direction of the ball in frame S??

Problem 3CQ a. Two balls move as shown in Figure Q27.3. What are the speed and direction of each ball in a reference frame that moves with ball 1? b. What are the speed and direction of each ball in a reference frame that moves with ball 2?

Problem 3P A boy on a skateboard coasts along at 5 m/s. He has a ball that he can throw at a speed of 10 m/s. What is the ball’s speed relative to the ground if he throws the ball (a) forward or (b) backward?

Problem 6CQ At the instant that a clock standing next to you reads t = 1.0 ?s, you look at a second clock, 300 m away, and see that it reads t = 0 ?s. Are the two clocks synchronized? If not, which one is ahead?

Problem 6P An assembly line has a staple gun that rolls to the left at 1.0 m/s while parts to be stapled roll past it to the right at 3.0 m/s. The staple gun fires 10 staples per second. How far apart are the staples in the finished part?

Problem 7CQ Firecracker 1 is 300 m from you. Firecracker 2 is 600 m from you in the same direction. You see both explode at the same time. Define event 1 to be “firecracker 1 explodes” and event 2 to be “firecracker 2 explodes.” Does event 1 occur before, after, or at the same time as event 2? Explain.

Problem 7P An out-of-control alien spacecraft is diving into a star at a speed of . At what speed, relative to the spacecraft, is the starlight approaching?

Problem 8P A starship blasts past the earth at . Just after passing the earth, the starship fires a laser beam out its back. With what speed does the laser beam approach the earth?

Problem 9CQ Your clocks and calendars are synchronized with the clocks and calendars in a star system exactly 10 ly from earth that is at rest relative to the earth. You receive a TV transmission from the star system that shows a date and time display. The date it shows is June 17, 2050. When you glance over at your own wall calendar, what date does it show?

Problem 10P Your job is to synchronize the clocks in a reference frame. You are going to do so by flashing a light at the origin at t = 0 s. To what time should the clock at (x, y, z) = (30 m, 40 m, 0 m) be preset?

Problem 11P Bjorn is standing at x = 600 m. Firecracker 1 explodes at the origin and firecracker 2 explodes at x = 900 m. The flashes from both explosions reach Bjorn’s eye at t = 3.0 ms. At what time did each firecracker explode?

Problem 12CQ

Problem 12P Bianca is standing at x = 600 m. Firecracker 1, at the origin, and firecracker 2, at x = 900 m, explode simultaneously. The flash from firecracker 1 reaches Bianca’s eye at t = 3.0 ms. At what time does she see the flash from firecracker 2?

Problem 14 CQ You’re passing a car on the highway. You want to know how much time is required to completely pass the car, from no overlap between the cars to no overlap between the cars. Call your car A, the car you are passing B. a. Specify two events that can be given spacetime coordinates. In describing the events, refer to cars A and B and to their front bumpers and rear bumpers. b. In either reference frame, is there one clock that is present at both events? c. Who, if anyone, measures the proper time between the events?

Problem 16P You are flying your personal rocketcraft at 0.90c from Star A toward Star B. The distance between the stars, in the stars’ reference frame, is 1.0 ly. Both stars happen to explode simultaneously in your reference frame at the instant you are exactly halfway between them. Do you see the flashes simultaneously? If not, which do you see first and what is the time difference between the two?

Problem 17CQ Dan picks up a 15-m-long pole and begins running very fast, holding the pole horizontally and pointing in the direction he’s running. He heads toward a barn that is 12 m long and has open doors at each end. Dan runs so fast that, to Farmer Brown standing by his barn, the ladder is only 5 m long. As soon as the pole is completely inside the barn, Farmer Brown closes both doors so that Dan and the pole are inside with both doors shut. Then, just before Dan reaches the far door, Farmer Brown opens both doors and Dan emerges, still moving at high speed. According to Dan, however, the barn is contracted to only 4 m and the pole has its full 15 m length. Farmer Brown sees the pole completely inside the barn with both doors closed. What does Dan see happening?

Problem 18CQ The rocket speeds shown in Figure Q27.18 are relative to the earth. Is the speed of A relative to B greater than, less than, or equal to 0.8c?

Problem 18P At what speed relative to a laboratory does a clock tick at half the rate of an identical clock at rest in the laboratory? Give your answer as a fraction of c.

Problem 19CQ Can a particle of mass m have total energy less than ? Explain. Step-by-step solution Step 1 of 1 ________________

Problem 19P An astronaut travels to a star system 4.5 ly away at a speed of 0.90c. Assume that the time needed to accelerate and decelerate is negligible. a. How long does the journey take according to Mission Control on earth? b. How long does the journey take according to the astronaut? c. How much time elapses between the launch and the arrival of the first radio message from the astronaut saying that she has arrived?

