Answer: Another approach to velocity transformations. In

A rod is to move at constant speed v along the x axis of reference frame S, with the rod's length parallel to that axis. An observer in frame S is to measure the length L of the rod. Which of the curves in Fig. 37-15 best gives length L (vertical axis of the graph) versus speed parameter f3?

Figure 37-16 shows a ship (attached to reference frame S') pass- 0 ing us (standing in reference frame S). A proton is fired at nearly the speed of light along the length of the ship, from the front to the rear. (a) Is 0.2 0.4 f3 0.6 Fig. 37-15 Questions 1 and 3. 0.8 the spatial separation !lX' between the point at which the proton is fired and the point at which it hits the ship's rear wall a positive or negative quantity? (b) Is the temporal separation M' between those events a positive or negative quantity?

Reference frame S' is to pass reference frame S at speed v along the common direction of the x' and x axes, as in Fig. 37-9.An observer who rides along with frame S' is to count off 25 s on his wristwatch. The corresponding time interval !It is to be measured by an observer in frame S. Which of the curves in Fig. 37-15 best gives M (vertical axis of the graph) versus speed parameter f3?

Figure 37-17 shows two clocks in stationary frame S' (they are synchronized in that frame) and one clock in moving frame S. Clocks C, and C; read zero S when they pass each other. When clocks C, and C2 pass each other, (a) which clock has the smaller reading and (b) which clock measures a proper time?

Figure 37-18 shows two clocks in stationary frame S (they are synchronized in that frame) and one clock in moving frame S'. Clocks C1 and C; read zero when they pass each other. When clocks C; and C2 pass each S other, (a) which clock has the smaller reading and (b) which clock measures a proper time?

Sam leaves Venus in a spaceship headed to Mars and passes Sally, who is on Earth, with a relative speed of O.5e. (a) Each measures the Venus-Mars voyage time. Who measures a Fig. 37-18 Question 5. proper time: Sam, Sally, or neither? (b) On the way, Sam sends a pulse of light to Mars. Each measures the travel time of the pulse. Who measures a proper time: Sam, Sally, or neither?

The plane of clocks and measuring rods in Fig. 37-19 is like that in Fig. 37-3. The clocks along the x axis are separated (center to cenQUESTIONS 1049 ter) by 1 light-second, as are the )' clocks along the y axis, and all the B~i'J=i~H:R clocks are synchronized via the procedure described in Section 37- 3. When the initial synchronizing signal of t = 0 from the origin reaches (a) clock A, (b) clock B, and (c) clock C, what initial time is then set on those clocks? An event z occurs at clock A when it reads 10 Fig.37-19 Question 7. s. (d) How long does the signal of that event take to travel to an observer stationed at the origin? (e) What time does that observer assign to the event?

The rest energy and total energy, respectively, of three particles, expressed in terms of a basic amount A are (1) A, 2A; (2) A, 3A; (3) 3A, 4A. Without written calculation, rank the particles according to their (a) mass, (b) kinetic energy, (c) Lorentz factor, and (d) speed, greatest first.

Figure 37-20 shows the triangle of Fig 37-14 for six particles; the slanted lines 2 and 4 have the same length. Rank the particles according to (a) mass, (b) momentum magnitUde, and (c) Lorentz factor, greatest first. (d) Identify which two particles have the same total energy. (e) Rank the three 3 4 Fig. 37-20 Question 9. lowest-mass particles according to kinetic energy, greatest first.

While on board a starship, you intercept signals from four shuttle craft that are moving either directly toward or directly away from you. The signals have the same proper frequency fo. The speed and direction (both relative to you) of the shuttle craft are (a) 0.3e toward, (b) 0.6e toward, (c) 0.3e away, and (d) 0.6e away. Rank the shuttle craft according to the frequency you receive, greatest first.

