1. Their velocities are NOT equal, because the two velocities have different directions.

Transcription

1 CHAPER 3: Kinematic in w Dimenin; Vectr Anwer t Quetin. heir elcitie are O equal, becaue the tw elcitie hae different directin.. (a) Durin ne ear, the Earth trael a ditance equal t the circumference f it rbit, but ha a diplacement f relatie t the Sun. (b) he pace huttle trael a lare ditance durin an fliht, but the diplacement m ne launch t the net i. (c) An ind f cr cuntr rund trip air trael wuld reult in a lare ditance traeled, but a diplacement f. (d) he diplacement a race car m the tart t the finih f the Ind 5 aut race i. 3. he diplacement can be thuht f a the traiht line path m the initial lcatin t the final lcatin. he lenth f path will alwa be reater than r equal t the diplacement, becaue the diplacement i the hrtet ditance between the tw lcatin. hu the diplacement can neer be lner than the lenth f path, but it can be le. r an path that i nt a inle traiht line ement, the lenth f path will be lner than the diplacement. 4. Since bth the batter and the ball tarted their mtin at the ame lcatin (where the ball wa hit) and ended their mtin at the ame lcatin (where the ball wa cauht), the diplacement f bth wa the ame. 5. he manitude f the ectr um need nt be larer than the manitude f either cntributin ectr. r eample, if the tw ectr bein added are the eact ppite f each ther, the ectr um will hae a manitude f. he manitude f the um i determined b the anle between the tw cntributin ectr. 6. If the tw ectr are in the ame directin, the manitude f their um will be a maimum, and will be 7.5 m. If the tw ectr are in the ppite directin, the manitude f their um will be a minimum, and will be.5 m. If the tw ectr are riented in an ther cnfiuratin, the manitude f their um will be between.5 m and 7.5 m. 7. w ectr f unequal manitude can neer add t ie the zer ectr. Hweer, three ectr f unequal manitude can add t ie the zer ectr. If their emetric um uin the tail-t-tip methd ie a cled trianle, then the ectr um will be zer. See the diaram, in which A B C A B C 8. (a) he manitude f a ectr can equal the lenth f ne f it cmpnent if the ther cmpnent f the ectr are all ; i.e. if the ectr lie aln ne f the crdinate ae. (b) he manitude f a ectr can neer be le than ne f it cmpnent, becaue each cmpnent cntribute a pitie amunt t the erall manitude, thruh the Ptharean relatinhip. he quare rt f a um f quare i neer le than the ablute alue f an indiidual term. 9. A particle with cntant peed can be acceleratin, if it directin i chanin. Driin n a cured radwa at cntant peed wuld be an eample. Hweer, a particle with cntant elcit cannt be acceleratin it acceleratin mut be zer. It ha bth cntant peed and cntant directin. 5 Pearn Educatin, Inc., Upper Saddle Rier, J. All riht reered. hi material i prtected under all cpriht law a the currentl eit. prtin f thi material ma be reprduced, in an m r b an mean, withut permiin in writin m the 4

