Relativity Worked Solutions

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1 Physis Option Problems - Relativisti Addition of Veloities (from Gianoli 6 th edition 43. (I A person on a roket traveling at.5 (with respet to the Earth observes a meteor ome from behind and pass her at a speed she measures as.5. How fast is the meteor moving with respet to the Earth? 44. (II Two spaeships leave Earth in opposite diretions, eah with a speed of.5 with respet to Earth. (a What is the veloity of spaeship relative to spaeship? (b What is the veloity of spaeship relative to spaeship? 45. (II A spaeship leaves Earth traveling at.7. A seond spaeship leaves the first at a speed of.7 with respet to the first. Calulate the speed of the seond ship with respet to Earth if it is fired (a in the same diretion the first spaeship is already moving, (b diretly bakward toward Earth. 46. (II An observer on Earth sees an alien vessel approah at a speed of.6. The Enterprise omes to the resue (Fig. 6, overtaking the aliens while moving diretly toward Earth at a speed of.9 relative to Earth. What is the relative speed of one vessel as seen by the other? 47. (II A spaeship in distress sends out two esape pods in opposite diretions. One travels at a speed v. 6 in one diretion, and the other travels at a speed v +. 7 in the other diretion, as observed from the spaeship.. What speed does the first esape pod measure for the seond esape pod? 4. (II Roket A passes Earth at a speed of.75. At the same time, roket B passes Earth moving.95 relative to Earth in the same diretion. How fast is B moving relative to A when it passes A? Answers 43. We find the speed in the frame of the Earth from ( (.5 + ( (a In the referene frame of the seond spaeship, the Earth is moving at.5, and the first spaeship is moving at.5 relative to the Earth. Thus the speed of the first spaeship relative to the seond is ( ( +.5(.5 (b In the referene frame of the first spaeship, the Earth is moving at.5, and the seond spaeship is moving at.5 relative to the Earth. Thus the speed of the seond spaeship relative to the first is.5 + (.5., as expeted. + (.5 + (.5

2 Physis Option 45. We take the positive diretion in the diretion of the first spaeship. (a In the referene frame of the Earth, the first spaeship is moving at +.7, and the seond spaeship is moving at +.7 relative to the first. Thus the speed of the seond spaeship relative to the Earth is ( ( +.7(.7 (b In the referene frame of the Earth, the first spaeship is moving at +.7, and the seond spaeship is moving at.7 relative to the first. Thus the speed of the seond spaeship relative to the Earth is ( ( +.7( We take the positive diretion in the diretion of the Enterprise. In the referene frame of the alien vessel, the Earth is moving at.6, and the Enterprise is moving at +.9 relative to the Earth. Thus the speed of the Enterprise relative to the alien vessel is ( (.6 + ( +.9 Note that the relative speed of the two vessels as seen on Earth is From the first pod s point of view, the spaeship and the seond pod are moving in the same diretion. Thus the speed of the seond pod relative to the first is v + u vu + + (.6(.7 4. The speed of roket B relative to Earth an be written as v + u u, vu + where v is the speed of Roket A relative to Earth and u is the speed of Roket B relative to Roket A. With the given values, then,.75 + u.95, whih gives u.7..75u +

