Chapter 35

The most important fact we learned about aether is that: a) It was experimentally proven not to exist. b) Its existence was proven experimentally. c) It transmits light in all directions equally. d) It transmits light faster in longitudinal direction. e) It transmits light slower in longitudinal direction.

A particle of rest mass \(m_{0}\) travels at a speed \(v=0.20 c\) How fast must the particle travel in order for its momentum to increase to twice its original momentum? a) \(0.40 c\) c) \(0.38 c\) e) \(0.99 c\) b) \(0.10 c\) d) \(0.42 c\)

Which quantity is invariant-that is, has the same value-in all reference frames? a) time interval, \(\Delta t\) d) space-time interval, b) space interval, \(\Delta x\) \(c^{2}(\Delta t)^{2}-(\Delta x)^{2}\) c) velocity, \(v\)

Two twins, \(A\) and \(B\), are in deep space on similar rockets traveling in opposite directions with a relative speed of \(c / 4\). After a while, twin A turns around and travels back toward twin \(\mathrm{B}\) again, so that their relative speed is \(c / 4\). When they meet again, is one twin younger, and if so which twin is younger? a) Twin A is younger. d) Each twin thinks b) Twin \(B\) is younger. the other is younger. c) The twins are the same age.

A proton with a momentum of \(3.0 \mathrm{GeV} / \mathrm{c}\) is moving with what velocity relative to the observer? a) \(0.31 c\) c) \(0.91 c\) e) \(3.2 c\) b) \(0.33 c\) d) \(0.95 c\)

A square of area \(100 \mathrm{~m}^{2}\) that is at rest in the reference frame is moving with a speed \((\sqrt{3} / 2) c\). Which of the following statements is incorrect? a) \(\beta=\sqrt{3} / 2\) b) \(\gamma=2\) c) To an observer at rest, it looks like another square with an area less than \(100 \mathrm{~m}^{2}\) d) The length along the moving direction is contracted by a factor of \(\frac{1}{2}\)

Consider a particle moving with a speed less than \(0.5 c\). If the speed of the particle is doubled, by what factor will the momentum increase? a) less than 2 b) equal to 2 c) greater than 2

In mechanics, one often uses the model of a perfectly rigid body to model and determine the motion of physical objects (see, for example, Chapter 10 on rotation). Explain how this model contradicts Einstein's special theory of relativity.

Use light cones and world lines to help solve the following problem. Eddie and Martin are throwing water balloons very fast at a target. At \(t=-13 \mu s,\) the target is at \(x=0,\) Eddie is at \(x=-2 \mathrm{~km},\) and Martin is at \(x=5 \mathrm{~km},\) and all three remain in these positions for all time. The target is hit at \(t=0 .\) Who made the successful shot? Prove this using the light cone for the target. When the target is hit, it sends out a radio signal. When does Martin know the target has been hit? When does Eddie know the target has been hit? Use the world lines to show this. Before starting to draw your diagrams, consider: If your \(x\) position is measured in \(\mathrm{km}\) and you are plotting \(t\) versus \(x / c,\) what units must \(t\) be in, to the first significant figure?

A gravitational lens should produce a halo effect and not arcs. Given that the light travels not only to the right and left of the intervening massive object but also to the top and bottom, why do we typically see only arcs?

Suppose you are explaining the theory of relativity to a friend, and you have told him that nothing can go faster than \(300,000 \mathrm{~km} / \mathrm{s}\). He says that is obviously false: Suppose a spaceship traveling past you at \(200,000 \mathrm{~km} / \mathrm{s}\), which is perfectly possible according to what you are saying, fires a torpedo straight ahead whose speed is \(200,000 \mathrm{~km} / \mathrm{s}\) relative to the spaceship, which is also perfectly possible; then, he says, the torpedo's speed is \(400,000 \mathrm{~km} / \mathrm{s}\). How would you answer him?

Consider a positively charged particle moving at constant speed parallel to a current-carrying wire, in the direction of the current. As you know (after studying Chapters 27 and 28), the particle is attracted to the wire by the magnetic force due to the current. Now suppose another observer moves along with the particle, so according to him the particle is at rest. Of course, a particle at rest feels no magnetic force. Does that observer see the particle attracted to the wire or not? How can that be? (Either answer seems to lead to a contradiction: If the particle is attracted, it must be by an electric force because there is no magnetic force, but there is no electric field from a neutral wire; if the particle is not attracted, you see that the particle is, in fact, moving toward the wire.)

