Mastering Physics Solutions Chapter 29 Relativity
Chapter 29 Relativity Q.1CQ
Some distant galaxies are moving away from us at speeds greater than 0.5c. What is the speed of the light received on Earth from these galaxies? Explain.
Solution:
According 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.
Chapter 29 Relativity Q.1P
CE 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:
- 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
- You will measure a speed less than 0.1c because of time dilation, which causes clocks to run slow.
- When you measure a speed you find if be between c and 0.9c.
Solution:
- As the signal passes our ship, we will measure speed of the signal greater than 0.1c
- 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.
Chapter 29 Relativity Q.2CQ
The speed of light in glass is less than c. Why is this not a violation of the second postulate of relativity?
Solution:
We 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.
Chapter 29 Relativity Q.2P
Albert is piloting his spaceship, heading east with a speed of 0.90c. Albert’s 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’s ship. With what speed does Isaac see?
Solution:
According 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’s light beam observed by Isaac is c
Chapter 29 Relativity Q.3CQ
How would velocities add if the speed of light were infinitely large? Justify your answer by considering Equation 29–4.
Solution:
Chapter 29 Relativity Q.3P
CE 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.
Solution:
The 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.
- (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
- (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
- (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
- (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
Chapter 29 Relativity Q.4CQ
Describe some of the everyday consequences that would follow if the speed of light were 35 mi/h.
Solution:
If the speed of light is 35 mi/h then we experience relativistic effects like
- A person in motion would age slowly than a person who stays at home due to time dilation.
- The length of a car in motion seems to be shorter due to length contraction.
- No object can move faster than light.
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
Chapter 29 Relativity Q.4P
СE 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:
- The clock will run slow, just as before. The rate of the clock depends only on relative speed, not on direction of motion.
- When the rocket reverses direction the rate of the clock reverses too, and this makes it run fast.
- Reversing the direction of the rocket undoes the the time dilation effect, and so the clock will now run at its normal rate.
Solution:
- The clock will still run slow as before.
- Answer (I)
The clock will run slow, just as before. The rate of the clock depends only on relative speed, not on direction of motion.
Chapter 29 Relativity Q.5CQ
When we view a distant galaxy, we notice that the light coming from it has a longer wavelength (it is “red-shifted”) than the corresponding light here on Earth. Is this consistent with the postulate that all observers measure the same speed of light? Explain.
Solution:
Yes
Here 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 “red shifted” light will be affected, this is because we know the formula. v = λf 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.
Chapter 29 Relativity Q.5P
СE 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:
- 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.
- Collect your pay according to the clock on the spaceship because according to you the clock on Earth has run slow.
- 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.
Solution:
- I would like to be paid according to the clock at Spacely Sprockets universal Headquarters on earth.
- 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.
Chapter 29 Relativity Q.6CQ
According 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.
Solution:
Chapter 29 Relativity Q.6P
A neon sign in front of a café 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?
Solution:
Chapter 29 Relativity Q.7CQ
Give an argument that shows that аn object of finite mass cannot be accelerated from rest to a speed greater than the speed of light in a vacuum.
Solution:
Chapter 29 Relativity Q.7P
A 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?
Solution:
Chapter 29 Relativity Q.8P
Refer to Example 29–1. How much does Benny age if he travels to Vega with a speed of 0.9995c?
Solution:
Chapter 29 Relativity Q.9P
As a spaceship flies past with speed v, you observe that 1.0000 s elapses on the ship’s 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.)
Solution:
Chapter 29 Relativity Q.10P
Donovan 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’s run and measured his time to be 15.44 s, find the time that was recorded on Earth.
Solution:
Chapter 29 Relativity Q.11P
Find the average distance (in the Earth’s frame of reference) covered by the muons in Example 29–2 if their speed relative to Earth is 0.750c.
Solution:
Chapter 29 Relativity Q.12P
The Pi Meson An elementary particle called a pi meson (or pion for short) has an average lifetime of 2.6 × 10–8 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?
Solution:
Chapter 29 Relativity Q.13P
The Σ– Particle The Σ– 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?
Solution:
Chapter 29 Relativity Q.14P
IP (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’s 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?
Solution:
Chapter 29 Relativity Q.15P
The 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?
Solution:
Chapter 29 Relativity Q.16P
An 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?
Solution:
Chapter 29 Relativity Q.17P
IP 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’s heart rate, is the result greater than, less than, or equal to 72 beats per minute? Explain. (b) Calculate the astronaut’s heart rate as measured on Earth.
