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1EExercise 2 Conservation of Linear Momentum Is Covariant Under the Galilean Transformation. Assume that two masses and are moving in the positive x direction with velocities and as measured by an observer in before a collision. After the collision, the two masses stick together and move with a velocity in . Show that if an observer in finds momentum to be conserved, so does an observer in S.
If the speed of the observer is increased by 5.0%, what is the period of the pendulum when measured by this observer?
If the ship moves past the observer at 0.01000c, what length will the observer measure?
5EWhat two measurements will two observers in relative motion always agree on?
A spaceship in the shape of a sphere moves past an observer on Earth with a speed of 0.5c. What shape will the observer see as the spaceship moves past?
An astronaut moves away from Earth at a speed close to the speed of light. If an observer on Earth could make measurements of the astronauts size and pulse rate, what changes (if any) would he or she measure? Would the astronaut measure any changes?Two identically constructed clocks are synchronized. One is put in an eastward orbit around Earth while the other remains on Earth. Which clock runs slower? When the moving clock returns to Earth, will the two clocks still be synchronized?Two lasers situated on a moving spacecraft are triggered simultaneously. An observer on the spacecraft claims to see the pulses of light simultaneously. What condition is necessary in order that another observer agrees that the two pulses are emitted simultaneously?
6QWhen we speak of time dilation, do we mean that time passes more slowly in moving systems or that it simply appears to do so?8Q9QIt 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?11QWhat happens to the density of an object as its speed increases, as measured by an Earth observer?
In a lab frame of reference, an observer finds Newton’s second law is valid in the form. Show that Newton’s second law is not valid in a reference frame moving past the laboratory frame of Problem 1 with a constant acceleration a1. Assume that mass is an invariant quantity and is constant in time.
2P3PAn airplane flying upwind, downwind, and crosswind shows the main principle of the MichelsonMorley experiment. A plane capable of flying at speed c in still air is flying in a wind of speed v. Suppose the plane flies upwind a distance L and then returns downwind to its starting point. (a) Find the time needed to make the round-trip and compare it with the time to fly crosswind a distance L and return. Before calculating these times, sketch the two situations. (b) Compute the time difference for the two trips if L = 100 mi, c = 500 mi/h, and v = 100 mi/h.5P6PA clock on a moving spacecraft runs 1 s slower per day relative to an identical clock on Earth. What is the relative speed of the spacecraft? (Hint: For v/c << 1, note that γ ≈ 1 + v2/2c2.)
A meter stick moving in a direction parallel to its length appears to be only 75 cm long to an observer. What is the speed of the meter stick relative to the observer?A spacecraft moves at a speed of 0.900c. If its length is L as measured by an observer on the spacecraft, what is the length measured by a ground observer?
The average lifetime of a pi meson in its own frame of reference is 2.6 × 10−8 s. If the meson moves with a speed of 0.95c, what is (a) its mean lifetime as measured by an observer on Earth and (b) the average distance it travels before decaying, as measured by an observer on Earth?
An atomic clock is placed in a jet airplane. The clock measures a time interval of 3600 s when the jet moves with a speed of 400 m/s. How much longer or shorter a time interval does an identical clock held by an observer on the ground measure? (Hint: For , γ ≈ 1 + v2/2c2.)
An astronaut at rest on Earth has a heartbeat rate of 70 beats/min. What will this rate be when she is traveling in a spaceship at 0.90c as measured (a) by an observer also in the ship and (b) by an observer at rest on the Earth?
The muon is an unstable particle that spontaneously decays into an electron and two neutrinos. If the number of muons at t = 0 is N0, the number at time t is given by , where τ is the mean lifetime, equal to 2.2 μs. Suppose the muons move at a speed of 0.95c and there are 5.0 × 104 muons at t = 0. (a) What is the observed lifetime of the muons? (b) How many muons remain after traveling a distance of 3.0 km?
A rod of length L0 moves with a speed v along the horizontal direction. The rod makes an angle of θ0 with respect to the x′-axis. (a) Show that the length of the rod as measured by a stationary observer is given by . (b) Show that the angle that the rod makes with the x-axis is given by the expression tan θ = γ tan θ0. 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.)
The classical Doppler shift for light. A light source recedes from an observer with a speed v that is small compared with c. (a) Show that in this case, Equation 1.15 reduces to ffvc (b) Also show that in this case vc (Hint: Differentiate f = c to show that / = f/f) (c) Spectroscopic measurements of an absorption line normally found at = 397 nm reveal a redshift of 20 nm for light coming from a galaxy in Ursa Major. What is the recessional speed of this galaxy?Calculate, for the judge, how fast you were going in miles per hour when you ran the red light because it appeared Doppler-shifted green to you. Take red light to have a wavelength of 650 nm and green to have a wavelength of 550 nm.