Problem 20CQ In your chemistry classes, you have probably learned that, in a chemical reaction, the mass of the products is equal to the mass of the reactants. That is, the mass of the substances produced in a chemical reaction is equal to the mass of the substances consumed in the reaction. Is this absolutely true, or is there actually a small difference? Explain.

Problem 20P A starship voyages to a distant planet 10 ly away. The explorers stay 1 yr, return at the same speed, and arrive back on earth 26 yr after they left. Assume that the time needed to accelerate and decelerate is negligible. a. What is the speed of the starship? b. How much time has elapsed on the astronauts’ chronometers?

Problem 21P At what speed, as a fraction of c, will a moving rod have a length 60% that of an identical rod at rest?

Problem 22MCQ A space cowboy wants to eject from his spacecraft 100,000 km after passing a space buoy, as seen by spectators at rest with respect to the buoy. To do this, the cowboy sets a timer on his craft that will start as he passes the buoy. He plans to cruise by the buoy at 0.300c. How much time should he allow between passing the buoy and ejecting? A. 1.01 s B. 1.06 s C. 1.11 s D. 1.33 s E. 1.58 s

Problem 23MCQ Event 1 occurs at (0 m, 0 s) in reference frame S. The following events occur, also measured in reference frame S. Which of them could have been caused by event 1? A. (lm, 2 ns) B. (?1 m, ?2 ns) C. (?lm, 2 ns) D. All of them E. None of them

Problem 23P A muon travels 60 km through the atmosphere at a speed of 0.9997c. According to the muon, how thick is the atmosphere?

Problem 24MCQ Energy in the sun is produced by the fusion of four protons into a helium nucleus. The process involves several steps, but the net reaction is simply . Given this, you can say that A. One helium atom has more mass than four hydrogen atoms. B. One helium atom has less mass than four hydrogen atoms. C. One helium atom has the same mass as four hydrogen atoms.

Problem 24P The Stanford Linear Accelerator (SLAC) accelerates electrons to v = 0.99999997c in a 3.2-km-long tube. If they travel the length of the tube at full speed (they don’t, because they are accelerating), how long is the tube in the electrons’ reference frame?

Problem 25MCQ A particle moving at speed 0.40c has momentum . The speed of the particle is increased to 0.80c. Its momentum is now

Problem 25P Our Milky Way galaxy is 100,000 ly in diameter. A spaceship crossing the galaxy measures the galaxy’s diameter to be a mere 1.0 ly. a. What is the speed of the spaceship relative to the galaxy? b. How long is the crossing time as measured in the galaxy’s reference frame?

Problem 26MCQ A particle moving at speed 0.40c has kinetic energy . The speed of the particle is increased to 0.80c. The kinetic energy is now

Problem 26P An optical interferometer can detect a displacement of about 50 nm. At what speed would a meter stick “shrink” by 50 nm? Hint: Use the binomial approximation.

Problem 27P A rocket cruising past earth at 0.800c shoots a bullet out the back door, opposite the rocket’s motion, at 0.900c relative to the rocket. What is the bullet’s speed relative to the earth?

Problem 28P A base on Planet X fires a missile toward an oncoming space fighter. The missile’s speed according to the base is 0.85c. The space fighter measures the missile’s speed as 0.96c. How fast is the space fighter traveling relative to Planet X?

Problem 29P A solar flare blowing out from the sun at 0.90c is overtaking a rocket as it flies away from the sun at 0.80c. According to the crew on board, with what speed is the flare gaining on the rocket?

Problem 30P A proton is accelerated to 0.999c. a. What is the proton’s momentum? b. By what factor does the proton’s momentum exceed its Newtonian momentum?

Problem 31P A 1.0 g particle has momentum 400,000 kg · m/s. What is the particle’s speed?

Problem 32P At what speed is a particle’s momentum twice its Newtonian value?

Problem 33P What is the speed of a particle whose momentum is mc?

Problem 34P What are the kinetic energy, the rest energy, and the total energy of a 1.0 g particle with a speed of 0.80c?

Problem 35P A quarter-pound hamburger with all the fixings has a mass of 200 g. The food energy of the hamburger (480 food calories) is 2 MJ. a. What is the energy equivalent of the mass of the hamburger? b. By what factor does the energy equivalent exceed the food energy?

Problem 36P How fast must an electron move so that its total energy is 10% more than its rest mass energy?

Problem 37P At what speed is a particle’s kinetic energy twice its rest energy?