Solution:
Chapter 29 Relativity Q.18P
BIO 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?
Solution:
Chapter 29 Relativity Q.19P
An 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.
Solution:
Chapter 29 Relativity Q.20P
As measured in Earth’s 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?
Solution:
Chapter 29 Relativity Q.21P
Captain 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?
Solution:
Chapter 29 Relativity Q.22P
IP 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.
Solution:
Chapter 29 Relativity Q.23P
CE 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.
Solution:
Chapter 29 Relativity Q.24P
How fast does a 250-m spaceship move relative to an observer who measures the ship’s length to be 150 m?
Solution:
Chapter 29 Relativity Q.25P
Suppose 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.
Solution:
Chapter 29 Relativity Q.26P
A rectangular painting is 124 cm wide and 80.5 cm high, as indicated in Figure 29–29. At what speed, v, must the painting move parallel to its width if it is to appear to be square?
Solution:
Chapter 29 Relativity Q.27P
The 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’s frame of reference?
Solution:
Chapter 29 Relativity Q.28P
A 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?
Solution:
Chapter 29 Relativity Q.29P
When 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’s frame of reference?
Solution:
Chapter 29 Relativity Q.30P
An astronaut travels to a distant star with a speed of 0.55c relative to Earth. From the astronaut’s 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.
Solution:
Chapter 29 Relativity Q.31P
IP 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’s speed? (c) In the electron’s frame of reference, how far did the laboratory travel?
Solution:
Chapter 29 Relativity Q.32P
You 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’s ship, (b) how long is your ship, and (c) what is the speed of your friend’s ship relative to yours?
Solution:
Chapter 29 Relativity Q.33P
A 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.
Solution:
Chapter 29 Relativity Q.34P
When 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?
Solution:
Chapter 29 Relativity Q.35P
IP 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.
Solution:
Chapter 29 Relativity Q.36P
A 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.
Solution:
Chapter 29 Relativity Q.37P
Suppose 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.
Solution:
Chapter 29 Relativity Q.38P
A 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 29–4 to calculate the speed of the beam of light relative to the observer.
Solution:
Chapter 29 Relativity Q.39P
Suppose 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 29–30. 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?
FIGURE 29–30
Solution:
Chapter 29 Relativity Q.40P
Two 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.
Solution:
Chapter 29 Relativity Q.41P
Two 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?
Solution:
Chapter 29 Relativity Q.42P
A 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?
Solution:
Chapter 29 Relativity Q.43P
An 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.
Solution:
Chapter 29 Relativity Q.44P
IP Two rocket ships are racing toward Earth, as shown in Figure 29–31. Ship A is in the lead, approaching the Earth at 0.80c and separating from ship В 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 В 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 В for the situation described in part (b).
Solution:
Chapter 29 Relativity Q.45P
IP 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.
(a) Construction details aside, should you invest in this scheme?
(b) What is the actual speed of the object relative to the ground?
Solution:
Chapter 29 Relativity Q.46P
A 4.5 × 106-kg spaceship moves away from Earth with a speed of 0.75c. What is the magnitude of the ship’s (a) classical and (b) relativistic momentum?
Solution:
Chapter 29 Relativity Q.47P
An asteroid with a mass of 8.2 × 1011 kg is observed to have a relativistic momentum of magnitude 7.74 × 1020 kg • m/s. What is the speed of the asteroid relative to the observer?
Solution:
Chapter 29 Relativity Q.48P
An object has a relativistic momentum that is 7.5 times greater than its classical momentum. What is its speed?
Solution:
Chapter 29 Relativity Q.49P
A 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.
Solution:
Chapter 29 Relativity Q.50P
In 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?
Solution:
Chapter 29 Relativity Q.51P
A space probe with a rest mass of 8.2 × 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?
Solution:
Chapter 29 Relativity Q.52P
At what speed does the classical momentum, p = mv, give an error, when compared with the relativistic momentum, of (a) 1.00% and (b)’5.00%?
Solution:
Chapter 29 Relativity Q.53P
A proton has 1836 times the rest mass of an electron. At what speed will an electron have the same momentum as a proton moving at 0.0100c?
Solution:
Chapter 29 Relativity Q.54P
СE Particles A through D have the following rest energies and total energies:
Rank these particles in order of increasing (a) rest mass, (b) kinetic energy, and (c) speed. Indicate ties where appropriate.
Solution:
Chapter 29 Relativity Q.55P
Find the work that must be done on a proton to accelerate it from rest to a speed of 0.90c.