At rest, a rocket has an overall length of \(L .\) A garage at rest (built for the rocket by the lowest bidder) is only \(L / 2\) in length. Luckily, the garage has both a front door and a back door, so that when the rocket flies at a speed of \(v=0.866 c\), the rocket fits entirely into the garage. However, according to the rocket pilot, the rocket has length \(L\) and the garage has length \(L / 4\). How does the rocket pilot observe that the rocket does not fit into the garage?

A rod at rest on Earth makes an angle of \(10^{\circ}\) with the \(x\) -axis. If the rod is moved along the \(x\) -axis, what happens to this angle, as viewed by an observer on the ground?

An astronaut in a spaceship flying toward Earth's Equator at half the speed of light observes Earth to be an oblong solid, wider and taller than it appears deep, rotating around its long axis. A second astronaut flying toward Earth's North Pole at half the speed of light observes Earth to be a similar shape but rotating about its short axis. Why does this not present a contradiction?

Consider two clocks carried by observers in a reference frame moving at speed \(v\) in the positive \(x\) -direction relative to ours. Assume that the two reference frames have parallel axes, and that their origins coincide when clocks at that point in both frames read zero. Suppose the clocks are separated by distance \(l\) in the \(x^{\prime}-\) direction in their own reference frame; for instance, \(x^{\prime}=0\) for one clock and \(x^{\prime}=I\) for the other, with \(y^{\prime}=z^{\prime}=0\) for both. Determine the readings \(t^{\prime}\) on both clocks as functions of the time coordinate \(t\) in our reference frame.

Prove that in all cases, two sub-light-speed velocities "added" relativistically will always yield a sub-light-speed velocity. Consider motion in one spatial dimension only.

A famous result in Newtonian dynamics is that if a particle in motion collides elastically with an identical particle at rest, the two particles emerge from the collision on perpendicular trajectories. Does the same hold in the special theory of relativity? Suppose a particle of rest mass \(m\) and total energy \(E\) collides with an identical particle at rest, the same two particles emerging from the collision with new velocities. Are those velocities necessarily perpendicular? Explain.

Find the speed of light in feet per nanosecond, to three significant figures.

Find the value of \(g\), the gravitational acceleration at Earth's surface, in light-years per year, to three significant figures.

Michelson and Morley used an interferometer to show that the speed of light is constant, regardless of Earth's motion through any perceived luminiferous aether. An analogy can be understood from the different times it takes for a rowboat to travel two different round-trip paths in a river that flows at a constant velocity \((u)\) downstream. Let one path be for a distance \(D\) directly across the river, then back again; and let the other path be the same distance \(D\) directly upstream, then back again. Assume that the rowboat travels at constant speed, \(v\) (with respect to the water), for both trips. Neglect the time it takes for the rowboat to turn around. Find the ratio of the cross-stream time divided by the upstream-downstream time, as a function of the given constants.

What is the value of \(\gamma\) for a particle moving at a speed of \(0.8 c ?\)

An astronaut on a spaceship traveling at a speed of \(0.50 c\) is holding a meter stick parallel to the direction of motion. a) What is the length of the meter stick as measured by another astronaut on the spaceship? b) If an observer on Earth could observe the meter stick, what would be the length of the meter stick as measured by that observer?

A spacecraft travels along a straight line from Earth to the Moon, a distance of \(3.84 \cdot 10^{8} \mathrm{~m}\). Its speed measured on Earth is \(0.50 c\). a) How long does the trip take, according to a clock on Earth? b) How long does the trip take, according to a clock on the spacecraft? c) Determine the distance between Earth and the Moon if it were measured by a person on the spacecraft.

If a muon is moving at \(90.0 \%\) of the speed of light, how does its measured lifetime compare to when it is in the rest frame of a laboratory, where its lifetime is \(2.2 \cdot 10^{-6}\) s?

In Jules Verne's classic Around the World in Eighty Days, Phileas Fogg travels around the world in, according to his calculation, 81 days. Due to crossing the International Date Line he actually made it only 80 days. How fast would he have to go in order to have time dilation make 80 days to seem like \(81 ?\) (Of course, at this speed, it would take a lot less than even 1 day to get around the world \(\ldots . .)\)

Suppose NASA discovers a planet just like Earth orbiting a star just like the Sun. This planet is 35 light-years away from our Solar System. NASA quickly plans to send astronauts to this planet, but with the condition that the astronauts would not age more than 25 years during this journey. a) At what speed must the spaceship travel, in Earth's reference frame, so that the astronauts age 25 years during this journey? b) According to the astronauts, what will be the distance of their trip?