3 Speial Relativity Part : Coneptual Questions from Gianoli. A young-looking woman astronaut has just arrived home from a long trip. She rushes up to an old gray-haired man and in the ensuing onversation refers to him as her son. How might this be possible? If the young-looking astronautt has been traveling at extremely high speeds (relative to Earth for many Earth years, sine her lok willl run slow ompared to the loks at rest on Earth, it is possible that her son ould now be muh older than she is after her trip. This is a twist on the twin paradox. 9. If you were traveling away from Earth at speed.5, would you notie a hange in your heartbeat? Would your mass, height, or waistline hange? What would observers on Earth using telesopes say about you? You would not notie a hange in your own heartbeat, mass, height or waistline. No matter how fast you are moving relative to Earth, you are at rest in your own referene frame. Thus, you would not notie any hanges in your own harateristis. To observers on Earth, you are moving away at.5, whih gives γ /.7. To these observers, it would appear that your heartbeat has slowed by a fator of.7, that your mass has inreased by a fator of /.7.5, and that your waistline has dereased by a fator of.7 (all due to the relativity equations for time dilation, mass inrease, and length ontration, but that your height would be unhanged (sine there is no relative motion between you and Earth in that diretion.. Do time dilation and length ontration our at ordinary speeds, say 9 km h? Time dilation and length ontration do our at 9 km/h, but the effets are so small that they are not measurable. 6. Can a partile of nonzero rest mass attain the speed of light? No, it is no ot possible for a non- inrease to as zero rest mass partile to attain a speed of. This is beause its relativisti mass would v, whih means it would take an infinite amount of energy to inrease the speed of this infinite mass partile to. Physis Option 7. Does the equation E m onflit with the onservation of energy priniple? Explain. No, E m does not onflit with the onservation of energy, it atually ompletes it. Sine this equation shows us that mass and energy are interonvertible, it says it is now neessary to inlude mass as a form of energy in the analysis of physial proesses.. If mass is a form of energy, does this mean that a spring has more mass when ompressed than when relaxed? Yes. One way to desribe the energy stored in the ompressed spring is to say it is a mass inrease (although it would be so small that it ould not be measured. This masss will onvert bak to energy when the spring is unompressed. 9. It is not orret to say that matter an neither be reated nor destroyed. What must we say instead? Matter and energy are interonvertible (matter an be onverted into energy and energy an be onverted into matter, thus we should say energy an neither be reated nor destroyed.. Is our intuitive notion that veloities simply add, ompletely wrong? No, our intuitive notion that veloities simply add is not ompletely wrong. Our intuition is based on our everyday experienes and at these everyday speeds our intuition is orret regarding how veloities add. Our intuition does break down, though, at very high speeds, where we have to take into aount relativisti effets. Relativity does not ontradit lassial mehanis, but it is a more general theory where lassial mehanis is a limiting ase.

4 Problems. (I A ertain hemial reation requires mass of the produts over the reatants?. We find the inrease in mass from 4 ( 4. J ( 3. m s 3 m 5.36 kg. Note that this is so small, most hemial reations are onsidered to have mass onserved. 4 Physis Option 4. J of energy input for it to go. What is the inrease in. (I When a uranium nuleus at rest breaks apart in the proess known as fission in a nulear reator, the resulting fragments have a total kineti energy of about MeV. How muh masss was lost in the proess?. We find the loss in mass from ( ( ( 3 MeV.6 J MeV m 3. m s. (I Calulate the rest energy of an eletron in joules and in MeV ( MeV.6. The rest energy of the eletron is E m ( ( 4 (. J 3 (.6 J MeV 3 9. kg kg. 4 m s. J.5MeV. 3 J. 3. (I Calulate the rest mass of a 3. The rest mass of the proton is ( 7 ( ( E.67 kg.9979 m s 93MeV 3..6 J MeV 4. (I The total annual energy onsumption in the United States is about J. How muh mass would have to be onverted to energy to fuel this need? 4. We find the neessary mass onversion from 9 ( J ( 3. m s m 5. (II How muh energy an be obtained from onversion of. gram of mass? How muh mass ould this energy raise to a height of.5 km above the Earth s surfae? 5. We find the energy equivalent of the mass from E ( ( 3 m. kg 3. proton in MeV. If this energy inreases the gravitational energy, we have E mgh; 9 kg. 3 m s 9. J. 9

5 ( ( J m 9. m s.5 m, 6. (II What is the speed of a partile when its kineti energy equals its rest energy? 6. If the kineti energy is equal to the rest energy, we have KE m m m, or m m. ( We find the speed from m ; v, whih gives v.66. v 7. (II At what speed will an objet s kineti energy be 5% of its rest energy? 7. If the kineti energy is 5% of the rest energy, we have KE m m.5 m, or m.5 m. ( whih gives r We find the speed from m ; v.5, whih gives v.6. v. (II How muh work is required to aelerate a proton from rest up to a speed of..997? Physis Option m 3.7 kg.. (a We find the work required from W KE ( m m m v ( 93 MeV (.9977 ( 3. MeV.GeV.3 9 J.

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