Solution:<\/strong><\/span>
\nAccording to second postulate of special theory of relativity the speed of light (c) is same in all inertial frames of references in vacuum. So light from these galaxies received on the earth moves with the speed as is true for all light in a vacuum.<\/p>\nChapter 29 Relativity Q.1P<\/strong>
\nCE Predict\/Explain You are in spaceship, traveling directly away from the Moon with a speed of 0.9c A light signal is sent in your direction from the surface of the Moon. (a) As the signal passes your ship, do you measure its speed to be greater than, among the following:<\/p>\n\n- the speed you measure will be greater then 0.1c in fact, it will be c, since all observers in inertial frames measure the same speed of light<\/li>\n
- You will measure a speed less than 0.1c because of time dilation, which causes clocks to run slow.<\/li>\n
- When you measure a speed you find if be between c and 0.9c.<\/li>\n<\/ul>\n
Solution:<\/strong><\/span><\/p>\n\n- As the signal passes our ship, we will measure speed of the signal greater than 0.1c<\/li>\n
- Answer (I) The speed you measure will be greater than 0.1c; in fact, it will be c, since all observers in inertial frames measures the same speed of light.<\/li>\n<\/ul>\n
Chapter 29 Relativity Q.2CQ<\/strong>
\nThe speed of light in glass is less than c. Why is this not a violation of the second postulate of relativity?
\nSolution:<\/strong><\/span>
\nWe know according to second postulate of special theory of relativity, the speed (c) of light is same in all inertial frames of references, and speed of light in vacuum is c. Here c is the speed of light in vacuum only. In all other media other than vacuum the speed of light is less than c. So the speed of light in glass is less than c.<\/p>\nChapter 29 Relativity Q.2P<\/strong>
\nAlbert is piloting his spaceship, heading east with a speed of 0.90c. Albert\u2019s ship sends a light beam in the forward (east-ward) direction, which travels away from his ship at a speed c. Meanwhile, Isaac is piloting his ship in the westward direction, also at 0.90c, toward Albert\u2019s ship. With what speed does Isaac see?
\nSolution:<\/strong><\/span>
\nAccording to second postulate of special theory of relativity, the speed of light in vacuum, , is same in all inertial frames of reference, independent of the motion of the source or the receiver. The speed of Albert\u2019s light beam observed by Isaac is c<\/p>\nChapter 29 Relativity Q.3CQ<\/strong>
\nHow would velocities add if the speed of light were infinitely large? Justify your answer by considering Equation 29\u20134.
\nSolution:<\/strong><\/span>
\n<\/p>\nChapter 29 Relativity Q.3P<\/strong>
\nCE A street performer tosses a ball straight up into the air (event 1) and then catches it in his mouth (event 2). For each of the following observers, state whether the time they measure between these two events is the proper time or the dilated time: (a) the street performer; (b) a stationary observer on the other side of the street; (c) a person sitting at home watching the performance on TV; (d) a person observing the performance from a moving car.
\nSolution:<\/strong><\/span>
\nThe definition of proper time is the time difference between the two events which occur at the same location by given observer or particular observer. Here in the given problem, the two events are a street performer tosses a ball straight up into the air (event 1) and then catches it in his mouth (event 2). Now we have to decide for the following observers, whether the time they measured between these two events is the proper time or dilated time.<\/p>\n\n- (a) Here the street performer is in the rest frame of reference. With respect to this street performer these two events occur at the same location. So he measures the time between these two events is proper time<\/li>\n
- (b) The stationary observer on the other side of the street is also in the rest frame. With respect to him these two events occur at the same location. So he measures the time between these two events is proper time<\/li>\n
- (c) The person sitting at home watching the performance on TV is in the rest frame. With respect to him these two events occur at the same location. So he measures the time difference between these two events is proper time<\/li>\n
- (d) A person moving in the car observed these two events are situated at different locations. So with respect to him, he measures the time between these two events is dilated time<\/li>\n<\/ul>\n
Chapter 29 Relativity Q.4CQ<\/strong>
\nDescribe some of the everyday consequences that would follow if the speed of light were 35 mi\/h.