Figure 37-21 shows one of four star cruisers that are in a race. As each cruiser passes the starting line, a shuttle craft leaves the cruiser and races toward the finish line. You, judging the race, are stationary relative to the starting and finish lines. The speeds Vc of the cruisers relative to you and the speeds Vs of the shuttle craft relative to their respective starships are, in that order, (1) 0.70e, OAOe; (2) OAOe, 0.70e; (3) 0.20e, 0.90e; (4) 0.50e, 0.60e. (a) Rank the shuttle craft according to their speeds relative to you, greatest first. (b) Rank the shuttle craft according to the distances their pilots measure from the starting line to the finish line, greatest first. (c) Each starship sends a signal to its shuttle craft at a certain frequency fo as measured on board the starship. Rank the shuttle craft according to the frequencies they detect, greatest first.

The length of a spaceship is measured to be exactly half its rest length. (a) To three significant figures, what is the speed parameter (3 of the spaceship relative to the observer's frame? (b) By what factor do the spaceship's clocks run slow relative to clocks in the observer's frame?

A space traveler takes off from Earth and moves at speed 0.9900e toward the star Vega, which is 26.00 ly distant. How much time will have elapsed by Earth clocks (a) when the traveler reaches Vega and (b) when Earth observers receive word from the traveler that she has arrived? (c) How much older will Earth observers calculate the traveler to be (measured from her frame) when she reaches Vega than she was when she started the trip?

A rod is to move at constant speed v along the x axis of reference frame S, with the rod's length parallel to that axis. An observer in frame S is to measure the length L of the rod. Figure 37-23 gives length L versus speed parameter (3 for a range of values for (3. The vertical axis scale is set by La = 1.00 m. What is L if v = 0.95e?

The center of our Milky Way galaxy is about 23 000 Iy away. (a) To eight significant figures, at what constant speed parameter would you need to travel exactly 23 000 Iy (measured in the Galaxy frame) in exactly 30 y (measured in your frame)? (b) Measured in your frame and in light-years, what length of the Galaxy would pass by you during the trip?

Observer S reports that an event occurred on the x axis of his reference frame at x = 3.00 X 108 m at time t = 2.50 s. Observer S' and her frame are moving in the positive direction of the x axis at a speed of 0.400c. Further, x = x' = 0 at t = t' = O. What are the (a) spatial and (b) temporal coordinate of the event according to S'? If S' were, instead, moving in the negative direction of the x axis, what would be the (c) spatial and (d) temporal coordinate of the event according to S'?

In Fig. 37-9, the origins of the two frames coincide at t = t' = 0 and the relative speed is 0.950c. Two micrometeorites collide at coordinates x = 100 km and t = 200 J.LS according to an observer in frame S. What are the (a) spatial and (b) temporal coordinate of the collision according to an observer in frame S'?

Inertial frame S' moves at a speed of 0.60c with respect to frame S (Fig. 37-9). Further, x = x' = 0 at t = t' = 0.1\\10 events are recorded. In frame S, event 1 occurs at the origin at t = 0 and event 2 occurs on the x axis at x = 3.0 km at t = 4.0 J.LS. According to observer S', what is the time of (a) event 1 and (b) event 2? (c) Do the two observers see the two events in the same sequence or the reverse sequence?

An experimenter arranges to trigger two flashbulbs simultaneously, producing a big flash located at the origin of his reference frame and a small flash at x = 30.0 km. An observer moving at a speed of 0.250c in the positive direction of x also views the flashes. (a) What is the time interval between them according to her? (b) Which flash does she say occurs first?

As in Fig. 37-9, reference frame S' passes reference frame S with a certain velocity. Events 1 and 2 are to have a certain temporal separation t:J.t' according to the S' observer. However, their spatial separation /lx' according to that observer has not been set yet. Figure 37-24 gives their temporal separation t:J.t according to the S observer as a function of /lx ' for a range of /lx' values. The vertical axis scale is set by Ma = 6.00 J.LS. What is M'?

