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Monochromatic light at 577 nm illuminates a diffraction grating with 325 lines/mm. Determine (a) the angle to the first-order maximum. (b) the highest order that can be observed with this grating at the given wavelength, and (c) the angle to this highest-order maximum.Light of wavelength 5.00 102 nm is incident normally on a diffraction grating. If the third-order maximum of the diffraction pattern is observed at 32.0, (a) what is the number of rulings per centimeter for the grating? (b) Determine the total number of primary maxima that can be observed in this situation.50PThe angle of incidence of a light beam in air onto a reflecting surface b continuously variable. The reflected ray is found to be completely polarized when the angle of incidence is 48.0. (a) What is the index of refraction of the reflecting material? (b) If some of the incident light (at an angle of 48.0) passes into the material below the surface, what is the angle of refraction?Unpolarized light passes through two Polaroid sheets. The transmission axis of the analyzer makes an angle of 53.0 with the axis of the polarizer. (a) What fraction of the original unpolarized light is transmitted through the analyzer? (b) What fraction of the original light is absorbed by the analyzer?The index of retraction of a glass plate is 1.52. What is the Brewsters angle when the plate is (a) in air and (b) in water? (Sec Problem 57.)At what angle above the horizon is the Sun if light from it is completely polarized upon reflection from water?55PThe critical angle for total internal reflection for sapphire surrounded by air is 34.4. Calculate the Brewsters angle for sapphire if the light is incident from the air.Equation 24.14 assumes the incident light is in air. If the light is incident from a medium of index n1 onto a medium of index n2, follow the procedure used to derive Equation 24.14 to show that tan p = n2/n1.58PThree polarizing plates whose planes are parallel are centered on a common axis. The directions of the transmission axes relative to the common vertical direction are shown in Figure P24.59. A linearly polarized beam of light with plane of polarization parallel to the vertical reference direction is incident from the left onto the first disk with intensity Ii = 10.0 units (arbitrary). Calculate the transmitted intensity It when 1 = 20.0, 2 = 40.0, and 3 = 60.0. Hint: Make repeated use of Maluslaw. Figure P24.59 Problems 59 and 70.Light of intensity I0 is polarized vertically and is incident on an analyzer rotated at an angle from the vertical. Find the angle if the transmitted light has intensity (a) I = (0.750)I0, (b) I = (0.500)I0, (c) I = (0.250)I0, and (d) I = 0.Light with a wavelength in vacuum of 546.1 nm falls perpendicularly on a biological specimen that is 1.000 m thick. The light splits into two beams polarized at right angles, for which the indices of refraction are 1.320 and 1.333, respectively. (a) Calculate the wavelength of each component of the light while it is traversing the specimen. (b) Calculate the phase difference between the two beams when they emerge from the specimen.Light from a helium-neon laser ( = 632.8 nm) is incident on a single slit. What is the maximum width of the slit for which no diffraction minima are observed?Laser light with a wavelength of 632.6 nm is directed through one slit or two slits and allowed to fall on a screen 2.60 in beyond. Figure P24.63 shows the pattern on the screen, with a centimeter ruler below it. Did the light pass through one slit or two slits? Explain how you can tell. If the answer is one slit, find its width. If the answer is two slits, find the distance between their centers. Figure P24.63In a Youngs interference experiment, the two slits are separated by 0.150 mm and the incident light includes two wavelengths: l = 5.40 102 nm (green) and 2 = 4.50 102 nm (blue). The overlapping interference patterns are observed on a screen 1.40 m from the slits. (a) Find a relationship between the orders m1, and m2, that determines where a bright fringe of the green light coincides with a bright fringe of the blue light. (The order m1 is associated with 1, and m2 is associated with 2.) (b) Find the minimum values of m1 and m2 associated with 2.) (b) Find the minimum values of m1 and m2 such that the overlapping of the bright fringes will occur and find the position of the overlap on the screen.Light of wavelength 546 nm (the intense green line from a mercury source) produces a Young's interference pattern in which the second minimum from the central maximum is along a direction that makes an angle of 18.0 min of are with the axis through the central maximum. What is the distance between the parallel slits?The two speakers are placed 35.0 cm apart. A single oscillator makes the speakers vibrate in phase at a frequency of 2.00 kHz. At what angles, measured from the perpendicular bisector of the line joining the speakers, would a distant observer hear maximum sound intensity? Minimum sound intensity? (Take the speed of sound to be 340. m/s.)Interference effects are produced at point P on a screen as a result of direct rays from a 5.00 102nm source and reflected rays off a mirror, as shown in Figure P24.67. If the source is