\nSolution:<\/strong><\/span>
\nIf the speed of light is 35 mi\/h then we experience relativistic effects like<\/p>\n\n- \u00a0A person in motion would age slowly than a person who stays at home due to time dilation.<\/li>\n
- \u00a0The length of a car in motion seems to be shorter due to length contraction.<\/li>\n
- \u00a0No object can move faster than light.<\/li>\n<\/ul>\n
Even if the engine of your car is very powerful and even if you try to accelerate it, your car will not be able to move faster than 35 mi\/h<\/p>\n
Chapter 29 Relativity Q.4P<\/strong>
\n\u0421E Predict\/Explain A clock in a moving rocket is observed to run slow, (a) If the rocket reverses direction, does the clock run slow, fast, or at its normal rate? (b) Choose the best explanation from among the following:<\/p>\n\n- The clock will run slow, just as before. The rate of the clock depends only on relative speed, not on direction of motion.<\/li>\n
- When the rocket reverses direction the rate of the clock reverses too, and this makes it run fast.<\/li>\n
- Reversing the direction of the rocket undoes the the time dilation effect, and so the clock will now run at its normal rate.<\/li>\n<\/ul>\n
Solution:<\/strong><\/span><\/p>\n\n- The clock will still run slow as before.<\/li>\n
- Answer (I)<\/li>\n<\/ul>\n
The clock will run slow, just as before. The rate of the clock depends only on relative speed, not on direction of motion.<\/p>\n
Chapter 29 Relativity Q.5CQ<\/strong>
\nWhen we view a distant galaxy, we notice that the light coming from it has a longer wavelength (it is \u201cred-shifted\u201d) than the corresponding light here on Earth. Is this consistent with the postulate that all observers measure the same speed of light? Explain.
\nSolution:<\/strong><\/span>
\nYes
\nHere the light rays are coming from the distant galaxy. In spite of not taking the wavelength of the light rays coming from the galaxy, all these light rays move with the same speed in vacuum. These light rays have longer wavelength and the frequency of this \u201cred shifted\u201d light will be affected, this is because we know the formula. v = \u03bbf From this formula the wavelength () and frequency (f) of the light rays are inversely proportional to each other. From this relation the longer wavelength implies a smaller frequency.<\/p>\nChapter 29 Relativity Q.5P<\/strong>
\n\u0421E Predict\/Explain Suppose you are a traveling salesman for SSC, the Spacely Sprockets Company. You travel on a spaceship that reaches speeds near the speed of light, and you are paid by the hour, (a) When you return to Earth after a sales trip, would you prefer to be paid according to the clock at Spacely Sprockets universal headquarters on Earth, according to the clock on the spaceship in which you travel, or would your pay be the same in either case? (b) Choose the best explanation from among the following:<\/p>\n\n- You want to be paid according to the clock on Earth, because the clock on the spaceship runs slow when it approaches the speed of light.<\/li>\n
- Collect your pay according to the clock on the spaceship because according to you the clock on Earth has run slow.<\/li>\n
- Your pay would be the same in either case because motion is relative, and all mertial observers will agree on the amount of time that has elapsed.<\/li>\n<\/ul>\n
Solution:<\/strong><\/span><\/p>\n\n- I would like to be paid according to the clock at Spacely Sprockets universal Headquarters on earth.<\/li>\n
- Answer (I) You want to be paid according to the clock on earth, because the clock on the space ship runs slow when it approaches the speed of light.<\/li>\n<\/ul>\n
Chapter 29 Relativity Q.6CQ<\/strong>
\nAccording to the theory of relativity, the maximum speed foi any particle is the speed of light. Is there a similar restriction on the maximum energy of a particle? Is there a maximum momentum? Explain.
\nSolution:<\/strong><\/span>
\n<\/p>\nChapter 29 Relativity Q.6P<\/strong>
\nA neon sign in front of a caf\u00e9 flashes on and off once every 4.1 s, as measured by the head cook. How much time elapses between flashes of the sign as measured by an astronaut in a spaceship moving toward Earth With a speed of 0.84c?
\nSolution:<\/strong><\/span>
\n<\/p>\nChapter 29 Relativity Q.7CQ<\/strong>
\nGive an argument that shows that \u0430n object of finite mass cannot be accelerated from rest to a speed greater than the speed of light in a vacuum.
