EXERCISES AND PROBLEMS

Transcription

1 Exercises and Problems 71 EXERCISES AND PROBLEMS The icon in fron of a problem indicaes ha he problem can be done on a Dnamics Workshee. Dnamics Workshees are found a he back of he Suden Workbook. If ou use a workshee, draw a moion diagram in he Phsical Represenaion secion, esablish our coordinae ssem and smbols in he Picorial Represenaion secion, hen solve he problem in he Mahemaical Represenaion secion. Exercises Secion.1 Moion in One Dimension 1. Figure Ex.1 shows a moion diagram of a car raveling down a sree. The camera ook one frame ever second. A disance scale is provided. a. Measure he x-value of he car a each do. Place our daa in a able, similar o Table.1, showing each posiion and he insan of ime a which i occurred. b. Make a posiion-versus-ime graph for he ball. Because ou have daa onl a cerain insans of ime, our graph should consis of dos ha are no conneced ogeher. 1 frame per second r v FIGURE EX.1. For each moion diagram, deermine he sign (posiive or negaive) of he posiion, he veloci, and he acceleraion. FIGURE EX. (a) (b) 3. Wrie a shor descripion of he moion of a real objec for which his would be a realisic posiion-versus-ime graph. 3 x (b) (c) Secion. Uniform Moion 5. A car sars a he origin and moves wih veloci v r 5 (1 m/s, norheas). How far from he origin will he car be afer raveling for 5 s? 6. Larr leaves home a 9:5 and runs a consan speed o he lamppos. He reaches he lamppos a 9:7, immediael urns, and runs o he ree. Larr arrives a he ree a 9:1. a. Wha is Larr s average veloci during each of hese wo inervals? b. Wha is he average veloci for Larr s enire run? FIGURE EX.6 7. Alan leaves Los Angeles a 8: A.M. o drive o San Francisco, mi awa. He ravels a a sead 5 mph. Beh leaves Los Angeles a 9: A.M. and drives a sead 6 mph. a. Who ges o San Francisco firs? b. How long does he firs o arrive have o wai for he second? 8. A bicclis has he posiion-versus-ime graph shown. Wha is he bicclis s veloci a 5 1 s, a 5 5 s, and a 5 35 s? FIGURE EX.8 x (ards) Julie drives 1 mi o Grandmoher s house. On he wa o Grandmoher s, Julie drives half he disance a mph and half he disance a 6 mph. On her reurn rip, she drives half he ime a mph and half he ime a 6 mph. a. Wha is Julie s average speed on he wa o Grandmoher s house? b. Wha is her average speed on he reurn rip? FIGURE EX.3. Wrie a shor descripion of he moion of a real objec for which his would be a realisic posiion-versus-ime graph. FIGURE EX. 1 6 x (mi) 1 3 (hours) 5 Secion.3 Insananeous Veloci Secion. Finding Posiion from Veloci 1. Figure Ex.1 shows he posiion graph of a paricle. a. Draw he paricle s veloci graph for he inerval s # # s. b. Does his paricle have a urning poin or poins? If so, a wha ime or imes? FIGURE EX

