| The diameter of the solar system is 10 light hours. A spaceship crosses the solar system in 15 hours, as measured on earth. How long, in hours, does the passage take according to passengers on the spaceship?

Hint: $c=1$ light hour per hour.

|| A 30-m-long rocket train car is traveling from Los Angeles to New York at 0.50c when a light at the center of the car flashes. When the light reaches the front of the car, it immediately rings a bell. Light reaching the back of the car immediately sounds a siren.

1. Are the bell and siren simultaneous events for a passenger seated in the car? If not, which occurs first and by how much time?

2. Are the bell and siren simultaneous events for a bicyclist waiting to cross the tracks? If not, which occurs first and by how much time?

|| Two events in reference frame S occur $10\mu \text{s}$ apart at the same point in space. The distance between the two events is 2400 m in reference frame $\text{S}\prime$.

1. What is the time interval between the events in reference frame $\text{S}\prime ?$

2. What is the velocity of $\text{S}\prime$ relative to S?

|| The star Alpha goes supernova. Ten years later and 100 ly away, as measured by astronomers in the galaxy, star Beta explodes.

1. Is it possible that the explosion of Alpha is in any way responsible for the explosion of Beta? Explain.

2. An alien spacecraft passing through the galaxy finds that the distance between the two explosions is 120 ly. According to the aliens, what is the time between the explosions?

|| A starship voyages to a distant planet 10 ly away. The explorers stay 1 year, return at the same speed, and arrive back on earth 26 years, as measured on earth, after they left. Assume that the time needed to accelerate and decelerate is negligible.

1. What is the speed of the starship?

2. How much time has elapsed on the astronauts’ chronometers?

| The Stanford Linear Accelerator (SLAC) accelerates electrons to $v=0.99999997c$ in a 3.2-km-long tube. If they travel the length of the tube at full speed (they don’t, because they are accelerating), how long is the tube in the electrons’ reference frame?

|| INT On a futuristic highway, a 15-m-long rocket travels so fast that a red stoplight, with a wavelength of 700 nm, appears to the pilot to be a green light with a wavelength of 520 nm. What is the length of the rocket to an observer standing at the intersection as the rocket speeds through?

Hint: The Doppler effect for light was covered in Chapter 16.

|| In an attempt to reduce the extraordinarily long travel times for voyaging to distant stars, some people have suggested traveling at close to the speed of light. Suppose you wish to visit the red giant star Betelgeuse, which is 430 ly away, and that you want your 20,000 kg rocket to move so fast that you age only 20 years during the round trip.

1. How fast, as a fraction of c, must the rocket travel relative to earth?

2. How much energy is needed to accelerate the rocket to this speed?

3. Compare this amount of energy to the total energy used by the United States in the year 2020, which was roughly $1.0\times {10}^{20}\text{J}$.

|| The star Delta goes supernova. One year later and 2.0 ly away, as measured by astronomers in the galaxy, star Epsilon explodes. Let the explosion of Delta be at $x_{\text{D}}=0$ and $t_{\text{D}}=0$. The explosions are observed by three spaceships cruising through the galaxy in the direction from Delta to Epsilon at velocities $v_1=0.30c$, $v_2=0.50c$, and $v_3=0.70c$. All three spaceships, each at the origin of its reference frame, happen to pass Delta as it explodes.

1. What are the times of the two explosions as measured by scientists on each of the three spaceships?

2. Does one spaceship find that the explosions are simultaneous? If so, which one?

3. Does one spaceship find that Epsilon explodes before Delta? If so, which one?

4. Do your answers to parts b and c violate the idea of causality? Explain.

|| CALC The quantity $dE/dv$, the rate of increase of energy with speed, is the amount of additional energy a moving object needs per 1 m/s increase in speed.

1. A 25,000 kg truck is traveling at 30 m/s. How much additional energy is needed to increase its speed by 1 m/s?

2. A 25,000 kg rocket is traveling at 0.90c. How much additional energy is needed to increase its speed by 1 m/s?

||| A rocket traveling at 0.50c sets out for the nearest star, Alpha Centauri, which is 4.3 ly away from earth. It will return to earth immediately after reaching Alpha Centauri. What distance will the rocket travel and how long will the journey last according to (a) stay-at-home earthlings and (b) the rocket crew? (c) Which answers are the correct ones, those in part a or those in part b?

| Two rockets approach each other. Each is traveling at 0.75c in the earth’s reference frame. What is the speed, as a fraction of c, of one rocket relative to the other?

|| Two rockets, A and B, approach the earth from opposite directions at speed 0.80c. The length of each rocket measured in its rest frame is 100 m. What is the length of rocket A as measured by the crew of rocket B?

|| A rocket fires a projectile at a speed of 0.95c while traveling past the earth. An earthbound scientist measures the projectile’s speed to be 0.90c. What was the rocket’s speed as a fraction of c?

|| INT Through what potential difference must an electron be accelerated, starting from rest, to acquire a speed of 0.99c?

|| INT What is the speed, in m/s, of a proton after being accelerated from rest through a $50\times {10}^6\text{V}$ potential difference?

|| The half-life of a muon at rest is $1.5\mu \text{s}$. Muons that have been accelerated to a very high speed and are then held in a circular storage ring have a half-life of $7.5\mu \text{s}$.