\nSolution:<\/strong><\/span>
\n
\n<\/p>\nChapter 29 Relativity Q.7P<\/strong>
\nA lighthouse sweeps its beam of light around in a circle once every 7.5 s. To an observer in a spaceship moving, away from Earth, the beam of light completes one full circle every 15 s. What is the speed of the spaceship relative to Earth?
\nSolution:<\/strong><\/span>
\n
\n<\/p>\nChapter 29 Relativity Q.8P<\/strong>
\nRefer to Example 29\u20131. How much does Benny age if he travels to Vega with a speed of 0.9995c?
\nSolution:<\/strong><\/span>
\n
\n<\/p>\nChapter 29 Relativity Q.9P<\/strong>
\nAs a spaceship flies past with speed v, you observe that 1.0000 s elapses on the ship\u2019s clock in the same time that 1.0000 min elapses on Earth. How fast is the ship traveling, relative to the Earth? (Express your answer as a fraction of the speed of light.)
\nSolution:<\/strong><\/span>
\n<\/p>\nChapter 29 Relativity Q.10P<\/strong>
\nDonovan Bailey set a world record for the 100-m dash on July 27, 1996. If observers on a spaceship moving with a speed of 0.7705c relative to Earth saw Donovan Bailey\u2019s run and measured his time to be 15.44 s, find the time that was recorded on Earth.
\nSolution:<\/strong><\/span>
\n<\/p>\nChapter 29 Relativity Q.11P<\/strong>
\nFind the average distance (in the Earth\u2019s frame of reference) covered by the muons in Example 29\u20132 if their speed relative to Earth is 0.750c.
\nSolution:<\/strong><\/span>
\n<\/p>\nChapter 29 Relativity Q.12P<\/strong>
\nThe Pi Meson An elementary particle called a pi meson (or pion for short) has an average lifetime of 2.6 \u00d7 10\u20138 s when at rest. If a pion moves with a speed of 0.99c relative to Earth, find (a) the average lifetime of the pion as measured by an observer on Earth and (b) the average distance traveled by the pion as measured by the same observer, (c) How far would the pion have traveled relative to Earth if relativistic time dilation did not occur?
\nSolution:<\/strong><\/span>
\n
\n<\/p>\nChapter 29 Relativity Q.13P<\/strong>
\nThe \u03a3\u2013 Particle The \u03a3\u2013 is an exotic particle that has a lifetime (when at rest) of 0.15 ns. How fast would it have to travel in order for its lifetime, as measured by laboratory clocks, to be 0.25 ns?
\nSolution:<\/strong><\/span>
\n<\/p>\nChapter 29 Relativity Q.14P<\/strong>
\nIP (a) Is it possible for you to travel far enough and fast enough so that when you return from a trip, you are younger than your stay-at-home sister, who was born 5.0 y after you? (b) Suppose you fly on a rocket with a speed v = 0.99c for 1 y, according to the ship\u2019s clocks and calendars. How much time elapses on Earth during your 1-y trip? (c) If you were 22 y old when you left home and your sister was 17, what are your ages when you return?
\nSolution:<\/strong><\/span>
\n<\/p>\nChapter 29 Relativity Q.15P<\/strong>
\nThe radar antenna on a navy ship rotates with an angular speed of 0.29 rad\/s. What is the angular speed of the antenna as measured by an observer moving away from the antenna with a speed of 0.82c?
\nSolution:<\/strong><\/span>
\n
\n<\/p>\nChapter 29 Relativity Q.16P<\/strong>
\nAn observer moving toward Earth with a speed of 0.95c notices that it takes 5.0 min for a person to fill her car with gas. Suppose, instead, that the observer had been moving away from Earth with a speed of 0.80c. How much time would the observer have measured for the car to be filled in this case?
\nSolution:<\/strong><\/span>
\n
\n<\/p>\nChapter 29 Relativity Q.17P<\/strong>
\nIP An astronaut moving with a speed of 0.65c relative to Earth measures her heart rate to be 72 beats per minute, (a) When an Earth-based observer measures the astronaut\u2019s heart rate, is the result greater than, less than, or equal to 72 beats per minute? Explain. (b) Calculate the astronaut\u2019s heart rate as measured on Earth.