Relativistic reversal of events. Figures 37-25a and b show the (usual) situation in which a primed reference frame passes an unprimed reference frame, in the common positive direction of the x PROBLEMS 1051 and x' axes, at a constant relative velocity of magnitude v. We are at rest in the unprimed frame; Bullwinkle, an astute student of relativity in spite of his cartoon upbringing, is at rest in the primed frame. The figures also indicate events A and B that occur at the following spacetime coordinates as measured in our unprimed frame and in Bullwinkle's primed frame: Event Unprimed Primed A B (xA, tA) (x~, t~) In our frame, event A occurs before event B, with temporal separation t:J.t = tB - tA = 1.00 J.LS and spatial separation Llx = XB - XA = 400 m. Let t:J.t' be the temporal separation of the events according to Bullwinkle. (a) Find an expression for t:J.t' in terms of the speed parameter f3 (= vie) and the given data. Graph t:J.t' versus f3 for the following two ranges of f3: (b) 0 to 0.01 (v is low, from 0 to O.Ole) (c) 0.1 to 1 (v is high, from O.le to the limit c) (d) At what value of f3 is t:J.t' = O? For what range of f3 is the sequence of events A and B according to Bullwinkle (e) the same as ours and (f) the reverse of ours? (g) Can event A cause event B, or vice versa? Explain.

For the passing reference frames in Fig. 37-25, events A and B occur at the following spacetime coordinates: according to the unprimed frame, (XA, tA) and (xn, tn); according to the primed frame, (xA, tA) and (x~, t~). In the unprimed frame, t:J.t = tn - tA = 1.00 J.LS and /lx = Xn - XA = 400 m. (a) Find an expression for /lx' in terms of the speed parameter f3 and the given data. Graph /lx' versus f3 for two ranges of f3: (b) 0 to 0.01 and (c) 0.1 to 1. (d) At what value of f3is /lx ' minimum, and (e) what is that minimum?

A clock moves along an x axis at a speed of 0.600e and reads zero as it passes the origin. (a) Calculate the clock's Lorentz factor. (b) What time does the clock read as it passes x = 180 m?

Bullwinkle in reference frame S' passes you in reference frame S along the common direction of the x' and x axes, as in Fig. 37-9. He carries three meter sticks: meter stick 1 is parallel to the x' axis, meter stick 2 is parallel to the y' axis, and meter stick 3 is parallel to the z' axis. On his wristwatch he counts off 15.0 s, which takes 30.0 s according to you. 1\\10 events occur during his passage. According to you, event 1 occurs at Xj = 33.0 m and tj = 22.0 ns, and event 2 occurs at X2 = 53.0 m and t2 = 62.0 ns. According to your measurements, what is the length of (a) meter stick 1, (b) meter stick 2, and (c) meter stick 3? According to Bullwinkle, what are (d) the spatial separation and (e) the temporal separation between events 1 and 2, and (f) which event occurs first?

In Fig. 37-9, observer S detects two flashes of light. A big flash occurs at Xl = 1200 m and, 5.00 fLs later, a small flash occurs at X2 = 4S0 m. As detected by observer S', the two flashes occur at a single coordinate x'. (a) What is the speed parameter of S', and (b) is S' moving in the positive or negative direction of the x axis? To S', (c) which flash occurs first and (d) what is the time interval between the flashes?

In Fig. 37-9, observer S detects two flashes of light. A big flash occurs at Xl = 1200 m and, slightly later, a small flash occurs at X2 = 4S0 m. The time interval between the flashes is t:.t = f2 - fl' What is the smallest value of t:.t for which observer S' will determine that the two flashes occur at the same x' coordinate?

A particle moves along the x' axis of frame S' with velocity 0.40e. Frame S' moves with velocity 0.60e with respect to frame S. What is the velocity of the particle with respect to frame S?

In Fig. 37-11, frame S' moves relative to frame S with velocity 0.62ci while a particle moves parallel to the common x and x' axes. An observer attached to frame S' measures the particle's velocity to be 0.47ei. In terms of e, what is the particle's velocity as measured by an observer attached to frame S according to the (a) relativistic and (b) classical velocity transformation? Suppose, instead, that the S' measure of the particle's velocity is -0.47ei. What velocity does the observer in Snow measure according to the (c) relativistic and (d) classical velocity transformation?

Galaxy A is reported to be receding from us with a speed of 0.35e. Galaxy B, located in precisely the opposite direction, is also found to be receding from us at this same speed. What mUltiple of e gives the recessional speed an observer on Galaxy A would find for (a) our galaxy and (b) Galaxy B?

Stellar system Ql moves away from us at a speed of O.SOOe. Stellar system Q2> which lies in the same direction in space but is closer to us, moves away from us at speed 0.400e. What multiple of e gives the speed of Q2 as measured by an observer in the reference frame OfQl?

A spaceship whose rest length is 350 m has a speed of 0.S2e with respect to a certain reference frame. A micrometeorite, also with a speed of 0.S2e in this frame, passes the spaceship on an anti parallel track. How long does it take this object to pass the ship as measured on the ship?

In Fig. 37-26a, particle P is to move parallel to the x and x' axes of reference frames Sand S', at a certain velocity relative to frame S. Frame S' is to move parallel to the x axis of frame S at velocity v. Figure 37-26b gives the velocity u' of the particle relative to frame S' for a range of values for v. The vertical axis scale is set by u~ = O.SOOe. What value will u' have if (a) v = 0.90e and (b) v ~ e?

frame S. Frame S' is to move parallel to the x axis of frame S at velocity v. Figure 37-26b gives the velocity u' of the particle relative to frame S' for a range of values for v. The vertical axis scale is set by u~ = O.SOOe. What value will u' have if (a) v = 0.90e and (b) v ~ e?

A sodium light source moves in a horizontal circle at a constant speed of 0.100e while emitting light at the proper wavelength of Ao = 5S9.00 nm. Wavelength A is measured for that light by a detector fixed at the center of the circle. What is the wavelength shift A - Ao?

A spaceship, moving away from Earth at a speed of 0.900e, reports back by transmitting at a frequency (measured in the spaceship frame) of 100 MHz. To what frequency must Earth receivers be tuned to receive the report?

Certain wavelengths in the light from a galaxy in the constellation Virgo are observed to be 0.4% longer than the corresponding light from Earth sources. (a) What is the radial speed of this galaxy with respect to Earth? (b) Is the galaxy approaching or receding from Earth?

Assuming that Eq. 37-36 holds, find how fast you would have to go through a red light to have it appear green. Take 620 nm as the wavelength of red light and 540 nm as the wavelength of green light.

Figure 37-27 is a graph of intensity versus wavelength for light reaching Earth from galaxy NGC 7319, which is about 3 X 108 light-years away. The most intense light is emitted by the oxygen in NGC 7319. In a laboratory that emission is at wavelength A = 513 nm, but in the light from NGC 7319 it has been shifted to 525 nm due to the Doppler effect (all the emissions from NGC 7319 have been shifted). (a) What is the radial speed of NGC 7319 relative to Earth? (b) Is the relative motion toward or away from our planet?

A spaceship is moving away from Earth at speed 0.20e. A source on the rear of the ship emits light at wavelength 450 nm according to someone on the ship. What (a) wavelength and (b) color (blue, green, yellow, or red) are detected by someone on Earth watching the ship?

How much work must be done to increase the speed of an electron from rest to (a) 0.500e, (b) 0.990e, and (c) 0.9990e?

The mass of an electron is 9.109 38188 X 10-31 kg. To six significant figures, find (a) y and (b) (3 for an electron with kinetic energy K = 100.000 Me V.

What is the minimum energy that is required to break a nucleus of 12C (of mass 11.996 71 u) into three nuclei of 4He (of mass 4.00151 u each)?

How much work must be done to increase the speed of an electron (a) from 0.18e to 0.1ge and (b) from 0.98e to 0.9ge? Note that the speed increase is O.Ole in both cases.

In the reaction p + 19F ~ a + 160, the masses are m(p) = 1.007825 u, m(F) = 18.998405 u, m(a) = 4.002603 u, m(O) = 15.994915 u. Calculate the Q of the reaction from these data.

In a high-energy collision between a cosmic-ray particle and a particle near the top of Earth's atmosphere, 120 km above sea level, a pion is created. The pion has a total energy E of 1.35 X 105 MeV and is traveling vertically downward. In the pion's rest frame, the pion decays 35.0 ns after its creation. At what altitude above sea level, as measured from Earth's reference frame, does the decay occur? The rest energy of a pion is 139.6 MeV.