Problem 38GP A firecracker explodes at x = 0 m, t = 0 ?s. A second explodes at x = 300 m, t = 2.0 ?s. What is the proper time between these events?

Problem 39GP You’re standing on an asteroid when you see your best friend rocketing by in her new spaceship. As she goes by, you notice that the front and rear of her ship coincide exactly with the 400-m-diameter of another nearby asteroid that is stationary with respect to you. However, you happen to know that your friend’s spaceship measured 500 m long in the showroom. What is your friend’s speed relative to you?

Problem 40GP A subatomic particle moves through the laboratory at 0.90c. Laboratory experimenters measure its lifetime, from creation to annihilation, to be 2.3 ps . According to the particle, how long did it live?

Problem 42GP A very fast-moving train car passes you, moving to the right at 0.50c. You measure its length to be 12 m. Your friend David flies past you to the right at a speed relative to you of 0.80c. How long does David measure the train car to be?

Problem 41GP You and Maria each own identical spaceships. As you fly past Maria, you measure her ship to be 90 m long and your own ship to be 100 m long. a. How long does Maria measure your ship to be? b. How fast is Maria moving relative to you?

Problem 43GP Two events in reference frame S occur 10 ?s apart at the same point in space. Frame S? travels at speed v = 0.90c relative to frame S. a. What is the time interval between the events in reference frame S?? b. What is the distance between the events in frame S??

Problem 44GP A 30-m-long rocket train car is traveling from Los Angeles to New York at 0.50c when a light at the center of the car flashes. When the light reaches the front of the car, it immediately rings a bell. Light reaching the back of the car immediately sounds a siren. a. Are the bell and siren simultaneous events for a passenger seated in the car? If not, which occurs first and by how much time? b. Are the bell and siren simultaneous events for a bicyclist waiting to cross the tracks? If not, which occurs first and by how much time?

Problem 45GP Because of the earth’s rotation, a person living on top of a mountain moves at a faster speed than someone at sea level. The mountain dweller’s clocks thus run slowly compared to those at sea level. If the average life span of a hermit is 80 years, on average how much longer would a hermit dwelling on the top of a 3000-m-high mountain live compared to a sea-level hermit?

Problem 46GP Two clocks are synchronized. One is placed in a race car that drives around a 2.0-km-diameter track at 100 m/s for 24 h. Afterward, by how much do the two clocks differ? Hint: Use the binomial approximation.

Problem 47GP You fly 5000 km across the United States on an airliner at 250 m/s. You return two days later at the same speed. a. Have you aged more or less than your friends at home? b. By how much? Hint: Use the binomial approximation.

Problem 48GP A cube has a density of while at rest in the laboratory. What is the cube’s density as measured by an experimenter in the laboratory as the cube moves through the laboratory at 90% of the speed of light in a direction perpendicular to one of its faces?

Problem 49GP In Section 27.6 we explained that muons can reach the ground because of time dilation. But how do things appear in the muon’s reference frame, where the muon’s half-life is only 1.5 µs? How can a muon travel the 60 km to reach the earth’s surface before decaying? Resolve this apparent paradox. Be as quantitative as you can in your answer.

Problem 51GP Marissa’s spaceship approaches Joseph’s at a speed of 0.99c. As Marissa passes Joseph, they synchronize their clocks to both read t = 0 s. When Marissa’s clock reads 100 s, she sends a light signal back to Joseph. According to his clock, when does he receive this signal?

Problem 52GP At a speed of 0.90c, a spaceship travels to a star that is 9.0 ly distant. a. According to a scientist on earth, how long does the trip take? b. According to a scientist on the spaceship, how long does the trip take? c. According to the scientist on the spaceship, what is the distance traveled during the trip? d. At what speed do observers on the spaceship see the star approaching them?

Problem 53GP In an attempt to reduce the extraordinarily long travel times for voyaging to distant stars, some people have suggested traveling at close to the speed of light. Suppose you wish to visit the red giant star Betelgeuse, which is 430 ly away, and that you want your 20,000 kg rocket to move so fast that you age only 20 years during the round trip. a. How fast must the rocket travel relative to earth? b. How much energy is needed to accelerate the rocket to this speed? c. How many times larger is this energy than the total energy used by the United States in the year 2012, which was roughly ?

Problem 54GP A rocket traveling at 0.500c sets out for the nearest star, Alpha Centauri, which is 4.25 ly away from earth. It will return to earth immediately after reaching Alpha Centauri. What distance will the rocket travel and how long will the journey last according to (a) stay-at-home earthlings and (b) the rocket crew? (c) Which answers are the correct ones, those in part a or those in part b?