\nSolution:<\/strong><\/span>
\n
\n<\/p>\nChapter 29 Relativity Q.18P<\/strong>
\nBIO Newly sprouted sunflowers can grow at the rate of 0.30 in. per day. One such sunflower is left on Earth, and an identical one is placed on a spacecraft that is traveling away from Earth with a speed of 0.94c. How tall is the sunflower on the spacecraft when a person on Earth says his is 2.0 in. high?
\nSolution:<\/strong><\/span>
\n<\/p>\nChapter 29 Relativity Q.19P<\/strong>
\nAn astronaut travels to Mars with a speed of 8350 m\/s. After a month (30.0 d) of travel, as measured by clocks on Earth, how much difference is there between the Earth clock and the spaceship clock? Give your answer in seconds.
\nSolution:<\/strong><\/span>
\n<\/p>\nChapter 29 Relativity Q.20P<\/strong>
\nAs measured in Earth\u2019s frame of reference, two planets are 424,000 km apart. A spaceship flies from one planet to the other with a constant velocity, and the clocks on the ship show that the trip lasts only 1.00 s. How fast is the ship traveling?
\nSolution:<\/strong><\/span>
\n
\n
\n<\/p>\nChapter 29 Relativity Q.21P<\/strong>
\nCaptain Jean-Luc is piloting the USS Enterprise XXIII at a constant speed v = 0.825c. As the Enterprise passes the planet Vulcan, he notices that Ms watch and the Vulcan clocks both read 1:00 p.m. At 3:00 p.m., according to his watch, the Enterprise passes the planet Endor. If the Vulcan and Endor clocks are synchronized with each other, what time do the Endor clocks read when the Enterprise passes by?
\nSolution:<\/strong><\/span>
\n
\n<\/p>\nChapter 29 Relativity Q.22P<\/strong>
\nIP A plane flies with a constant velocity of 222 m\/s. The clocks on the plane show that it takes exactly 2.00 h to travel a certain distance, (a) According to ground-based clocks, will the flight take slightly more or slightly less than 2.00 h? (b) Calculate how much longer or shorter than 2.00 h this flight will last, according to clocks on the ground.
\nSolution:<\/strong><\/span>
\n
\n<\/p>\nChapter 29 Relativity Q.23P<\/strong>
\nCE Tf the universal speed of light in a vacuum were larger than 3.00 X 108 m\/s, would the effects of length contraction be greater or less than they are now? Explain.
\nSolution:<\/strong><\/span>
\n<\/p>\nChapter 29 Relativity Q.24P<\/strong>
\nHow fast does a 250-m spaceship move relative to an observer who measures the ship\u2019s length to be 150 m?
\nSolution:<\/strong><\/span>
\n<\/p>\nChapter 29 Relativity Q.25P<\/strong>
\nSuppose the speed of light in a vacuum were only 25.0 mi\/h. Find the length of a bicycle being ridden at a speed of 20.0 mi\/h as measured by an observer sitting on a park bench, given that its proper length is 1.89 m.
\nSolution:<\/strong><\/span>
\n<\/p>\nChapter 29 Relativity Q.26P<\/strong>
\nA rectangular painting is 124 cm wide and 80.5 cm high, as indicated in Figure 29\u201329. At what speed, v, must the painting move parallel to its width if it is to appear to be square?
\n
\nSolution:<\/strong><\/span>
\n<\/p>\nChapter 29 Relativity Q.27P<\/strong>
\nThe Linac portion of the Fermilab Tevatron contains a high-vacuum tube that is 64 m long, through which protons travel with an average speed v = 0.65c. How long is the Linac tube, as measured in the proton\u2019s frame of reference?
\nSolution:<\/strong><\/span>
\n<\/p>\nChapter 29 Relativity Q.28P<\/strong>
\nA cubical box is 0.75 m on a side, (a) What are the dimensions of the box as measured by an observer moving with a speed of 0.88c parallel to one of the edges of the box? (b) What is the volume of the box, as measured by this observer?
\nSolution:<\/strong><\/span>
\n<\/p>\nChapter 29 Relativity Q.29P<\/strong>
\nWhen parked, your car is 5.0 m long. Unfortunately, your garage is only 4.0 m long, (a) How fast would your car have to be moving for an observer on the ground to find your car shorter than your garage? (b) When you are driving at this speed, how long is your garage, as measured in tire car\u2019s frame of reference?
