Chapter 37

The mean lifetime of stationary muons is measured to be \(2.2000 \mu \mathrm{s}\). The mean lifetime of high-speed muons in a burst of cosmic rays observed from Earth is measured to be \(16.000 \mu \mathrm{s}\). To five significant figures, what is the speed parameter \(\beta\) of these cosmic-ray muons relative to Earth?

To eight significant figures, what is speed parameter \(\beta\) if the Lorentz factor \(\gamma\) is (a) \(1.0100000\), (b) \(10.000000\), (c) \(100.00000\), and (d) \(1000.0000 ?\)

(Come) back to the future. Suppose that a father is \(20.00 \mathrm{y}\) older than his daughter. He wants to travel outward from Earth for \(2.000 \mathrm{y}\) and then back for another \(2.000 \mathrm{y}\) (both intervals as he measures them) such that he is then \(20.00 \mathrm{y}\) younger than his daughter. What constant speed parameter \(\beta\) (relative to Earth) is required?

An unstable high-energy particle enters a detector and leaves a track of length \(1.05 \mathrm{~mm}\) before it decays. Its speed relative to the detector was \(0.992 c .\) What is its proper lifetime? That is, how long would the particle have lasted before decay had it been at rest with respect to the detector?

The premise of the Planet of the Apes movies and book is that hibernating astronauts travel far into Earth's future, to a time when human civilization has been replaced by an ape civilization. Considering only special relativity, determine how far into Earth's future the astronauts would travel if they slept for \(120 \mathrm{y}\) while traveling relative to Earth with a speed of \(0.9990 c\), first outward from Earth and then back again.

An electron of \(\beta=0.999987\) moves along the axis of an evacuated tube that has a length of \(3.00 \mathrm{~m}\) as measured by a laboratory observer \(S\) at rest relative to the tube. An observer \(S^{\prime}\) who is at rest relative to the electron, however, would see this tube moving with speed \(v(=\beta c) .\) What length would observer \(S^{\prime}\) measure for the tube?

A spaceship of rest length \(130 \mathrm{~m}\) races past a timing station at a speed of \(0.740 c .\) (a) What is the length of the spaceship as measured by the timing station? (b) What time interval will the station clock record between the passage of the front and back ends of the ship?

A meter stick in frame \(S^{\prime}\) makes an angle of \(30^{\circ}\) with the \(x^{\prime}\) axis. If that frame moves parallel to the \(x\) axis of frame \(S\) with speed \(0.90 c\) relative to frame \(S\), what is the length of the stick as measured from \(S ?\)

A rod lies parallel to the \(x\) axis of reference frame \(S\), moving along this axis at a speed of \(0.630 c\). Its rest length is \(1.70 \mathrm{~m}\). What will be its measured length in frame \(S\) ?

The length of a spaceship is measured to be exactly half its rest length. (a) To three significant figures, what is the speed parameter \(\beta\) of the spaceship relative to the observer's frame? (b) By what factor do the spaceship's clocks run slow relative to clocks in the observer's frame?

A space traveler takes off from Earth and moves at speed \(0.9900 \mathrm{c}\) toward the star Vega, which is \(26.00\) ly distant. How much time will have elapsed by Earth clocks (a) when the traveler reaches Vega and (b) when Earth observers receive word from the traveler that she has arrived? (c) How much older will Earth observers calculate the traveler to be (measured from her frame) when she reaches Vega than she was when she started the trip?

A rod is to move at constant speed \(v\) along the \(x\) axis of reference frame \(S\), with the rod's length parallel to that axis. An observer in frame \(S\) is to measure the length \(L\) of the rod. Figure \(37-23\) gives length \(L\) versus speed parameter \(\beta\) for a range of values for \(\beta\). The vertical axis scale is set by \(L_{a}=1.00 \mathrm{~m}\). What is \(L\) if \(v=0.95 c\) ?

