Chapter 37

The positive muon \(\left(\mu^{+}\right),\) an unstable particle, lives on average \(2.20 \times 10^{-6} \mathrm{s}\) (measured in its own frame of reference) before decaying. (a) If such a particle is moving, with respect to the laboratory, with a speed of \(0.900 c,\) what average lifetime is measured in the laboratory? (b) What average distance, measured in the laboratory, does the particle move before decaying?

How fast must a rocket travel relative to the earth so that time in the rocket "slows down" to half its rate as measured by earth-based observers? Do present-day jet planes approach such speeds?

A spaceship flies past Mars with a speed of 0.985 c relative to the surface of the planet. When the spaceship is directly overhead, a signal light on the Martian surface blinks on and then off. An observer on Mars measures that the signal light was on for 75.0\(\mu\) s. (a) Does the observer on Mars or the pilot on the spaceship measure the proper time? (b) What is the duration of the light pulse measured by the pilot of the spaceship?

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 \(4.20 \times 10^{-7} \mathrm{s}\) . Calculate the speed of the pion expressed as a fraction of \(c .\) (b) What distance, measured in the laboratory, does the pion travel during its average lifetime?

As you pilot your space utility vehicle at a constant speed toward the moon, a race pilot flies past you in her spaceracer at a constant speed of 0.800 c relative to you. At the instant the space-racer passes you, both of you start timers at zero. (a) At the instant when you measure that the spaceracer has traveled \(1.20 \times 10^{8} \mathrm{in}\) past you, what does the race pilot read on her timer? (b) When the race pilot reads the value calculated in part (a) on her timer, what does she measure to be your distance from her? (c) At the instant when the race pilot reads the value calculated in part (a) on her timer, what do you read on yours?

A spaceraft ties 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 shortest elapsed time?

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 spacecraft measures that the searchlight is on for 12.0 \(\mathrm{ms}\) . (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 of the Trade Federation flies past the planet Coruscant at a speed of 0.600 \(\mathrm{c}\) . A scientist on Coruscant measures the length of the moving spacecraft to be 74.0 \(\mathrm{m}\) . The spacecraft later lands on Coruscant, and the same scientist measures the length of the now stationary spacecraft. What value does she get?

A meter stick 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 meter stick to be \(1.00 \mathrm{ft}(1 \mathrm{ft}=0.3048 \mathrm{m})-\) for example, by comparing it to a 1 -foot ruler that is at rest relative to you \(-\) at what speed is the meter stick moving relative to you?

Muons are unstable subatomic particles that decay to electrons with a mean lifetime of 2.2\(\mu s\) . 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 -\mus lifetime? (b) According to your answer in part (a), it would seem that muons could never make it to the ground. But the \(2.2-\mu\) s lifetime is measured in the frame of the muon, and muons are moving very fast. At a speed of 0.999 \(\mathrm{c}\) . What is the mean lifetime of a muon as measured by an observer at rest on the earth? How far would the muon travel in this time? Does this result explain why we find muons in cosmic 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? It is now clear how the muon is able to reach the ground?

An unstable particle is created in the upper atmosphere from a cosmic ray and travels straight down toward the surface of the earth with a speed of 0.99540 c relative to the earth. A scientist at rest on the carth's surface measures that the particle is created at an altitude of 45.0 \(\mathrm{km}\) (a) As measured by the scientist, how much time does it take the particle to travel the 45.0 \(\mathrm{km}\) to the surface of the earth? ( b) Use the length- contraction formula to calculate the distance from where the particle is created to the surface of the earth as measured in the particle's frame. (c) In the particle's frame, how much time does it take the particle to travel from where it is created to the surface of the earth? Calculate this time both by the time dilation formula and from the distance calculated in part (b). Do the two results agree?

As measured by an observer on the earth, a spacecraft runway on earth has a length of 3600 \(\mathrm{m}\) (a) What is the length of the runway as measured by a pilot of a spacecraft flying past at a speed of \(4.00 \times 10^{7} \mathrm{m} / \mathrm{s}\) relative to the earth? (b) An observer on earth measures the time interval from when the spacecraft is directly over one end of the runway until it is directly over the other end. What result does she get? (c) The pilot of the spacecraft measures the time it takes him to travel from one end of the runway to the other end. What value does he get?

An observer in frame \(S^{\prime}\) is moving to the right \((+x-\text { direction })\) at speed \(u=0.600 \mathrm{c}\) away from a stationary observer in frame \(S\) . The observer in \(S^{\prime}\) measures the speed \(v^{\prime}\) of a particle moving to the right away from her. What speed \(v\) does the observer in S measure for the particle if \((a) v^{\prime}=0.400 c ;(b) v^{\prime}=0.900 c\) (c) \(v^{\prime}=0.990 c ?\)

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 \(\mathrm{c}\) . The pursuit ship is traveling at a speed of 0.800 \(\mathrm{c}\) relative to Tatooine, in the same direction as the cruiser. (a) For the pursuit ship to catch the cruiser, should the speed of the cruiser relative to the pursuit ship be positive or negative? (b) What is the speed of the cruiser relative to the pursuit ship?