\nSolution:<\/strong><\/span>
\n
\n<\/p>\nChapter 29 Relativity Q.30P<\/strong>
\nAn astronaut travels to a distant star with a speed of 0.55c relative to Earth. From the astronaut\u2019s point of view, the star is 7.5 ly from Earth. On the return trip, the astronaut travels with a speed of 0.89c relative to Earth. What is the distance covered on the return trip, as measured by the astronaut? Give your answer in light-years.
\nSolution:<\/strong><\/span>
\n
\n<\/p>\nChapter 29 Relativity Q.31P<\/strong>
\nIP Laboratory measurements show that an electron traveled 3.50 cm in a time of 0.200 ns. (a) In the rest frame of the electron, did the lab travel a distance greater than or less than 3.50 cm? Explain, (b) What is the electron\u2019s speed? (c) In the electron\u2019s frame of reference, how far did the laboratory travel?
\nSolution:<\/strong><\/span>
\n<\/p>\nChapter 29 Relativity Q.32P<\/strong>
\nYou and a friend travel through space in identical spaceships. Your friend informs you that he has made some length measurements and that his ship is 150 m long but that yours is only 120 m long. From your point of view, (a) how long is your friend\u2019s ship, (b) how long is your ship, and (c) what is the speed of your friend\u2019s ship relative to yours?
\nSolution:<\/strong><\/span>
\n
\n<\/p>\nChapter 29 Relativity Q.33P<\/strong>
\nA ladder 5.0 m long leans against a wall inside a spaceship. From the point of view of a person on the ship, the base of the ladder is 3.0 m from the wall, and the top of the ladder is 4.0 m above the floor. The spaceship moves past the Earth with a speed of 0.90c in a direction parallel to the floor of the ship. Find the angle the ladder makes with the floor as seen by an observer on Earth.
\nSolution:<\/strong><\/span>
\n
\n<\/p>\nChapter 29 Relativity Q.34P<\/strong>
\nWhen traveling past an observer with a relative speed v, a rocket is measured to be 9.00 m long. When the rocket moves with a relative speed 2v, its length is measured to be 5.00 m. (a) What is the speed v? (b) What is the proper length of the rocket?
\nSolution:<\/strong><\/span>
\n
\n
\n<\/p>\nChapter 29 Relativity Q.35P<\/strong>
\nIP The starships Picard and La Forge are traveling in the same direction toward the Andromeda galaxy. The Picard moves with a speed of 0.90c relative to the La Forge. A person on the La Forge measures the length of the two ships and finds the same value, (a) Tf a person on the Picard also measures the lengths of the two ships, which of the following is observed : (i) the Picard is longer; (ii) the La Forge is longer; or (iii) both ships have the same length? Explain, (b) Calculate the ratio of the proper length of the Picard to the proper length of the La Forge.
\nSolution:<\/strong><\/span>
\n
\n<\/p>\nChapter 29 Relativity Q.36P<\/strong>
\nA spaceship moving toward Earth with a speed of 0.90c launches a probe in the forward direction with a speed of 0.10c relative to the ship. Find the speed of the probe relative to Earth.
\nSolution:<\/strong><\/span>
\n<\/p>\nChapter 29 Relativity Q.37P<\/strong>
\nSuppose the probe in Problem 36 is launched in the opposite direction to the motion of the spaceship. Find the speed of the probe relative to Earth in this case.
\nSolution:<\/strong><\/span>
\n<\/p>\nChapter 29 Relativity Q.38P<\/strong>
\nA spaceship moving relative to an observer with a speed of 0.70c shines a beam of light in the forward direction, directly toward the observer. Use Equation 29\u20134 to calculate the speed of the beam of light relative to the observer.
\nSolution:<\/strong><\/span>
\n<\/p>\nChapter 29 Relativity Q.39P<\/strong>
\nSuppose the speed of light is 35 mi\/h. A paper girl riding a bicycle at 22 mi\/h throws a rolled-up newspaper in the forward direction, as shown in Figure 29\u201330. If the paper is thrown with a speed of 19 mi \/h relative to the bike, what is its speed, v, with respect to the ground?