2 Chapter 3 Kinematic in w Dimenin; Vectr. find the initial peed, ue the linht t ht the rc directl hrizntall (n initial ertical peed) m a heiht f meter. he ertical diplacement f the rc can be related t the time f fliht b Eq. -b. ae dwnward t be pitie. t at m t t m 9.8 m.45. Meaure the hrizntal rane R f the rc with the meter tic. hen, if we meaure the hrizntal rane R, we nw that R t.45, and R.45. he nl meaurement are the heiht f fall and the rane, bth f which can be meaured b a meter tic.. Aume that the bullet wa fired m behind and belw the airplane. A the bullet re in the air, it ertical peed wuld be lwed b bth rait and air reitance, and it hrizntal peed wuld be lwed b air reitance. If the altitude f the airplane wa lihtl belw the maimum heiht f the bullet, then at the altitude f the airplane, the bullet wuld be min quite lwl in the ertical directin. If the bullet hrizntal peed had al lwed enuh t apprimatel match the peed f the airplane, then the bullet elcit relatie t the airplane wuld be mall. With the bullet min lwl, it culd afel be cauht b hand.. he min walwa will be min at the ame peed a the car. hu, if u are n the walwa, u are min the ame peed a the car. Yur elcit relatie t the car i, and it i ea t et int the car. But it i er difficult t eep ur balance while trin t it dwn int a min car m a tatinar platm. It i eaier t eep ur balance b teppin n t the min platm while walin, and then ettin int the car with a elcit f relatie t the car. 3. Yur reference ame i that f the train u are ridin. If u are traelin with a relatiel cntant elcit (nt er a hill r arund a cure r draticall chanin peed), then u will interpret ur reference ame a bein at ret. Since u are min ward fater than the ther train, the ther train i min bacward relatie t u. Seein the ther train pat ur windw m nt t rear mae it l lie the ther train i in bacward. hi i imilar t pain a emi truc n the intertate ut f a paener windw, it l lie the truc i in bacward. 4. When u tand till under the umbrella in a ertical rain, u are in a clinder-haped lume in which there i n rain. he rain ha n hrizntal cmpnent f elcit, and the rain cannt me m utide that clinder int it. Yu ta dr. But a u run, u hae a ward hrizntal elcit relatie t the rain, and the rain ha a bacward hrizntal elcit relatie t u. It i the ame a if u were tandin till under the umbrella but the rain had me hrizntal cmpnent f elcit tward u. he perfectl ertical umbrella wuld nt cmpletel hield u. 5. (a) he ball land at the ame pint m which it wa thrwn inide the train car bac in the thrwer hand. (b) If the car accelerate, the ball will land behind the pint m which it wa thrwn. (c) If the car decelerate, the ball will land in nt f the pint m which it wa thrwn. (d) If the car rund a cure (aume it cure t the riht), then the ball will land t the left f the pint m which it wa thrwn. (e) he ball will be lwed b air reitance, and will land behind the pint m which it wa thrwn. 6. Bth rwer need t cer the ame "cr rier" ditance. he rwer with the reatet peed in the "cr rier" directin will be the ne that reache the ther ide firt. he current ha n bearin n the prblem becaue the current den't help either f the bat me acr the rier. hu the rwer headin traiht acr will reach the ther ide firt. All f hi rwin eft ha ne int 5 Pearn Educatin, Inc., Upper Saddle Rier, J. All riht reered. hi material i prtected under all cpriht law a the currentl eit. prtin f thi material ma be reprduced, in an m r b an mean, withut permiin in writin m the 4

3 Giancli Phic: Principle with Applicatin, 6 th Editin crin the rier. r the uptream rwer, me f hi rwin eft e int battlin the current, and hi "cr rier" peed will be nl a actin f hi rwin peed. 7. he baeball i hit and cauht at apprimatel the ame heiht, and the rane mula f R in i particularl applicable. hu the baeball plaer i judin the initial peed f the ball and the initial anle at which the ball wa hit. 8. he arrw huld be aimed abe the taret, becaue rait will deflect the arrw dwnward m a hrizntal fliht path. he anle f aim (abe the hrizntal) huld increae a the ditance m the taret increae, becaue rait will hae mre time t act in deflectin the arrw m a traiht-line path. If we aume that the arrw when ht i at the ame heiht a the taret, then the rane mula i applicable: R in in R. A the rane and hence the arument f the inere ine functin increae, the anle increae. 9. he hrizntal cmpnent f the elcit ta cntant in prjectile mtin, aumin that air reitance i neliible. hu the hrizntal cmpnent f elcit. ecnd after launch will be the ame a the hrizntal cmpnent f elcit. ecnd after launch. In bth cae the hrizntal elcit will be ien b c 3 m c 3 6 m.. (a) Cannnball A, with the larer anle, will reach a hiher eleatin. It ha a larer initial ertical elcit, and b Eq. -c, will rie hiher bee the ertical cmpnent f elcit i. (b) Cannnball A, with the larer anle, will ta in the air lner. It ha a larer initial ertical elcit, and tae mre time t decelerate t and tart t fall. (c) he cannnball with a launch anle clet t 45 will trael the farthet. he rane i a maimum a launch anle f 45, and decreae anle either larer r maller than 45. Slutin t Prblem. he reultant ectr diplacement f the car i ien b D D D. he wetward diplacement i 5 85c m and the uth diplacement i 85in m. he reultant diplacement ha a manitude f 8 m. he directin i R wet uthwet Duth- wet tan uth f wet.. he truc ha a diplacement f 8 6 blc nrth and blc D wet D R m eat. he reultant ha a manitude f f tan nrth f eat. blc and a directin D nrth D eat D uth D R 3. Label the ICORREC ectr a ectr X. hen i. 3-6 (c) illutrate the relatinhip V+X=V ia the tail-t-tip methd. hu X V V. V X 5 Pearn Educatin, Inc., Upper Saddle Rier, J. All riht reered. hi material i prtected under all cpriht law a the currentl eit. prtin f thi material ma be reprduced, in an m r b an mean, withut permiin in writin m the 43 V

