Chapter 27

A spaceship is traveling toward earth from the space colony on Asteroid 1040 \(\mathrm{A}\) . The ship is at the halfway point of the trip, passing Mars at a speed of 0.9\(c\) relative to Mars's frame of reference. At the same instant, a passenger on the spaceship receives a radio message from her boyfriend on 1040 \(\mathrm{A}\) and another from her hairdresser on earth. According to the passenger on the ship, were these messages sent simultane- ously or at different times. If at different times, which one was sent first? Explain your reasoning.

\(\bullet\) A rocket is moving to the right at half the speed of light relative to the earth. A lightbulb in the center of a room inside the rocket suddenly turns on. Call the light hitting the front end of the room event \(A\) and the light hitting the back of the room event \(B\) . (See Figure \(27.23 . )\) Which event occurs first, \(A\) or \(B\) , or are they simultaneous, as viewed by (a) an astronaut riding in the rocket and (b) a person at rest on the earth?

A futuristic spaceship flies past Pluto with a speed of 0.964 \(\mathrm{c}\) relative to the surface of the planet. When the spaceship is directly overhead at an altitude of \(1500 \mathrm{km},\) a very bright signal light on the surface of Pluto blinks on and then off. An observer on Pluto measures the signal light to be on for 80.0\(\mu\) y. What is the duration of the light pulse as measured by the pilot of the spaceship?

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

\(\bullet\) You take a trip to Pluto and back (round trip 11.5 billion km), traveling at a constant speed (except for the turnaround at Pluto) of \(45,000 \mathrm{km} / \mathrm{h}\) . (a) How long does the trip take, in hours, from the point of view of a friend on earth? About how many years is this? (b) When you return, what will be the difference between the time on your atomic wristwatch and the time on your friend's? (Hint: Assume the distance and speed are highly precise, and carry a lot of significant digits in your calculation!)

\(\bullet\) The negative pion \(\left(\pi^{-}\right)\) is an unstable particle with an average lifetime of \(2.60 \times 10^{-8} \mathrm{s}\) (measured in the rest frame of the pion).(a) If the pion is made to travel at very high speed relative to a laboratory, its average lifetime is measured in the laboratory to be \(4.20 \times 10^{-7}\) s. Calculate the speed of the pion expressed as a fraction of \(c .\) (b) What distance, as measured in the laboratory, does the pion travel during its average lifetime?

\(\bullet\) An alien spacecraft is flying overhead at a great distance as you stand in your backyard. You see its searchlight blink on for 0.190 s. The first officer on the craft measures the searchlight to be on for 12.0 ms. (a) Which of these two measured times is the proper time? (b) What is the speed of the spacecraft relative to the earth, expressed as a fraction of the speed of light, c?

A spacecraft flies away from the earth with a speed of \(4.80 \times 10^{6} \mathrm{m} / \mathrm{s}\) relative to the earth and then returns at the same speed. The spacecraft carries an atomic clock that has been carefully synchronized with an identical clock that remains at rest on earth. The spacecraft returns to its starting point 365 days \((1\) year) later, as measured by the clock that remained on earth. What is the difference in the elapsed times on the two clocks, measured in hours? Which clock, the one in the spacecraft or the one on earth, shows the smaller elapsed time?

\(\bullet\) You measure the length of a futuristic car to be 3.60 \(\mathrm{m}\) when the car is at rest relative to you. If you measure the length of the car as it zooms past you at a speed of \(0.900 c,\) what result do you get?

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

In the year \(2084,\) a spacecraft flies over Moon Station III at a speed of 0.800\(c .\) A scientist on the moon measures the length of the moving spacecraft to be 140 \(\mathrm{m} .\) The spacecraft later lands on the moon, and the same scientist measures the length of the now stationary spacecraft. What value does she get?