Solution:
Chapter 29 Relativity Q.56P
If a neutron moves with a speed of 0.99c, what are its (a) total energy, (b) rest energy, and (c) kinetic energy?
Solution:
Chapter 29 Relativity Q.57P
A spring with a force constant of 584 N/m is compressed a distance of 39 cm. Find the resulting increase in the spring’s mass.
Solution:
Chapter 29 Relativity Q.58P
When a certain capacitor is charged, its mass increases by 8.3 × 10−16 kg. How much energy is stored in the capacitor?
Solution:
Chapter 29 Relativity Q.59P
What minimum energy must a gamma ray have to create an electron-antielectron pair?
Solution:
Chapter 29 Relativity Q.60P
When a proton encounters an antiproton, the two particles annihilate each other, producing two gamma rays. Assuming the particles were at rest when they annihilated, find the energy of each of the two gamma rays produced. (Note: Tire rest energies of an antiproton and a proton are identical.)
Solution:
Chapter 29 Relativity Q.61P
A rocket with a mass of 2.7 × 106 kg has a relativistic kinetic energy of 2.7 × 1023 J. How fast is the rocket moving?
Solution:
Chapter 29 Relativity Q.62P
A rocket with a mass of 2.7 × 106 kg has a relativistic kinetic energy of 2.7 × 1023 J. How fast is the rocket moving?
Solution:
Chapter 29 Relativity Q.63P
A nuclear power plant produces an average of 1.0 × 103 MW of power during a year of operation. Find the corresponding change in mass of reactor fuel, assuming all of the energy released by the fuel can be converted directly to electrical energy. (In a practical reactor, only a relatively small fraction of the energy can be converted to electricity.)
Solution:
Chapter 29 Relativity Q.64P
A helium atom has a rest mass of mHe = 4.002603 u. When disassembled into its constituent particles (2 protons, 2 neutrons, 2 electrons), the well-separated individual particles have the following masses: mp = 1.007276 u, mn = 1.008665 u, me = 0.000549 u. How much work is required to completely disassemble a helium atom? (Note: 1 u of mass has a rest energy of 931.49 MeV.)
Solution:
Chapter 29 Relativity Q.65P
Solution:
Chapter 29 Relativity Q.66P
A proton has 1836 times the rest mass of an electron. At what speed will an electron have the same kinetic energy as a proton moving at 0.0250c?
Solution:
Chapter 29 Relativity Q.67P
IP Consider a baseball with a rest mass of 0.145 kg. (a) How much work is required to increase the speed of the baseball from 25.0 m/s to 35.0 m/s? (b) Is the work required to increase the speed of the baseball from 200,000,025 m/s to 200,000,035 m/s greater than, less than, or the same as the amount found in part (a)? Explain, (c) Calculate the work required for the increase in speed indicated in part (b).
Solution:
Chapter 29 Relativity Q.68P
IP A particle has a kinetic energy equal to its rest energy, (a) What is the speed of this particle? (b) If the kinetic energy of this particle is doubled, does its speed increase by a more than, less than, or exactly a factor of 2? Explain. (c) Calculate the speed of a particle whose kinetic energy is twice its rest energy.
Solution:
Chapter 29 Relativity Q.69P
A lump of putty with a mass of 0.240 kg and a speed of 0.980c collides head-on and sticks to an identical lump of putty moving with the same speed. After the collision the system is at rest. What is the mass of the system after the collision?
Solution:
Chapter 29 Relativity Q.70P
Find the radius to which the Sun must be compressed for it to become a black hole.
Solution:
Chapter 29 Relativity Q.71P
The Black Hole in the Center of the Milky Way Recent measurements show that the black hole at the center of the Milky Way galaxy, which is believed to coincide with the powerful radio source Sagittarius A*, is 2.6 million times more massive than the Sun; that is, M = 5.2 × 1036 kg. (a) What is the maximum radius of this black hole? (b) Find the acceleration of gravity at the Schwarzschild radius of this black hole, using the expression for R given in Equation 29–10. (c) How does your answer to part (b) change if the mass of the black hole is doubled? Explain.
Solution:
Chapter 29 Relativity Q.72GP
CE Two observers are moving relative to one another. Which of the following quantities will they always measure to have the same value: (a) their relative speed; (b) the time between two events; (c) the length of an object; (d) the speed of light in a vacuum; (e) the speed of a third observer?