1. What is the speed, as a fraction of c, of the muons in the storage ring?

2. What is the total energy of a muon in the storage ring? The mass of a muon is 207 times the mass of an electron.

|| This chapter has assumed that lengths perpendicular to the direction of motion are not affected by the motion. That is, motion in the x-direction does not cause length contraction along the y- or z-axes. To find out if this is really true, consider two spray-paint nozzles attached to rods perpendicular to the x-axis. It has been confirmed that, when both rods are at rest, both nozzles are exactly 1 m above the base of the rod. One rod is placed in the S reference frame with its base on the x-axis; the other is placed in the $\text{S}\prime$ reference frame with its base on the $x\prime \text{-}\text{axis}$. The rods then swoop past each other and, as FIGURE P36.60 shows, each paints a stripe across the other rod.

FIGURE P36.60

We will use proof by contradiction. Assume that objects perpendicular to the motion are contracted. An experimenter in frame S finds that the $\text{S}\prime$ nozzle, as it goes past, is less than 1 m above the x-axis. The principle of relativity says that an experiment carried out in two different inertial reference frames will have the same outcome in both.

1. Pursue this line of reasoning and show that you end up with a logical contradiction, two mutually incompatible situations.

2. What can you conclude from this contradiction?

|| Derive the Lorentz transformations for t and $t\prime$.

Hint: See the comment following Equation 36.22.

||

1. Derive a velocity transformation equation for $u_y$ and $u^{\prime }{}_y$. Assume that the reference frames are in the standard orientation with motion parallel to the x- and $x\prime \text{-}\text{axes}$.

2. A rocket passes the earth at 0.80c. As it goes by, it launches a projectile at 0.60c perpendicular to the direction of motion. What is the particle’s speed, as a fraction of c, in the earth’s reference frame?

|| A rocket is fired from the earth to the moon at a speed of 0.990c. Let two events be “rocket leaves earth” and “rocket hits moon.”

1. In the earth’s reference frame, calculate $\text{Δ}x$, $\text{Δ}t$, and the spacetime interval s for these events.

2. In the rocket’s reference frame, calculate $\text{Δ}x\prime$, $\text{Δ}t\prime$, and the spacetime interval $s\prime$ for these events.

3. Repeat your calculations of part a if the rocket is replaced with a laser beam.

|| INT Let’s examine whether or not the law of conservation of momentum is true in all reference frames if we use the Newtonian definition of momentum: $p_x=m{u}_x$. Consider an object A of mass $3m$ at rest in reference frame S. Object A explodes into two pieces: object B, of mass m, that is shot to the left at a speed of $c/2$ and object C, of mass $2m$, that, to conserve momentum, is shot to the right at a speed of $c/4$. Suppose this explosion is observed in reference frame $\text{S}\prime$ that is moving to the right at half the speed of light.

1. Use the Lorentz velocity transformation to find the velocity and the Newtonian momentum of A in $\text{S}\prime$.

2. Use the Lorentz velocity transformation to find the velocities and the Newtonian momenta of B and C in $\text{S}\prime$.

3. What is the total final momentum in $\text{S}\prime ?$

4. Newtonian momentum was conserved in frame S. Is it conserved in frame $\text{S}\prime ?$

||

1. What are the momentum and total energy of a proton with speed 0.99c?

2. What is the proton’s momentum in a different reference frame in which $E\prime =5.0\times {10}^{-10}\text{J}?$

|| At what speed, as a fraction of c, is the kinetic energy of a particle twice its Newtonian value?

| INT A typical nuclear power plant generates electricity at the rate of 1000 MW. The efficiency of transforming thermal energy into electric energy is $\frac{1}{3}$ and the plant runs at full capacity for 80% of the year. (Nuclear power plants are down about 20% of the time for maintenance and refueling.)

1. How much thermal energy does the plant generate in one year?

2. What mass of uranium is transformed into energy in one year?

|| Many science fiction spaceships are powered by antimatter reactors. Suppose a 20-m-long spaceship, with a mass of 15,000 kg when empty, carries 2000 kg of fuel: 1000 kg each of matter and antimatter. The matter and antimatter are slowly combined, and the energy of their total annihilation is used to propel the ship. After consuming all the fuel and reaching top speed, the spaceship flies past a space station that is stationary with respect to the planet from which the ship was launched. What is the length of the spaceship as measured by astronauts on the space station?

|| The sun radiates energy at the rate $3.8\times {10}^{26}\text{W}$. The source of this energy is fusion, a nuclear reaction in which mass is transformed into energy. The mass of the sun is $2.0\times {10}^{30}\text{kg}$.