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

A spaceship makes the long trip from earth to the nearest star system, Alpha Centauri, at a speed of 0.955\(c .\) The star is about 4.37 light years from earth, as measured in earth's frame of reference \((1\) light year is the distance light travels in a year). (a) How many years does the trip take, according to an observer on earth? (b) How many years does the trip take according to a passenger on the spaceship? (c) How many light years distant is Alpha Centauri from earth, as measured by a passenger on the speeding spacecraft? (Note that, in the ship's frame of reference, the passengers are at rest, while the space between earth and Alpha Centauri goes rushing past at 0.955\(c .\) (d) Use your answer from part (c) along with the speed of the spacecraft to calculate another answer for part (b). Do your two answers for that part agree? Should they?

A A muon is created 55.0 \(\mathrm{km}\) above the surface of the earth (as measured in the earth's frame). The average life- time of a muon, measured in its own rest frame, is 2.20\(\mu \mathrm{s}\) , and the muon we are considering has this lifetime. In the frame of the muon, the earth is moving toward the muon with a speed of 0.9860\(c\) . (a) In the muon's frame, what is its initial height above the surface of the earth? (b) In the muon's frame, how much closer does the earth get during the lifetime of the muon? What fraction is this of the muon's original height, as measured in the muon's frame? (c) In the earth's frame, what is the lifetime of the muon? In the earth's frame, how far does the muon travel during its life- time? What fraction is this of the muon's original height in the earth's frame?

\(\cdot\) An enemy spaceship is moving toward your starfighter with a speed of \(0.400 c,\) as measured in your reference frame. The enemy ship fires a missile toward you at a speed of 0.700\(c\) relative to the enemy ship. (See Figure \(27.24 . )\) (a) What is the speed of the missile relative to you? Express your answer in terms of the speed of light. (b) If you measure the enemy ship to be \(8.00 \times 10^{6} \mathrm{km}\) away from you when the missile is fired, how much time, measured in your frame, will it take the missile to reach you?

\(\bullet\) An imperial spaceship, moving at high speed relative to the planet Arrakis, fires a rocket toward the planet with a speed of 0.920\(c\) relative to the spaceship. An observer on Arrakis measures the rocket to be approaching with a speed of 0.360\(c .\) What is the speed of the spaceship relative to Arrakis? Is the spaceship moving toward or away from Arrakis?

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

A pursuit spacecraft from the planet Tatooine is attempting to catch up with a Trade Federation cruiser. As measured by an observer on Tatooine, the cruiser is traveling away from the planet with a speed of 0.600\(c .\) The pursuit ship is traveling at a speed of 0.800\(c\) relative to Tatooine, in the same direction as the cruiser. What is the speed of the cruiser relative to the pur- suit ship?

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

\(\bullet\) Neutron stars are the remains of exploded stars, and they rotate at very high rates of speed. Suppose a certain neutron star has a radius of 10.0 \(\mathrm{km}\) and rotates with a period of 1.80 \(\mathrm{ms}\) . (a) Calculate the surface rotational speed at the equator of the star as a fraction of \(c .\) (b) Assuming the star's surface is an iner- tial frame of reference (which it isn't, because of its rotation), use the Lorentz velocity transformation to calculate the speed of a point on the equator with respect to a point directly oppo- site it on the star's surface.

At what speed is the momentum of a particle three times as great as the result obtained from the nonrelativistic expression \(m v ?\)

\(\bullet\) (a) At what speed does the momentum of a particle differ by 1.0\(\%\) from the value obtained with the nonrelativistic expression \(m v ?\) (b) Is the correct relativistic value greater or less than that obtained from the nonrelativistic expression?

\(\bullet\) Sketch a graph of (a) the nonrelativistic Newtonian momentum as a function of speed \(v\) and (b) the relativistic momentum as a function of \(v .\) In both cases, start from \(v=0\) and include the region where \(v \rightarrow c .\) Does either of these graphs extend beyond \(v=c ?\)

An electron is acted upon by a force of \(5.00 \times 10^{-15} \mathrm{N}\) due to an electric field. Find the acceleration this force produces in each case: (a) The electron's speed is 1.00 \(\mathrm{km} / \mathrm{s}\) . (b) The electron's speed is \(2.50 \times 10^{8} \mathrm{m} / \mathrm{s}\) and the force is parallel to the velocity.

