1 Million+ Step-by-step solutions

A 2 000-kg car moving at 20.0 m/s collides and locks together with a 1 500-kg car at rest at a stop sign. Show that momentum is conserved in a reference frame moving at 10.0 m/s in the direction of the moving car.

A ball is thrown at 20.0 m/s inside a boxcar moving along the tracks at 40.0 m/s. What is the speed of the ball relative to the ground if the ball is thrown?

(a) Forward

(b) Backward

(c) Out the side door?

(a) Forward

(b) Backward

(c) Out the side door?

In a laboratory frame of reference, an observer notes that Newton’s second law is valid. Show that it is also valid for an observer moving at a constant speed, small compared with the speed of light, relative to the laboratory frame.

Show that Newton’s second law is not valid in a reference frame moving past the laboratory frame of Problem 3 with a constant acceleration.

How fast must a meter stick be moving if its length is measured to shrink to 0.500 m?

At what speed does a clock move if it is measured to run at a rate that is half the rate of a clock at rest with respect to an observer?

An astronaut is traveling in a space vehicle that has a speed of 0.500c relative to the Earth. The astronaut measures her pulse rate at 75.0 beats per minute. Signals generated by the astronaut’s pulse are radioed to Earth when the vehicle is moving in a direction perpendicular to the line that connects the vehicle with an observer on the Earth.

(a) What pulse rate does the Earth observer measure?

(b) What If? What would be the pulse rate if the speed of the space vehicle were increased to 0.990c?

(a) What pulse rate does the Earth observer measure?

(b) What If? What would be the pulse rate if the speed of the space vehicle were increased to 0.990c?

An astronomer on Earth observes a meteoroid in the southern sky approaching the Earth at a speed of 0.800c. At the time of its discovery the meteoroid is 20.0 ly from the Earth. Calculate

(a) The time interval required for the meteoroid to reach the Earth as measured by the earthbound astronomer,

(b) This time interval as measured by a tourist on the meteoroid, and

(c) The distance to the Earth as measured by the tourist.

(a) The time interval required for the meteoroid to reach the Earth as measured by the earthbound astronomer,

(b) This time interval as measured by a tourist on the meteoroid, and

(c) The distance to the Earth as measured by the tourist.

An atomic clock moves at 1 000 km/h for 1.00 h as measured by an identical clock on the Earth. How many nanoseconds slow will the moving clock be compared with the Earth clock, at the end of the 1.00-h interval?

A muon formed high in the Earth’s atmosphere travels at speed v = 0.990c for a distance of 4.60 km before it decays into an electron, a neutrino, and an antineutrino (μ− → e− + v + v).

(a) How long does the muon live, as measured in its reference frame?

(b) How far does the earth travel, as measured in the frame of the muon?

(a) How long does the muon live, as measured in its reference frame?

(b) How far does the earth travel, as measured in the frame of the muon?

A spacecraft with a proper length of 300 m takes 0.750 μs to pass an Earth observer. Determine the speed of the spacecraft as measured by the Earth observer.

(a) An object of proper length Lp takes a time interval ∆t to pass an Earth observer. Determine the speed of the object as measured by the Earth observer.

(b) A column of tanks, 300 m long, takes 75.0 s to pass a child waiting at a street corner on her way to school. Determine the speed of the armored vehicles.

(c) Show that the answer to part (a) includes the answer to Problem 11 as a special case, and includes the answer to part (b) as another special case.

(b) A column of tanks, 300 m long, takes 75.0 s to pass a child waiting at a street corner on her way to school. Determine the speed of the armored vehicles.

(c) Show that the answer to part (a) includes the answer to Problem 11 as a special case, and includes the answer to part (b) as another special case.

In 1963 Mercury astronaut Gordon Cooper orbited the Earth 22 times. The press stated that for each orbit he aged 2 millionths of a second less than he would have if he had remained on the Earth.

(a) Assuming that he was 160 km above the Earth in a circular orbit, determine the time difference between someone on the Earth and the orbiting astronaut for the 22 orbits. You will need to use the approximation √1 – x ≈ 1 – x/2, for small x.

(b) Did the press report accurate information? Explain.

(a) Assuming that he was 160 km above the Earth in a circular orbit, determine the time difference between someone on the Earth and the orbiting astronaut for the 22 orbits. You will need to use the approximation √1 – x ≈ 1 – x/2, for small x.

(b) Did the press report accurate information? Explain.