L = 1.00 102m to the left of the screen and h = 1.00 cm above the mirror, find the distance y (in millimetres) to the first dark band above the mirror. Figure P24.6768APFigure P24.69 shows a radio-wave transmitter and a receiver, both h = 50.0 m above the ground and d = 6.00 102 m apart. The receiver can receive signals directly from the transmitter and indirectly from signals that bounce off the ground. If the ground is level between the transmitter and receiver and a /2 phase shift occurs upon reflection, determine the longest wavelengths that interior (a) constructively and (b) destructively. Figure P24.69Three polarizers, centered on a common axis and with their planes parallel to one another, have transmission axes oriented at angles of 1, 2, and 3 from the vertical, as shown in Figure P24.59. Light of intensity Ii, polarized with its plane of polarization oriented vertically, is incident from the left onto the first polarizer. What is the ratio If/Ii of the final transmitted intensity to the incident intensity if (a) 1 = 45, 2 = 90, and 3 = 0? (b) 1 = 0, 2 = 45, and 3 = 90?71APA plano-convex lens (flat on one side, convex on the other) with index of refraction n rests with its curved side (radius of curvature R) on a flat glass surface of the same index of refraction with a film of index nfilm between them. The lens is illuminated from above by light of wavelength . Show that the dark Newton rings that appear have radii of rmR/nfilm where m is an integer.A diffraction pattern is produced on a screen 1.40 m from a single slit, using monochromatic light of wavelength 5.00 102 nm. The distance from the center of the central maximum to the first-order maximum is 3.00 mm. Calculate the slit width. Hint: Assume that the first-order maximum is halfway between the first- and second-order minima.74APTwo campers wish to start a fire during the day. One camper is nearsighted and one is farsighted. Whose glasses should be used to focus the Suns rays onto some paper to start the fire? (a) either campers (b) the nearsighted campers (c) the farsighted campersSuppose you are observing a binary star with a telescope and are having difficulty resolving the two stars. Which color filter will better help resolve the stars? (a) blue (b) red (c) neither because colored filters have no effect on resolutionA lens is used to examine an object across a room. Is the lens probably being used as a simple magnifier? Explain in terms of focal length, the image, and magnification.A CCD camera is equipped with a lens with constant focal-length. As the f-number is decreased, determine whether the following quantities increase, decrease, or remain unchanged. Indicate your answers with I, D, or U. (a) the aperture (b) the depth of field (c) the intensity of light reaching the sensor (d) the appropriate exposure timeThe optic nerve and the brain invert the image formed on the retina. Why dont we see everything upside down?Suppose you are observing the interference pattern formed by a Michelson interferometer in a laboratory and a joking colleague holds a lit match in the light path of one arm of the interferometer. Will this match have an effect on the interference patient?If you want to examine the fine detail of an object with a magnifying glass with a power of +20.0 diopters, where should the object be placed so as to observe a magnified image of the object?Compare and contrast the eye and a camera. What parts of the camera correspond to the iris, the retina, and the cornea of the eye?Choose the option from each pair that makes the following statement correct. For a nearsighted person, the [(a) near point; (b) far point] is always located closer than [(c) infinity; (d) 25 cm] from the eye and the corrective lens is [(e) converging; (f) diverging].Choose the option from each pair that makes the following statement correct. For a farsighted person, the [(a) near point; (b) far point] is always located farther than 1 (c) 1 m; (d) 25 cm] from the eye and lens corrective lens is [(e) converging; (f) diverging).Explain why it is theoretically impossible to see an object as small as an atom regardless of the quality of the light microscope being used.Large telescopes are usually reflecting rather than refracting. List some reasons for this choice.A patient has a near point of 1.25 m. Is she nearsighted or farsighted? Should the corrective lens be converging or diverging?A lens with a certain power is used as a simple magnifier. If the power of the lens is doubled, does the angular magnification increase or decrease?Suppose a microscopes resolution is diffraction limited. Which one of the following changes would provide the greatest improvement to its resolution? (a) Observing at a longer wavelength through a smaller aperture. (b) Observing at a shorter wavelength through a larger aperture. (c) Decreasing the object distance. (d) Using a CCD sensor instead of a standard eyepiece.During LASIK eye surgery (laser-assisted in situ keratomileusis), the shape of the cornea is modified by vaporizing some of its material. If the surgery is performed to correct for nearsightedness, how does the cornea need to be reshaped?If you increase the aperture diameter of a camera by a factor of 3, how is the intensity of the light striking the