Solution:
Here two observers are moving relative to one another so they measure that they have the same speed relative to one another. Also according to the second postulate of special theory of relativity the speed of light is c in vacuum in all inertial frames of references. So the two observers will always measure
(a) The relative speed and
(d) The speed of light in vacuum have the same value, the remaining are the time between two events, length of an object and speed of a third observer are different for different observers.
Chapter 29 Relativity Q.73GP
CE You are standing next to a runway as an airplane kinds, (a) If you and the pilot observe a clock in the cockpit, which of you measures the proper time? (b) If you and the pilot observe a large clock on the control tower, which of you measures the proper time? (c) Which of you measures the proper length of the airplane? (d) Which of you measures the proper length of the runway?
Solution:
(a) The pilot will measure the correct proper time of the cockpit clock because the he is in rest frame of reference with respect to the clock. So pilot will observe correct proper time in the clock which is in cock pit.
(b) You measure the correct proper time in the large clock on the control tower, because you are in the rest frame of reference with respect to the clock on the control tower.
(c) The pilot measures the correct proper length of the aero plane. Because pilot is in the rest frame of reference with respect to the aero plane.
(d) You measure the correct proper length of the runway because you are in the rest frame with respect to the runway.
Chapter 29 Relativity Q.74GP
CE Which clock runs slower relative to a clock on the North Pole: clock on an airplane flying from New York to Los Angeles, or clock 2 on an airplane flying from Los Angeles to New York? Assume each plane has the same speed relative to the surface of the Earth. Explain.
Solution:
Clock 2 on an airplane flying from Los Angeles to New York runs slower than clock 1 on an airplane flying from New York to Los Angeles because its speed relative to the axis of the spinning earth is greater than the speed of other airplane.
Here the airplane which has the clock 2 has greater speed because the direction of motion of this airplane is in the same direction of the axis of the spinning earth.
Chapter 29 Relativity Q.75GP
CE An apple drops from the bough of a tree to the ground. Is the mass of the apple near the top of its fall greater than, less than, or the same as its mass after it has landed? Explain.
Solution:
Let h be the height of apple in the bough of a tree from the ground.
If the apple is at the near top of its fall, i.e. at greater height h then t the earth- apple system has more gravitational potential energy. Hence this increased energy is equivalent to the increased mass according to the Einstein’s mass-energy equivalence relation. E = m\({ c }^{ 2 }\)
So the mass of the apple near the top of its fall is greater
Chapter 29 Relativity Q.76GP
CE Predict/Explain Consider two apple pies that are identical in every respect, except that pie 1 is piping hot and pie 2 is at room temperature, (a) If identical forces are applied to the two pies, is the acceleration of pie 1 greater than, less than, or equal to the acceleration of pie 2? (b) Choose the best explanation from among the following:
- The acceleration of pie 1 is greater because the fact that it is hot means it has the greater energy.
- The fact that pie 1 is hot means it behaves as if it has more mass than pie 2, and therefore it has a smaller acceleration.
- The pies have the same acceleration regardless of their temperature because they have identical rest masses.
Solution:
a) Acceleration of pie 1 is less than pie 2
(b) Here pie1 is hotter than the pie2. That means the pie 1 has more energy than the pie 2. Therefore the pie1 is more massive than the pie 2.
So when we apply same force to the two pies, the massive pie will have less acceleration. Therefore acceleration of pie 1 is less than the acceleration of pie 2. Therefore option II is correct.
Chapter 29 Relativity Q.77GP
CE Is the mass of a warm cup of tea greater than, less than, or the same as the mass of the same cup of tea when it has cooled? Explain.
Solution:
The Einstein’s mass – energy equivalence relationship E = m\({ c }^{ 2 }\)
From this formula we can say that the energy (E) and mass (m) of a particle are directly proportional to each other. The warm cup of tea has more energy than the same cup of tea when it has cooled. So by using the above relation the mass of warm cup of tea is greater than the mass of the same cup of tea when it has cooled.
Chapter 29 Relativity Q.78GP
CE Predict/Explain An uncharged capacitor is charged by moving some electrons from one plate of the capacitor to the other plate, (a) Is the mass of the charged capacitor greater than, less than, or the same as the mass of the uncharged capacitor? (b) Choose the best explanation from among the following:
- The charged capacitor has more mass because it is storing energy within it, just like a compressed spring.
- The charged capacitor has less mass because some of its mass now appears as the energy of the electric field between its plates.
- The capacitor has the same mass whether it is charged or not because charging it only involves moving electrons from one plate to the other without changing the total number of electrons.