A friend passes by you in a spacecraft traveling at a high speed. He tells you that his craft is 20.0 m long and that the identically constructed craft you are sitting in is 19.0 m long. According to your observations,

(a) How long is your spacecraft?

(b) How long is your friend’s craft, and

(c) What is the speed of your friend’s craft?

(a) How long is your spacecraft?

(b) How long is your friend’s craft, and

(c) What is the speed of your friend’s craft?

The identical twins Speedo and Goslo join a migration from the Earth to Planet X. It is 20.0 ly away in a reference frame in which both planets are at rest. The twins, of the same age, depart at the same time on different spacecraft. Speedo’s craft travels steadily at 0.950c and Goslo’s at 0.750c. Calculate the age difference between the twins after Goslo’s spacecraft lands on Planet X. Which twin is the older?

An interstellar space probe is launched from the Earth. After a brief period of acceleration it moves with a constant velocity, with a magnitude of 70.0% of the speed of light. Its nuclear-powered batteries supply the energy to keep its data transmitter active continuously. The batteries have a lifetime of 15.0 yr as measured in a rest frame.

(a) How long do the batteries on the space probe last as measured by Mission Control on the Earth?

(b) How far is the probe from the Earth when its batteries fail, as measured by Mission Control?

(c) How far is the probe from the Earth when its batteries fail, as measured by its built-in trip odometer?

(d) For what total time interval after launch are data received from the probe by Mission Control? Note that radio waves travel at the speed of light and fill the space between the probe and the Earth at the time of battery failure.

(a) How long do the batteries on the space probe last as measured by Mission Control on the Earth?

(b) How far is the probe from the Earth when its batteries fail, as measured by Mission Control?

(c) How far is the probe from the Earth when its batteries fail, as measured by its built-in trip odometer?

(d) For what total time interval after launch are data received from the probe by Mission Control? Note that radio waves travel at the speed of light and fill the space between the probe and the Earth at the time of battery failure.

An alien civilization occupies a brown dwarf, nearly stationary relative to the Sun, several light years away. The extraterrestrials have come to love original broadcasts of I Love Lucy, on our television channel 2, at carrier frequency 57.0 MHz. Their line of sight to us is in the plane of the Earth’s orbit. Find the difference between the highest and lowest frequencies they receive due to the Earth’s orbital motion around the Sun.

Police radar detects the speed of a car (Fig. P39.19) as follows. Microwaves of a precisely known frequency are broadcast toward the car. The moving car reflects the microwaves with a Doppler shift. The reflected waves are received and combined with an attenuated version of the transmitted wave. Beats occur between the two microwave signals. The beat frequency is measured.

(a) For an electromagnetic wave reflected back to its source from a mirror approaching at speed v, show that the reflected f=fsource c + v/ c – v where fsource is the source frequency.

(b) When v is much less than c, the beat frequency is much smaller than the transmitted frequency. In this case use the approximation f + fsource + 2 fsource and show that the beat frequency can be written as fbeat = 2v/A).

(c) What beat frequency is measured for a car speed of 30.0 m/s if the microwaves have frequency 10.0 GHz?

(d) If the beat frequency measurement is accurate to +5 Hz, how accurate is the velocity measurement?

(a) For an electromagnetic wave reflected back to its source from a mirror approaching at speed v, show that the reflected f=fsource c + v/ c – v where fsource is the source frequency.

(b) When v is much less than c, the beat frequency is much smaller than the transmitted frequency. In this case use the approximation f + fsource + 2 fsource and show that the beat frequency can be written as fbeat = 2v/A).

(c) What beat frequency is measured for a car speed of 30.0 m/s if the microwaves have frequency 10.0 GHz?

(d) If the beat frequency measurement is accurate to +5 Hz, how accurate is the velocity measurement?

The red shift A light source recedes from an observer with a speed vsource that is small compared with c.

(a) Show that the fractional shift in the measured wavelength is given by the approximate expression ∆A/A ≈ vsource/c this phenomenon is known as the red shift, because the visible light is shifted toward the red.

(b) Spectroscopic measurements of light at A = 397 nm coming from a galaxy in Ursa Major reveal a red shift of 20.0 nm. What is the recessional speed of the galaxy?

(a) Show that the fractional shift in the measured wavelength is given by the approximate expression ∆A/A ≈ vsource/c this phenomenon is known as the red shift, because the visible light is shifted toward the red.

(b) Spectroscopic measurements of light at A = 397 nm coming from a galaxy in Ursa Major reveal a red shift of 20.0 nm. What is the recessional speed of the galaxy?

A physicist drives through a stop light. When he is pulled over, he tells the police officer that the Doppler shift made the red light of wavelength 650 nm appear green to him, with a wavelength of 520 nm. The police officer writes out a traffic citation for speeding. How fast was the physicist traveling, according to his own testimony?

Suzanne observes two light pulses to be emitted from the same location, but separated in time by 3.00 μs. Mark sees the emission of the same two pulses separated in time by 9.00 μs.

(a) How fast is Mark moving relative to Suzanne?

(b) According to Mark, what is the separation in space of the two pulses?

(a) How fast is Mark moving relative to Suzanne?

(b) According to Mark, what is the separation in space of the two pulses?

A moving rod is observed to have a length of 2.00 m and to be oriented at an angle of 30.0° with respect to the direction of motion, as shown in Figure P39.23. The rod has a speed of 0.995c.

(a) What is the proper length of the rod?

(b) What is the orientation angle in the proper frame?

(a) What is the proper length of the rod?

(b) What is the orientation angle in the proper frame?

An observer in reference frame S sees two events as simultaneous. Event A occurs at the point (50.0 m, 0, 0) at the instant 9:00:00 Universal time, 15 January 2004. Event B occurs at the point (150 m, 0, 0) at the same moment. A second observer, moving past with a velocity of 0.800c i, also observes the two events. In her reference frame S", which event occurred first and what time interval elapsed between the events?

A red light flashes at position xR = 3.00 m and time tR = 1.00 x 10-9 s, and a blue light flashes at xB = 5.00 m and tB = 9.00 x 10-9 s, all measured in the S reference frame. Reference frame S" has its origin at the same point as S at t = t = 0; frame S" moves uniformly to the right. Both flashes are observed to occur at the same place in S".

(a) Find the relative speed between S and S".

(b) Find the location of the two flashes in frame S".

(c) At what time does the red flash occur in the S" frame?

(a) Find the relative speed between S and S".

(b) Find the location of the two flashes in frame S".

(c) At what time does the red flash occur in the S" frame?

A Klingon spacecraft moves away from the Earth at a speed of 0.800c (Fig. P39.26). The starship Enterprise pursues at a speed of 0.900c relative to the Earth. Observers on the Earth see the Enterprise overtaking the Klingon craft at a relative speed of 0.100c. With what speed is the Enterprise overtaking the Klingon craft as seen by the crew of the Enterprise?

Two jets of material from the center of a radio galaxy are ejected in opposite directions. Both jets move at 0.750c relative to the galaxy. Determine the speed of one jet relative to the other.

A spacecraft is launched from the surface of the Earth with a velocity of 0.600c at an angle of 50.0° above the horizontal positive x axis. Another spacecraft is moving past, with a velocity of 0.700c in the negative x direction. Determine the magnitude and direction of the velocity of the first spacecraft as measured by the pilot of the second spacecraft.

Calculate the momentum of an electron moving with a speed of

(a) 0.010 0c,

(b) 0.500c, and

(c) 0.900c.

(a) 0.010 0c,

(b) 0.500c, and

(c) 0.900c.

The non-relativistic expression for the momentum of a particle, p = mu, agrees with experiment if u << c. For what speed does the use of this equation give an error in the momentum of

(a) 1.00% and

(b) 10.0%?

(a) 1.00% and

(b) 10.0%?

A golf ball travels with a speed of 90.0 m/s. By what fraction does its relativistic momentum magnitude p differ from its classical value mu? That is, find the ratio (p - mu)/mu.

Show that the speed of an object having momentum of magnitude p and mass m is u= c/√1 + (mc/p) 2.

An unstable particle at rest breaks into two fragments of unequal mass. The mass of the first fragment is 2.50 x 10-28 kg, and that of the other is 1.67 x 10-27 kg. If the lighter fragment has a speed of 0.893c after the breakup, what is the speed of the heavier fragment?

Determine the energy required to accelerate an electron from

(a) 0.500c to 0.900c and

(b) 0.900c to 0.990c.

(a) 0.500c to 0.900c and

(b) 0.900c to 0.990c.

A proton in a high-energy accelerator moves with a speed of c/2. Use the work–kinetic energy theorem to find the work required to increase its speed to

(a) 0.750c and

(b) 0.995c.

(a) 0.750c and

(b) 0.995c.

Show that, for any object moving at less than one-tenth the speed of light, the relativistic kinetic energy agrees with the result of the classical equation K = ½ mu2 to within less than 1%. Thus for most purposes, the classical equation is good enough to describe these objects, whose motion we call non-relativistic.

Find the momentum of a proton in MeV/c units assuming its total energy is twice its rest energy.

Find the kinetic energy of a 78.0-kg spacecraft launched out of the solar system with speed 106 km/s by using

(a) The classical equation K = ½ mu2

(b) What If? Calculate its kinetic energy using the relativistic equation.

(a) The classical equation K = ½ mu2

(b) What If? Calculate its kinetic energy using the relativistic equation.

A proton moves at 0.950c. Calculate its

(a) Rest energy,

(b) Total energy, and

(c) Kinetic energy.

(a) Rest energy,

(b) Total energy, and

(c) Kinetic energy.

A cube of steel has a volume of 1.00 cm3 and a mass of 8.00 g when at rest on the Earth. If this cube is now given a speed u = 0.900c, what is its density as measured by a stationary observer? Note that relativistic density is defined as Eg/x2V.

An unstable particle with a mass of 3.34 x 10-27 kg is initially at rest. The particle decays into two fragments that fly off along the x axis with velocity components 0.987c and -0.868c. Find the masses of the fragments. (Suggestion: Conserve both energy and momentum.)

An object having mass 900 kg and traveling at speed 0.850c collides with a stationary object having mass 1 400 kg. The two objects stick together. Find

(a) The speed and

(b) The mass of the composite object.

(a) The speed and

(b) The mass of the composite object.

Show that the energy–momentum relationship E2 = p2c2 + (mc2)2 follows from the expressions E = y mc2 and p = y mu.

In a typical color television picture tube, the electrons are accelerated through a potential difference of 25 000 V.

(a) What speed do the electrons have when they strike the screen?

(b) What is their kinetic energy in joules?

(a) What speed do the electrons have when they strike the screen?

(b) What is their kinetic energy in joules?

Consider electrons accelerated to an energy of 20.0 GeV in the 3.00-km-long Stanford Linear Accelerator.

(a) What is the y factor for the electrons?

(b) What is their speed?

(c) How long does the accelerator appear to them?

(a) What is the y factor for the electrons?

(b) What is their speed?

(c) How long does the accelerator appear to them?

Compact high-power lasers can produce a 2.00-J light pulse of duration 100 fs, focused to a spot 1 +m in diameter. (See Mourou and Umstader, “Extreme Light,” Scientific American, May 2002, page 81.) The electric field in the light accelerates electrons in the target material to near the speed of light.

(a) What is the average power of the laser during the pulse?

(b) How many electrons can be accelerated to 0.999 9c if 0.0100% of the pulse energy is converted into energy of electron motion?

(a) What is the average power of the laser during the pulse?

(b) How many electrons can be accelerated to 0.999 9c if 0.0100% of the pulse energy is converted into energy of electron motion?

A pion at rest (mπ = 273me) decays to a muon (mμ ≈ 207me) and an antineutrino. The reaction is written π− → μ− + v. Find the kinetic energy of the muon and the energy of the antineutrino in electron volts. (Suggestion: Conserve both energy and momentum.)

According to observer A, two objects of equal mass and moving along the x axis collide head on and stick to each other. Before the collision, this observer measures that object 1 move to the right with a speed of 3c/4, while object 2 moves to the left with the same speed. According to observer B, however, object 1 is initially at rest.

(a) Determine the speed of object 2 as seen by observer B.

(b) Compare the total initial energy of the system in the two frames of reference.

(a) Determine the speed of object 2 as seen by observer B.

(b) Compare the total initial energy of the system in the two frames of reference.

Make an order-of-magnitude estimate of the ratio of mass increase to the original mass of a flag, as you run it up a flagpole. In your solution explain what quantities you take as data and the values you estimate or measure for them.

When 1.00 g of hydrogen combines with 8.00 g of oxygen, 9.00 g of water is formed. During this chemical reaction, 2.86 x 105 J of energy is released. How much mass do the constituents of this reaction lose? Is the loss of mass likely to be detectable?

In a nuclear power plant the fuel rods last 3 yr before they are replaced. If a plant with rated thermal power 1.00 GW operates at 80.0% capacity for 3.00 yr, what is the loss of mass of the fuel?