film affected? (a) It increases by a factor of 3. (b) It decreases by a factor of 3. (c) 11 increases by a factor of 9. (d) It decreases by a factor of 9. (e) Increasing the aperture doesnt affect the intensity.A lens has a focal length of 28 cm and a diameter of 4.0 cm. What is the f-number of the lens?A certain camera has f-numbers that range from 1.2 to 22. If the focal length of the lens is 55 mm, what is the range of aperture diameters for the camera?An f/2.80 CCD camera has a 105-mm focal length lens and can focus on objects from infinity to as near as 30.0 cm from the lens, (a) Determine the cameras aperture diameter. Determine the (b) minimum and (c) maximum distances from the CCD sensor over which the lens must be able to travel during focusing. Note: f/2.80 means an f-number of 2.80.A digital camera equipped with an f = 50.0-mm lens uses a CCD sensor of width 8.70 mm and height 14.0 mm. Find the closest distance from the camera to a 1.80-m-tall person if the persons full image is to 6t on the CCD sensor.A camera is being used with a correct exposure at f/4 and a shutter speed of 115 s. In addition to the f-numbers listed in Section 25.1, this camera has f-numbers f/1, f/1.4, and f/2. To photograph a rapidly moving subject, the shutter speed is changed to 1/125 s. Find the new f-number setting needed on this camera to maintain satisfactory exposure.(a) Use conceptual arguments to show that the intensity of light (energy per unit area per unit time) reaching the film in a camera is proportional to the square of the reciprocal of the f-number as I1(f/D)2 (b) The correct exposure time for a camera set to f/1.8 is (1/500) s. Calculate the correct exposure time if the f-number is changed to f/4 under the same lighting conditions. Note: f/4, on a camera, means an f-number of 4.A certain type of film requires an exposure time of 0.010 s with an f/11 lens setting. Another type of film requires twice the light energy to produce the same level of exposure. What f-number does the second type of film need with the 0.010-s exposure time?A certain camera lens has a focal length of 175 mm. Its position can be adjusted to produce images when the lens is between 180. mm and 210. mm from the plane of the film. Over what range of object distances is the lens useful?The near point of a persons eye is 60.0 cm. To see objects clearly at a distance of 25.0 cm, what should be the (a) focal length and (b) power of the appropriate corrective lens? (Neglect the distance from the lens to the eye.)A patient cant see objects closer than 40.0 cm and wishes to clearly see objects that are 20.0 cm from his eye. (a) Is the patient nearsighted or farsighted? (b) If the eye-lens distance is 2.00 cm, what is the minimum object distance p from the lens? (c) What image position with respect to the lens will allow the patient to see the object? (d) Is the image real or virtual? Is the image distance q positive or negative? (e) Calculate the required focal length. (f) Find the power of the lens in diopters. (g) If a contact lens is to be prescribed instead, find p, q, and f and the power of the lens.The accommodation limits for Nearsighted Nicks eyes are 18.0 cm and 80.0 cm. When he wears his glasses, he is able to see faraway objects clearly. At what minimum distance is he able to see objects clearly?12PAn individual is nearsighted; his near point is 18.0 cm and his far point is 50.0 cm. (a) What lens power is needed to correct his nearsightedness? (b) When the lenses are in use, what is this persons near point?A particular nearsighted patient cant see objects clearly beyond 15.0 cm from their eye. Determine (a) the lens power required to correct the patients vision and (b) the type of lens required (converging or diverging). Neglect the distance between the eye and the corrective lens.A particular patients eyes are unable to focus on objects closer than 35.0 cm and corrective lenses are to be prescribed so that the patient can focus on objects 20.0 cm from their eyes. (a) Is the patient nearsighted or farsighted? (b) If contact lenses are to lie prescribed, determine the required lens power. (c) If eyeglasses are to be prescribed instead and the distance between the eyes and the lenses is 2.00 cm, determine the power of the required corrective lenses. (d) Are the required lenses converging or diverging?A patient has a near point of 45.0 cm and far point of 85.0 cm. (a) Can a single lens correct the patients vision? Explain the patients options. (b) Calculate the power lens needed to correct the near point so that the patient can see objects 25.0 cm away. Neglect the eye-lens distance. (c) Calculate the power lens needed to correct the patients far point, again neglecting the eye-lens distance.17P18P19PA person sees clearly wearing eyeglasses that have a power of 4.00 diopters when the lenses are 2.00 cm in front of the eyes. (a) What is the focal length of the lens? (b) Is the person nearsighted or farsighted? (c) If the person wants to switch to contact lenses placed directly on the eyes, what lens power should be prescribed?A stamp collector uses a lens with 7.5-cm focal length as a simple magnifier. The virtual image is produced at the normal near point (25 cm). (a) How far from the lens should the stamp be placed? (b) What