buku physics11

Another way to look at time dilation is by using an analogy. Let’s say Fig.9.11that you are blasting off from Earth towards a distant planet that light willtake five minutes to reach. You are travelling at four-fifths the speed of lightand you depart at midnight. Five minutes into your trip, you look at yourwatch and it reads 12:05 a.m. But the image of the time on your watch trav-els at speed c, faster than you. For a stationary observer on the distantplanet you are approaching, the time on your watch would read 12:04 a.m.because light from the 12:05 a.m. image on your watch hasn’t reached heryet. She would judge your watch to be running slow.Watch image vship ϭ 4 ctravels at c 5StationaryobserverLength Contraction DEDUCTION OF LENGTH CONTRACTIONWhen objects are moving at speeds near the speed of light, time becomes rel-ative, not absolute. Similarly with the dimensions of objects. As the object Ί๶t ϭ ᎏLvᎏo and t0 ϭ t 1 Ϫ ᎏcvᎏ22 fromapproaches the speed of light, its length in the line of motion decreases,according to the equation the time dilation formula Ί๶L ϭ Lo 1 Ϫ ᎏvcᎏ22 Ί๶so L ϭ vto ϭ vt 1 Ϫ ᎏcvᎏ22 Ί๶But Lo ϭ vt so L ϭ Lo 1 Ϫ ᎏcvᎏ22where Lo is the rest length or the normal length of the object at v ϭ 0, L is therelativistic length as measured by a stationary observer at the speed, v, of theobject, and c is the speed of light.e x a m p l e 2 Length contraction at speeds close to the speed of lightWhat is the relativistic length of the car that the student is driving as seenby the principal in Example 1?Solution and Connection to TheoryGiven Lo ϭ 4.2 m v ϭ 2.5 ϫ 108 m/sto ϭ 2.0 s mo ϭ 1.50 ϫ 103 kgc ϭ 3.0 ϫ 108 m/s L ϭ ?Ί๶ Ί๶๶๶L ϭ Lo 1 Ϫ ᎏvcᎏ22 ϭ 4.2 m 1 Ϫ ᎏ((23..50 ϫϫᎏ110088))22L ϭ 4.2 m(0.553) ϭ 2.32 mTo the principal, the car would appear to be only 2.3 m long. chapter 9: Special Relativity and Rest Energy 289

Mass Increase Just as the “absolutes” of time and length are affected at relativistic speeds, so too is mass. Like time, mass appears dilated or increased to a stationary observer viewing objects travelling at relativistic speeds. Ί๶๶mϭ ᎏ1mϪᎏ0 ᎏvcᎏ22 where m is the relativistic mass, mo is the “rest” mass, v is the velocity of the object, and c is the speed of light.In every relativistic equation, the e x a m p l e 3 Mass increase at speeds close to the speed of light3relativistic factor always appears: In Examples 1 and 2, the student’s car would also appear to the principal Ί๶1 Ϫ ᎏcvᎏ22 to gain mass. What would the relativistic mass of the student’s car be at the speed of 2.5 ϫ 108 m/s?In the time dilation and massincrease formulas, this component is Solution and Connection to Theorydivided into the rest values becauserelativistic time and mass are larger. GivenWith length contraction, this compo-nent is multiplied by the rest length to ϭ 2.0 s mo ϭ 1.50 ϫ 103 kg Lo ϭ 4.2 m v ϭ 2.5 ϫ 108 m/sbecause relativistic length is shorter c ϭ 3.0 ϫ 108 m/s m ϭ ?than rest length. ᎏ1mϪᎏ0 ᎏvcᎏ22 ϭ ᎏ1.5ᎏϫ 103ᎏkg 1 Ϫ ᎏ((23..50 ϫϫᎏ110088))22 Ί๶๶ Ί๶๶๶m ϭ m ϭ ᎏ1.5 ϫ ᎏ103 kg ϭ 2.7 ϫ 103 kg 0.553 The principal would observe the student’s car to be 2.7 ϫ 103 kg. Fig.9.12 Relativistic Effectsconnecti Same view of ts each other ngtheConcep Stationary observer Two observers notice Moving observer sees other with slower no change in own sees other with slower time, mass, length time, more mass, time, more mass, shorter length shorter length Same view of each other290 u n i t B : Wo r k , E n e rg y, a n d Powe r