Solution:
(a) Mass of charged capacitor is greater than the mass of uncharged capacitor.
(b) Option I is correct
The charged capacitor has more mass because it is storing energy within it, just like a compressed spring.
Chapter 29 Relativity Q.79GP
Cosmic Rays Protons in cosmic rays have been observed with kinetic energies as large as 1.0 × 1020 eV. (a) How fast are these protons moving? Give your answer as a fraction of the speed of light, (b) Show that the kinetic energy of a single one of these protons is much greater than the kinetic energy of a 15-mg ant walking with a speed of 8.8 mm/s.
Solution:
Chapter 29 Relativity Q.80GP
An apple falls from a tree, landing on the ground 3.7 m below. How long is the apple in the air, as measured by an observer moving toward Earth with a speed of 0.89c?
Solution:
Chapter 29 Relativity Q.81GP
What is the momentum of a proton with 1.50 × 103 MeV of kinetic energy? (Note: The rest energy of a proton is 938 MeV.)
Solution:
Chapter 29 Relativity Q.82GP
IP A container holding 2.00 moles of an ideal monatomic gas is heated at constant volume until the temperature of the gas increases by 112 F°. (a) Does the mass of the gas increase, decrease, or stay the same? Explain, (b) Calculate the change in mass of the gas, if any.
Solution:
Chapter 29 Relativity Q.83GP
Al4C nucleus, initially at rest, emits a beta particle. The beta particle is an electron with 156 keV of kinetic energy, (a) What is the speed of the beta particle? (b) What is the momentum of the beta particle? (c) What is the momentum of the nucleus after it emits the beta particle? (d) What is the speed of the nucleus after it emits the beta particle?
Solution:
Chapter 29 Relativity Q.84GP
A clock at rest has a rectangular shape, with a width of 24 cm and a height of 12 cm. When this clock moves parallel to its width with a certain speed v its width and height are the same. Relative to a clock at rest, how long does it take for the moving clock to advance by 1.0 s?
Solution:
Chapter 29 Relativity Q.85GP
A starship moving toward Earth with a speed of 0.75c launches a shuttle craft in the forward direction. Tire shuttle, which has a proper length of 12.5 m, is only 6.25 m long as viewed from Earth. What is the speed of the shuttle relative to the starship?
Solution:
Chapter 29 Relativity Q.86GP
When a particle of charge q and momentum p enters a uniform magnetic field at right angles it follows a circular path of radius R = p/qB, as shown in Figure 29–32. What radius does this expression predict for a proton traveling with a speed v = 0.99c through a magnetic field В = 0.20 T if you use (a) the nonrelativistic momentum (p = mv) or (b) the relativistic momentum
Solution:
Chapter 29 Relativity Q.87GP
IP A starship moving away from Earth with a speed of 0.75c launches a shuttle craft in the reverse direction, that is, toward Earth. (a) If the speed of the shuttle relative to the starship is 0.40c, and its proper length is 13 m, how long is the shuttle as measured by an observer on Earth? (b) If the shuttle had been launched in the forward direction instead, would its length as measured by an observer on Earth be greater than, less than, or the same as the length found in part (a)? Explain. (c) Calculate the length for the case described in part (b).
Solution:
Chapter 29 Relativity Q.88GP
A 2.5-m titanium rod in a moving spacecraft is at an angle of 45° with respect to the direction of motion. The craft moves directly toward Earth at 0.98c. As viewed from Earth, (a) how long is the rod and (b) what angle does the rod make with the direction of motion?
Solution:
Chapter 29 Relativity Q.89GP
Electrons are accelerated from rest through a potential difference of 276,000 V. What is the final speed predicted (a) classically and (b) relativistically?
Solution:
Chapter 29 Relativity Q.90GP
IP In Conceptual Checkpoint 29-2 we considered an astronaut at rest on an inclined bed inside a moving spaceship. From the point of view of observer 1, on board the ship, the astronaut has a length L0 and is inclined at an angle θ0 above the floor. Observer 2 sees the spaceship moving to the right with a speed v.
Suppose a pion (a subatomic particle) is observed to have a kinetic energy K = 35.0 MeV and a momentum p = 5.61 × 1020 kg3 m/s = 105 MeV/c. What is the rest energy of the pion? Give your answer in MeV.
Solution:
Chapter 29 Relativity Q.91GP
A small star of mass m orbits a supermassive black hole of mass M. (a) Find the orbital speed of the star if its orbital radius is 2R, where R is the Schwarzschild radius (Equation 29–10). (b) Repeat part (a) for an orbital radius equal to R.