is the expected angular magnification?When a drop of water is placed on a flat, clear surface such as a glass slide or plastic sheet, surface tension pulls the top surface into a curved, lens-like shape so that the drop functions as a simple magnifier. Suppose a drop of water has a maximum angular magnification of 3.50. (a) Find the drops focal length. (b) Assuming the bottom surface of the drop is flat, use the lens-makers equation from Topic 23 to calculate the radius of curvature of the top surface.A biology student uses a simple magnifier to examine the structural features of an insects wing. The wing is held 3.50 cm in front of the lens, and the image is formed 25.0 cm from the eye. (a) What is the focal length of the lens? (b) What angular magnification is achieved?A jewelers lens of focal length 5.0 cm is used as a magnifier. With the lens held near the eye, determine (a) the angular magnification when the object is at the focal point of the lens and (b) the angular magnification when the image formed by the lens is at the near point of the eye (25 cm). (c) What is the object distance giving the maximum magnification?A leaf of length h is positioned 71.0 cm in front of a converging lens with a focal length of 39.0 cm. An observer views the image of the leaf from a position 1.26 in behind the lens, as shown in Figure P25.25. (a) What is the magnitude of the lateral magnification (the ratio of the image size to the object size) produced by the lens? (b) What angular magnification is achieved by viewing the image of the leaf rather than viewing the loaf directly? Figure P25.25(a) What is the maximum angular magnification of an eyeglass lens having a focal length of 18.0 cm when used as a simple magnifier? (b) What is the magnification of this lens when the eye is relaxed?The desired overall magnification of a compound microscope is 140X. The objective alone produces a lateral magnification of 12X. Determine the required focal length of the eyepiece.28P29PA microscope has an objective lens with a focal length of 16.22 mm and an eyepiece with a focal length of 9.50 mm. With the length of the barrel set at 29.0 cm, the diameter of a red blood cells image subtends an angle of 1.43 mrad with the eye. It the final image distance is 29.0 cm from the eyepiece, what is the actual diameter of the red blood cell? Hint: To solve this question, go back to basics and use the thin-lens equation.The two lenses of a compound microscope are separated by a distance of 20.0 cm. If the objective lens produces a lateral magnification of 10.0X and the overall magnification is 115X, determine (a) the angular magnification of the eyepiece, (b) the focal length of the eyepiece, and (c) the focal length of the objective lens.32P33P34PSuppose an astronomical telescope is being designed to have an angular magnification of 34.0. If the focal length of the objective lens being used is 86.0 cm, find (a) the required focal length of the eyepiece and (b) the distance between the two lenses for a relaxed eye. Hint: For a relaxed eye, the image formed by the objective lens is at the focal point of the eyepiece.36P37PAn elderly sailor is shipwrecked on a desert Island, but manages to save his eyeglasses. The lens for one eye has a power of +1.20 diopters, and the other lens has a power of +9.00 diopters. (a) What is the magnifying power of the telescope he can construct with these lenses? (b) How far apart are the lenses when the telescope is adjusted for minimum eyestrain?39P40PA converging lens with a diameter of 30.0 cm forms an image of a satellite passing overhead. The satellite has two green lights (wavelength 500. nm) spaced 1.00 m apart. If the lights can just be resolved according to the Rayleigh criterion, what is the altitude of the satellite?42P43P(a) Calculate the limiting angle of resolution for the eye, assuming a pupil diameter of 2.00 mm, a wavelength of 500 nm in air, and an index of refraction for the eye of 1.33. (b) What is the maximum distance from the eye at which two points separated by 1.00 cm could he resolved?45P46P47PA spy satellite circles Earth at an altitude of 200. km and carries out surveillance with a special high-resolution telescopic camera having a lens diameter of 35 cm. If the angular resolution of this camera is limited by diffraction, estimate the separation of two small objects on Earths surface that are just resolved in yellow-green light ( = 550 nm).A diffraction grating has a second-order resolving power of 1 250. (a) Find the number of illuminated lines on the grating. (b) Calculate the smallest difference in wavelengths surrounding 525 nm that can be resolved in the first-order diffraction pattern.50P51PMonochromatic light is beamed into a Michelson interferometer. The movable mirror is displaced 0.382 mm, causing the central spot in the interferometer pattern to change from bright to dark and back to bright N = 1 700 times. (a) Determine the wavelength of the light. What color is it? (b) If monochromatic red light is used instead and the mirror is moved the same distance, would N be larger or smaller? Explain.Light of wavelength 550. nm is used to calibrate a Michelson interferometer. With the use of a micrometer screw, the platform on which one mirror is mounted is moved 0.180 mm. How many fringe shifts are counted?54PAn interferometer is used to measure the length of a bacterium. The wavelength of the light used is 650. nm. As one arm of the interferometer is moved from one end of the cell to the other, 310. fringe shifts are counted. How long is the bacterium?The Michelson interferometer can be used to measure the index of refraction of a gas by placing an evacuated transparent tube in the light path along one arm of the device. Fringe shifts occur as the gas is slowly added to the tube. Assume 600.