Observer \(S\) reports that an event occurred on the \(x\) axis of his reference frame at \(x=3.00 \times 10^{8} \mathrm{~m}\) at time \(t=2.50 \mathrm{~s}\). Observer \(S^{\prime}\) and her frame are moving in the positive direction of the \(x\) axis at a speed of \(0.400 c\). Further, \(x=x^{\prime}=0\) at \(t=t^{\prime}=0 .\) What are the (a) spatial and (b) temporal coordinate of the event according to \(S^{\prime} ?\) If \(S^{\prime}\) were, instead, moving in the negative direction of the \(x\) axis, what would be the (c) spatial and (d) temporal coordinate of the event according to \(S^{\prime} ?\)

Inertial frame \(S^{\prime}\) moves at a speed of \(0.60 c\) with respect to frame \(S\) (Fig. 37-9). Further, \(x=x^{\prime}=0\) at \(t=t^{\prime}=0\). Two events are recorded. In frame \(S\), event 1 occurs at the origin at \(t=0\) and event 2 occurs on the \(x\) axis at \(x=3.0 \mathrm{~km}\) at \(t=4.0 \mu \mathrm{s}\). According to observer \(S^{\prime}\), what is the time of (a) event 1 and (b) event \(2 ?\) (c) Do the two observers see the same sequence or the reverse?

An experimenter arranges to trigger two flashbulbs simultaneously, producing a big flash located at the origin of his reference frame and a small flash at \(x=30.0 \mathrm{~km}\). An observer moving at a speed of \(0.250 \mathrm{c}\) in the positive direction of \(x\) also views the flashes. (a) What is the time interval between them according to her? (b) Which flash does she say occurs first?

A clock moves along an \(x\) axis at a speed of \(0.600 \mathrm{c}\) and reads zero as it passes the origin of the axis. (a) Calculate the clock's Lorentz factor. (b) What time does the clock read as it passes \(x=180 \mathrm{~m} ?\)

A particle moves along the \(x^{\prime}\) axis of frame \(S^{\prime}\) with velocity \(0.40 c\). Frame \(S^{\prime}\) moves with velocity \(0.60 c\) with respect to frame \(S\). What is the velocity of the particle with respect to frame \(S ?\)

Galaxy A is reported to be receding from us with a speed of \(0.35 c .\) Galaxy B, located in precisely the opposite direction, is also found to be receding from us at this same speed. What multiple of \(c\) gives the recessional speed an observer on Galaxy A would find for (a) our galaxy and (b) Galaxy B?

Stellar system \(Q_{1}\) moves away from us at a speed of \(0.800 c\). Stellar system \(Q_{2}\), which lies in the same direction in space but is closer to us, moves away from us at speed \(0.400 c\). What multiple of c gives the speed of \(Q_{2}\) as measured by an observer in the reference frame of \(Q_{1}\) ?

A spaceship whose rest length is \(350 \mathrm{~m}\) has a speed of \(0.82 c\) with respect to a certain reference frame. \(A\) micrometeorite, also with a speed of \(0.82 c\) in this frame, passes the spaceship on an antiparallel track. How long does it take this object to pass the ship as measured on the ship?

(a) An armada of spaceships that is \(1.00\) ly long (as measured in its rest frame) moves with speed \(0.800 c\) relative to a ground station in frame \(S .\) A messenger travels from the rear of the armada to the front with a speed of \(0.950 c\) relative to \(S .\) How long does the trip take as measured (a) in the rest frame of the messenger, (b) in the rest frame of the armada, and (c) by an observer in the ground frame \(S ?\)

A sodium light source moves in a horizontal circle at a constant speed of \(0.100 c\) while emitting light at the proper wavelength of \(\lambda_{0}=589.00 \mathrm{~nm}\). Wavelength \(\lambda\) is measured for that light by a detector fixed at the center of the circle. What is the wavelength shift \(\lambda-\lambda_{0}\) ?

