Show that an object with momentum p and rest mass m0 has a

Problem 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?

Problem 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.

Problem 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: I. the speed you measure will be greater than 0.1c in fact, it will be c, since all observers in inertial frames measure the same speed of light II. You will measure a speed less than 0.1c because of time dilation, which causes clocks to run slow. III. When you measure a speed you find if be between c and 0.9c.

Problem 2CQ The speed of light in glass is less than c. Why is this not a violation of the second postulate of relativity?

Problem 3CQ How would velocities add if the speed of light were infinitely large? Justify your answer by considering Equation 29–4.

Problem 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?

Problem 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.

Problem 4CQ Describe some of the everyday consequences that would follow if the speed of light were 35 mi/h.

Problem 4P • ?E 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: I. The clock will run slow, just as before. The rate of the clock depends only on relative speed, not on direction of motion. II. When the rocket reverses direction the rate of the clock reverses too, and this makes it run fast. III. Reversing the direction of the rocket undoes the the time dilation effect, and so the clock will now run at its normal rate.

Problem 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.

Problem 5P • ?E 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: I. 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. II. Collect your pay according to the clock on the spaceship because according to you the clock on Earth has run slow. III. 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.

Problem 6CQ According to the theory of relativity, the maximum speed for 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.

Problem 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?

Problem 7CQ Give an argument that shows that ?n object of finite mass cannot be accelerated from rest to a speed greater than the speed of light in a vacuum.

Problem 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?

Problem 8P • Refer to Example 29–1. How much does Benny age if he travels to Vega with a speed of 0.9995c?

Problem 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.)

Problem 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.

Problem 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.

Problem 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?

Problem 13P •• The ?– Particle The ?– 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?

Problem 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?

Problem 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?

Problem 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.

Problem 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?

Problem 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?

Problem 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.

Problem 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?

Problem 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.

Problem 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?

Problem 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.

Problem 24P • How fast does a 250-m spaceship move relative to an observer who measures the ship’s length to be 150 m?

Problem 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.

A rectangular painting is \(124 \mathrm{~cm}\) wide and \(80.5 \mathrm{~cm}\) high, as indicated in Figure 29–29. At what speed, , must the painting move parallel to its width if it is to appear to be square? Equation Transcription: Text Transcription: 124 cm 80.5 cm

Problem 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?

Problem 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?

Problem 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?

Problem 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.

Problem 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?

Problem 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?

Problem 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.

Problem 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?

Problem 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.

Problem 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.

Problem 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.

Problem 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.

Problem 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.

Suppose the speed of light is \(35 \mathrm{mi} / \mathrm{h}\). A paper girl riding a bi-cycle at \(22 \mathrm{mi} / \mathrm{h}\) throws a rolled-up newspaper in the forward direction, as shown in Figure 29–30. If the paper is thrown with a speed of \(19 \mathrm{mi} / \mathrm{h}\) relative to the bike, what is its speed, , with respect to the ground? Equation Transcription: Text Transcription: 35 mi/h 22 mi/h 19 mi/h

Problem 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?

Problem 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?

Problem 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.

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 B 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 B 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 B for the situation described in part (b).

Problem 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?

Problem 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?

Problem 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?

Problem 48P •• An object has a relativistic momentum that is 7.5 times greater than its classical momentum. What is its speed?

Problem 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.

Problem 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?

Problem 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?

Problem 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%?

Problem 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?

Problem 54P ?E Particles A through D have the following rest energies and total energies: Particle Rest Energy Total Energy A 6E 6 E ? 2E 4 E ? 4 E 6 £ D 3 E 4E Rank these particles in order of increasing (a) rest mass, (b) kinetic energy, and (c) speed. Indicate ties where appropriate.

Problem 55P Find the work that must be done on a proton to accelerate it from rest to a speed of 0.90c.

Problem 56P If a neutron moves with a speed of 0.99c, what are its (a) total energy, (b) rest energy, and (c) kinetic energy?

Problem 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.

Problem 58P When a certain capacitor is charged, its mass increases by 8.3 × 10?16 kg. How much energy is stored in the capacitor?

Problem 59P What minimum energy must a gamma ray have to create an electron-antielectron pair?

Problem 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.)

Problem 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?

Problem 62P An object has a total energy that is 5.5 times its rest energy. What is its speed?

Problem 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.)

Problem 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.)