(a) If m is a particle's mass, p is its momentum magnitude, and K is its kinetic energy, show that (pc? - K2 m = 2Ke2 (b) For low particle speeds, show that the right side of the equation reduces to m. (c) If a particle has K = 55.0 MeV when p = 121 Me VIc, what is the ratio mIme of its mass to the electron mass?

A 5.00-grain aspirin tablet has a mass of 320 mg. For how many kilometers would the energy equivalent of this mass power an automobile? Assume 12.75 km/L and a heat of combustion of 3.65 X 107 JIL for the gasoline used in the automobile.

The mass of a muon is 207 times the electron mass; the average lifetime of muons at rest is 2.20 p,s. In a certain experiment, muons moving through a laboratory are measured to have an average lifetime of 6.90 p,s. For the moving muons, what are (a) (3, (b) K, and (c) p (in MeV/e)?

As you read this page (on paper or monitor screen), a cosmic ray proton passes along the left-right width of the page with relative speed v and a total energy of 14.24 nl According to your measurements, that left-right width is 21.0 cm. (a) What is the width according to the proton's reference frame? How much time did the passage take according to (b) your frame and (c) the proton's frame?

To four significant figures, find the following when the kinetic energy is 10.00 MeV: (a) y and (b) (3 for an electron (Eo = 0.510 998 MeV), (c) yand (d) (3 for a proton (Eo = 938.272 MeV), and (e) yand (f) (3 for an a particle (Eo = 3727.40 MeV).

What must be the momentum of a particle with mass m so that the total energy of the particle is 3.00 times its rest energy?

Apply the binomial theorem (Appendix E) to the last part of Eq. 37-52 for the kinetic energy of a particle. (a) Retain the first two terms of the expansion to show the kinetic energy in the form K = (first term) + (second term). The first term is the classical expression for kinetic energy. The second term is the first-order correction to the classical expression. Assume the particle is an electron. If its speed v is el20, what is the value of (b) the classical expression and (c) the first -order correction? If the electron's speed is 0.80e, what is the value of (d) the classical expression and (e) the first-order correction? (f) At what speed parameter (3 does the first-order correction become 10% or greater of the classical expression?

In Section 28-6, we showed that a particle of charge q and mass m will move in a circle of radius r = mvl'.!iIB when its velocity v is perpendicular to a uniform magnetic field B. We also found that the period T of the motion is independent of speed v. These two results are approximately correct if v ; e. For relativistic speeds, we must use the correct equation for the radius: p ymv r = IqlB = IqIB' (a) Using this equation and the definition of period (T = 21T1'lv), find the correct expression for the period. (b) Is T independent of v? If a 10.0 MeV electron moves in a circular path in a uniform magnetic field of magnitude 2.20 T, what are (c) the radius according to Chapter 28, (d) the correct radius, (e) the period according to Chapter 28, and (f) the correct period?

What is (3 for a particle with (a) K = 2.00Eo and (b) E = 2.00Eo?

A certain particle of mass m has momentum of magnitude me. What are (a) (3, (b) y, and (c) the ratio KIEo?

(a) The energy released in the explosion of 1.00 mol of TNT is 3.40 Ml The molar mass of TNT is 0.227 kg/mol. What weight of TNT is needed for an explosive release of 1.80 X 1014 J? (b) Can you carry that weight in a backpack, or is a truck or train required? (c) Suppose that in an explosion of a fission bomb, 0.080% of the fissionable mass is converted to released energy. What weight of fissionable material is needed for an explosive release of 1.80 X 1014 J? (d) Can you carry that weight in a backpack, or is a truck or train required?

Quasars are thought to be the nuclei of active galaxies in the early stages of their formation. A typical quasar radiates energy at the rate of 1041 W. At what rate is the mass of this quasar being reduced to supply this energy? Express your answer in solar mass units per year, where one solar mass unit (1 smu = 2.0 X 1030 kg) is the mass of our Sun.

The mass of an electron is 9.109 38188 X 10-31 kg. To eight significant figures, find the following for the given electron kinetic energy: (a) y and (b) (3 for K = 1.000 000 0 ke V, (c) y and (d) (3 for K = 1.0000000 MeV, and then (e) yand (f) (3 for K = 1.000 000 0 Ge V.

An alpha particle with kinetic energy 7.70 MeV collides with an 14N nucleus at rest, and the two transform into an 170 nucleus and a proton. The proton is emitted at 90 to the direction of the incident alpha particle and has a kinetic energy of 4.44 MeV. The masses of the various particles are alpha particle, 4.00260 u; 14N, 14.00307 u; proton, 1.007825 u; and 170,16.99914 u.

Temporal separation between two events. Events A and B occur with the following spacetime coordinates in the reference frames of Fig. 37-25: according to the unprimed frame, (XA' tA) and (XB' tB); according to the primed frame, (xA' tA) and (x~, t~). In the unprimed frame, I::J.t = tB - tA = 1.00}J.S and LU = XB - XA = 240 ill. (a) Find an expression for I:!.t' in terms of the speed parameter f3 and the given data. Graph 1::J.t' versus f3 for the following two ranges of f3: (b) 0 to 0.01 and (c) 0.1 to 1. (d) At what value of f3 is M minimum and (e) what is that minimum? (f) Can one of these events cause the other? Explain.

Spatial separation between two events. For the passing reference frames of Fig. 37-25, events A and B occur with the following spacetime coordinates: according to the unprimed frame, (XA' tA) and (XB' tB); according to the primed frame, (xA' tA) and (x~, t~). In the unprimed frame, I::J.t = tB - tA = 1.00 f-LS and I:!.x = XB - XA = 240 m. (a) Find an expression for I:!.x' in terms of the speed parameter f3 and the given data. Graph I:!.x' versus f3 for two ranges of f3: (b) 0 to 0.01 and (c) 0.1 to 1. (d) At what value of f3 is I:!.x' = O?

In Fig. 37-28a, particle P is to move parallel to the x and x' axes of reference frames Sand S', at a certain velocity relative to frame S. Frame S' is to move parallel to the x axis of frame S at velocity v. Figure 37-28b gives the velocity u' of the particle relative to frame S' for a range of values for v. The vertical axis scale is set by u~ = -0.800c. What value will u' have if (a) v = 0.80c and (b) v ~ c?

Superluminal jets. Figure 37-29a shows the path taken by a knot in a jet of ionized gas that has been expelled from a galaxy. The knot travels at constant velocity v at angle e from the direction of Earth. The knot occasionally emits a burst of light, which is eventually detected on Earth. Tho bursts are indicated in Fig. 37-29a, separated by time t as measured in a stationary frame near the bursts. The bursts are shown in Fig. 37-29b as if they were photographed on the same piece of film, first when light from burst 1 arrived on Earth and then later when light from burst 2 arrived. The apparent distance Dapp traveled by the knot between the two bursts is the distance across an Earth-observer's view of the knot's path. The apparent time Tapp between the bursts is the difference in the arrival times of the light from them. The apparent speed of the knot is then Vapp = DapplTapp. In terms of v, t, and e, what are (a) Dapp and (b) Tapp? (c) Evaluate Vapp for v = 0.980c and e = 30.0. When superluminal (faster than light) jets were first observed, they seemed to defy special relativity-at least until the correct geometry (Fig. 37-29a) was understood.

When superluminal (faster than light) jets were first observed, they seemed to defy special relativity-at least until the correct geometry (Fig. 37-29a) was understood.

Another approach to velocity transformations. In Fig. 37-31, reference frames Band C move past reference frame A in the common direction of their x axes. Represent the x components of the velocities of one frame relative to another with a two-letter subscript. For example, v AB is the x component of the velocity of A relative to B. Similarly, represent the corresponding speed parameters with two-letter subscripts. For example, f3AB (= v AB/C) is the speed parameter corresponding to VAB. (a) Show that f3AC = f3AB + f3BC . 1 + f3ABf3BC Let MAB represent the ratio (1 - f3AB)/(l + f3AB)' and let M Bc and MAC represent similar ratios. (b) Show that the relation MAc = MABMBc is true by deriving the equation of part (a) from it.

Continuation of Problem 65. Use the result of part (b) in Problem 65 for the motion along a single axis in the following situation. Frame A in Fig. 37-31 is attached to a particle that moves with velocity +0.500c past frame B, which moves past frame C with a velocity of +0.500c. What are (a) MAc, (b) f3AC, and (c) the velocity of the particle relative to frame C?

Continuation of Problem 65. Let reference frame C in Fig. 37-31 move past reference frame D (not shown). (a) Show that MAD = MABMBcMcD' (b) Now put this general result to work: Three particles move parallel to a single axis on which an observer is stationed. Let plus and minus signs indicate the directions of motion along that axis. Particle A moves past particle B at f3AB = +0.20. Particle B moves past particle Cat f3BC = -0.40. Particle C moves past observer D at f3CD = +0.60. What is the velocity of particle A relative to observer D? (The solution technique here is much faster than using Eq. 37-29.)

Figure 37-16 shows a ship (attached to reference frame S') passing us (standing in reference frame S) with velocity v = 0.950d. A proton is fired at speed 0.980c relative to the ship from the front of the ship to the rear. The proper length of the ship is 760 m. What is the temporal separation between the time the proton is fired and the time it hits the rear wall of the ship according to (a) a passenger in the ship and (b) us? Suppose that, instead, the proton is fired from the rear to the front. What then is the temporal separation between the time it is fired and the time it hits the front wall according to (c) the passenger and (d) us?

The car-in-the-garage problem. Carman has just purchased the world's longest stretch limo, which has a proper length of Lc = 30.5 m. In Fig. 37-32a, it is shown parked in front of a garage with a proper length of Lg = 6.00 m. The garage has a front door (shown open) and a back door (shown closed). The limo is obviously longer than the garage. Still, Garageman, who owns the garage and knows something about relativistic length contraction, makes a bet with Carman that the limo can fit in the garage with both doors closed. Carman, who dropped his physics course before reaching special relativity, says such a thing, even in principle, is impossible. To analyze Garageman's scheme, an Xc axis is attached to the limo, with Xc = 0 at the rear bumper, and an Xg axis is attached to the garage, with Xg = 0 at the (now open) front door. Then Carman is to drive the limo directly toward the front door at a velocity of 0.9980c (which is, of course, both technically and financially impossible). Carman is stationary in the Xc reference frame; Garageman is stationary in the Xg reference frame. There are two events to consider. Event 1: When the rear bumper clears the front door, the front door is closed. Let the time of this event be zero to both Carman and Garageman: tgl = tel = O. The event occurs at Xc = Xg = O. Figure 37-32b shows event 1 according to the Xg reference frame. Event 2: When the front bumper reaches the back door, that door opens. Figure 37-32c shows event 2 according to the Xg reference frame. According to Garageman, (a) what is the length of the limo, and what are the spacetime coordinates (b) Xg2 and (c) tg2 of event 2? (d) For how long is the limo temporarily "trapped" inside the garage with both doors shut? Now consider the situation from the Xc reference frame, in which the garage comes racing past the limo at a velocity of -0.9980c. According to Carman, (e) what is the length of the passing garage, what are the spacetime coordinates (f) Xc2 and (g) tc2 of event 2, (h) is the limo ever in the garage with both doors shut, and (i) which event occurs first? (j) Sketch events 1 and 2 as seen by Carman. (k) Are the events causally related; that is, does one of them cause the other? (1) Finally, who wins the bet?

An airplane whose rest length is 40.0 m is moving at uniform velocity with respect to Earth, at a speed of 630 m/s. (a) By what fraction of its rest length is it shortened to an observer on Earth? (b) How long would it take, according to Earth clocks, for the airplane's clock to fall behind by 1.00 j.Ls?

An airplane whose rest length is 40.0 m is moving at uniform velocity with respect to Earth, at a speed of 630 m/s. (a) By what fraction of its rest length is it shortened to an observer on Earth? (b) How long would it take, according to Earth clocks, for the airplane's clock to fall behind by 1.00 j.Ls?