A spaceship, moving away from Earth at a speed of \(0.900 c\), reports back by transmitting at a frequency (measured in the spaceship frame) of \(100 \mathrm{MHz}\). To what frequency must Earth receivers be tuned to receive the report?

Certain wavelengths in the light from a galaxy in the constellation Virgo are observed to be \(0.4 \%\) longer than the corresponding light from Earth sources. (a) What is the radial speed of this galaxy with respect to Earth? (b) Is the galaxy approaching on receding from Earth?

A spaceship is moving away from Earth at speed \(0.20 c\). A source on the rear of the ship emits light at wavelength \(450 \mathrm{~nm}\) according to someone on the ship. What (a) wavelength and (b) color (blue, green, yellow, or red) are detected by someone on Earth watching the ship?

How much work must be done to increase the speed of an electron (a) from \(0.18 c\) to \(0.19 c\) and (b) from \(0.98 c\) to \(0.99 c\) ? Note that the speed increase is \(0.01 c\) in both cases.

What is the minimum energy that is required to break a nucleus of \({ }^{12} \mathrm{C}\) (of mass \(11.99671 \mathrm{u}\) ) into three nuclei of \({ }^{4} \mathrm{He}\) (of mass \(4.00151\) u each \() ?\)

In the reaction \(\mathrm{p}+{ }^{19} \mathrm{~F} \rightarrow \alpha+{ }^{16} \mathrm{O}\), the masses are $$ \begin{array}{ll} m(\mathrm{p})=1.007825 \mathrm{u}, & m(\alpha)=4.002603 \mathrm{u}, \\ m(\mathrm{~F})=18.998405 \mathrm{u}, & m(\mathrm{O})=15.994915 \mathrm{u} . \end{array} $$ Calculate the \(Q\) of the reaction from these data.

In a high-energy collision between a cosmic-ray particle and a particle near the top of Earth's atmosphere, \(120 \mathrm{~km}\) above sea level, a pion is created. The pion has a total energy \(E\) of \(1.35 \times 10^{5}\) \(\mathrm{MeV}\) and is traveling vertically downward. In the pion's rest frame, the pion decays \(35.0 \mathrm{~ns}\) after its creation. At what altitude above sea level, as measured from Earth's reference frame, does the decay occur? The rest energy of a pion is \(139.6 \mathrm{MeV}\).

A \(5.00\) -grain aspirin tablet has a mass of \(320 \mathrm{mg}\). For how many kilometers would the energy equivalent of this mass power an automobile? Assume \(12.75 \mathrm{~km} / \mathrm{L}\) and a heat of combustion of \(3.65 \times 10^{7} \mathrm{~J} / \mathrm{L}\) for the gasoline used in the automobile.

The mass of a muon is 207 times the electron mass; the average lifetime of muons at rest is \(2.20 \mu \mathrm{s}\). In a certain experiment, muons moving through a laboratory are measured to have an average lifetime of \(6.90 \mu \mathrm{s}\). For the moving muons, what are (a) \(\beta\), (b) \(K\), and (c) \(p\) (in \(\mathrm{MeV} / \mathrm{c}\) )?

To four significant figures, find the following when the kinetic energy is \(10.00 \mathrm{MeV}:\) (a) \(\gamma\) and (b) \(\beta\) for an electron \(\left(E_{0}=\right.\) \(0.510998 \mathrm{MeV}\) ), (c) \(\gamma\) and (d) \(\beta\) for a proton \(\left(E_{0}=938.272 \mathrm{MeV}\right)\) and (e) \(\gamma\) and (f) \(\beta\) for an \(\alpha\) particle \(\left(E_{0}=3727.40 \mathrm{MeV}\right)\).

What must be the momentum of a particle with mass \(m\) so that the total energy of the particle is \(3.00\) times its rest energy?