1. Explain the concept of time dilation for two people, one stationary gpplyin and the other moving at close to the speed of light. Co the a ncep 2. What would the consequence be if the speed of light was not always ts the same for all observers? 291 3. For an object moving in the horizontal direction at near the speed of light, select the aspects of the object that change as it changes ref- erence frames (time, length in the x direction, length in the y direc- tion, mass, and speed). 4. a) Two good friends on Earth say good bye to each other as one of them takes off on a trip into space at 85% the speed of light. If the trip lasts one year, how many years have gone by on Earth? b) What would the mass of the spaceship appear to be if it had a 1200 kg rest mass? c) What is the observed length of the ship if it was 5.6 m long while sitting on Earth? 9.4 Strange Effects of Special RelativityA Fountain of Youth?A paradox is a statement or situation that seems to be contradictory to pop-ular belief but which is true. One of the most interesting aspects of relativitythat has made it into the realm of science fiction is that of time travel. Onethought experiment that describes the consequences of relativistic speeds isthe twin paradox. One twin blasts off from Earth at a speed close to c, whilethe other remains on Earth. After an extensive period of time according tothe stationary twin, the traveller returns to Earth, appearing much youngerthan her earthbound sister because the moving clock had slowed. Are relativistic effects reversible for the two observers? One might expectthat from the point of view of the space traveller, it is Earth that moves awayat a high speed. Therefore, when Earth “returns,” the twin on Earth wouldappear younger. However, this is not the case. The relativistic effects wouldonly be witnessed by the earthbound twin because she is observing the eventfrom an inertial frame of reference. In contrast, the travelling twin acceler-ates, decelerates, and stops before returning to Earth, that is, she is in a non-inertial frame of reference.Stopping TimeIf time slows as speed increases, then it must be possible to stop time. Whenthe speed of a clock reaches c, the speed of light, the relativistic termbecomes zero. chapter 9: Special Relativity and Rest Energy

Ί๶1 Ϫ ᎏvcᎏ22Requiring division by zero, this makes the time dilation formula Ί๶t ϭ ᎏ1 tϪo ᎏvcᎏ22mathematically undefined. In physics, this means that time becomes infi-nitely slow and would represent the clock stopping. Figure 9.13 summarizeswhat happens to time, length, and mass as well as kinetic energy as speedsapproach the speed of light. This figure illustrates that the mass, and there-fore the kinetic energy, of objects would become infinitely large at the speedof light. It seems a terrible shame that although many interesting thingsoccur when objects reach the speed of light, they can never be achievedpractically. An infinite amount of energy would have to be transferred to anobject to push its speed to c, since its mass would also have to be increasedto an infinite value. This idea implies an absolute speed limit — no objectwith rest mass can ever reach the speed of light. Also, there is no way thattime could ever stop.Fig.9.13 Effects on length, time, mass, and energyas an object approaches the speed of light 1.2Ratio L 1.0L0 0.8 0.6 0.2 0.4 0.6 0.8 1.0 1.2 0.4 Velocity as fraction of c 0.2(a) 0 6.0E 5.0E0 4.0 3.0, 2.0 1.0t t0(b) 0,m m0Ratios 0.2 0.4 0.6 0.8 1.0 1.2 Velocity as fraction of c292 u n i t B : Wo r k , E n e rg y, a n d Powe r

Experimental Confirmations of Special Relativity g pplyin Co theAlthough the effects of relativity seem very strange, they have been a ncepconfirmed many times. In 1971, J.C. Haffele and R.E. Keating trans- tsported a cesium clock (based on the decay rate of the isotope). They Fig.9.14 LEP, the world’s mostflew one clock in a jet plane for 45 hours and compared the time of theclock on the plane to the time of an identical clock on the ground. powerful electron-positron colliderAlthough the effect of time dilation was small, there was a measurable (Geneva, Switzerland)difference, one that was predicted by the time dilation formula. For image These effects are now commonly seen in high-energy accelerators see student(Fig. 9.14), where radioactive particles are accelerated to speeds nearc. The amount of material remaining after travelling in the accelerator text.is greater than that predicted by the half-life calculation (see Chapter19) from a stationary reference frame. If one takes into account theeffect of time dilation, then the correct value of the remaining materialis obtained. To us, the material appears to have a longer half-life.1. The average lifetime of a mu-meson (muon) is 2.2 ϫ 10Ϫ6 s. Mu- mesons are created in the upper atmosphere, thousands of metres above the ground, by cosmic rays. If the speed of a mu-meson is 2.994 ϫ 108 m/s, a) calculate the distance it can travel in its average lifetime (2.2 ϫ 10Ϫ6 s). b) calculate its relativistic distance using the answer from part a). c) calculate the apparent lifetime of the meson from the reference point of an observer on the ground. d) calculate the distance the muon travels using the new time from part c).The muon is in fact observed on Earth due to relativistic effects. 9.5 Mass and Energy 293Figure 9.13 implies that as more and more work is done to bring an objectcloser to the speed of light, the mass of the object also increases. This observa-tion suggests that mass is somehow a storehouse of energy. Einstein (Fig. 9.15)summarized the mass_energy relationship with the equation E ϭ mc2where the constant, c, is the speed of light. This equation predicts that if any small mass could be converted toenergy, the amount of energy released would be incredibly large. The fol-lowing example illustrates the amount of energy Einstein’s equation pre-dicts would be created from the complete transformation of 1 g of matter. chapter 9: Special Relativity and Rest Energy