The total volume of water in the oceans is approximately 1.40 x 109 km3. The density of sea water is 1 030 kg/m3, and the specific heat of the water is 4 186 J/(kg 0 °C). Find the increase in mass of the oceans produced by an increase in temperature of 10.0°C.

The power output of the Sun is 3.77 x 1026 W. How much mass is converted to energy in the Sun each second?

An Earth satellite used in the global positioning system moves in a circular orbit with period 11 h 58 min.

(a) Determine the radius of its orbit.

(b) Determine its speed.

(c) The satellite contains an oscillator producing the principal nonmilitary GPS signal. Its frequency is 1 575.42 MHz in the reference frame of the satellite. When it is received on the Earth’s surface, what is the fractional change in this frequency due to time dilation, as described by special relativity?

(d) The gravitational blue shift of the frequency according to general relativity is a separate effect. The magnitude of that fractional change is given by where ∆Ug is the change in gravitational potential energy of an object–Earth system when the object of mass m is moved between the two points at which the signal is observed. Calculate this fractional change in frequency.

(e) What is the overall fractional change in frequency? Superposed on both of these relativistic effects is a Doppler shift that is generally much larger. It can be a red shift or a blue shift, depending on the motion of a particular satellite relative to a GPS receiver (Fig. P39.55)

(a) Determine the radius of its orbit.

(b) Determine its speed.

(c) The satellite contains an oscillator producing the principal nonmilitary GPS signal. Its frequency is 1 575.42 MHz in the reference frame of the satellite. When it is received on the Earth’s surface, what is the fractional change in this frequency due to time dilation, as described by special relativity?

(d) The gravitational blue shift of the frequency according to general relativity is a separate effect. The magnitude of that fractional change is given by where ∆Ug is the change in gravitational potential energy of an object–Earth system when the object of mass m is moved between the two points at which the signal is observed. Calculate this fractional change in frequency.

(e) What is the overall fractional change in frequency? Superposed on both of these relativistic effects is a Doppler shift that is generally much larger. It can be a red shift or a blue shift, depending on the motion of a particular satellite relative to a GPS receiver (Fig. P39.55)

An astronaut wishes to visit the Andromeda galaxy, making a one-way trip that will take 30.0 yr in the spacecraft’s frame of reference. Assume that the galaxy is 2.00 x 106 ly away and that the astronaut’s speed is constant.

(a) How fast must he travel relative to the Earth?

(b) What will be the kinetic energy of his 1 000-metric-ton spacecraft?

(c) What is the cost of this energy if it is purchased at a typical consumer price for electric energy: $0.130/kWh?

(a) How fast must he travel relative to the Earth?

(b) What will be the kinetic energy of his 1 000-metric-ton spacecraft?

(c) What is the cost of this energy if it is purchased at a typical consumer price for electric energy: $0.130/kWh?

The cosmic rays of highest energy are protons that have kinetic energy on the order of 1013 MeV.

(a) How long would it take a proton of this energy to travel across the Milky Way galaxy, having a diameter ~105 ly, as measured in the proton’s frame?

(b) From the point of view of the proton, how many kilometers across is the galaxy?

(a) How long would it take a proton of this energy to travel across the Milky Way galaxy, having a diameter ~105 ly, as measured in the proton’s frame?

(b) From the point of view of the proton, how many kilometers across is the galaxy?

An electron has a speed of 0.750c.

(a) Find the speed of a proton that has the same kinetic energy as the electron.

(b) What If? Find the speed of a proton that has the same momentum as the electron.

(a) Find the speed of a proton that has the same kinetic energy as the electron.

(b) What If? Find the speed of a proton that has the same momentum as the electron.

Ted and Mary are playing a game of catch in frame S", which is moving at 0.600c with respect to frame S, while Jim, at rest in frame S watches the action (Fig. P39.59). Ted throws the ball to Mary at 0.800c (according to Ted) and their separation (measured in S") is 1.80 x 1012 m.

(a) According to Mary, how fast is the ball moving?

(b) According to Mary, how long does it take the ball to reach her?

(c) According to Jim, how far apart are Ted and Mary, and how fast is the ball moving?

(d) According to Jim, how long does it take the ball to reach Mary?

(a) According to Mary, how fast is the ball moving?

(b) According to Mary, how long does it take the ball to reach her?

(c) According to Jim, how far apart are Ted and Mary, and how fast is the ball moving?

(d) According to Jim, how long does it take the ball to reach Mary?

A rechargeable AA battery with a mass of 25.0 g can supply a power of 1.20 W for 50.0 min.

(a) What is the difference in mass between a charged and an uncharged battery?

(b) What fraction of the total mass is this mass difference?

(a) What is the difference in mass between a charged and an uncharged battery?

(b) What fraction of the total mass is this mass difference?

The net nuclear fusion reactions inside the Sun can b written as 41H → 4He + ∆E. The rest energy of each hydrogen atom is 938.78 MeV and the rest energy of the helium-4 atom is 3 728.4 MeV. Calculate the percentage of the starting mass that is transformed to other forms of energy.

An object disintegrates into two fragments. One of the fragments has mass 1.00 MeV/c2 and momentum 1.75 MeV/c in the positive x direction. The other fragment has mass 1.50 MeV/c2 and momentum 2.00 MeV/c in the positive y direction. Find

(a) The mass and

(b) The speed of the original object.

(a) The mass and

(b) The speed of the original object.

An alien spaceship traveling at 0.600c toward the Earth launches a landing craft with an advance guard of purchasing agents and physics teachers. The lander travels in the same direction with a speed of 0.800c relative to the mother ship. As observed on the Earth, the spaceship is 0.200 ly from the Earth when the lander is launched.

(a) What speed do the Earth observers measure for the approaching lander?

(b) What is the distance to the Earth at the time of lander launch, as observed by the aliens?

(c) How long does it take the lander to reach the Earth as observed by the aliens on the mother ship?

(d) If the lander has a mass of 4.00 x 105 kg, what is its kinetic energy as observed in the Earth reference frame?

(a) What speed do the Earth observers measure for the approaching lander?

(b) What is the distance to the Earth at the time of lander launch, as observed by the aliens?

(c) How long does it take the lander to reach the Earth as observed by the aliens on the mother ship?

(d) If the lander has a mass of 4.00 x 105 kg, what is its kinetic energy as observed in the Earth reference frame?

A physics professor on the Earth gives an exam to her students, who are in a spacecraft traveling at speed v relative to the Earth. The moment the craft passes the professor, she signals the start of the exam. She wishes her students to have a time interval T0 (spacecraft time) to complete the exam. Show that she should wait a time interval (Earth time) of before sending a light signal telling them to stop. (Suggestion: Remember that it takes some time for the second light signal to travel from the professor to the students.)

Spacecraft I, containing students taking a physics exam, approaches the Earth with a speed of 0.600c (relative to the Earth), while spacecraft II, containing professors proctoring the exam, moves at 0.280c (relative to the Earth) directly toward the students. If the professors stop the exam after 50.0 min have passed on their clock, how long does the exam last as measured by

(a) The students

(b) An observer on the Earth?

(a) The students

(b) An observer on the Earth?

Energy reaches the upper atmosphere of the Earth from the Sun at the rate of 1.79 x 1017 W. If all of this energy were absorbed by the Earth and not re-emitted, how much would the mass of the Earth increase in 1.00 yr?

A super train (proper length 100 m) travels at a speed of 0.950c as it passes through a tunnel (proper length 50.0 m). As seen by a trackside observer, is the train ever completely within the tunnel? If so with how much space to spare?

Imagine that the entire Sun collapses to a sphere of radius Rg such that the work required to remove a small mass m from the surface would be equal to its rest energy mc 2. This radius is called the gravitational radius for the Sun. Find Rg . (It is believed that the ultimate fate of very massive stars is to collapse beyond their gravitational radii into black holes.)

A particle with electric charge q moves along a straight line in a uniform electric field E with a speed of u. The electric force exerted on the charge is qE. The motion and the electric field are both in the x direction.

(a) Show that the acceleration of the particle in the x direction is given by

(b) Discuss the significance of the dependence of the acceleration on the speed.

(c) What If? If the particle starts from rest at x = 0 at t = 0, how would you proceed to find the speed of the particle and its position at time t?

(a) Show that the acceleration of the particle in the x direction is given by

(b) Discuss the significance of the dependence of the acceleration on the speed.

(c) What If? If the particle starts from rest at x = 0 at t = 0, how would you proceed to find the speed of the particle and its position at time t?

An observer in a coasting spacecraft moves toward a mirror at speed v relative to the reference frame labeled by S in Figure P39.70. The mirror is stationary with respect to S. A light pulse emitted by the spacecraft travels toward the mirror and is reflected back to the craft. The front of the craft is a distance d from the mirror (as measured by observers in S) at the moment the light pulse leaves the craft. What is the total travel time of the pulse as measured by observers in?