-nm light is used, the tube is 5.00 cm long, and 160 fringe shifts occur as the pressure of the gas in the tube increases to atmospheric pressure. What is the index of refraction of the gas? Hint: The fringe shifts occur because the wavelength of the light changes inside the gas-filled tube.A thin sheet of transparent material has an index of refraction of 1.40 and is 15.0 m thick. When it is inserted in the light path along one arm of an interferometer, how many fringe shifts occur in the pattern? Assume the wavelength (in a vacuum) of the light used is 600 nm. Hint: The wavelength will change within the material.58AP59APA person with a nearsighted eye has near and far points of 16 cm and 25 cm, respectively. (a) Assuming a lens is placed 2.0 cm from the eye, what power must the lens have to correct this condition? (b) Suppose contact lenses placed directly on the cornea are used to correct the person's eyesight. What is the power of the lens required in this case, and what is the new near point? Hint: The contact lens and the eyeglass lens require slightly different powers because they are at different distances from the eye.An American standard analog television picture (non-HDTV), also known as NTSC, is composed of approximately 485 visible horizontal lines of varying light intensity. Assume your ability to resolve the lines is limited only by the Rayleigh criterion, the pupils of your eyes are 5.00 min in diameter, and the average wavelength of the light coming from the screen is 550. nm. Calculate the ratio of the minimum viewing distance to the vertical dimension of the picture such that you will not be able to resolve the lines.62APThe near point of an eye is 75.0 cm. (a) What should be the power of a corrective lens prescribed to enable the eye to see an object clearly at 25.0 cm? (b) If, using the corrective lens, the person can see an object clearly at 26 0 cm but not at 25.0 cm, by how many diopters did the lens grinder miss the prescription?64APA cataract-impaired lens in an eye may be surgically removed and replaced by a manufactured lens. The focal length required for the new lens is determined by the lens-lo-retina distance, which is measured by a sonar-like device, and by the requirement that the implant provide for correct distance vision. (a) If the distance from lens to retina is 22.4 mm, calculate the power of the implanted lens in diopters. (b) Since there is no accommodation and the implant allows for correct distance vision, a corrective lens for close work or reading must be used. Assume a reading distance of 33.0 cm, and calculate the power of the lens in the reading glasses.A laboratory (astronomical) telescope is used to view a scale that is 300 cm from the objective, which has a focal length of 20.0 cm; the eyepiece has a focal length of 2.00 cm. Calculate the angular magnification when the telescope is adjusted for minimum eyestrain. Note: The object is not at infinity, so the simple expression m = fo/fe is not sufficiently accurate for this problem. Also, assume small angles, so that tan = .67AP26.1QQSuppose youre an astronaut being paid according to the time you spend traveling in space. You take a long voyage traveling at a speed near that of light. Upon your return to Earth, youre asked how youd like to be paid: according to the time elapsed on a clock on Earth or according to your ships clock. To maximize your paycheck, which should you choose? (a) The Earth clock (b) The ship's clock (c) Either clock because it doesnt make a differenceTrue or False: People traveling near the speed of light relative to Earth would measure their lifespans and find them, on the average, longer than the average human lifespan as measured on Earth.You are packing for a trip to another star, and on your journey you will be traveling at a speed of 0.99c. Can you sleep in a smaller cabin than usual, because you will be shorter when you lie down? Explain your answer.You observe a locket moving away from you. (i) Compared with its length when it was at rest on the ground, will you measure its length to be (a) shorter, (b) longer, or (c) the same? Compared to the passage of time measured by the watch on your wrist, is the passage of time on the rocket's c lock (d) faster, (e) slower, or (f) the same? (ii) Answer the same questions from part (i) if the rocket turns around and comes toward you.26.6QQ26.7QQChoose the option from each pair that makes the following statement correct. According to an observer at rest, moving clocks run more [(a) slowly; (b) quickly] than stationary clocks and moving rods are [ (c) longer; (d) shorter] than stationary rods.Choose the option that makes the following statement correct. Two events at a single location define a