2 7 CHAPTER. Kinemaics: The Mahemaics of Moion 11. A paricle sars from x 5 1 m a 5 and moves wih he veloci graph shown in Figure Ex.11. a. Wha is he objec s posiion a 5 s, 3 s, and s? b. Does his paricle have a urning poin? If so, a wha ime? FIGURE EX.11 Secion.5 Moion wih Consan Acceleraion 1. Figure Ex.1 shows he veloci graph of a paricle. Draw he paricle s acceleraion graph for he inerval s # # s. Give boh axes an appropriae numerical scale. FIGURE EX Figure Ex.13 shows he veloci graph of a rain ha sars from he origin a 5 s. a. Draw posiion and acceleraion graphs for he rain. b. Find he acceleraion of he rain a 5 3. s. FIGURE EX Figure Ex.1 shows he veloci graph of a paricle moving along he x-axis. Is iniial posiion is x 5 m a 5 s. A 5 s, wha are he paricle s (a) posiion, (b) veloci, and (c) acceleraion? FIGURE EX a. Wha consan acceleraion, in SI unis, mus a car have o go from zero o 6 mph in 1 s? b. Wha fracion of g is his? c. How far has he car raveled when i reaches 6 mph? Give our answer boh in SI unis and in fee. 16. A je plane is cruising a 3 m/s when suddenl he pilo urns he engines up o full hrole. Afer raveling. km, he je is moving wih a speed of m/s. a. Wha is he je s acceleraion, assuming i o be a consan acceleraion? b. Is our answer reasonable? Explain. 17. A speed skaer moving across fricionless ice a 8. m/s his a 5.-m-wide pach of rough ice. She slows seadil, hen coninues on a 6. m/s. Wha is her acceleraion on he rough ice? 18. a. How man das will i ake a spaceship o accelerae o he speed of ligh ( m/s) wih he acceleraion g? b. How far will i ravel during his inerval? c. Wha fracion of a ligh ear is our answer o par b? A ligh ear is he disance ligh ravels in one ear. NOTE We know, from Einsein s heor of relaivi, ha no objec can ravel a he speed of ligh. So his problem, while ineresing and insrucive, is no realisic. Secion.6 Free Fall 19. Ball bearings are made b leing spherical drops of molen meal fall inside a all ower called a sho ower and solidif as he fall. a. If a bearing needs. s o solidif enough for impac, how high mus he ower be? b. Wha is he bearing s impac veloci?. A ball is hrown vericall upward wih a speed of 19.6 m/s. a. Wha is he ball s veloci and is heigh afer 1,, 3, and s? b. Draw he ball s veloci-versus-ime graph. Give boh axes an appropriae numerical scale. 1. A suden sanding on he ground hrows a ball sraigh up. The ball leaves he suden s hand wih a speed of 15 m/s when he hand is. m above he ground. How long is he ball in he air before i his he ground? (The suden moves her hand ou of he wa.). A rock is ossed sraigh up wih a speed of m/s. When i reurns, i falls ino a hole 1 m deep. a. Wha is he rock s veloci as i his he boom of he hole? b. How long is he rock in he air, from he insan i is released unil i his he boom of he hole? Secion.7 Moion on an Inclined Plane 3. A car raveling a 3 m/s runs ou of gas while raveling up a slope. How far up he hill will i coas before saring o roll back down?. A skier is gliding along a 3. m/s on horizonal, fricionless snow. He suddenl sars down a 1 incline. His speed a he boom is 15 m/s. a. Wha is he lengh of he incline? b. How long does i ake him o reach he boom? Secion.8 Insananeous Acceleraion 5. A paricle moving along he x-axis has is posiion described b he funcion x 5 ( 3 1 1) m, where is in s. A 5 s, wha are he paricle s (a) posiion, (b) veloci, and (c) acceleraion? 6. A paricle moving along he x-axis has is veloci described b he funcion 5 m/s, where is in s. Is iniial posiion is x 5 1 m a 5 s. A 5 1 s, wha are he paricle s (a) posiion, (b) veloci, and (c) acceleraion?