A certain particle of mass \(m\) has momentum of magnitude \(m c\). What are (a) \(\beta\), (b) \(\gamma\), and (c) the ratio \(K / E_{0}\) ?

(a) The energy released in the explosion of \(1.00\) mol of TNT is \(3.40 \mathrm{MJ}\). The molar mass of TNT is \(0.227 \mathrm{~kg} / \mathrm{mol}\). What weight of TNT is needed for an explosive release of \(1.80 \times 10^{14} \mathrm{~J} ?\) (b) Can you carry that weight in a backpack, or is a truck or train required? (c) Suppose that in an explosion of a fission bomb, \(0.080 \%\) of the fissionable mass is converted to released energy. What weight of fissionable material is needed for an explosive release of \(1.80 \times\) \(10^{14} \mathrm{~J}\) ? (d) Can you carry that weight in a backpack, or is a truck or train required?

Quasars are thought to be the nuclei of active galaxies in the early stages of their formation. A typical quasar radiates energy at the rate of \(10^{41} \mathrm{~W}\). At what rate is the mass of this quasar being reduced to supply this energy? Express your answer in solar mass units per year, where one solar mass unit (1 smu \(=2.0 \times 10^{30} \mathrm{~kg}\) ) is the mass of our Sun.

The mass of an electron is \(9.10938188 \times 10^{-31} \mathrm{~kg}\). To eight significant figures, find the following for the given electron kinetic energy: (a) \(\gamma\) and (b) \(\beta\) for \(K=1.0000000 \mathrm{keV},(\mathrm{c}) \gamma\) and \((\) d \() \beta\) for \(K=\) \(1.0000000 \mathrm{MeV}\), and then (e) \(\gamma\) and (f) \(\beta\) for \(K=1.0000000 \mathrm{GeV}\).

An alpha particle with kinetic energy \(7.70 \mathrm{MeV}\) collides with an \({ }^{14} \mathrm{~N}\) nucleus at rest, and the two transform into an \({ }^{17} \mathrm{O}\) nucleus and a proton. The proton is emitted at \(90^{\circ}\) to the direction of the incident alpha particle and has a kinetic energy of \(4.44 \mathrm{MeV}\). The masses of the various particles are alpha particle, \(4.00260 \mathrm{u} ;{ }^{14} \mathrm{~N}\). \(14.00307\) u; proton, \(1.007825 \mathrm{u}\); and \({ }^{17} \mathrm{O}, 16.99914 \mathrm{u}\). In \(\mathrm{MeV}\), what are (a) the kinetic energy of the oxygen nucleus and (b) the \(Q\) of the reaction? (Hint: The speeds of the particles are much less than \(c .\) )

The car-in-the-garage problem. Carman has just purchased the world's longest stretch limo, which has a proper length of \(L_{c}=30.5 \mathrm{~m}\). In Fig. \(37-32 a\), it is shown parked in front of a garage with a proper length of \(L_{g}=6.00 \mathrm{~m} .\) The garage has a front door (shown open) and a back door (shown closed). The limo is obviously longer than the garage. Still, Garageman, who owns the garage and knows something about relativistic length contraction, makes a bet with Carman that the limo can fit in the garage with both doors closed. Carman, who dropped his physics course before reaching special relativity, says such a thing, even in principle, is impossible. To analyze Garageman's scheme, an \(x_{c}\) axis is attached to the limo, with \(x_{c}=0\) at the rear bumper, and an \(x_{g}\) axis is attached to the garage, with \(x_{g}=0\) at the (now open) front door. Then Carman is to drive the limo directly toward the front door at a velocity of \(0.9980 \mathrm{c}\) (which is, of course, both technically and financially impossible). Carman is stationary in the \(x_{c}\) reference frame; Garageman is stationary in the \(x_{g}\) reference frame. There are two events to consider. Event \(1:\) When the rear bumper clears the front door, the front door is closed. Let the time of this event be zero to both Carman and Garageman: \(t_{g 1}=t_{c 1}=0\). The event occurs at \(x_{c}=x_{g}=0\). Figure \(37-32 b\) shows event 1 according to the \(x_{g}\) reference frame. Event 2 : When the front bumper reaches the back door, that door opens. Figure \(37-32 c\) shows event 2 according to the \(\bar{x}_{8}\) reference frame. According to Garageman, (a) what is the length of the limo, and what are the spacetime coordinates (b) \(x_{g^{2}}\) and (c) \(t_{g 2}\) of event \(2 ?\) (d) For how long is the limo temporarily "trapped" inside the garage with both doors shut? Now consider the situation from the \(x_{c}\) reference frame, in which the garage comes racing past the limo at a velocity of \(-0.9980 c\). According to Carman, (e) what is the length of the passing garage, what are the spacetime coordinates (f) \(x_{c 2}\) and (g) \(t_{c 2}\) of event \(2,(\mathrm{~h})\) is the limo ever in the garage with both doors shut, and (i) which event occurs first? (j) Sketch events 1 and 2 as seen by Carman. (k) Are the events causally related; that is, does one of them cause the other? (I) Finally, who wins the bet?