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 \(\mathrm{c}\) , and the speed of each partcle relative to the other is 0.950 \(\mathrm{c}\) . What is the speed of the second particle, as measured in the laboratory?

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?

An enemy spaceship is moving toward your starfighter with a speed, as measured in your frame, of 0.400 \(\mathrm{c}\) . The enemy ship fires a missile toward you at a speed of 0.700\(c\) relative to the enemy ship (Fig. 37.28\()\) . (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 that the enemy ship is \(8.00 \times 10^{6} \mathrm{km}\) away from you when the missile is fired, how much time, measured in your frame, will it take the missile to reach you?

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 that the rocket is approaching with a speed of 0.360 \(\mathrm{c}\) . What is the speed of the spaceship relative to Arrakis? Is the spaceship moving toward or away from Arrakis?

(a) How fast must you be spproaching a red traffic light \((\lambda=675 \mathrm{nm})\) for it to appear yellow \((\lambda=575 \mathrm{nm}) ?\) Express your answer in terms of the speed of light. (b) If you used this as a reason not to get a ticket for running a red light, how much of a fine would you get for speeding? Assume that the fine is \(\$ 1.00\) for each kilometer per hour that your speed exceds the posted limit of 90 \(\mathrm{km} / \mathrm{h}\).

Show that when the source of electromagnetic waves moves away from us at 0.600 \(\mathrm{c}\) , the frequency we measure is half the value measured in the rest frame of the source.

As you have seen, relativistic calculations usually involve the quantity \(\gamma .\) When \(\gamma\) is appreciably greater than \(1,\) we must use relativistic formulas instead of Newtonian ones. For what speed \(v\) (in terms of \(c )\) is the value of \(\gamma(\mathrm{a}) 1.0 \%\) greater than \(1 ;\) (b) 10\(\%\) greater than 1 ; (c) 100\(\%\) greater than 1\(?\)

(a) At what speed is the momentum of a particle twice as great as the result obtained from the nonrelativistic expression \(m v ?\) Express your answer in terms of the speed of light. (b) A force is apphed to a particle along its direction of motion. At what speed is the magnitude of force required to produce a given acceleration twice as great as the force required to produce the same acceleration when the particle is at rest? Express your answer in terms of the speed of light.

Calculate the magnitude of the force required to give a \(0.145-\mathrm{kg}\) baseball an acceleration \(a=1.00 \mathrm{m} / \mathrm{s}^{2}\) in the direction of the baseball's initial velocity when this velocity has a magnitude of (a) \(10.0 \mathrm{m} / \mathrm{s} ;\) (b) 0.900 \(\mathrm{c}\) (c) \(0.990 c .(\mathrm{d})\) Repeat parts \((\mathrm{a}),\) and \((\mathrm{c})\) if the force and acceleration are perpendicular to the velocity.

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

In proton-antiproton annihilation a proton and an antiproton (a negatively charged proton) collide and disappear, producing electromagnetic radiation. If cach 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.

A proton (rest mass \(1.67 \times 10^{ \times 27} \mathrm{kg} )\) 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?

(a) How much work must be done on a particle with mass \(m\) to accelerate it (a) from rest to a speed of 0.090\(c\) and (b) from a speed of 0.900\(c\) to a speed of 0.990\(c ?\) (Express the answers in terms of \(m c^{2}-)(c)\) How do your answers in parts \((a)\) and \((b)\) compare?

(a) By what percentage does your rest mass increase when you climb 30 \(\mathrm{m}\) to the top of a ten-story building? Are you aware of this increase? Explain. (b) By how many grams does the mass of a \(120-\mathrm{g}\) spring with force constant 200 \(\mathrm{N} / \mathrm{cm}\) change when you compress it by 6.0 \(\mathrm{cm} \%\) Does the mass increase or decrease? Would you notice the change in mass if you were holding the spring? Explain.

A \(60.0-\mathrm{kg}\) person is standing at rest on level ground. How fast would she have to run to (a) double her total energy and (b) increase her total energy by a factor of 10\(?\)

When a particle meets its antiparticle, they annihilate each other and their mass is converted to light energy. The United States uses approximately \(1.0 \times 10^{19} \mathrm{J}\) of energy per year (a) If all this energy came from a futuristic ant-matter reactor, how much mass of matter and antimatter fuel would be consumed yearly? (b) If this fuel had the density of iron \(\left(7.86 \mathrm{g} / \mathrm{cm}^{3}\right)\) 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 \(\psi\) ( psi) particle has mass \(5.52 \times 10^{-2} \mathrm{kg}\) . Compute the rest energy of the \(\psi\) particle in MeV.