4 Chapter 3 Kinematic in w Dimenin; Vectr 4. Gien that V 6.8 unit and V 7.4 unit, the manitude f V i ien b V V V unit. he directin V i ien b an anle f ai. 7.4 tan , r 47 belw the pitie - V V 5. he ectr the prblem are drawn apprimatel t cale. he reultant ha a lenth f 58 m and a directin 48 nrth f eat. If calculatin are dne, the actual reultant huld be 57.4 m at 47.5 nrth f eat. V V V R V 3 V 3 V V 6. he um i fund b addin the cmpnent f ectr V and V V=V +V 8., 3.7,. 3.9, 8., 4.4.9,.8, 4.4 V V (a) See the accmpanin diaram (b) V 4.3c unit V 4.3in unit (c) V V V unit 8.6 tan 34.8 abe the ai.7 V 34.8 V V 8. (a) V 6.6 unit V unit V 8.5c unit (b) V +V V V V V V +V V,.6, in unit unit tan 84.6 he um ha a manitude f 6. unit, and i 84 clcwie m the neatie -ai, r 96 cunterclcwie m the pitie -ai. V V V V 9. (a) nrth 735 m h c m h wet 735 m h in m h (b) d t 55 m h 3. h 65 m nrth nrth d t 487 m h 3. h 46 m wet wet 5 Pearn Educatin, Inc., Upper Saddle Rier, J. All riht reered. hi material i prtected under all cpriht law a the currentl eit. prtin f thi material ma be reprduced, in an m r b an mean, withut permiin in writin m the 44

5 Giancli Phic: Principle with Applicatin, 6 th Editin A B C 44.c A 44.in c B 6.5in c 7. C 3.in 7 3. (a) A+B+C (b) A C A+B+C A+B+C c A 44.in c 7. C 3.in tan A C A C A A B C tan c A 44. in c B 6.5in (a) B A B A te that ince the cmpnent i neatie and the cmpnent i pitie, the ectr i in the nd quadrant..3 B A tan.4 abe ai B A (b) A B A B A B C te that ince the cmpnent i pitie and the cmpnent i neatie, the ectr i in the 4 th quadrant..3 A B tan.4 belw ai 53.7 Cmparin the reult hw that B A i the ppite f A B. 44.c A 44.in c B 6.5in c 7. C 3.in 7 3. (a) A B C A B C te that ince the cmpnent i pitie and the cmpnent i neatie, the ectr i in the 4 th quadrant. 3.3 A B C tan 3. belw ai Pearn Educatin, Inc., Upper Saddle Rier, J. All riht reered. hi material i prtected under all cpriht law a the currentl eit. prtin f thi material ma be reprduced, in an m r b an mean, withut permiin in writin m the 45

6 Chapter 3 Kinematic in w Dimenin; Vectr (b) A B C A B C A B C tan (c) C A B C A B te that ince bth cmpnent are neatie, the ectr i in the 3 rd quadrant C A B tan 7.9 belw ai 4.3 te that the anwer t (c) i the eact ppite f the anwer t (b). 4. A B C 44.c A 44.in c B 6.5in c 7. C 3.in 7 3. (a) B A B A te that ince bth cmpnent are neatie, the ectr i in the 3 rd quadrant B A tan.8 belw ai 9.5 (b) A 3B C A 3B C te that ince the cmpnent i pitie and the cmpnent i neatie, the ectr i in the 4 th quadrant A 3B C tan 35.3 belw ai.6 5. he cmpnent i neatie and the cmpnent i pitie, ince the ummit i t the wet f nrth. he anle meaured cunterclcwie m the pitie ai wuld be.4. hu the cmpnent are fund t be 458in m 458 c m z 45 m r 45 m,387 m,45 m r m unit 7. Che dwnward t be the pitie directin. he riin will be at the pint where the tier leap m the rc. In the hrizntal directin, 3.5 m and a. In the ertical directin,, a 9.8 m,, and the final lcatin 6.5 m. he time the tier t reach the rund i fund m applin Eq. -b t the ertical mtin. 5 Pearn Educatin, Inc., Upper Saddle Rier, J. All riht reered. hi material i prtected under all cpriht law a the currentl eit. prtin f thi material ma be reprduced, in an m r b an mean, withut permiin in writin m the 46