1. How much mass does the sun lose each year?

2. What percent is this of the sun’s total mass?

3. Fusion takes place in the core of a star, where the temperature and pressure are highest. A star like the sun can sustain fusion until it has transformed about 0.10% of its total mass into energy, then fusion ceases and the star slowly dies. Estimate the sun’s lifetime, giving your answer in billions of years.

|| The radioactive element radium (Ra) decays by a process known as alpha decay, in which the nucleus emits a helium nucleus. (These high-speed helium nuclei were named alpha particles when radioactivity was first discovered, long before the identity of the particles was established.) The reaction is ${}^{226}\text{Ra}{\to }^{222}\text{Rn}{+}^4\text{He}$ where Rn is the element radon. The accurately measured atomic masses of the three atoms are 226.0254 u, 222.0176 u, and 4.0026 u. How much energy is released in each decay? (The energy released in radioactive decay is what makes nuclear waste “hot.”)

|| The nuclear reaction that powers the sun is the fusion of four protons into a helium nucleus. The process involves several steps, but the net reaction is simply $4\text{p}{\to }^4\text{He}+\text{energy}$. The mass of a proton, to four significant figures, is $1.673\times {10}^{-27}\text{kg,}$ and the mass of a helium nucleus is known to be $6.644\times {10}^{-27}\text{kg}$.

1. How much energy is released in each fusion?

2. What fraction of the initial rest mass energy is this energy?

|| Consider the inelastic collision $\text{e}{}^{-}+{\text{e}}^{-}\to {\text{e}}^{-}+{\text{e}}^{-}+{\text{e}}^{-}+{\text{e}}^{+}$ in which an electron-positron pair is produced in a head-on collision between two electrons moving in opposite directions at the same speed. This is similar to Figure 36.39, but both of the initial electrons are moving.

1. What is the threshold kinetic energy? That is, what minimum kinetic energy must each electron have to allow this process to occur?

2. What is the speed of an electron with this kinetic energy?

||| An electron moving to the right at 0.90c collides with a positron moving to the left at 0.90c. The two particles annihilate and produce two gamma-ray photons. What is the wavelength of the photons?

||| Two rockets are each 1000 m long in their rest frame. Rocket Orion, traveling at 0.80c relative to the earth, is overtaking rocket Sirius, which is poking along at a mere 0.60c. According to the crew on Sirius, how long does Orion take to completely pass? That is, how long is it from the instant the nose of Orion is at the tail of Sirius until the tail of Orion is at the nose of Sirius?

||| Some particle accelerators allow protons $(\text{p}{}^{+})$ and antiprotons $(\text{p}{}^{-})$ to circulate at equal speeds in opposite directions in a device called a storage ring. The particle beams cross each other at various points to cause $\text{p}{}^{+}+{\text{p}}^{-}$ collisions. In one collision, the outcome is $\text{p}{}^{+}+{\text{p}}^{-}\to {\text{e}}^{+}+{\text{e}}^{-}+\gamma +\gamma$, where $\gamma$ represents a high-energy gamma-ray photon. The electron and positron are ejected from the collision at 0.9999995c and the gamma-ray photon wavelengths are found to be $1.0\times {10}^{-6}\text{nm}$. What were the proton and antiproton speeds, as a fraction of c, prior to the collision?

||| A ball of mass m traveling at a speed of $0.80c$ has a perfectly inelastic collision with an identical ball at rest. If Newtonian physics were correct for these speeds, momentum conservation would tell us that a ball of mass $2m$ departs the collision with a speed of $0.40c$. Let’s do a relativistic collision analysis to determine the mass and speed of the ball after the collision.

1. What is ${\gamma }_{\text{p}}$, written as a fraction like $a/b?$

2. What is the initial total momentum? Give your answer as a fraction times $mc$.

3. What is the initial total energy? Give your answer as a fraction times $m{c}^2$. Don’t forget that there are two balls.

4. Because energy can be transformed into mass, and vice versa, you cannot assume that the final mass is 2m. Instead, let the final state of the system be an unknown mass M traveling at the unknown speed $u_{\text{f}}$. You have two conservation laws. Find M and $u_{\text{f}}$.

||| A very fast pole vaulter lives in the country. One day, while practicing, he notices a 10.0-m-long barn with the doors open at both ends. He decides to run through the barn at 0.866c while carrying his 16.0-m-long pole. The farmer, who sees him coming, says, “Aha! This guy’s pole is length contracted to 8.0 m. There will be a short interval of time when the pole is entirely inside the barn. If I’m quick, I can simultaneously close both barn doors while the pole vaulter and his pole are inside.” The pole vaulter, who sees the farmer beside the barn, thinks to himself, “That farmer is crazy. The barn is length contracted and is only 5.0 m long. My 16.0-m-long pole cannot fit into a 5.0-m-long barn. If the farmer closes the doors just as the tip of my pole reaches the back door, the front door will break off the last 11.0 m of my pole.”

Can the farmer close the doors without breaking the pole? Show that, when properly analyzed, the farmer and the pole vaulter agree on the outcome. Your analysis should contain both quantitative calculations and written explanation.

FIGURE CP36.77