Find the speed parameter of a particle that takes 2.0 y longer than light to travel a distance of 6.0 ly.

How much work is needed to accelerate a proton from a speed of 0.9850c to a speed of 0.9860c?

A pion is created in the higher reaches of Earth's atmosphere when an incoming high-energy cosmic-ray particle collides with an atomic nucleus. A pion so formed descends toward Earth with a speed of 0.99c. In a reference frame in which they are at rest, pions decay with an average life of 26 ns. As measured in a frame fixed with respect to Earth, how far (on the average) will such a pion move through the atmosphere before it decays?

If we intercept an electron having total energy 1533 MeV that came from Vega, which is 26ly from us, how far in lightyears was the trip in the rest frame of the electron?

The total energy of a proton passing through a laboratory apparatus is 10.611 nl What is its speed parameter (3? Use the proton mass given in Appendix B under "Best Value," not the commonly remembered rounded number.

A spaceship at rest in a certain reference frame S is given a speed increment of 0.50c. Relative to its new rest frame, it is then given a further 0.50c increment. This process is continued until its speed with respect to its original frame S exceeds 0.999c. How many increments does this process require?

In the red shift of radiation from a distant galaxy, a certain radiation, known to have a wavelength of 434 nm when observed in the laboratory, has a wavelength of 462 nm. (a) What is the radial speed of the galaxy relative to Earth? (b) Is the galaxy approaching or receding from Earth?

What is the momentum in MeV/c of an electron with a kinetic energy of 2.00 MeV?

The radius of Earth is 6370 km, and its orbital speed about the Sun is 30 km/s. Suppose Earth moves past an observer at this speed. To the observer, by how much does Earth's diameter contract along the direction of motion?

A particle with mass m has speed c!2 relative to inertial frame S. The particle collides with an identical particle at rest relative to frame S. Relative to S, what is the speed of a frame S' in which the total momentum of these particles is zero? This frame is called the center of momentum frame.

An elementary particle produced in a laboratory experiment travels 0.230 mm through the lab at a relative speed of 0.960c before it decays (becomes another particle). (a) What is the proper lifetime of the particle? (b) What is the distance the particle travels as measured from its rest frame?

What are (a) K, (b) E, and (c) p (in GeV/c) for a proton moving at speed 0.990c? What are (d) K, (e) E, and (f) p (in MeV/c) for an electron moving at speed 0.990c?

A radar transmitter T is fixed to a reference frame S' that is moving to the right with speed v relative to reference frame S (Fig. 37-33). A mechanical timer (essentially a clock) in frame S', having a period TO (measured in S'), causes transmitter T to emit timed radar pulses, which travel at the speed of light and are received by R, a receiver fixed in frame S. (a) What is the period Tof the timer as detected by observer A, who is fixed in frame S? (b) Show that at receiver R the time interval between pulses arriving from Tis not Tor TO, but (c) Explain why receiver R and observer A, who are in the same reference frame, measure a different period for the transmitter. (Hint: A clock and a radar pulse are not the same thing.)

One cosmic-ray particle approaches Earth along Earth's north-south axis with a speed of 0.80c toward the geographic north pole, and another approaches with a speed of 0.60c toward the geographic south pole (Fig. 37-34). What is the relative speed of approach of one particle with respect to the other?

(a) How much energy is released in the explosion of a fission bomb containing 3.0 kg of fissionable material? Assume that 0.10% of the mass is converted to released energy. (b) What mass of TNT would have to explode to provide the same energy release? Assume that each mole of TNT liberates 3.4 MJ of energy on exploding. The molecular mass of TNT is 0.227 kg/mol. (c) For the same mass of explosive, what is the ratio of the energy released in a nuclear explosion to that released in a TNT explosion?

(a) What potential difference would accelerate an electron to speed c according to classical physics? (b) With this potential difference, what speed would the electron actually attain?

A Foron cruiser moving directly toward a Reptulian scout ship fires a decoy toward the scout ship. Relative to the scout ship, the speed of the decoy is 0.980c and the speed of the Foron cruiser is 0.900c. What is the speed of the decoy relative to the cruiser?