3 Exercises and Problems Figure Ex.7 shows he acceleraion-versus-ime graph of a paricle moving along he x-axis. Is iniial veloci is 5 8. m/s a 5 s. Wha is he paricle s veloci a 5. s? (m/s ) 3. The veloci-versus-ime graph is shown for a paricle moving along he x-axis. Is iniial posiion is x 5. m a 5 s. a. Wha are he paricle s posiion, veloci, and acceleraion a 5 1. s? b. Wha are he paricle s posiion, veloci, and acceleraion a 5 3. s? FIGURE EX.7 FIGURE P Problems 8. Figure P.8 shows he moion diagram, made a wo frames of film per second, of a ball rolling along a rack. The rack has a 3.-m-long sick secion. a. Use he meer sick o measure he posiions of he cener of he ball. Place our daa in a able, similar o Table.1, showing each posiion and he insan of ime a which i occurred. b. Make a posiion-versus-ime graph for he ball. Because ou have daa onl a cerain insans of ime, our graph should consis of dos ha are no conneced ogeher. c. Wha is he change in he ball s posiion from 5 s o 5 1. s? d. Wha is he change in he ball s posiion from 5. s o 5. s? e. Wha is he ball s veloci before reaching he sick secion? f. Wha is he ball s veloci afer passing he sick secion? g. Deermine he ball s acceleraion on he sick secion of he rack. frames per second 5 s Sick secion of rack Meers FIGURE P A paricle s posiion on he x-axis is given b he funcion x 5 ( 1 ) m, where is in s. a. Make a posiion-versus-ime graph for he inerval s # # 5 s. Do his b calculaing and ploing x ever.5 s from s o 5 s, hen drawing a smooh curve hrough he poins. b. Deermine he paricle s veloci a 5 1. s b drawing he angen line on our graph and measuring is slope. c. Deermine he paricle s veloci a 5 1. s b evaluaing he derivaive a ha insan. Compare his o our resul from par b. d. Are here an urning poins in he paricle s moion? If so, a wha posiion or posiions? e. Where is he paricle when 5. m/s? f. Draw a moion diagram for he paricle. 31. Three paricles move along he x-axis, each saring wih 5 1 m/s a 5 s. The graph for A is a posiion-versusime graph; he graph for B is a veloci-versus-ime graph; he graph for C is an acceleraion-versus-ime graph. Find each paricle s veloci a 5 7. s. Work wih he geomer of he graphs, no wih kinemaic equaions FIGURE P The veloci graph is shown for a paricle having iniial posiion x 5 m a 5 s. a. A wha ime or imes is he paricle found 5 35 m? Work wih he geomer of he graph, no wih kinemaic equaions. b. Draw a moion diagram for he paricle. FIGURE P The acceleraion graph is shown for a paricle ha sars from res a 5 s. Deermine he objec s veloci a imes 5 s, s, s, 6 s, and 8 s. FIGURE P.33 Paricle A (m/s ) (m/s ) Paricle C Paricle B 6 8

4 7 CHAPTER. Kinemaics: The Mahemaics of Moion 3. A block is suspended from a spring, pulled down, and released. The block s posiion-versus-ime graph is shown in Figure P.3. a. A wha imes is he veloci zero? A wha imes is he veloci mos posiive? Mos negaive? b. Draw a reasonable veloci-versus-ime graph. FIGURE P The acceleraion graph is shown for a paricle ha sars from res a 5 s. a. Draw he paricle s veloci graph over he inerval s # # 1 s. Include an appropriae numerical scale on boh axes. b. Describe, in words, how he veloci graph would differ if he paricle had an iniial veloci of. m/s. FIGURE P (m/s ) Drop a rubber ball or a ennis ball from a heigh of abou 5 cm (<1 f) and wach carefull as i bounces. Then draw a posiion graph, a veloci graph, and an acceleraion graph from he insan ou drop he ball unil i reurns o is maximum heigh. Sack our hree graphs vericall so ha he ime axes are aligned wih each oher. Pa paricular aenion o he ime when he ball is in conac wih he ground. This is a shor inerval of ime, bu i s no zero. 37. The posiion of a paricle is given b he funcion x 5 ( ) m, where is in s. a. A wha ime or imes is 5 m/s? b. Wha are he paricle s posiion and is acceleraion a his ime(s)? 