Solution:
Chapter 29 Relativity Q.92GP
IP Consider a “relativistic air track” on which two identical air carts undergo a completely inelastic collision. One cart is initially at rest; the other has an initial speed of 0.650c. (a) In classical physics, the speed of the carts after the collision would be 0.325c. Do you expect the final speed in this relativistic collision to be greater than or less than 0.325c? Explain, (b) Use relativistic momentum conservation to find the speed of the carts after they collide and stick together.
Solution:
Chapter 29 Relativity Q.93GP
IP In Conceptual Checkpoint 29-2 we considered an astronaut at rest on an inclined bed inside a moving spaceship. From the point of view of observer 1, on board the ship, the astronaut has a length L0 and is inclined at an angle θ0 above the floor. Observer 2 sees the spaceship moving to the right with a speed v.
Solution:
Chapter 29 Relativity Q.94GP
A pulsar is a collapsed, rotating star that sends out a narrow beam of radiation, like the light from a lighthouse. With each revolution, we see a brief, intense pulse of radiation from the pulsar. Suppose a pulsar is receding directly away from Earth with a speed of 0.800c, and the starship Endeavor is sent out toward the pulsar with a speed of 0.950c relative to Earth. If an observer or Earth finds that 153 pulses are emitted by the pulsar every second, at what rate does an observer on the Endeavor see pulses emitted?
Solution:
Chapter 29 Relativity Q.95GP
Show that the total energy of an object is related to its momentum by the relation E2 = p2c2 + (m0c2)2.
Solution:
Chapter 29 Relativity Q.96GP
Show that if 0<v1<с and 02<с are two velocities pointing in the same direction, the relativistic sum of these velocities, v, is greater than v1 and greater than v2 but less than c. In particular, show that this is true even if v1 and v2 are greater than 0.5c.
Solution:
Chapter 29 Relativity Q.97GP
Show that an object with momentum p and rest mass m0 has a speed given by
Solution:
Chapter 29 Relativity Q.98GP
Decay of the ∑− Particle When at rest, the ∑− particle has a lifetime of 0.15 ns before it decays into a neutron and a pion. One particular particle is observed to travel 3.0 cm in the lab before decaying. What was its speed?
Solution:
Chapter 29 Relativity Q.99PP
Find the speed of an electron accelerated through a voltage of 25.0 kV—ignoring relativity. Express your answer as a fraction times the speed of light. (Speeds over about 0.1c are generally regarded as relativistic.)
A. 0.221c
B. 0.281c
C. 0.312c
D. 0.781c
Solution:
Chapter 29 Relativity Q.100PP
When relativistic effects are included, do you expect the speed of the electrons to be greater than, less than, or the same as the result found in the previous problem?
Solution:
When the relativistic effects are induced then the speed of the electrons will be less than the speed we found in problem 99.
Chapter 29 Relativity Q.101PP
Find the speed of the electrons in Problem 99, this time using a correct relativistic calculation. As before, express your answer as a fraction times the speed of light.
A. 0.301c
B. 0.312c
C. 0.412c
D. 0.953c
Solution:
Chapter 29 Relativity Q.102PP
Suppose the accelerating voltage in Problem 99 is increased by a factor of 10. What is the correct relativistic speed of an electron in this case?
A. 0.205c
B. 0.672c
C. 0.740c
D. 0.862c
Solution:
Chapter 29 Relativity Q.103IP
Referring to Example 29?4 The Picard approaches star- base Faraway Point with a speed of 0.806c, and the La Forge approaches the starbase with a speed of 0.906c. Suppose the Picard now launches a probe toward tire starbase. (a) What velocity must the probe have relative to the Picard if it is to be at rest relative to the la Forge? (b) Win at velocity must the probe have relative to the Picard if its velocity relative to the La Forge is to be 0.100c? (c) For the situation described in part (b), what is the velocity of the probe relative to the Faraway Point starbase?
Solution:
Chapter 29 Relativity Q.104IP
Referring to Example 29-4 Faraway Point starbase launches a probe toward the approaching starships. The probe has a velocity relative to the Picard of −0.906c. The Picard approaches starbase Faraway Point with a speed of 0.806c, and the La Forge approaches the starbase with a speed of 0.906c. (a) What is the velocity of the probe relative to the La Forge? (b) What is the velocity of the probe relative to Faraway Point starbase?
Solution:
