- 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
- 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
- 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
- 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
- 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
- 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
- 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−
- 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,
- 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 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
- 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
- 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
- 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
- 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 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
- 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
- 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)
- 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)
- 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
- 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
- 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
- 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
- Calculate the momentum of an electron moving with a speed of (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
- 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
- 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
- Determine the energy required to accelerate an electron from (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.
- 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
- 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
- A proton moves at 0.950c. Calculate its (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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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?
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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

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