(a) The S frame and

(b) The front of the spacecraft?

(a) The S frame and

(b) The front of the spacecraft?

The creation and study of new elementary particles is an important part of contemporary physics. Especially interesting is the discovery of a very massive particle. To create a particle of mass M requires energy Mc2. With enough energy, an exotic particle can be created by allowing a fast moving particle of ordinary matter, such as a proton, to collide with a similar target particle. Let us consider a perfectly inelastic collision between two protons: an incident proton with mass mp, kinetic energy K, and momentum magnitude p joins with an originally stationary target proton to form a single product particle of mass M. You might think that the creation of a new product particle, nine times more massive than in a previous experiment, would require just nine times more energy for the incident proton. Unfortunately not all of the kinetic energy of the incoming proton is available to create the product particle, since conservation of momentum requires that after the collision the system as a whole still must have some kinetic energy. Only a fraction of the energy of the incident particle is thus available to create a new particle. You will determine how the energy available for particle creation depends on the energy of the moving proton. Show that the energy available to create a product particle is given by from this result, when the kinetic energy K of the incident proton is large compared to its rest energy mpc2, we see that M approaches (2mpK) 1/2/c. Thus if the energy of the incoming proton is increased by a factor of nine, the mass you can create increases only by a factor of three. This disappointing result is the main reason that most modern accelerators, such as those at CERN (in Europe), at Fermilab (near Chicago), at SLAC (at Stanford), and at DESY (in Germany), use colliding beams. Here the total momentum of a pair of interacting particles can be zero. The center of mass can be at rest after the collision, so in principle all of the initial kinetic energy can be used for particle creation, according to where K is the total kinetic energy of two identical colliding particles. Here if K >> mc2, we have M directly proportional to K, as we would desire. These machines are difficult to build and to operate, but they open new vistas in physics.

A particle of mass m moving along the x axis with a velocity component +u collides head-on and sticks to a particle of mass m/3 moving along the x axis with the velocity component -u. What is the mass M of the resulting particle?

A rod of length L0 moving with a speed v along the horizontal direction makes an angle 4 0 with respect to the x" axis.

(a) Show that the length of the rod as measured by a stationary observer is L = L0 [1 - (v2/c2) cos2 4 0]1/2.

(b) Show that the angle that the rod makes with the x axis is given by tan 4 = μ tan 4 0. These results show that the rod is both contracted and rotated.

(Take the lower end of the rod to be at the origin of the primed coordinate system.)

(a) Show that the length of the rod as measured by a stationary observer is L = L0 [1 - (v2/c2) cos2 4 0]1/2.

(b) Show that the angle that the rod makes with the x axis is given by tan 4 = μ tan 4 0. These results show that the rod is both contracted and rotated.

(Take the lower end of the rod to be at the origin of the primed coordinate system.)

Suppose our Sun is about to explode. In an effort to escape, we depart in a spacecraft at v = 0.800c and head toward the star Tau Ceti, 12.0 ly away. When we reach the midpoint of our journey from the Earth, we see our Sun explode and, unfortunately, at the same instant we see Tau Ceti explode as well.

(a) In the spacecraft’s frame of reference, should we conclude that the two explosions occurred simultaneously? If not, which occurred first?

(b) What If? In a frame of reference in which the Sun and Tau Ceti are at rest, did they explode simultaneously? If not, which exploded first?

(a) In the spacecraft’s frame of reference, should we conclude that the two explosions occurred simultaneously? If not, which occurred first?

(b) What If? In a frame of reference in which the Sun and Tau Ceti are at rest, did they explode simultaneously? If not, which exploded first?

A 57Fe nucleus at rest emits a 14.0-keV photon. Use conservation of energy and momentum to deduce the kinetic energy of the recoiling nucleus in electron volts. (Use Mc2 = 8.60 x 10-9 J for the final state of the 57Fe nucleus.)

Prepare a graph of the relativistic kinetic energy and the classical kinetic energy, both as a function of speed, for an object with a mass of your choice. At what speed does the classical kinetic energy underestimate the experimental value by 1%? by 5%? by 50%?

What two speed measurements do two observers in relative motion always agree on?

A spacecraft with the shape of a sphere moves past an observer on Earth with a speed 0.5c. What shape does the observer measure for the spacecraft as it moves past?