What is the percent difference between the classical kinetic energy, \(K_{c l}=\frac{1}{2} m_{0} v^{2}\), and the correct relativistic kinetic energy, \(K=m_{0} c^{2} / \sqrt{1-v^{2} / c^{2}-m_{0} c^{2}}\), at a speed of (a) \(0.10 c\) and (b) \(0.90 c\)? Equation Transcription: Text Transcription: K_c l=\frac{1 2 m_0 v^2 K=m_0 c^2 / \sqrt{1-v^2 / c^2-m_0 c^2 0.10 c 0.90 c

Problem 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).

Problem 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?

Problem 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.

Problem 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?

Problem 70P Find the radius to which the Sun must be compressed for it to become a black hole.

Problem 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.

Problem 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?

Problem 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?

Problem 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.

Problem 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.

Problem 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: I. The acceleration of pie 1 is greater because the fact that it is hot means it has the greater energy. II. 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. III. The pies have the same acceleration regardless of their temperature because they have identical rest masses.

Problem 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.

Problem 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: I. The charged capacitor has more mass because it is storing energy within it, just like a compressed spring. II. The charged capacitor has less mass because some of its mass now appears as the energy of the electric field between its plates. III. 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.

Problem 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.

Problem 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.)

Problem 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.

Problem 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?

Problem 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?

Problem 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?

Problem 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?

Problem 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).

Problem 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?

When a particle of charge q and momentum enters a uniform magnetic field at right angles it follows a circular path of radius \(R=p / q B\), as shown in Figure 29-32. What radius does this expression predict for a proton traveling with a speed \(v=0.99 c\) through a magnetic field \(B=0.20 T\) if you use (a) the nonrelativistic momentum \((p=m v)\) or (b) the relativistic momentum \(\left(p=m v / \sqrt{1-v^{2} / c^{2}}\right)\)? Equation Transcription: Text Transcription: R=p/qB v=0.99 c B=0.20 T (p=mv) ( p=mv/ \sqrt 1 - v2/c2 )

Problem 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?

The rest energy, \(m_{0} c^{2}\), of a particle with a kinetic energy and a momentum can be determined as follows: \(m_{0} c^{2}=\frac{(p c)^{2}-K^{2}}{2 K}\) Suppose a pion (a subatomic particle) is observed to have a kinetic energy \(K=35.0 \mathrm{MeV}\) and a momentum \(p=5.61 \times 10^{-20} \mathrm{~kg}^{3} \mathrm{~m} / \mathrm{s}=105 \mathrm{MeV} / \mathrm{c}\). What is the rest energy of the pion? Give your answer in . Equation Transcription: Text Transcription: m0c2 m0c2=(pc)2 - K2 over 2K K=35.0MeV p=5.61 x 10-20 kg3 m/s=105MeV/c

Problem 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.

Problem 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.

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 \(L_{0}\) and is inclined at an angle \(\theta_{0}\) above the floor. Ob-server 2 sees the spaceship moving to the right with a speed \(v\) (a) Show that the length of the astronaut as measured by observer 2 is \(L=L_{0} \sqrt{1-\left(\frac{v^{2}}{c^{2}}\right) \cos ^{2} \theta_{0}}\) (b) Show that the angle \theta\) the astronaut makes with the floor of the ship, as measured by observer 2, is given by \(\tan \theta=\frac{\tan \theta_{0}}{\sqrt{1-v^{2} / c^{2}}}\) Equation Transcription: Text Transcription: L_0 phi_0 v L=L_0 sqrt 1-(v^2/c^2))cos^2 theta_0 theta tan =tan theta_0 sqrt 1-v^2/c^2

Problem 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?

Problem 95GP Show that the total energy of an object is related to its momentum by the relation E2 = p2c2 + (m0c2)2.

Problem 96GP Show that if 0<v1<? and 02<? 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.

Show that an object with momentum and rest mass \(m_{0}\) has a speed given by \(v=\frac{c}{\sqrt{1+\left(m_{0} c / p\right)^{2}}}\) Equation Transcription: Text Transcription: m_{0} v=\frac c{\sqrt{1+(m_0 c / p)^2

Decay of the \(\Sigma^{-}\) Particle When at rest, the \(\Sigma^{-}\) particle has a lifetime of ns before it decays into a neutron and a pion. One particular \(\Sigma^{-}\) particle is observed to travel \(3.0 \mathrm{~cm}\) in the lab before decaying. What was its speed? (Hint: Its speed was not \(\frac{2}{3} c\).) Equation Transcription: Text Transcription: \Sigma^{-} \Sigma^{-} \Sigma^{-} 3.0 cm 2 over 3c

Problem 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

Problem 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?

Problem 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

Problem 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

Problem 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?