38. An objec sars from res 5 m a ime 5 s. Five seconds laer, a 5 5. s, he objec is observed o be a x 5. m and o have veloci 5 11 m/s. a. Was he objec s acceleraion uniform or nonuniform? Explain our reasoning. b. Skech he veloci-versus-ime graph implied b hese daa. Is he graph a sraigh line or curved? If curved, is i concave upward or downward? 39. A paricle s veloci is described b he funcion 5 k m/s, where k is a consan and is in s. The paricle s posiion a 5 s is x 59. m. A s, he paricle is a x m. Deermine he value of he consan k. Be sure o include he proper unis.. A paricle s acceleraion is described b he funcion 5 (1 ) m/s, where is in s. Is iniial condiions are x 5 m and 5 m/s a 5 s. a. A wha ime is he veloci again zero? b. Wha is he paricle s posiion a ha ime? 1. A ball rolls along he fricionless rack shown. Each segmen of he rack is sraigh, and he ball passes smoohl from one segmen o he nex wihou changing speed or leaving he rack. Draw hree vericall sacked graphs showing posiion, veloci, and acceleraion versus ime. Each graph should have he same ime axis, and he proporions of he graph should be qualiaivel correc. Assume ha he ball has enough speed o reach he op. FIGURE P.1. Draw posiion, veloci, and acceleraion graphs for he ball shown here. See Problem 1 for more informaion. FIGURE P. 3. Draw posiion, veloci, and acceleraion graphs for he ball shown here. See Problem 1 for more informaion. The ball changes direcion bu no speed as i bounces from he reflecing wall. Reflecing wall v s. FIGURE P.3. Figure P. shows a se of kinemaic graphs for a ball rolling on a rack. All segmens of he rack are sraigh lines, bu some ma be iled. Draw a picure of he rack and also indicae he ball s iniial condiion. FIGURE P. 5. Figure P.5 shows a se of kinemaic graphs for a ball rolling on a rack. All segmens of he rack are sraigh lines, bu some ma be iled. Draw a picure of he rack and also indicae he ball s iniial condiion. FIGURE P.5 v s. s v s a s s v s a s v s 5

5 Exercises and Problems Figure P.6 shows a se of kinemaic graphs for a ball rolling on a rack. All segmens of he rack are sraigh lines, bu some ma be iled. Draw a picure of he rack and also indicae he ball s iniial condiion. FIGURE P.6 s v s a s 7. The akeoff speed for an Airbus A3 jeliner is 8 m/s. Veloci daa measured during akeoff are as follows: (s) v s a. Wha is he akeoff speed in miles per hour? b. Is he jeliner s acceleraion consan during akeoff? Explain. c. A wha ime do he wheels leave he ground? d. For safe reasons, in case of an abored akeoff, he runwa mus be hree imes he akeoff disance. Can an A3 ake off safel on a.5-mi-long runwa? 8. Does a real auomobile have consan acceleraion? Measured daa for a Porsche 9 Turbo a maximum acceleraion is as follows: (s) v s (mph) a. Make a graph of veloci versus ime. Based on our graph, is he acceleraion consan? Explain. b. Draw a smooh curve hrough he poins on our graph, hen use our graph o esimae he car s acceleraion a. s and 8. s. Give our answer in SI unis. c. Use our graph o esimae he disance raveled in he firs 1 s. Hin: Approximae he curve wih five seps, each of widh D 5 s. Le he heigh of each sep be he average of he velociies a he beginning and end of he sep. 9. A driver has a reacion ime of.5 s, and he maximum deceleraion of her car is 6. m/s. She is driving a m/s when suddenl she sees an obsacle in he road 5 m in fron of her. Can she sop he car in ime o avoid a collision? 