A wedge-shaped spaceship has a width of \(20.0 \mathrm{~m}\) a length of \(50.0 \mathrm{~m},\) and is shaped like an isosceles triangle. What is the angle between the base of the ship and the side of the ship as measured by a stationary observer if the ship is traveling by at a speed of \(0.400 c\) ? Plot this angle as a function of the speed of the ship.

In your physics class you have just learned about the relativistic frequency shift, and you decide to amaze your friends at a party. You tell them that once you drove through a stop light and that when you were pulled over you did not get ticketed because you explained to the police officer that the relativistic Doppler shift made the red light of wavelength \(650 \mathrm{nm}\) appear green to you, with a wavelength of \(520 \mathrm{nm}\). If your story had been true, how fast would you have been traveling?

A meteor made of pure kryptonite (Yes, we know: There really isn't such a thing as kryptonite ...) is moving toward Earth. If the meteor eventually hits Earth, the impact will cause severe damage, threatening life as we know it. If a laser hits the meteor with wavelength \(560 \mathrm{nm}\), the entire meteor will blow up. The only laser powerful enough on Earth has a \(532-\mathrm{nm}\) wavelength. Scientists decide to launch the laser in a spacecraft and use special relativity to get the right wavelength. The meteor is moving very slowly, so there is no correction for relative velocities. At what speed does the spaceship need to move so the laser has the right wavelength, and should it travel toward or away from the meteor?

Radar-based speed detection works by sending an electromagnetic wave out from a source and examining the Doppler shift of the reflected wave. Suppose a wave of frequency \(10.6 \mathrm{GHz}\) is sent toward a car moving away at a speed of \(32.0 \mathrm{~km} / \mathrm{h}\). What is the difference between the frequency of the wave emitted by the source and the frequency of the wave an observer in the car would detect?

A HeNe laser onboard a spaceship moving toward a remote space station emits a beam of red light toward the space station. The wavelength of the beam, as measured by a wavelength meter on board the spaceship, is \(632.8 \mathrm{nm}\). If the astronauts on the space station see the beam as a blue beam of light with a measured wavelength of \(514.5 \mathrm{nm},\) what is the relative speed of the spaceship with respect to the space station? What is the shift parameter \(z\) in this case?

Sam sees two events as simultaneous: (i) Event \(A\) occurs at the point (0,0,0) at the instant 0: 00: 00 universal time; (ii) Event \(B\) occurs at the point \((500, \mathrm{~m}, 0,0)\) at the same moment. Tim, moving past Sam with a velocity of \(0.999 c \hat{x}\), also observes the two events. a) Which event occurred first in Tim's reference frame? b) How long after the first event does the second event happen in Tim's reference frame?

Use the relativistic velocity addition to reconfirm that the speed of light with respect to any inertial reference frame is \(c\). Assume one-dimensional motion along a common \(x\) -axis.

You are driving down a straight highway at a speed of \(v=50.0 \mathrm{~m} / \mathrm{s}\) relative to the ground. An oncoming car travels with the same speed in the opposite direction. With what relative speed do you observe the oncoming car?

A rocket ship approaching Earth at \(0.90 c\) fires a missile toward Earth with a speed of \(0.50 c,\) relative to the rocket ship. As viewed from Earth, how fast is the missile approaching Earth?

In the twin paradox example, Alice boards a spaceship that flies to a space station 3.25 light-years away and then returns with a speed of \(0.65 c\). a) Calculate the total distance Alice traveled during the trip, as measured by Alice. b) With the aforementioned total distance, calculate the total time duration for the trip, as measured by Alice.

In the twin paradox example, Alice boards a spaceship that flies to a space station 3.25 light-years away and then returns with a speed of \(0.650 c .\) This can be viewed in terms of Alice's reference frame. a) Show that Alice must travel with a speed of \(0.914 c\) to establish a relative speed of \(0.650 c\) with respect to Earth when Alice is returning back to Earth. b) Calculate the time duration for Alice's return flight toward Earth with the aforementioned speed.