A particle has rest mass \(6.64 \times 10^{-27} \mathrm{kg}\) and momentum \(2.10 \times 10^{-18} \mathrm{kg} \cdot \mathrm{m} / \mathrm{s}\) . (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?

Compute the kinctic energy of a proton (mass \(1.67 \times$$10^{-27} \mathrm{kg}\) using both the nonrelativistic and relativistic expressions, and compute the ratio of the two results (relativistic divided by nonrelativistic) for speeds of (a) \(8.00 \times 10^{7} \mathrm{m} / \mathrm{s}\) and (b) \(2.85 \times 10^{8} \mathrm{m} / \mathrm{s}\) .

(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 electron volts.

Two protons (each with rest mass \(M=1.67 \times 10^{-27} \mathrm{kg}\) ) are initially moving with equal speeds in opposite directions. The protons continue to exist after a collision that also produces an \(\eta^{0}\) particle (see Chapter \(44 ) .\) The rest mass of the \(\eta^{0}\) is \(m=9.75 \times 10^{-28} \mathrm{kg}\) . (a) If the two protons and the \(\eta^{0}\) are all at rest after the collision, find the initial speed of the protons, expressed as a fraction of the speed of light. (b) What is the kinetic energy of each proton? Express your answer in MeV. (c) What is the rest energy of the \(\eta^{0},\) expressed in MeV? (d) Discuss the relationship between the answers to parts (b) and (c).

Find the speed of a particle whose relativistic kinetic energy is 50\(\%\) greater than the Newtonian value for the same speed.

A \(0.100-\mu g\) speck of dust is accelerated from rest to a speed of 0.900\(c\) by a constant \(1.00 \times 10^{6} \mathrm{N}\) force. (a) If the nonrelativistic form of Newton's second law \((\Sigma F=m a)\) is used, how far does-the object travel to reach its final speed? (b) Using the correct relativistic treatment of Section 37.8 , how far does the object travel to reach its final speed? (c) Which distance is greater? Why?

After being produced in a collision between elementary particles, a positive pion \(\left(\pi^{+}\right)\) must travel down a \(1.20-\mathrm{km}\) -long thibe to reach an experimental area. A \(\pi^{+}\) particle has an average life-time (measured in its rest frame) of \(2.60 \times 10^{-8} \mathrm{s}\) ; the \(\pi^{+}\) we are considering has this lifetime. (a) How fast must the \(\pi^{+}\) travel if it is not to decay before it reaches the end of the tube? (Since u will be very close to \(c,\) write \(u=(1-\Delta) c\) and give your answer in terms of \(\Delta\) rather than \(u . )\) (b) The \(\pi^{+}\) has a rest energy of 139.6 \(\mathrm{MeV}\) . What is the total energy of the \(\pi^{+}\) at the speed calculated in part (a)?

A cube of metal with sides of length \(a\) sits at rest in a frame \(S\) with one edge parallel to the \(x\) -axis. Therefore, in \(S\) the cube has volume \(a^{3} .\) Frame \(S^{\prime}\) moves along the \(x\) -axis with a speed \(u\) . As measured by an observer in frame \(S^{\prime},\) what is the volume of the metal cube?

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 \(\mathrm{c}\) . 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?

Two atomic clocks are carefully synchronized. One remains in New York, and the other is loaded on an airliner that travels at an average speed of 250 \(\mathrm{m} / \mathrm{s}\) and then returns to New York. When the plane returns, the elapsed time on the clock that stayed behind is 4.00 \(\mathrm{h}\) . By how much will the readings of the two clocks differ, and which clock will show the shorter elapsed time? (Hint: Since \(u \ll c,\) you can simplify \(\sqrt{1-u^{2} / c^{2} \text { by a binomial expansion. }}\))

A nuclear bomb containing 8.00 \(\mathrm{kg}\) of plutonium explodes. The sum of the rest masses 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\(\mu \mathrm{s}\) , 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 \(\mathrm{km} ?\)

A photon with energy \(E\) is emitted by an atom with mass \(m\) which recoils in the opposite direction. (a) Assuming that the motion of the atom can be treated nonrelativistically, compute the recoil speed of the atom. (b) From the result of part (a), show that the recoil speed is much less than \(c\) whenever \(E\) is much less than the rest energy \(m c^{2}\) of the atom.

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?