17P18PTwo spaceships approach each other, each moving with the same speed as measured by an observer on the Earth. If their relative speed is 0.70c, what is the speed of each spaceship?20PAn observer on Earth observes two spacecraft moving in the same direction toward the Earth. Spacecraft A appears to have a speed of 0.50c, and spacecraft B appears to have a speed of 0.80c. What is the speed of spacecraft A measured by an observer in spacecraft B?
Speed of light in a moving medium. The motion of a medium such as water influences the speed of light. This effect was first observed by Fizeau in 1851. Consider a light beam passing through a horizontal column of water moving with a speed v. (a) Show that if the beam travels in the same direction as the flow of water, the speed of light measured in the laboratory frame is given by
where n is the index of refraction of the water. (Hint: Use the inverse Lorentz velocity transformation and note that the speed of light with respect to the moving frame is given by c/n.) (b) Show that for v << c, the preceding expression is in good agreement with Fizeau’s experimental result:
This proves that the Lorentz velocity transformation and not the Galilean velocity transformation is correct for light.
An observer in frame S sees lightning simultaneously strike two points 100 m apart. The first strike occurs at x1 = y1 = z1 = t1 = 0 and the second at x2 = 100 m, y2 = z2 = t2 = 0. (a) What are the coordinates of these two events in a frame S moving in the standard configuration at 0.70c relative to S? (b) How far apart are the events in S? (c) Are the events simultaneous in S? If not, what is the difference in time between the events, and which event occurs first?As seen from Earth, two spaceships A and B are approaching along perpendicular directions. If A is observed by an Earth observer to have velocity uy = 0.90c and B to have a velocity ux = +0.90c, find the speed of ship A as measured by the pilot of B.25PThe proper length of one spaceship is three times that of another. The two spaceships are traveling in the same direction and, while both are passing overhead, an Earth observer measures the two spaceships to have the same length. If the slower spaceship is moving with a speed of 0.35c, determine the speed of the faster spaceship.
27P28PA spaceship moves away from Earth at a speed v and fires a shuttle craft in the forward direction at a speed v relative to the ship. The pilot of the shuttle craft launches a probe at speed v relative to the shuttle craft. Determine (a) the speed of the shuttle craft relative to Earth, and (b) the speed of the probe relative to Earth.
An observer in a rocket moves toward a mirror at speed v relative to the reference frame labeled by S in Figure P1.30. The mirror is stationary with respect to S. A light pulse emitted by the rocket travels toward the mirror and is reflected back to the rocket. The front of the rocket is a distance d from the mirror (as measured by observers in S) at the moment the light pulse leaves the rocket. What is the total travel time of the pulse as measured by observers in (a) the S frame and (b) the front of the rocket?
Figure P1.30
A physics professor on Earth gives an exam to her students who are on a spaceship traveling at speed v relative to Earth. The moment the ship passes the professor, she signals the start of the exam. If she wishes her students to have time T0 (spaceship time) to complete the exam, show that she should wait a time (Earth time) of
before sending a light signal telling them to stop. (Hint: Remember that it takes some time for the second light signal to travel from the professor to the students.)
A yet-to-be-built spacecraft starts from Earth moving at constant speed to the yet-to-be-discovered planet Retah, which is 20 lighthours away from Earth. It takes 25 h (according to an Earth observer) for a spacecraft to reach this planet. Assuming that the clocks are synchronized at the beginning of the journey, compare the time elapsed in the spacecraft’s frame for this one-way journey with the time elapsed as measured by an Earth-based clock.
Suppose our Sun is about to explode. In an effort to escape, we depart in a spaceship at v = 0.80c and head toward the star Tau Ceti, 12 lightyears 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 spaceship’s frame of reference, should we conclude that the two explosions occurred simultaneously? If not, which occurred first? (b) In a frame of reference in which the Sun and Tau Ceti are at rest, did they explode simultaneously? If not, which exploded first?
Two powerless rockets are on a collision course. The rockets are moving with speeds of 0.800c and 0.600c and are initially 2.52 × 1012 m apart as measured by Liz, an Earth observer, as shown in Figure P1.34. Both rockets are 50.0 m in length as measured by Liz. (a) What are their respective proper lengths? (b) What is the length of each rocket as measured by an observer in the other rocket? (c) According to Liz, how long before the rockets collide? (d) According to rocket 1, how long before they collide? (e) According to rocket 2, how long before they collide? (f) If both rocket crews are capable of total evacuation within 90 min (their own time), will there be any casualties?