5. You are driving o he grocer sore a m/s. You are 11 m from an inersecion when he raffic ligh urns red. Assume ha our reacion ime is.5 s and ha our car brakes wih consan acceleraion. a. How far are ou from he inersecion when ou begin o appl he brakes? b. Wha acceleraion will bring ou o res righ a he inersecion? c. How long does i ake ou o sop? 51. The minimum sopping disance for a car raveling a a speed of 3 m/s is 6 m, including he disance raveled during he driver s reacion ime of.5 s. a. Wha is he minimum sopping disance for he same car raveling a a speed of m/s? b. Draw a posiion-versus-ime graph for he moion of he car in par a. Assume he car is 5 m when he driver firs sees he emergenc siuaion ahead ha calls for a rapid hal. 5. You re driving down he highwa lae one nigh a m/s when a deer seps ono he road 35 m in fron of ou. Your reacion ime before sepping on he brakes is.5 s, and he maximum deceleraion of our car is 1 m/s. a. How much disance is beween ou and he deer when ou come o a sop? b. Wha is he maximum speed ou could have and sill no hi he deer? 53. A kg weaher rocke is loaded wih 1 kg of fuel and fired sraigh up. I acceleraes upward a 3 m/s for 3 s, hen runs ou of fuel. Ignore an air resisance effecs. a. Wha is he rocke s maximum aliude? b. How long is he rocke in he air? c. Draw a veloci-versus-ime graph for he rocke from lifoff unil i his he ground. 5. A 1kg weaher rocke is launched sraigh up. The rocke moor provides a consan acceleraion for 16 s, hen he moor sops. The rocke aliude s afer launch is 51 m. You can ignore an effecs of air resisance. a. Wha was he rocke s acceleraion during he firs 16 s? b. Wha is he rocke s speed as i passes hrough a cloud 51 m above he ground? 55. A lead ball is dropped ino a lake from a diving board 5. m above he waer. Afer enering he waer, i sinks o he boom wih a consan veloci equal o he veloci wih which i hi he waer. The ball reaches he boom 3. s afer i is released. How deep is he lake? 56. A hoel elevaor ascends m a maximum speed of 5 m/s. Is acceleraion and deceleraion boh have a magniude of 1. m/s. a. How far does he elevaor move while acceleraing o full speed from res? b. How long does i ake o make he complee rip from boom o op? 57. A car sars from res a a sop sign. I acceleraes a. m/s for 6 seconds, coass for s, and hen slows down a a rae of 3. m/s for he nex sop sign. How far apar are he sop signs?

6 76 CHAPTER. Kinemaics: The Mahemaics of Moion 58. A car acceleraes a. m/s along a sraigh road. I passes wo marks ha are 3 m apar a imes 5. s and 5 5. s. Wha was he car s iniial veloci? 59. Sana loses his fooing and slides down a fricionless, snow roof ha is iled a an angle of 3. If Sana slides 1 m before reaching he edge, wha is his speed as he leaves he roof? 6. Ann and Carol are driving heir cars along he same sraigh road. Carol is locaed 5. mi a 5 hours and drives a a sead 36 mph. Ann, who is raveling in he same direcion, is locaed 5. mi a 5.5 hours and drives a a sead 5 mph. a. A wha ime does Ann overake Carol? b. Wha is heir posiion a his insan? c. Draw a posiion-versus-ime graph showing he moion of boh Ann and Carol. 61. A ball rolls along he smooh rack shown in he figure wih an iniial speed of 5. m/s. Assume ha he ball urns all he corners smoohl, wih no loss of speed. a. Wha is he ball s speed as i goes over he op? b. Wha is is speed when i reaches he level rack on he righ side? c. B wha percenage does he ball s final speed differ from is iniial speed? Is his surprising? FIGURE P.61 1 m A o rain is pushed forward and released 5. m wih a speed of. m/s. I rolls a a sead speed for. s, hen one wheel begins o sick. The rain comes o a sop 6. m from is saring poin. Wha is he rain s acceleraion afer is wheel begins o sick? 63. Bob is driving he geawa car afer he big bank robber. He s going 5 m/s when his headlighs suddenl reveal a nail srip ha he cops have placed across he road 15 m in fron of him. If Bob can sop in ime, he can hrow he car ino reverse and escape. Bu if he crosses he nail srip, all his ires will go fla and he will be caugh. Bob s reacion ime before he can hi he brakes is.6 s, and his car s maximum deceleraion is 1 m/s. Is Bob in jail? 6. One game a he amusemen park has ou push a puck up a long, fricionless ramp. You win a suffed animal if he puck, a is highes poin, comes o wihin 1 cm of he end of he ramp wihou going off. You give he puck a push, releasing i wih a speed of 5. m/s when i is 8.5 m from he end of he ramp. The puck s speed afer raveling 3. m is. m/s. Are ou a winner? 