time interval. The proper time interval tp, is measured by an observer [(a) at rest; (b) moving] relative to the location where the two events occur.Choose the option that makes the following statement correct. An objects proper length is measured by an observer [(a) who measures the length of the moving object as it passes; (b) who is at rest relative to the object.]Choose the option from each pair that makes the following statement correct. Because for an object in relative motion is always [(a) greater, (b) less] than 1, a proper time interval tp is always [(c) longer; (d) shorter] than the time interval t and a proper length Lp is always [(e) longer; (f) shorter] than the length L.A spacecraft with the shape of a sphere of diameter D moves past an observer on Earth with a speed of 0.5c. What shape does the observer measure for the spacecraft as it moves past?What two speed measurements will two observers in relative motion always agree upon?The speed of light in water is 2.30 108 m/s. Suppose an electron is moving through water at 2.50 108 m/s. Does this particle speed violate the principle of relativity?With regard to reference frames, how does general relativity differ from special relativity?Give a physical argument that shows it is impossible to accelerate an object of mass m to the speed of light, even with a continuous force acting on it.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 a speed near that of light? How would you answer this question?List some ways our day-to-day lives would change if the speed of light were only 50 m/s.Two identically constructed clocks are synchronized. One is put into orbit around Earth, and the other remains on Earth. (a) Which clock runs more slowly? (b) When the moving clock returns to Earth, will the two clocks still be synchronized? Discuss from the standpoints of both special and general relativity.13CQImagine an astronaut on a trip to Sirius, which lies 8 light-years from Earth. Upon arrival at Sirius, the astronaut finds that the trip lasted 6 years. If the trip was made at a constant speed of 0.8c, how can the 8-light-year distance be reconciled with the 6-year duration?Explain why, when defining the length of a rod, it is necessary to specify that the positions of the ends of the rod are to be measured simultaneously.16CQThe control panel on a spaceship contains a light that blinks every 2.00 s as observed by an astronaut in the ship. If the spaceship is moving past Earth with a speed of 0.750c, determine (a) the proper time interval between blinks and (b) the time interval between blinks as observed by a person on Earth.A spaceship moves past Earth with a speed of 0.900c. As it is passing, a person on Earth measures the spaceship's length to be 75.0 m. (a) Determine the spaceships proper length. (b) Determine the time required for the spaceship to pass a point on Earth as measured by a person on Earth and (c) by an astronaut onboard the spaceship.If astronauts could travel at v = 0.950c, we on Earth would say it takes (4.20/0.950) = 4.42 years to reach Alpha Centauri, 4.20 light-years away. The astronauts disagree. (a) How much time passes on the astronauts clocks? (b) What is the distance to Alpha Centauri as measured by the astronauts?a meterstick moving at 0.900c relative to the Earths surface approaches an observer at rest with respect to the Earths surface. (a) What is the metersticks length as measured by the observer? (b) Qualitatively, how would the answer to part (a) change if the observer started running toward the meterstick?The length of a moving spaceship is 28.0 m according to an astronaut on the spaceship. If the spaceship is contracted by 15.0 cm according to an Earth observer, what is the speed of the spaceship?An astronaut at rest on Earth has a heart rate of 70. beats/min. When the astronaut is traveling in a spaceship at 0.90c, what will this rate be as measured by (a) an observer also in the ship and (b) an observer at rest on Earth?The average lifetime of a pi meson in its own frame of reference (i.e., the proper lifetime) is 2.6 108 s. If the meson moves with a speed of 0.98c, 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? (c) What distance would it travel if time dilation did not occur?An astronaut is traveling in a space vehicle that has a speed of 0.500c relative to Earth. The astronaut measures his pulse rate at 75.0 beats per minute. Signals generated by the astronauts pulse are radioed to Earth when the vehicle is moving perpendicular to a line that connects the vehicle with an Earth observer. (a) What pulse rate does the Earth observer measure? (b) What would be the pulse rate if the speed of the space vehicle were increased to 0.990c?A muon formed high in Earth's atmosphere travels toward Earth at a speed v = 0.990c for a distance of 4.60 km as measured by an observer at rest with respect to Earth. It then decays into an electron, a neutrino, and an antineutrino. (a) How long does the muon survive according to an observer at rest on Earth? (b) Compute the gamma factor associated with the muon. (c) How much time passes according to an observer traveling with the muon? (d) What distance does the muon travel according to an observer traveling with the muon? (e) A third observer traveling toward the