Problem 55GP A distant quasar is found to be moving away from the earth at 0.80c. A galaxy closer to the earth and along the same line of sight is moving away from us at 0.20c. What is the recessional speed of the quasar as measured by astronomers in the other galaxy?

Problem 56GP Two rockets approach each other. Each is traveling at 0.75c in the earth’s reference frame. What is the speed of one rocket relative to the other?

Problem 57GP A military jet traveling at 1500 m/s has engine trouble and the pilot must bail out. Her ejection seat shoots her forward at 300 m/s relative to the jet. According to the Lorentz velocity transformation, by how much is her velocity relative to the ground less than the 1800 m/s predicted by Galilean relativity? Hint: Use the binomial approximation.

Problem 58GP James, Daniella, and Tara all possess identical clocks. As Daniella passes James in her rocket, James observes that her clock runs at 80% the rate of his clock. As Tara passes in her rocket, in the same direction as Daniella, James observes that her clock runs at 70% the rate of his clock. At what rate, relative to her clock, does Daniella observe Tara’s clock to run?

Problem 59GP Two rockets approach earth from opposite directions at equal speeds relative to the earth. A scientist on earth notes that it takes 1 h and 10 min, according to her watch, for the clocks on the rockets to advance by 1 h. a. How fast are the rockets moving with respect to the earth? b. According to an astronaut on one rocket, how long does it take the clock on the other rocket to advance by 1 h?

Problem 60GP Two rockets, A and B, approach the earth from opposite directions at speed 0.800c. The length of each rocket measured in its rest frame is 100 m. What is the length of rocket A as measured by the crew of rocket B?

Problem 61GP The highest-energy cosmic ray ever detected had an energy of about . Assume that this cosmic ray was a proton. a. What was the proton’s speed as a fraction of c? b. If this proton started at the same time and place as a photon traveling at the speed of light, how far behind the photon would it be after traveling for 1 ly?

Problem 62GP What is the speed of an electron after being accelerated from rest through a potential difference?

Problem 63GP What is the speed of a proton after being accelerated from rest through a potential difference?

Problem 64GP The half-life of a muon at rest is 1.5 ?s. Muons that have been accelerated to a very high speed and are then held in a circular storage ring have a half-life of 7.5 ?s. a. What is the speed of the muons in the storage ring? b. What is the total energy of a muon in the storage ring? The mass of a muon is 207 times the mass of an electron.

Problem 66GP What is the momentum of a particle whose total energy is four times its rest energy? Give your answer as a multiple of mc.

Problem 65GP What is the momentum of a particle with speed 0.95c and total energy ?

Problem 67GP What is the total energy, in MeV, of a. A proton traveling at 99.0% of the speed of light? b. An electron traveling at 99.0% of the speed of light?

Problem 68GP What is the velocity, as a fraction of c, of a. A proton with 500 GeV total energy? b. An electron with 2.0 GeV total energy?

Problem 69GP At what speed is the kinetic energy of a particle twice its Newtonian value?

Problem 71GP The factor ? appears in many relativistic expressions. A value ? = 1.01 implies that relativity changes the Newtonian values by approximately 1% and that relativistic effects can no longer be ignored. At what kinetic energy, in MeV, is ? = 1.01 for (a) an electron, and (b) a proton?

Problem 70GP What is the speed of an electron whose total energy equals the rest energy of a proton?

Problem 72GP The chemical energy of gasoline is 46 MJ/kg. If gasoline’s mass could be completely converted into energy, what mass of gasoline would be needed to equal the chemical energy content of 1.0 kg of gasoline?

Problem 73GP The sun radiates energy at the rate . The source of this energy is fusion, a nuclear reaction in which mass is transformed into energy. The mass of the sun is . a. How much mass does the sun lose each year? b. What percentage is this of the sun’s total mass? c. Estimate the lifetime of the sun.

Problem 74GP The radioactive element radium (Ra) decays by a process known as alpha decay, in which the nucleus emits a helium nucleus. (These high-speed helium nuclei were named alpha particles when radioactivity was first discovered, long before the identity of the particles was established.) The reaction is , where Rn is the element radon. The accurately measured atomic masses of the three atoms are 226.025, 222.017, and 4.003. How much energy is released in each decay? (The energy released in radioactive decay is what makes nuclear waste “hot.”)

Problem 75GP The nuclear reaction that powers the sun is the fusion of four protons into a helium nucleus. The process involves several steps, but the net reaction is simply . The mass of a helium nucleus is known to be . a. How much energy is released in each fusion? b. What fraction of the initial rest mass energy is this energy?

