(a) How tall is Mount Everest in a reference frame

CONTEXT-RICH You are standing on a corner when you see a friend drive past you in a car. Each of you is wearing a wristwatch. Both of you note the times when the car passes two different intersections and determine from your watch readings the time that elapses between the two events. Have either of you determined the proper time interval? Explain your answer.

CONTEXT-RICH In Problem 1, suppose your friend in the car measures the width of the car door to be 90 cm. You also measure the width as he goes by you. (a) Does either of you measure the proper width of the door? Explain your answer. (b) How will your value for the door width compare to his? (1) Your value will be smaller. (2) Your value will be larger. (3) Your value will be the same. (4) You cannot compare the widths, as the answer depends on the cars speed.

If event Aoccurs at a different location than event B in some reference frame, might it be possible for there to be a second reference frame in which they occur at the same location? If so, give an example. If not, explain why not.

If event A occurs prior to event B in some reference frame, might it be possible for there to be a second reference frame in which event B occurs prior to event A? If so, give an example. If not, explain why not.

Two events are simultaneous in a reference frame in which they also occur at the same location. Are they simultaneous in all other reference frames?

Two inertial observers are in relative motion. Under what circumstances can they agree on the simultaneity of two different events?

The approximate total energy of a particle of mass m moving at speed is (a) (b) (c) (d)

True or false: (a) The speed of light is the same in all reference frames. (b) The proper time interval is the shortest time interval between two events. (c) Absolute motion can be determined by means of length contraction. (d) The light-year is a unit of distance. (e) For two events to form a spacetime coincidence, they must occur at the same place. (f) If two events are not simultaneous in one reference frame, they cannot be simultaneous in any other reference frame.

(a) Show that pc has dimensions of energy. (b) There is a geometrical interpretation of Equation R-17 based on the Pythagorean theorem. Draw a picture of a triangle illustrating this interpretation.

A wad of putty of mass strikes and sticks to a second wad of putty of mass which is initially at rest. Do you expect that after the collision the combined putty mass will be (a) greater than, (b) less than, (c) the same as Explain your answer.

BIOLOGICAL APPLICATION Many nuclei of atoms are unstable; for example, an isotope of carbon, has a half-life of 5700 years. (By definition, the half-life is the time it takes for any given number of unstable particles to decay to half that number of particles.) This fact is used extensively for archeological and biological dating of old artifacts. Such unstable nuclei decay into several decay products, each with significant kinetic energy. Which of the following is true? (a) The mass of the unstable nucleus is larger than the sum of the masses of the decay products. (b) The mass of the unstable nucleus is smaller than the sum of the masses of the decay products. (c) The mass of the unstable nucleus is the same as the sum of the masses of the decay products. Explain your choice.

BIOLOGICAL APPLICATION Positron emission tomography (PET) scans are common in modern medicine. During this procedure, positrons (a positron has the same mass but the opposite charge of an electron) are emitted by radioactive nuclei that have been introduced into the body. Assume that an emitted positron, traveling slowly (with negligible kinetic energy), collides with an electron traveling at the same slow speed in the opposite direction. They undergo annihilation and two quanta of light (photons) are formed. You are in charge of designing detectors to receive these photons and measure their energies. (a) Explain why you would expect these two photons to come off in exactly opposite directions. (b) In terms of the electron mass how much energy would each photon have? (1) less than (2) greater (3) exactly Explain your choice.

In 1975, an airplane carrying an atomic clock flew back and forth at low altitude for 15 hours at an average speed of as part of a time-dilation experiment. The time on the clock was compared to the time on an atomic clock kept on the ground. What is the time difference between the atomic clock on the airplane and the atomic clock on the ground? (Ignore any effects that accelerations of the airplane have on the atomic clock that is on the airplane. Also assume that the airplane travels at costant speed.) SSM

(a) By making any necessary assumptions and finding certain stellar distances, estimate the speed at which a spaceship would have to travel for its passengers to make a trip to the nearest star (not the Sun!) and back to Earth in 1.0 Earth years, as measured by an observer on the spaceship. Assume that the passengers make the outgoing and return trips at constant speed, and ignore any effects due to the spaceship stopping and starting. (b) How much time would elapse on Earth during their round-trip? Include 2.0 Earth years for a low-speed exploration of the planets in the vicinity of the star.