\(\bullet\) Using both the nonrelativistic and relativistic expressions, compute the kinetic energy of an electron and the ratio of the two results (relativistic divided by nonrelativistic), for speeds of (a) \(5.00 \times 10^{7} \mathrm{m} / \mathrm{s},\left(\) b) \(2.60 \times 10^{8} \mathrm{m} / \mathrm{s}\) . \right.

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

\(\bullet\) Particle annihilation. In proton-antiproton annihilation, a proton and an antiproton (a negatively charged particle with the mass of a proton) collide and disappear, producing electromagnetic radiation. If each particle has a mass of \(1.67 \times 10^{-27} \mathrm{kg}\) and they are at rest just before the annihilation, find the total energy of the radiation. Give your answers in joules and in electron volts.

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

\(\bullet\) In a hypothetical nuclear-fusion reactor, two deuterium nuclei combine or "fuse" to form one helium nucleus. The mass of a deuterium nucleus, expressed in atomic mass units (u), is 2.0136 u; that of a helium nucleus is 4.0015 u. \(\left(1 \mathrm{u}=1.661 \times 10^{-27} \mathrm{kg} .\right)\) (a) How much energy is released when 1.0 \(\mathrm{kg}\) of deuterium undergoes fusion? (b) The annual consumption of electrical energy in the United States is on the order of \(1.0 \times 10^{19} \mathrm{J} .\) How much deuterium must react to pro- duce this much energy?

An antimatter reactor. When a particle meets its antipar- ticle (more about this in Chapter 30 , they annihilate each other and their mass is converted to light energy. The United States uses approximately 1.0 \(\times 10^{20} \mathrm{J}\) of energy per year. (a) If all this energy came from a futuristic antimatter reactor, how much mass would be consumed yearly? (b) If this antimatter fuel had the density of Fe \(\left(7.86 / \mathrm{cm}^{3}\right)\) and were stacked in bricks to form a cubical pile, how high would it be? (Before you get your hopes up, antimatter reactors are a long way in the future-if they ever will be feasible.)

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

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

A space probe is sent to the vicinity of the star Capella, which is 42.2 light years from the earth. (A light year is the distance light travels in a year.) The probe travels with a speed of 0.9910\(c\) relative to the earth. An astronaut recruit on board is 19 years old when the probe leaves the earth. What is her biological age when the probe reaches Capella, as measured by (a) the astronaut and (b) someone on earth?

\(\bullet\) Why are we bombarded by muons? Muons are unstable subatomic particles (more on them in Chapter 30 ) that decay to electrons with a mean lifetime of 2.2\(\mu \mathrm{s}\) . They are produced when cosmic rays bombard the upper atmosphere about 10 \(\mathrm{km}\) above the earth's surface, and they travel very close to the speed of light. The problem we want to address is why we see any of them at the earth's surface. (a) What is the greatest distance a muon could travel during its 2.2\(\mu\) s lifetime? (b) According to your answer in part (a), it would seem that muons could never make it to the ground. But the 2.2\(\mu\) lifetime is measured in the frame of the muon, and they are moving very fast. At a speed of \(0.999 c,\) what is the mean lifetime of a muon as measured by an observer at rest on the earth? How far could the muon travel in this time? Does this result explain why we find muons in cos- mic rays? (c) From the point of view of the muon, it still lives for only \(2.2 \mu s,\) so how does it make it to the ground? What is the thickness of the 10 \(\mathrm{km}\) of atmosphere through which the muon must travel, as measured by the muon? Is it now clear how the muon is able to reach the ground?

\(\cdot\) A cube of metal with sides of length \(a\) sits at rest in the laboratory with one edge parallel to the \(x\) axis. Therefore, in the laboratory frame, its volume is \(a^{3} .\) A rocket ship flies past the laboratory parallel to the \(x\) axis with a velocity \(v .\) To an observer in the rocket, what is the volume of the metal cube?