65. A professional skier s iniial acceleraion on fresh snow is 9% of he acceleraion expeced on a fricionless, inclined plane, he loss being due o fricion. Due o air resisance, his acceleraion slowl decreases as he picks up speed. The speed record on a mounain in Oregon is 18 kilomeers per hour a he boom of a 5 slope ha drops m. a. Wha exi speed could a skier reach in he absence of air resisance? b. Wha percenage of his ideal speed is los o air resisance? 1 m 66. Heaher and Jerr are sanding on a bridge 5 m above a river. Heaher hrows a rock sraigh down wih a speed of m/s. Jerr, a exacl he same insan of ime, hrows a rock sraigh up wih he same speed. Ignore air resisance. a. How much ime elapses beween he firs splash and he second splash? b. Which rock has he faser speed as i his he waer? 67. Nicole hrows a ball sraigh up. Chad waches he ball from a window 5. m above he poin where Nicole released i. The ball passes Chad on he wa up, and i has a speed of 1 m/s as i passes him on he wa back down. How fas did Nicole hrow he ball? 68. A mooris is driving a m/s when she sees ha a raffic ligh m ahead has jus urned red. She knows ha his ligh sas red for 15 s, and she wans o reach he ligh jus as i urns green again. I akes her 1. s o sep on he brakes and begin slowing. Wha is her speed as she reaches he ligh a he insan i urns green? 69. David is driving a sead 3 m/s when he passes Tina, who is siing in her car a res. Tina begins o accelerae a a sead. m/s a he insan when David passes. a. How far does Tina drive before passing David? b. Wha is her speed as she passes him? 7. A Porsche challenges a Honda o a -m race. Because he Porsche s acceleraion of 3.5 m/s is larger han he Honda s 3. m/s, he Honda ges a 5-m head sar. Boh cars sar acceleraing a he same insan. Who wins? 71. A ca is sleeping on he floor in he middle of a 3.-m-wide room when a barking dog eners wih a speed of 1.5 m/s. As he dog eners, he ca (as onl cas can do) immediael acceleraes a.85 m/s oward an open window on he opposie side of he room. The dog (all bark and no bie) is a bi sarled b he ca and begins o slow down a.1 m/s as soon as i eners he room. Does he dog cach he ca before he ca is able o leap hrough he window? 7. You wan o visi our friend in Seale during spring break. To save mone, ou decide o ravel here b rain. Unforunael, our phsics final exam ook he full 3 hours, so ou are lae in arriving a he rain saion. You run as fas as ou can, bu jus as ou reach he plaform ou see our rain, 3 m ahead of ou down he plaform, begin o accelerae a 1. m/s. You chase afer he rain a our maximum speed of 8. m/s, bu here s a barrier 5 m ahead. Will ou be able o leap ono he back sep of he rain before ou crash ino he barrier? 73. A rocke is launched sraigh up wih consan acceleraion. Four seconds afer lifoff, a bol falls off he side of he rocke. The bol his he ground 6. s laer. Wha was he rocke s acceleraion? In Problems 7 hrough 77 ou are given he kinemaic equaion or equaions ha are used o solve a problem. For each of hese, ou are o a. Wrie a realisic problem for which his is he correc equaion(s). Be sure ha he answer our problem requess is consisen wih he equaion(s) given. b. Draw he moion diagram and he picorial represenaion for our problem. c. Finish he soluion of he problem.