An airplane has rest length \(40.0 \mathrm{~m}\) and speed \(630 \mathrm{~m} / \mathrm{s}\). To a ground observer, (a) by what fraction is its length contracted and (b) how long is needed for its clocks to be \(1.00 \mu \mathrm{s}\) slow.

To circle Earth in low orbit, a satellite must have a speed of about \(2.7 \times 10^{4} \mathrm{~km} / \mathrm{h}\). Suppose that two such satellites orbit Earth in opposite directions. (a) What is their relative speed as they pass, according to the classical Galilean velocity transformation equation? (b) What fractional error do you make in (a) by not using the (correct) relativistic transformation equation?

Find the speed parameter of a particle that takes \(2.0 \mathrm{y}\) longer than light to travel a distance of \(6.0\) ly.

How much work is needed to accelerate a proton from a speed of \(0.9850 c\) to a speed of \(0.9860 c ?\)

A pion is created in the higher reaches of Earth's atmosphere when an incoming high-energy cosmic-ray particle collides with an atomic nucleus. A pion so formed descends toward Earth with a speed of \(0.99 c .\) In a reference frame in which they are at rest, pionsdecay with an average life of \(26 \mathrm{~ns}\). As measured in a frame fixed with respect to Earth, how far (on the average) will such a pion move through the atmosphere before it decays?

If we intercept an electron having total energy 1533 \(\mathrm{MeV}\) that came from Vega, which is 26 ly from us, how far in lightyears was the trip in the rest frame of the electron?

The total energy of a proton passing through a laboratory apparatus is \(10.611 \mathrm{~nJ} .\) What is its speed parameter \(\beta ?\) Use the proton mass given in Appendix B under "Best Value," not the commonly remembered rounded number.

A spaceship at rest in a certain reference frame \(S\) is given a speed increment of \(0.50 c .\) Relative to its new rest frame, it is then given a further \(0.50 c\) increment. This process is continued until its speed with respect to its original frame \(S\) exceeds \(0.999 c .\) How many increments does this process require?

In the red shift of radiation from a distant galaxy, a certain radiation, known to have a wavelength of \(434 \mathrm{~nm}\) when observed in the laboratory, has a wavelength of \(462 \mathrm{~nm}\). (a) What is the radial speed of the galaxy relative to Earth? (b) Is the galaxy approaching or receding from Earth?

What is the momentum in MeV/c of an electron with a kinetic energy of \(2.00 \mathrm{MeV} ?\)

The radius of Earth is \(6370 \mathrm{~km}\), and its orbital speed about the Sun is \(30 \mathrm{~km} / \mathrm{s}\). Suppose Earth moves past an observer at this speed. To the observer, by how much does Earth's diameter contract along the direction of motion?