muon at c/2 measures the lifetime of the particle. According to this observer, is the muons lifetime shorter or longer than the lifetime measured by the observer at rest with respect to Earth? Explain.A star is 15.0 light-years (ly) from Earth. (a) At what constant speed must a spacecraft travel on its journey to the star so that the Earth-star distance measured by an astronaut onboard the spacecraft is 3.00 ly? (b) What is the journeys travel time in years as measured by a person on Earth and (c) by the astronaut?The 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 has a speed of 0.350c with respect to Earth, determine the speed of the faster spaceship.A car traveling at 35.0 m/s takes 26.0 minutes to travel a certain distance according to the drivers clock in the car. How long does the trip take according to an observer at rest on Earth? Hint: The following approximation is helpful: [1x]121+12xforx1.A supertrain of proper length 1.00 102 m travels at a speed of 0.95c as it passes through a tunnel having proper length 50.0 m. As seen by a trackside observer, is the train ever completely within the tunnel? If so, by how much?A box is cubical with sides of proper lengths L1 = L2 = L3, as shown in Figure P26.14, when viewed in its own rest frame. If this block moves parallel to one of its edges with a speed of 0.80c past an observer, (a) what shape does it appear to have to this observer? (b) What is the length of each side as measured by the observer? Figure P26.1415P16P17P18PAn unstable particle at rest breaks up into two fragments of unequal mass. The mass of the lighter fragment is equal to 2.50 1028 kg and that of the heavier fragment is 1.67 1027 kg. If the lighter fragment has a speed of 0.893c after the breakup, what is the speed of the heavier fragment?Spaceship R is moving to the right at a speed of 0.70c with respect to Earth. A second spaceship, L, moves to the left at the same speed with respect to Earth. What is the speed of L with respect to R?An electron moves to the right with a speed of 0.90c relative to the laboratory frame. A proton moves to the left with a speed of 0.70c relative to the electron. Find the speed of the proton relative to the laboratory frame.A spaceship travels at 0.750c relative to Earth. If the spaceship fires a small rocket in the forward direction, how fast (relative to the ship) must it be fired for it to travel at 0.950c relative to Earth?A spaceship is moving away from Earth at 0.900c when it fires a small rocket in the forward direction at 0.500c relative to the spaceship. Calculate the rockets speed relative to Earth.Two identical spaceships with proper lengths of 175 m are launched from Earth. Spaceship A is launched in one direction at 0.500c and spaceship B is launched in the opposite direction at 0.750c. (a) What is the speed of spaceship B relative to spaceship A? (b) What is the length of spaceship A as measured by astronauts on spaceship B?Spaceship A moves away from Earth at a speed of 0.800c (Fig. P26.25). Spaceship B pursues at a speed of 0.900c relative to Earth. Observers on Earth see B overtaking A at a relative speed of 0.100c. With what speed is B overtaking A as seen by the crew of spaceship B? Figure P26.25A pulsar is a stellar object that emits light in short bursts. Suppose a pulsar with a speed of 0.950c approaches Earth, and a rocket with a speed of 0.995c heads toward the pulsar. (Both speeds are measured in Earth's frame of reference.) If the pulsar emits 10.0 pulses per second in its own frame of reference, at what rate are the pulses emitted in the rockets frame of reference?A rocket moves with a velocity of 0.92c to the right with respect to a stationary observer A. An observer B moving relative to observer A finds that the rocket is moving with a velocity of 0.95c to the left. What is the velocity of observer B relative to observer A? (Hint: Consider observer Bs velocity in the frame of reference of the rocket.)A proton moves with a speed of 0.950c. Calculate (a) its rest energy, (b) its total energy, and (c) its kinetic energy.Protons in an accelerator at the Fermi National Laboratory near Chicago are accelerated to a total energy that is 400 times their rest energy. (a) What is the speed of these protons in terms of c? (b) What is their kinetic energy in MeV?30P31P32PA chain of nuclear reactions in the Suns core converts four protons into a helium nucleus. (a) What is the mass difference between four protons and a helium nucleus? (b) How much energy in MeV is released during the conversion of four protons into a helium nucleus?An unstable particle with a mass equal to 3.34 1027 kg is initially at rest. The particle decay's into two fragments that fly off with velocities of 0.987c and 0.868c, respectively. Find the masses of the fragments. Hint: Conserve both mass-energy and momentum.35P36P37AP38AP39APA spring of force constant k is compressed by a distance x from its equilibrium length. (a) Does the mass of the spring change when the spring is compressed? Explain. (b) Find an expression for the change in mass of the spring in terms of k, x, and c. (c) What is the change in mass if the force constant is 2.0 102 N/m and x = 15 cm?A star is 5.00 ly from the Earth. At what speed must a space-craft travel on its journey