The speed of light in water is 230 Mm/s. Suppose an electron is moving through water at 250 Mm/s. Does this violate the principle of relativity?

Two identical clocks are synchronized. One is then put in orbit directed eastward around the Earth while the other remains on the Earth. Which clock runs slower? When the moving clock returns to the Earth, are the two still synchronized?

Explain why it is necessary, when defining the length of a rod, to specify that the positions of the ends of the rod are to be measured simultaneously.

A train is approaching you at very high speed as you stand next to the tracks. Just as an observer on the train passes you, you both begin to play the same Beethoven symphony on portable compact disc players.

(a) According to you, whose CD player finishes the symphony first?

(b) What If? According to the observer on the train, whose CD player finishes the symphony first?

(c) Whose CD player really finishes the symphony first?

(a) According to you, whose CD player finishes the symphony first?

(b) What If? According to the observer on the train, whose CD player finishes the symphony first?

(c) Whose CD player really finishes the symphony first?

List some ways our day-to-day lives would change if the speed of light were only 50 m/s.

Does saying that a moving clock runs slower than a stationary one imply that something is physically unusual about the moving clock?

How is acceleration indicated on a space–time graph?

A particle is moving at a speed less than c/2. If the speed of the particle is doubled, what happens to its momentum?

Give a physical argument that shows that it is impossible to accelerate an object of mass m to the speed of light, even with a continuous force acting on it.

The upper limit of the speed of an electron is the speed of light c. Does that mean that the momentum of the electron has an upper limit?

Because mass is a measure of energy, can we conclude that the mass of a compressed spring is greater than the mass of the same spring when it is not compressed?

It is said that Einstein, in his teenage years, asked the question, “What would I see in a mirror if I carried it in my hands and ran at the speed of light?” How would you answer this question?

Some distant astronomical objects, called quasars, are receding from us at half the speed of light (or greater). What is the speed of the light we receive from these quasars?

Photons of light have zero mass. How is it possible that they have momentum?

“Newtonian mechanics correctly describes objects moving at ordinary speeds and relativistic mechanics correctly describes objects moving very fast.” “Relativistic mechanics must make a smooth transition as it reduces to Newtonian mechanics in a case where the speed of an object becomes small compared to the speed of light.” Argue for or against each of these two statements.

Two cards have straight edges. Suppose that the top edge of one card crosses the bottom edge of another card at a small angle, as in Figure Q39.18a. A person slides the cards together at a moderately high speed. In what direction does the intersection point of the edges move? Show that it can move at a speed greater than the speed of light. A small flashlight is suspended in a horizontal plane and set into rapid rotation. Show that the spot of light it produces on a distant screen can move across the screen at a speed greater than the speed of light. (If you use a laser pointer, as in Figure Q39.18b, make sure the direct laser light cannot enter a person’s eyes.) Argue that these experiments do not invalidate the principle that no material, no energy, and no information can move faster than light moves in a vacuum.

Describe how the results of Example 39.7 would change if, instead of fast space vehicles, two ordinary cars were approaching each other at highway speeds.

Two objects are identical except that one is hotter than the other. Compare how they respond to identical forces.

With regard to reference frames, how does general relativity differ from special relativity?

Two identical clocks are in the same house, one upstairs in a bedroom, and the other downstairs in the kitchen. Which clock runs more slowly? Explain.

A thought experiment Imagine ants living on a merry goround turning at relativistic speed, which is their two-dimensional world. From measurements on small circles they are thoroughly familiar with the number 2. When they measure the circumference of their world, and divide it by the diameter, they expect to calculate the number π 2 = 3.141 59. We see the merry-go-round turning at relativistic speed. From our point of view, the ants’ measuring rods on the circumference are experiencing length contraction in the tangential direction; hence the ants will need some extra rods to fill that entire distance. The rods measuring the diameter, however, do not contract, because their motion is perpendicular to their lengths. As a result, the computed ratio does not agree with the number 2. If you were an ant, you would say that the rest of the universe is spinning in circles, and your disk is stationary. What possible explanation can you then give for the discrepancy, in light of the general theory of relativity?

You are in a windowless car in as exceptionally smooth train moving at constant velocity. It there any physical experiment you can do in the train car to determine whether you are moving? Explain.

You might have had the experience of being at a red light when, out of the corner od your eye, you see the car beside you creep forward. Instinctively you stomp on the brake pedal, thinking that you are rolling backward. What does this say about absolute relative motion?

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