Problem 76GP When antimatter (which we’ll learn more about in Chapter 30) interacts with an equal mass of ordinary matter, both matter and antimatter are converted completely into energy in the form of photons. In an antimatter-fueled spaceship, a staple of science fiction, the newly created photons are shot from the back of the ship, propelling it forward. Suppose such a ship has a mass of , and carries a mass of fuel equal to 1% of its mass, or of matter and an equal mass of antimatter. a. What is the final speed of the ship, assuming it starts from rest, if all energy released in the matter-antimatter annihilation is transformed into the kinetic energy of the ship? b. Not only do photons have energy, as you learned in Chapter 25, they also have momentum. Explain why, when energy and momentum conservation are both considered, the final speed of the ship will be less than you calculated in part a.

Problem 77PP Pion Therapy Subatomic particles called pions are created when protons, accelerated to speeds very near c in a particle accelerator, smash into the nucleus of a target atom. Charged pions are unstable particles that decay into muons with a half-life of . Pions have been investigated for use in cancer treatment because they pass through tissue doing minimal damage until they decay, releasing significant energy at that point. The speed of the pions can be adjusted so that the most likely place for the decay is in a tumor. Suppose pions are created in an accelerator, then directed into a medical bay 30 m away. The pions travel at the very high speed of 0.99995c. Without time dilation, half of the pions would have decayed after traveling only 5.4 m, not far enough to make it to the medical bay. Time dilation allows them to survive long enough to reach the medical bay, enter tissue, slow down, and then decay where they are needed, in a tumor. What is the half-life of a pion in the reference frame of the patient undergoing pion therapy?

Problem 78PP Pion Therapy Subatomic particles called pions are created when protons, accelerated to speeds very near c in a particle accelerator, smash into the nucleus of a target atom. Charged pions are unstable particles that decay into muons with a half-life of . Pions have been investigated for use in cancer treatment because they pass through tissue doing minimal damage until they decay, releasing significant energy at that point. The speed of the pions can be adjusted so that the most likely place for the decay is in a tumor. Suppose pions are created in an accelerator, then directed into a medical bay 30 m away. The pions travel at the very high speed of 0.99995c. Without time dilation, half of the pions would have decayed after traveling only 5.4 m, not far enough to make it to the medical bay. Time dilation allows them to survive long enough to reach the medical bay, enter tissue, slow down, and then decay where they are needed, in a tumor. According to the pion, what is the distance it travels from the accelerator to the medical bay? A. 0.30 m B. 3.0 m C. 30 m D. 3000 m

Problem 79PP Pion Therapy Subatomic particles called pions are created when protons, accelerated to speeds very near c in a particle accelerator, smash into the nucleus of a target atom. Charged pions are unstable particles that decay into muons with a half-life of . Pions have been investigated for use in cancer treatment because they pass through tissue doing minimal damage until they decay, releasing significant energy at that point. The speed of the pions can be adjusted so that the most likely place for the decay is in a tumor. Suppose pions are created in an accelerator, then directed into a medical bay 30 m away. The pions travel at the very high speed of 0.99995c. Without time dilation, half of the pions would have decayed after traveling only 5.4 m, not far enough to make it to the medical bay. Time dilation allows them to survive long enough to reach the medical bay, enter tissue, slow down, and then decay where they are needed, in a tumor. The proton collision that creates the pion also creates a gamma-ray photon traveling in the same direction as the pion. The photon will get to the medical bay first because it is moving faster. What is the speed of the photon in the pion’s reference frame? A. 0.00005c B. 0.5c C. 0.99995c D. c

Problem 80PP Pion Therapy Subatomic particles called pions are created when protons, accelerated to speeds very near c in a particle accelerator, smash into the nucleus of a target atom. Charged pions are unstable particles that decay into muons with a half-life of . Pions have been investigated for use in cancer treatment because they pass through tissue doing minimal damage until they decay, releasing significant energy at that point. The speed of the pions can be adjusted so that the most likely place for the decay is in a tumor. Suppose pions are created in an accelerator, then directed into a medical bay 30 m away. The pions travel at the very high speed of 0.99995c. Without time dilation, half of the pions would have decayed after traveling only 5.4 m, not far enough to make it to the medical bay. Time dilation allows them to survive long enough to reach the medical bay, enter tissue, slow down, and then decay where they are needed, in a tumor. If the pion slows down to 0.99990c, about what percentage of its kinetic energy is lost? A. 0.03% B. 0.3% C. 3% D. 30%

Problem 50GP A spaceship flies past an experimenter who measures its length to be one-half the length he had measured when the spaceship was at rest. An astronaut aboard the spaceship notes that his clock ticks at 1-second intervals. What is the time between ticks as measured by the experimenter?