Figure P1.34
35PSuzanne observes two light pulses to be emitted from the same location, but separated in time by 3.00 s. Mark sees the emission of the same two pulses separated in time by 9.00 s. (a) How fast is Mark moving relative to Suzanne? (b) According to Mark, what is the separation in space of the two pulses?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 2001. Event B occurs at the point (150 m, 0, 0) at the same moment. A second observer, moving past with a velocity of , also observes the two events. In her reference frame S′, which event occurred first and what time elapsed between the events?
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.
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 Earths surface, what is the fractional change in this frequency due to time dilation, as described by special relativity? (d) The gravitational blueshift of the frequency according to general relativity is a separate effect. The magnitude of that fractional change is given by ff=Ugmc2 where Ug/m is the change in gravitational potential energy per unit mass 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 redshift or a blueshift, depending on the motion of a particular satellite relative to a GPS receiver (Fig. P1.39).40PA particle is moving at a speed of less than c/2. If the speed of the particle is doubled, what happens to its momentum?
Give a physical argument showing that it is impossible to accelerate an object of mass m to the speed of light, even with a continuous force acting on it.
3Q4Q5Q6Q8Q9QCalculate the momentum of a proton moving with a speed of (a) 0.010c, (b) 0.50c, (c) 0.90c. (d) Convert the answers of (a)(c) to MeV/c.2PConsider the relativistic form of Newtons second law. Show that when F is parallel to v, F=m(1v2c2)3/2dvdt where m is the mass of an object and v is its speed.A charged particle moves along a straight line in a uniform electric field E with a speed v. If the motion and the electric field are both in the x direction, (a) show that the magnitude of the acceleration of the charge q is given by a=dvdt=qEm(1v2c2)3/2 (b) Discuss the significance of the dependence of the acceleration on the speed. (c) If the particle starts from rest at x = 0 at t = 0, find the speed of the particle and its position after a time t has elapsed. Comment on the limiting values of v and x as t .5P6P7PA proton moves at a speed of 0.95c. Calculate its (a) rest energy, (b) total energy, and (c) kinetic energy.
An electron has a kinetic energy 5 times greater than its rest energy. Find (a) its total energy and (b) its speed.Find the speed of a particle whose total energy is 50% greater than its rest energy.
A proton in a high-energy accelerator is given a kinetic energy of 50 GeV. Determine the (a) momentum and (b) speed of the proton.An electron has a speed of 0.75c. Find the speed of a proton that has (a) the same kinetic energy as the electron and (b) the same momentum as the electron.
Protons in an accelerator at the Fermi National Laboratory near Chicago are accelerated to an energy of 400 times their rest energy. (a) What is the speed of these protons? (b) What is their kinetic energy in MeV?How long will the Sun shine, Nellie? The Sun radiates about 4.0 × 1026 J of energy into space each second. (a) How much mass is released as radiation each second? (b) If the mass of the Sun is 2.0 × 1030 kg, how long can the Sun survive if the energy release continues at the present rate?
Electrons in projection television sets are accelerated through a total potential difference of 50,000 V. (a) Calculate the speed of the electrons using the relativistic form of kinetic energy assuming the electrons start from rest. (b) Calculate the speed of the electrons using the classical form of kinetic energy. (c) Is the difference in speed significant in the design of this set in your opinion?16P17P18P19P20PAn electron having kinetic energy K = 1.000 MeV makes a head-on collision with a positron at rest. (A positron is an antimatter particle that has the same mass as the electron but opposite charge.) In the collision the two particles annihilate each other and are replaced by two γ rays of equal energy, each traveling at equal angles θ with the electron’s direction of motion. (Gamma rays are massless particles of electromagnetic radiation having energy E = pc.) Find the energy E, momentum p, and angle of emission θ of the γ rays.
The K0 meson is an uncharged member of the particle “zoo” that decays into two charged pions according to K0 → π+ + π−. The pions have opposite charges, as indicated, and the same mass, mπ = 140 MeV/c2. Suppose that a K0 at rest decays into two pions in a bubble chamber in which a magnetic field of 2.0 T is present (see Fig. P2.22). If the radius of curvature of the pions is 34.4 cm, find (a) the momenta and speeds of the pions and (b) the mass of the K0 meson.