Robert, standing at the rear end of a railroad car of length \(100 . \mathrm{m},\) shoots an arrow toward the front end of the car. He measures the velocity of the arrow as \(0.300 c\). Jenny, who was standing on the platform, saw all of this as the train passed her with a velocity of \(0.750 c .\) Determine the following as observed by Jenny: a) the length of the car b) the velocity of the arrow c) the time taken by arrow to cover the length of the car d) the distance covered by the arrow

Consider motion in one spatial dimension. For any velocity \(v,\) define parameter \(\theta\) via the relation \(v=c \tanh \theta\) where \(c\) is the vacuum speed of light. This quantity is variously called the velocity parameter or the rapidity corresponding to velocity \(v\). a) Prove that for two velocities, which add according to the Lorentzian rule, the corresponding velocity parameters simply add algebraically, that is, like Galilean velocities. b) Consider two reference frames in motion at speed \(v\) in the \(x\) -direction relative to one another, with axes parallel and origins coinciding when clocks at the origin in both frames read zero. Write the Lorentz transformation between the two coordinate systems entirely in terms of the velocity parameter corresponding to \(v\), and the coordinates.

An electron's rest mass is \(0.511 \mathrm{MeV} / \mathrm{c}^{2}\) a) How fast must an electron be moving if its energy is to be 10 times its rest energy? b) What is the momentum of the electron at this speed?

The Relativistic Heavy Ion Collider (RHIC) can produce colliding beams of gold nuclei with beam kinetic energy of \(A \cdot 100 .\) GeV each in the center-of- mass frame, where \(A\) is the number of nucleons in gold (197). You can approximate the mass energy of a nucleon as approximately \(1.00 \mathrm{GeV}\). What is the equivalent fixed-target beam energy in this case?

How much work is required to accelerate a proton from rest up to a speed of \(0.997 c ?\)

In proton accelerators used to treat cancer patients, protons are accelerated to \(0.61 c\). Determine the energy of the proton, expressing your answer in MeV.

In some proton accelerators, proton beams are directed toward each other for head-on collisions. Suppose that in such an accelerator, protons move with a speed relative to the lab of \(0.9972 c\). a) Calculate the speed of approach of one proton with respect to another one with which it is about to collide head on. Express your answer as a multiple of \(c\), using six significant digits. b) What is the kinetic energy of each proton beam (in units of \(\mathrm{MeV}\) ) in the laboratory reference frame? c) What is the kinetic energy of one of the colliding protons (in units of \(\mathrm{MeV}\) ) in the rest frame of the other proton?

The hot filament of the electron gun in a cathode ray tube releases electrons with nearly zero kinetic energy. The electrons are next accelerated under a potential difference of \(5.00 \mathrm{kV}\), before being steered toward the phosphor on the screen of the tube. a) Calculate the kinetic energy acquired by the electron under this accelerating potential difference. b) Is the electron moving at relativistic speed? c) What is the electron's total energy and momentum? (Give both values, relativistic and nonrelativistic, for both quantities.)

Consider a one-dimensional collision at relativistic speeds between two particles with masses \(m_{1}\) and \(m_{2}\). Particle 1 is initially moving with a speed of \(0.700 c\) and collides with particle \(2,\) which is initially at rest. After the collision, particle 1 recoils with speed \(0.500 c\), while particle 2 starts moving with a speed of \(0.200 c\). What is the ratio \(m_{2} / m_{1} ?\)

In an elementary-particle experiment, a particle of mass \(m\) is fired, with momentum \(m c\), at a target particle of mass \(2 \sqrt{2} m\). The two particles form a single new particle (completely inelastic collision). Find: a) the speed of the projectile before the collision b) the mass of the new particle c) the speed of the new particle after the collision

Show that momentum and energy transform from one inertial frame to another as \(p_{x}^{\prime}=\gamma\left(p_{x}-v E / c^{2}\right) ; p_{y}^{\prime}=p_{y}\) \(p_{z}^{\prime}=p_{p} ; E^{\prime}=\gamma\left(E-v p_{x}\right) .\) Hint: Look at the derivation for the space-time Lorentz transformation.

Show that \(E^{2}-p^{2} c^{2}=E^{2}-p^{2} c^{2},\) that is, that \(E^{2}-p^{2} c^{2}\) is a Lorentz invariant. Hint: Look at derivation showing that the space-time interval is a Lorentz invariant.