7 Exercises and Problems m 5 m 1 (3 m/s)( s s) 1 1 ( s s) 75. (1 m/s) 5 v (9.8 m/s )(1 m m) 76. ( m/s) 5 (5 m/s) (9.8 m/s )(sin 1 )(x 1 m) 77. v 1x 5 m/s 1 ( m/s )(5 s s) x 1 5 m 1 ( m/s)(5 s s) 1 1 ( m/s )(5 s s) x 5 x 1 1 v 1x (1 s 5 s) Challenge Problems 78. Jill has jus goen ou of her car in he grocer sore parking lo. The parking lo is on a hill and is iled 3. Fif meers downhill from Jill, a lile old lad les go of a full loaded shopping car. The car, wih fricionless wheels, sars o roll sraigh downhill. Jill immediael sars o sprin afer he car wih her op acceleraion of. m/s. How far has he car rolled before Jill caches i? 79. As a science projec, ou drop a waermelon off he op of he Empire Sae Building, 3 m above he sidewalk. I so happens ha Superman flies b a he insan ou release he waermelon. Superman is headed sraigh down wih a speed of 35 m/s. How fas is he waermelon going when i passes Superman? 8. Your school science club has devised a special even for homecoming. You ve aached a rocke o he rear of a small car ha has been decoraed in he blue-and-gold school colors. The rocke provides a consan acceleraion for 9. s. As he rocke shus off, a parachue opens and slows he car a a rae of 5. m/s. The car passes he judges box in he cener of he grandsand, 99 m from he saring line, exacl 1 s afer ou fire he rocke. Wha is he car s speed as i passes he judges? 81. Careful measuremens have been made of Olmpic spriners in he 1-meer dash. A simple bu reasonabl accurae model is ha a spriner acceleraes a 3.6 m/s for s, hen runs a consan veloci o he finish line. a. Wha is he race ime for a spriner who follows his model? b. A spriner could run a faser race b acceleraing faser a he beginning, hus reaching op speed sooner. If a spriner s op speed is he same as in par a, wha acceleraion would he need o run he 1-meer dash in 9.9 s? c. B wha percen did he spriner need o increase his acceleraion in order o decrease his ime b 1%? 8. A spriner can accelerae wih consan acceleraion for. s before reaching op speed. He can run he 1-meer dash in 1 s. Wha is his speed as he crosses he finish line? 83. The Sarship Enerprise reurns from warp drive o ordinar space wih a forward speed of 5 km/s. To he crew s grea surprise, a Klingon ship is 1 km direcl ahead, raveling in he same direcion a a mere km/s. Wihou evasive acion, he Enerprise will overake and collide wih he Klingons in jus slighl over 3. s. The Enerprise s compuers reac insanl o brake he ship. Wha acceleraion does he Enerprise need o jus barel avoid a collision wih he Klingon ship? Assume he acceleraion is consan. Hin: Draw a posiion-versus-ime graph showing he moions of boh he Enerprise and he Klingon ship. Le x 5 km be he locaion of he Enerprise as i reurns from warp drive. How do ou show graphicall he siuaion in which he collision is barel avoided? Once ou decide wha i looks like graphicall, express ha siuaion mahemaicall. STOP TO THINK ANSWERS Sop o Think.1: d. The paricle sars wih posiive x and moves o negaive x. Sop o Think.: c. The veloci is he slope of he posiion graph. The slope is posiive and consan unil he posiion graph crosses he axis, hen posiive bu decreasing, and finall zero when he posiion graph is horizonal. Sop o Think.3: b. A consan posiive corresponds o a linearl increasing x, saring from x i 51 m. The consan negaive hen corresponds o a linearl decreasing x. Sop o Think.: a or b. The veloci is consan while a 5, i decreases linearl while a is negaive. Graphs a, b, and c all have he same acceleraion, bu onl graphs a and b have a posiive iniial veloci ha represens a paricle moving o he righ. Sop o Think.5: d. The acceleraion vecor a r poins downhill (negaive s-direcion) and has he consan value g sin u hroughou he moion. Sop o Think.6: c. Acceleraion is he slope of he graph. The slope is zero a B. Alhough he graph is seepes a A, he slope a ha poin is negaive, and so a A, a B. Onl C has a posiive slope, so a C. a B.