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

\(\bullet\) By what minimum amount does the mass of 4.00 \(\mathrm{kg}\) of ice increase when the ice melts at \(0.0^{\circ} \mathrm{C}\) to form water at that same temperature? (The heat of fusion of water is \(3.34 \times 10^{5} \mathrm{J} / \mathrm{kg.} )\)

In certain radioactive beta decay processes (more about ese in Chapter \(30,\) the beta particle (an electron) leaves the omic nucleus with a speed of 99.95\(\%\) the speed of light relave to to the decaying nucleus. If this nucleus is moving at 5.00\(\%\) the speed of light, find the speed of the emitted electron relative to the laboratory reference frame if the electron is emitted (a) in the same direction that the nucleus is moving, (b) in the opposite direction from the nucleus's velocity. (c) In each case in parts (a) and (b), find the kinetic energy of the electron as measured in (i) the laboratory frame and (ii) the reference frame of the decaying nucleus.

Space travel? Travel to the stars requires hundreds or thousands of years, even at the speed of light. Some people have suggested that we can get around this difficulty by accelerating the rocket (and its astronauts) to very high speeds so that they will age less due to time dilation. The fly in this ointment is that it takes a great deal of energy to do this. Suppose you want to go to the immense red giant Betelgeuse, which is about 500 light years away. (A light year is the distance that light travels in one year.) You plan to travel at constance that in a 1000 kg rocket ship (a little over a ton), which, in reality, is far too small for this purpose. In each case that follows, calculate the time for the trip, as measured by people on earth and by astronauts in the rocket ship, the energy needed in joules, and the energy needed as a percent of U.S. yearly use (which is \(1.0 \times 10^{20} \mathrm{J} ) .\) For comparison, arrange your results in a table showing \(v_{\text { Rocket }}, t_{\text { Earth }}, t_{\text { Rocket }}, E(\) in \(\mathrm{J}),\) and \(E\) (as \(\%\) of U.S. use \() .\) The rocket ship's speed is (a) \(0.50 c,\) (b) 0.99\(c\) , and (c) 0.9999\(c .\) On the basis of your results, does it seem likely that any government will invest in such high- speed space travel any time soon?

A A nuclear device containing 8.00 kg of plutonium explodes. The rest mass of the products of the explosion is less than the original rest mass by one part in \(10^{4} .(\) a) How much energy is released in the explosion? (b) If the explosion takes place in \(4.00 \mu s,\) what is the average power developed by the bomb? (c) What mass of water could the released energy lift to a height of 1.00 \(\mathrm{km}\) ?

\(\bullet\) Electrons are accelerated through a potential difference of \(750 \mathrm{kV},\) so that their kinetic energy is \(7.50 \times 10^{5} \mathrm{eV}\) . (a) What is the ratio of the speed \(v\) of an electron having this energy to the speed of light, \(c ?\) (b) What would the speed be if it were computed from the principles of classical mechanics?

\(\bullet\) The distance to a particular star, as measured in the earth's frame of reference, is 7.11 light years \((1\) light year is the distance light travels in 1 year). A spaceship leaves earth headed for the star, and takes 3.35 years to arrive, as measured by passengers on the ship. (a) How long does the trip take, according to observers on earth? (b) What distance for the trip do passengers on the spacecraft measure? (Hint: What is the speed of light in units of 1\(y / y ? )\)

An airplane has a length of 60 \(\mathrm{m}\) when measured at rest. When the airplane is moving at 180 \(\mathrm{m} / \mathrm{s}(400 \mathrm{mph})\) in the alternate uni- verse, how long would it appear to be to a stationary observer? A. 24 \(\mathrm{m}\) B. 36 \(\mathrm{m}\) C. 48 \(\mathrm{m}\) D. 60 \(\mathrm{m}\) E. 75 \(\mathrm{m}\)