to the star such that the Earth-star distance measured in the frame of the spacecraft is 2.00 ly?An electron has a total energy equal to five times its rest energy. (a) What is its momentum? (b) Repeat for a proton.An astronaut wishes to visit the Andromeda galaxy, making a one-way trip that will take 30.0 years in the space-ships frame of reference. Assume the galaxy is 2.00 million light-years away and his speed is constant. (a) How fast must he travel relative to Earth? (b) What will be the kinetic energy of his spacecraft, which has mass of 1.00 106 kg? (c) What is the cost of this energy if it is purchased at a typical consumer price for electric energy, 13.0 cents per kWh? The following approximation will prove useful: 11+x1x2forx1An alarm clock is set to sound in 10.0 h. At t = 0, the clock is placed in a spaceship moving with a speed of 0.75c (relative to Earth). What distance, as determined by an Earth observer, does the spaceship travel before the alarm clock sounds?Owen and Dina are at rest in frame S, which is moving with a speed of 0.600c with respect to frame S. They play a game of catch while Ed, at rest in frame S, watches the action (Fig. P26.45). Owen throws the ball to Dina with a speed of 0.800c (according to Owen) and their separation (measured in S) is equal to 1.80 1012 m. (a) According to Dina, how fast is the ball moving? (b) According to Dina, what time interval is required for the ball to reach her? According to Ed, (c) how far apart are Owen and Dina, and (d) how fast is the ball moving? Figure. P26.45An observer in a coasting spacecraft moves toward a mirror at speed v relative to the reference frame labeled by S in Figure P26.46. 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 spacecraft. The spacecraft is a distance d from the mirror (as measured by observers in S) at the moment the light pulse leaves the spacecraft. What is the total travel time of the pulse as measured by observers in (a) the S frame and (b) the spacecraft? Figure P26.46A spaceship of proper length 300. m takes 0.75 s to pass an Earth observer. Determine the speed of this spaceship as measured by the Earth observer.The cosmic rays of highest energy are protons that have kinetic energy on the order of 1013 MeV. (a) From the point of view of the proton, how many kilometers across is the galaxy? (b) How long would it take a proton of this energy to travel across the Milky Way galaxy, having a diameter 105 light-years, as measured in the protons frame?49AP50APThe muon is an unstable particle that spontaneously decays into an electron and two neutrinos. In a reference frame in which the muons are stationary, if the number of muons at t = 0 is N0, the number at time t is given by N0 = N0et/, where is the mean lifetime, equal to 2.2 s. 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?52APThe identical twins Speedo and Goslo join a migration from Earth to Planet X, which is 20.0 light-years 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. Speedos craft travels steadily at 0.950c, Goslos at 0.750c. Calculate the age difference between the twins after Goslos spacecraft lands on Planet X. Which twin is the older?An interstellar space probe is launched from Earth. After a brief period of acceleration, it moves with a constant velocity, 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 years as measured in a rest frame. (a) How long do the batteries on the space probe last as measured by mission control on Earth? (b) How far is the probe from Earth when its batteries fail as measured by mission control? (c) How far is the probe from Earth as measured by its built-in trip odometer when its batteries fail? (d) For what total time after launch are data received from the probe by mission control? Note dial radio waves travel at the speed of light and fill the space between the probe and Earth at the time the battery fails.An observer moving at a speed of 0.995c relative to a rod (Fig. P26.55) measures its length to be 2.00 m and sees its length to be oriented at 30.0 with respect to its direction of motion. (a) What is the proper length of the rod? (b) What is the orientation angle in a reference frame moving with the rod? Figure P26.55An alien spaceship traveling 0.600c toward Earth launches a landing craft with an advance guard of purchasing agents. The lander travels in the same direction with a velocity 0.800c relative to the spaceship. As observed on Earth, the spaceship is 0.200 light-years from Earth when the lander is launched. (a) With what velocity is the lander observed to be approaching by observers on Earth? (b) What is the distance to Earth at the time of landcr launch, as observed by the aliens on the mother ship? (c) How long does it take the lander to reach Earth as observed by the aliens on the mother ship? (d) If the lander has a mass of 4.00 105 kg, what is its kinetic energy as observed in Earths reference frame?27.1QQ27.2QQ27.3QQ27.4QQ27.5QQ1CQ2CQ3CQ4CQ5CQ6CQ7CQ8CQ9CQ10CQ11CQ12CQ13CQ14CQ15CQ16CQ1P2P3P4P5P6P7P8P9P10P11P12P13P14P15P16P17P18P19P20P21P22P23P24P25P26P27P28P29P30P31P32P33P34P35P36P37P38P39P40P41AP42AP43AP44AP45AP46AP47AP48AP49AP50AP51AP52AP28.1QQ28.2QQ28.3QQ1CQ2CQ3CQ4CQ5CQ6CQ7CQ8CQ9CQ10CQ11CQ12CQ13CQ14CQ15CQ1P2P3P4P5P6P7P8P9P10P11P12P13P14P15P16P17P18P19P