(a) Compare the kinetic energy of a moving car to its rest energy. (b) Compare the total energy of a moving car to its rest energy. (c) Estimate the error made in computing the kinetic energy of a moving car using nonrelativistic expressions compared to the relativistically correct expressions. Hint: Use of the binomial expansion may help.

The proper average (or mean) lifetime of a pion (a subatomic particle) is (A neutral pion has a much shorter lifetime. See Chapter 41.) A beam of pions has a speed of 0.85c relative to a laboratory. (a) What would be their mean lifetime as measured in that laboratory? (b) On average, how far would they travel in that laboratory before they decay? (c) What would your answer to Part (b) be if you had neglected time dilation?

In the reference frame of a pion in Problem 16, how far does the laboratory travel in

The proper average (or mean) lifetime of a muon (a subnuclear particle) is Muons in a beam are traveling at 0.999c relative to a laboratory. (a) What is their lifetime as measured in that laboratory? (b) On average, how far do they travel in that laboratory before they decay?

In the reference frame of the muon in Problem 18, how far does the laboratory travel in

CONTEXT-RICH You have been posted to a remote region of space to monitor traffic. Toward the end of a quiet shift, a spacecraft goes by and you measure its length using a laser device. This device reports a length of 85.0 m. You flip open your handy reference catalog and identify the craft as a CCCNX-22, which has a proper length of 100 m. When you phone in your report, what speed should you give for this spacecraft?

A spaceship travels from Earth to a star 95 light-years away at a speed of How long does the spaceship take to get to the star (a) as measured on Earth and (b) as measured by a passenger on the a spaceship?

The average lifetime of a beam of subatomic particles called pions (see Problem 16 for details on these particles) traveling at high speed is measured to be \(7.5 \times 10^{-8}\)s. Their average lifetime at rest is known to be \(2.6 \times 10^{-8}\)s. How fast is this pion beam traveling?

A meterstick moves with speed 0.80c relative to you in the direction parallel to the stick. (a) Find the length of the stick as measured by you. (b) How long does it take for the stick to pass you?

Recall that the half-life is the time it takes for any given amount of unstable particles to decay to half that amount of particles. The proper half-life of a species of charged subatomic particles called pions is (See Problem 16 for details on pions.) Suppose a group of these pions are produced in an accelerator and emerge with a speed of 0.998c. How far do these particles travel in the accelerators laboratory before half of them have decayed?

Your friend, who is the same age as you, travels to the star Alpha Centauri, which is 4.0 light-years away, and returns immediately. He claims that the entire trip took just 6.0 y. What was his speed? Ignore any accelerations of your friends spaceship and assume that the spaceship traveled at the same speed during the entire trip.

Two spaceships pass each other traveling in opposite directions. A passenger in ship A, knows that her ship is 100 m long. She notes that ship B is moving with a speed of 0.92c relative to ship A and that the length of B is 36 m. What are the lengths of the two spaceships as measured by a passenger in ship B?

Supersonic jets achieve maximum speeds of about (a) By what percentage would a jet traveling at this speed contract in length? (b) During a time of exactly one year or on your clock, how much time would elapse on the pilots clock? How many minutes are lost by the pilots clock in one year of your time? Assume that you are on the ground and the pilot is flying at the specified speed for the entire year.

The proper mean lifetime of a muon (see Problems 18 and 19 for details regarding muons) is Consider a muon, created in Earths upper atmosphere, speeding toward the surface 8.00 km below, at a speed of 0.980c. (a) What is the likelihood that the muon will survive its trip to Earths surface before decaying? The probability of a muon decaying is given by where is the time interval as measured in the reference frame in question. (b) Calculate the probability from the point of view of an observer moving with the muon. Show that the answer is the same from the point of view of an observer on Earth.