An unstable particle having a mass of 3.34 1027 kg is initially at rest. The particle decays into two fragments that fly off with velocities of 0.987c and 0.868c. Find the rest masses of the fragments.As measured by observers in a reference frame S, a particle having charge q moves with velocity v in a magnetic field B and an electric field E. The resulting force on the particle is then measured to be F = q(E + v × B). Another observer moves along with the charged particle and measures its charge to be q also but measures the electric field to be E′. If both observers are to measure the same force, F, show that E′ = E + v × B.
An object having mass of 900 kg and traveling at a speed of 0.850c collides with a stationary object having mass 1400 kg. The two objects stick together. Find (a) the speed and (b) the mass of the composite object.
27P28P29P30PA 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?
32PEnergy reaches the upper atmosphere of the Earth from the Sun at the rate of 1.79 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?Calculate the quantum number, n, for this pendulum with E = 1.5 102 J.An object of mass m on a spring of stiffness k oscillates with an amplitude A about its equilibrium position. Suppose that m = 300 g, k = 10 N/m, and A = 10 cm. (a) Find the total energy. (b) Find the mechanical frequency of vibration of the mass. (c) Calculate the change in amplitude when the system loses one quantum of energy.
1Q2Q3Q4Q5Q6Q7Q8Q9Q10Q11Q1P2P3P4P5P6P7P8P9P10P11P12P13P14P15P16P17P18P19P20P21P22P23P24P25P26P27P28P29P30P31P32P33P34P35P36P37PAs a single crystal is rotated in an x-ray spectrometer (Fig. 3.22a), many parallel planes of atoms besides AA and BB produce strong diffracted beams. Two such planes are shown in Figure P3.38. (a) Determine geometrically the interplanar spacings d1 and d2 in terms of d0. (b) Find the angles (with respect to the surface plane AA) of the n = 1, 2, and 3 intensity maxima from planes with spacing d1. Let = 0.626 and d0 = 4.00 . Note that a given crystal structure (for example, cubic) has interplanar spacings with characteristic ratios, which produce characteristic diffraction patterns. In this way, measurement of the angular position of diffracted x-rays may be used to infer the crystal structure. Figure P3.38 Atomic planes in a cubic lattice.39P40P41P42P43P44P46P47P48PExercise 1 Find the horizontal speed vx for this case.2E3E4E5E1Q2Q3Q4Q5Q6Q7Q8Q9Q10Q11Q1P2PA mystery particle enters the region between the plates of a Thomson apparatus as shown in Figure 4.6. The deflection angle θ is measured to be 0.20 radians (downwards) for this particle when V = 2000 V, ℓ = 10.0 cm, and d = 2.00 cm. If a perpendicular magnetic field of magnitude 4.57 × 10−2 T is applied simultaneously with the electric field, the particle passes through the plates without deflection. (a) Find q/m for this particle. (b) Identify the particle. (c) Find the horizontal speed with which the particle entered the plates. (d) Must we use relativistic mechanics for this particle?
4PA Thomson-type experiment with relativistic electrons. One of the earliest experiments to show that p = mv (rather than p = mv) was that of Neumann. [G. Neumann, Ann. Physik 45:529 (1914)]. The apparatus shown in Figure P4.5 is identical to Thomsons except that the source of high-speed electrons is a radioactive radium source and the magnetic field B is arranged to act on the electron over its entire trajectory from source to detector. The combined electric and magnetic fields act as a velocity selector, only passing electrons with speed v, where v = V/Bd (Equation 4.6), while in the region where there is only a magnetic field the electron moves in a circle of radius r, with r given by p = Bre. This latter region (E = 0, B = constant) acts as a momentum selector because electrons with larger momenta have paths with larger radii. (a) Show that the radius of the circle described by the electron is given by r = (l2 + y2)/2y. (b) Typical values for the Neumann experiment were d = 2.51 104 m, B = 0.0177 T, and l = 0.0247 m. For V = 1060 V, y, the most critical value, was measured to be 0.0024 0.0005 m. Show that these values disagree with the y value calculated from p = mv but agree with the y value calculated from p = mv within experimental error. (Hint: Find v from Equation 4.6, use mv = Bre or mv = Bre to find r, and use r to find y.) Figure P4.5 The Neumann apparatus.6P8P9P10P11P12P13P14P15P16P17P18P19P20P21P22P23P24P25P26P27P28P29P30P31P32P33P34P35P36P37P38P39P40P41P42P43P44PA 0.20-kg ball is thrown upward. How much work is done on the ball by gravity as the ball rises between heights of 2.0 m and 3.0 m?5E1Q2Q3Q4Q5Q7Q8Q9Q10Q11Q1P2P3P4P5P6P7P8P9P10P11P12P13P14PShow that the group velocity for a nonrelativistic free electron is also given by , where is the electron’s velocity.
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