Frame \(S^{\prime}\) has an \(x\) -component of velocity \(u\) relative to frame S, and at \(t=t^{\prime}=0\) the two frames coincide (see Fig. 37.3\()\) . A light pulse with a spherical wave front is emitted at the origin of \(S^{\prime}\) at time \(t^{\prime}=0 .\) Its distance \(x^{\prime}\) from the origin after a time \(t^{\prime}\) is given by \(x^{\prime 2}=c^{2} t^{\prime 2} .\) Use the Lorentz coordinate ransformation to S, and at \(t=t^{\prime}=0\) the two frames coincide (see Fig. 37.3\()\) . A light pulse with a spherical wave front is emitted at the origin of \(S^{\prime}\) at time \(t^{\prime}=0 .\) Its distance \(x^{\prime}\) from the origin after a time \(t^{\prime}\) is given by \(x^{\prime 2}=c^{2} t^{\prime 2} .\) Use the Lorentz coordinate ransformation to be spherical in both frames.

In certain radioactive beta decay processes, the beta particle (an electron) leaves the atomic nucleus with a speed of 99.95\(\%\) the speed of light relative to the decaying nucleus. If this nucleus is moving at 75.00\(\%\) 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 and (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.

A particle with mass \(m\) accelerated from rest by a constant force \(F\) will, according to Newtonian mechanics, continue to accelerate without bound; that is, as \(t \rightarrow \infty, v \rightarrow \infty .\) Show that according to relativistic mechanics, the particle's speed approaches \(c\) as \(t \rightarrow \infty\) . I Note: Auseful integralis \(\int\left(1-x^{2}\right)^{-3 / 2} d x=x / \sqrt{1-x^{2}} \cdot 1\)

Two events observed in a frame of reference Shave positions and times given by \(\left(x_{1}, t_{1}\right)\) and \(\left(x_{2}, t_{2}\right),\) respectively. (a) Frame \(S^{\prime}\) moves along the \(x\) -axis just fast enough that the two events occur at the same position in \(S^{\prime} .\) Show that in \(S^{\prime},\) the time interval \(\Delta t^{\prime}\) between the two events is given by $$\Delta t^{\prime}=\sqrt{(\Delta t)^{2}-\left(\frac{\Delta x}{c}\right)^{2}}$$ where \(\Delta x=x_{2}-x_{1}\) and \(\Delta t=t_{2}-t_{1}\) . Hence show that if \(\Delta x>c \Delta t,\) there is \(n o\) frame \(S^{\prime}\) in which the two events occur at the same point. The interval \(\Delta t^{\prime}\) is sometimes called the proper time interval for the events. Is this term appropriate? (b) Show that if \(\Delta x>c \Delta t,\) there is a different frame of reference \(S\) in which the two events occur simultaneously. Find the distance between the two events in \(S^{\prime} ;\) express your answer in terms of \(\Delta x, \Delta t,\) and \(c\). This distance is sometimes called a proper length. Is this term appropriate? (c) Two events are observed in a frame of reference \(S^{\prime}\) to occur simultancously at points separated by a distance of 2.50 \(\mathrm{m}\) . In a second frame \(S\) moving relative to \(S^{\prime}\) along the line joining the two points in \(S^{\prime},\) the two events appear to be separated by 5.00 \(\mathrm{m}\) . What is the time interval between the events as measured in \(S ?[\text { Hint: Apply the result obtained in part (b).1 }\)

Einstein and Lorentz, being avid tennis players, play a fast-paced game on a court where they stand 20.0 \(\mathrm{m}\) from each other. Being very skilled players, they play with- out a net. The tennis ball has mass 0.0580 \(\mathrm{kg}\) . You can ignore gravity and assume that the ball travels parallel to the ground as it travels between the two players. Unless otherwise specified, all measurements are made by the two men. (a) Lorentz serves the ball at 80.0 \(\mathrm{m} / \mathrm{s}\) . What is the ball's kinetic energy? (b) Einstein slams a return at \(1.80 \times 10^{8} \mathrm{m} / \mathrm{s}\) . What is the ball's kinetic energy? (c) During Einstein's return of the ball in part (a), a white rabbit runs beside the court in the direction from Einstein to Lorentz. The rabbit has a speed of \(220 \times 10^{8} \mathrm{m} / \mathrm{s}\) relative to the two men. What is the speed of the rabbit relative to the ball? (d) What does the rabbit measure as the distance from Einstein to Lorentz? (e) How much time does it take for the rabbit to run \(20.0 \mathrm{m},\) according to the players? (f) The white rabbit carries a pocket watch. He uses this watch to measure the time (as he sees a it) for the distance from Einstein to Lorentz to pass by under him. What time does he measure?

One of the wavelengths of light emitted by hydrogen atoms under normal laboratory conditions is \(\lambda=656.3 \mathrm{nm},\) in the red portion of the electromagnetic spectrum. In the light emitted from a distant galaxy this same spectral line is observed to be Doppler-shifted to \(\lambda=953.4 \mathrm{nm}\) , in the infrared portion of the spectrum. How fast are the emitting atoms moving relative to the earth? Are they approaching the earth or receding from it?