A spaceship commander traveling to the Magellanic Clouds travels at a uniform speed of 0.800c. When leaving the Kuiper belt, whose outer edge is 50.0 AU from Earth (Note: 1 AU 150,000,000 km and represents the average distance between Earth and the Sun; AU astronomical unit), he sends a message to ground control in Houston, Texas, saying that he is fine. Fifteen minutes later (according to him), he realizes he has made a typo, so he sends a correction. How much time passes at Houston between the receipt of his initial message and the receipt of second message?

According to Jamal: (a) What is the distance between the flashbulb and clock A? (b) How far does the flash travel to reach clock A? (c) How far does clock A travel while the flash is traveling from the flashbulb to it?

According to Jamal, how long does it take the flash to travel to clock A, and what does clock C read as the flash reaches clock A.

Show that clock C reads 100 min as the light flash reaches clock B, which is traveling away from clock C with speed 0.600c.

According to Jamal, the reading on clock C advances from 25 min to 100 min between the reception of the flashes by clocks A and B in Problems 31 and 32. According to Jamal, how much will the reading on clock A advance during this 75-min interval?

According to Jamal, the advance of clock A calculated in Problem 33 is the amount that clock A leads clock B. Compare this result with where

In an inertial reference frame S, event B occurs after and 1.50 km distant from event A. How fast must an observer be moving along the line joining the two events so that the two events occur simultaneously? For an observer traveling fast enough, is it possible for event B to precede event A?

A large flat space platform has an x axis painted on it. Afirecracker explodes on the x axis at and a second firecracker explodes on the x axis later at In the reference frame of a train traveling alongside the x axis at speed relative to the platform, these two explosions occur at the same place on that axis. What is the separation in time between the two explosions in the reference frame of the train?

Herb and Randy are twin jazz musicians who perform as a trombone-saxophone duo. At the age of twenty, however, Randy got an irresistible offer to perform on a star 15 light-years away. To celebrate his good fortune, he bought a new vehicle for the tripa deluxe space-coup that travels at 0.99c. Each of the twins promises to practice diligently, so they can reunite afterward. However, Randys gig goes so well that he stays for a full 10 years before returning to Herb. After their reunion, (a) how many years of practice will Randy have had, and (b) how many years of practice will Herb have had?

Al and Bert are twins. Al travels at \(0.600 \mathrm{c}\) to Alpha Centauri (which is \(4.00 \mathrm{c} \cdot \mathrm{y}\) from Earth, as measured in the reference frame of Earth) and returns immediately. Each twin sends the other a light signal every \(0.0100 \mathrm{y}\), as measured in his own reference frame. (a) At what rate does Bert receive signals as Al is moving away from him? (b) How many signals does Bert receive at this rate? (c) How many total signals are received by Bert before Al returns to Earth? (d) At what rate does Al receive signals as Bert is moving away from him? (e) How many signals does Al receive at this rate? (f) How many total signals are received by Al before Al returns to Earth? (g) Which twin is younger at the end of the trip and by how many years?

Find the ratio of the total energy to the rest energy of a particle of mass moving with speed (a) 0.100c, (b) 0.500c, (c) 0.800c, and (d) 0.990c.

A proton of rest energy 938 MeV has a total energy of 1400 MeV. (a) What is its speed? (b) What is its momentum?

How much energy would be required to accelerate a particle of mass m from rest to (a) 0.500c, (b) 0.900c, and (c) 0.990c? Express your answers as multiples of the rest energy,

If the kinetic energy of a particle equals its rest energy, what percentage error is made by using for its momentum? Is the nonrelativistic expression always smaller or larger than the relativistically correct expression for momentum?

What is the total energy of a proton whose momentum is

SPREADSHEET, ESTIMATION Using a spreadsheet program or graphing calculator, make a graph of the kinetic energy of a particle with rest energy of 100 MeV for speeds between 0 and c. On the same graph, plot by way of comparison. Using the graph, estimate what speed of the nonrelativistic expression is no longer a good approximation of the kinetic energy. As a suggestion, plot the energy in units of MeV and the speed in the dimensionless form

(a) Show that the speed of a particle of mass m and total energy E is given by and that when E is much greater than this can be approximated by Find the speed of an electron with kinetic energy of (b) 0.510 MeV, and (c) 10.0 MeV.