\nFIGURE 29\u201330
\n
\nSolution:<\/strong><\/span>
\n<\/p>\nChapter 29 Relativity Q.40P<\/strong>
\nTwo asteroids head straight for Earth from the same direction. Their speeds relative to Earth are 0.80c for asteroid 1 and 0.60c for asteroid 2. Find the speed of asteroid 1 relative to asteroid 2.
\nSolution:<\/strong><\/span>
\n
\n<\/p>\nChapter 29 Relativity Q.41P<\/strong>
\nTwo rocket ships approach Earth from opposite directions, each with a speed of 0.8c relative to Earth. What is the speed of one ship relative to the other?
\nSolution:<\/strong><\/span>
\n<\/p>\nChapter 29 Relativity Q.42P<\/strong>
\nA spaceship and an asteroid are moving in the same direction away from Earth with speeds of 0.77c and 0.41c, respectively. What is the relative speed between the spaceship and the asteroid?
\nSolution:<\/strong><\/span>
\n
\n<\/p>\nChapter 29 Relativity Q.43P<\/strong>
\nAn electron moves to the right in a laboratory accelerator with a speed of 0.84c. A second electron in a different accelerator moves to the left with a speed of 0.43c relative to the first electron. Find the speed of the second electron relative to the lab.
\nSolution:<\/strong><\/span>
\n<\/p>\nChapter 29 Relativity Q.44P<\/strong>
\nIP Two rocket ships are racing toward Earth, as shown in Figure 29\u201331. Ship A is in the lead, approaching the Earth at 0.80c and separating from ship \u0412 with a relative speed of 0.50c. (a) As seen from Earth, what is the speed, v, of Ship B? (b) If ship A increases its speed by 0.10c relative to the Earth, does the relative speed between ship A and ship \u0412 increase by 0.10c, by more than 0.10c, or by less than 0.10c? Explain, (c) Find the relative speed between ships A and \u0412 for the situation described in part (b).
\n
\nSolution:<\/strong><\/span>
\n
\n<\/p>\nChapter 29 Relativity Q.45P<\/strong>
\nIP An inventor has proposed a device that will accelerate objects to speeds greater than c. He proposes to place the object to be accelerated on a conveyor belt whose speed is 0.80c. Next, the entire system is to be placed on a second conveyor belt that also has a speed of 0.80c, thus producing a final speed of 1.6c.
\n(a) Construction details aside, should you invest in this scheme?
\n(b) What is the actual speed of the object relative to the ground?
\nSolution:<\/strong><\/span>
\n<\/p>\nChapter 29 Relativity Q.46P<\/strong>
\nA 4.5 \u00d7 106-kg spaceship moves away from Earth with a speed of 0.75c. What is the magnitude of the ship\u2019s (a) classical and (b) relativistic momentum?
\nSolution:<\/strong><\/span>
\n<\/p>\nChapter 29 Relativity Q.47P<\/strong>
\nAn asteroid with a mass of 8.2 \u00d7 1011 kg is observed to have a relativistic momentum of magnitude 7.74 \u00d7 1020 kg \u2022 m\/s. What is the speed of the asteroid relative to the observer?
\nSolution:<\/strong><\/span>
\n
\n<\/p>\nChapter 29 Relativity Q.48P<\/strong>
\nAn object has a relativistic momentum that is 7.5 times greater than its classical momentum. What is its speed?
\nSolution:<\/strong><\/span>
\n<\/p>\nChapter 29 Relativity Q.49P<\/strong>
\nA football player with a mass of 88 kg and a speed of 2.0 m\/s collides head-on with a player from the opposing team whose mass is 120 kg. The players stick together and are at rest after the collision. Find the speed of the second player, assuming the speed of light is 3.0 m\/s.
\nSolution:<\/strong><\/span>
\n
\n
\n
\n<\/p>\nChapter 29 Relativity Q.50P<\/strong>
\nIn the previous problem, suppose the speed of the second player is 1.2 m\/s. What is the speed of the players after the collision?
\nSolution:<\/strong><\/span>
\n
\n
\n<\/p>\nChapter 29 Relativity Q.51P<\/strong>
\nA space probe with a rest mass of 8.2 \u00d7 107 kg and a speed of 0.50c smashes into an asteroid at rest and becomes embedded within it. If the speed of the probe-asteroid system is 0.26c after the collision, what is the rest mass of the asteroid?
\nSolution:<\/strong><\/span>