Use the binomial expansion and Equation R-17 to show that when the total energy is given approximately by E _ mc2 _ (p2>2m).

Derive the equation (Equation R-17) by eliminating from Equations R-10 and R-16.

The rest energy of a proton is about 938 MeV. If its kinetic energy is also 938 MeV, find (a) its momentum, and (b) its speed.

What percentage error is made in using for the kinetic energy of a particle if its speed is (a) 0.10c, and (b) 0.90c?

A spaceship departs from Earth for the star Alpha Centauri, which is away in the reference frame of Earth. The spaceship travels at 0.75c. How long does it take to get there (a) as measured on Earth, and (b) as measured by a passenger on the spaceship?

The total energy of a particle is three times its rest energy. (a) Find for the particle. (b) Show that its momentum is given by

A spaceship travels past Earth moving at 0.70c relative to Earth. Five minutes after the spaceship is closest to Earth, a message is sent from the control center at Houston, Texas, to the craft. (Ignore any effects of the rotational motion of Earth.) (a) How long does it take for the signal to arrive? (b) The spaceship and the control center agree on the time when the ship is closest to Earth. Five minutes after the message is received aboard the ship, a return message is sent by the ship back to Houston. What is the time interval in Houston between the time their message is sent, and the time the return message is received?

Particles called muons traveling at 0.99995c are detected at the surface of Earth. One of your fellow students claims that the detected muons might have originated from the Sun. Prove him wrong. (The proper mean lifetime of the muon is )

(a) How tall is Mount Everest in a reference frame traveling with a cosmic ray muon that is traveling straight down, relative to Earth, at 0.99c? Take the height of Mount Everest according to an Earth-based observer to be 8846 m. (b) How long does it take the muon to travel the height of the mountain from the reference frame traveling with the muon? (c) How long does it take the muon to travel the height of the mountain from the Earth-based reference frame?

A gold nucleus has a radius of and a mass of 197 AMU. (One AMU has a rest energy of 932 MeV.) During experiments at Brookhaven National Laboratory, these nuclei are routinely accelerated to a kinetic energy of (a) How much less than the speed of light are they traveling? (b) At these energies, how long does it take them to travel 100 m in the laboratory reference frame?

APPROXIMATION Consider the flight of a beam of neutrons produced in a nuclear reactor. These neutrons have kinetic energies of up to 1.00 MeV. The rest energy of a neutron is 939 MeV. (a) What is the speed of 1.00 MeV neutrons? Express your answer in terms of (b) If the average lifetime of such a neutron is 15.0 min (in the laboratory reference frame), what is the maximum length of a beam of such neutrons (in a vacuum, in the absence of any interactions between the neutrons and other material)? Estimate this maximum range by calculating the length corresponding to five mean lifetimes. After five mean lifetimes, only or 0.007 (0.7%) of the neutrons remain. (c) Compare this range to the range of so-called thermally moderated neutrons, whose kinetic energies are around 0.025 eV. Express your answer as a percentage. That is, what percent of the 1.00 MeV-neutron range is the thermally moderated-neutron range? (Note that our assumption of a vacuum continues; however, in reality neutrons of this energy interact readily with matter, such as air or water, and real ranges are much shorter.)

CONTEXT-RICH You and Ernie are trying to fit a 15-ft-long ladder into a 10-ft-long shed with doors at each end. You suggest to Ernie that you open the front door to the shed and that he run toward it with the ladder at a speed such that the length contraction of the ladder shortens it enough so that it fits in the shed. As soon as the back end of the ladder passes through the door, you will slam it shut. (a) What is the minimum speed at which Ernie must run to fit the ladder into the shed? Express it as a fraction of the speed of light. (b) As Ernie runs toward the shed at a speed of 0.866c, he realizes that in the reference frame of the ladder, it is the shed that is shorter, not the ladder. How long is the shed in the rest frame of the ladder? (c) In the reference frame of the ladder, is there any instant that both ends of the ladder are simultaneously inside the shed? Examine this from the point of view of relativistic simultaneity.