7 Giancli Phic: Principle with Applicatin, 6 th Editin 6.5m t a t 6.5m 9.8 m t t.5 ec 9.8 m he hrizntal diplacement i calculated m the cntant hrizntal elcit. t 3.5 m.5 ec 4. m 8. Che dwnward t be the pitie directin. he riin will be at the pint where the dier die m the cliff. In the hrizntal directin,.8 m and a. In the ertical directin,, a 9.8 m,, and the time f fliht i t 3. m applin Eq. -b t the ertical mtin. t a t 9.8 m m. he heiht f the cliff i fund he ditance m the bae f the cliff t where the dier hit the water i fund m the hrizntal mtin at cntant elcit: t.8 m m 9. Appl the rane mula m Eample 3-8. in R R. m 9.8 m in m in.439 3,77 here are tw anle becaue each anle ie the ame rane. If ne anle i 45, then 45 i al a lutin. he tw path are hwn in the raph Che upward t be the pitie directin. he riin i the pint m which the pebble are releaed. In the ertical directin, a 9.8 m, the elcit at the windw i, and the ertical diplacement i 4.5 m. he initial elcit i fund m Eq. -c. a a 9.8m 4.5 m 9.39m ind the time the pebble t trael t the windw m Eq. -a. 9.4m at t.958 a 9.8 m ind the hrizntal peed m the hrizntal mtin at cntant elcit. t t 5. m m hi i the peed f the pebble when the hit the windw.. Che dwnward t be the pitie directin. he riin will be at the pint where the ball i thrwn m the rf f the buildin. In the ertical directin,, a 9.8 m,, and the diplacement i 45. m. he time f fliht i fund m applin Eq. -b t the ertical mtin. 5 Pearn Educatin, Inc., Upper Saddle Rier, J. All riht reered. hi material i prtected under all cpriht law a the currentl eit. prtin f thi material ma be reprduced, in an m r b an mean, withut permiin in writin m the 47

8 Chapter 3 Kinematic in w Dimenin; Vectr 45. m t a t 45. m 9.8 m t t 3.3 ec 9.8 m he hrizntal peed (which i the initial peed) i fund m the hrizntal mtin at cntant elcit: t t 4. m m.. Che the pint at which the ftball i iced the riin, and che upward t be the pitie directin. When the ftball reache the rund aain, the diplacement i. r the ftball, 8.in 35. m, a 9.8 m and the final elcit will be the ppite f the tartin elcit (reference prblem 3-8). Ue Eq. -a t find the time f fliht. 8.in 35. m 8.in 35. m at t. a 9.8 m 3. Che dwnward t be the pitie directin. he riin i the pint where the ball i thrwn m the rf f the buildin. In the ertical directin,,, and a 9.8 m. he initial hrizntal elcit i. m/ and the hrizntal rane i 36. m. he time f fliht i fund m the hrizntal mtin at cntant elcit. t t 36. m. m.6 he ertical diplacement, which i the heiht f the buildin, i fund b applin Eq. -b t the ertical mtin. t a t 9.8 m.6.9 m 4. (a) Ue the Leel hrizntal rane mula m Eample 3-8. (b) R in R 7.8 m 9.8 m in in m w increae the peed b 5.% and calculate the new rane. he new peed wuld be 9.6 m.5.m and the new rane wuld be R in.m in m an increae f.8 m % increae. 8.6 m, in 5. Calculate the rane a deried in Eample 3-8: R. If the launchin peed and anle are held cntant, the rane i inerel prprtinal t the alue f. he acceleratin due t rait n the Mn i /6 th that n Earth. in in R R R R Earth Mn Earth Earth Mn Mn Earth R R 6R Earth Mn Earth Earth Mn Mn 5 Pearn Educatin, Inc., Upper Saddle Rier, J. All riht reered. hi material i prtected under all cpriht law a the currentl eit. prtin f thi material ma be reprduced, in an m r b an mean, withut permiin in writin m the 48

9 Giancli Phic: Principle with Applicatin, 6 th Editin hu n the Mn, the pern can jump 6 time farther. 6. (a) Che dwnward t be the pitie directin. he riin i the pint where the bullet leae the un. In the ertical directin,,, and a 9.8 m. In the (b) hrizntal directin, 75. m and 8 m. he time f fliht i fund m the hrizntal mtin at cntant elcit. t t 75. m 8 m.467 hi time can nw be ued in Eq. -b t find the ertical drp f the bullet. t a t 9.8 m m r the bullet t hit the taret at the ame leel, the leel hrizntal rane mula f Eample 3-8 applie. he rane i 75. m, and the initial elcit i 8 m/. Slin the anle f launch reult in the fllwin. in R 75. m 9.8 m R in in.65 8 m Becaue f the mmetr f the rane mula, there i al an anwer f the cmplement f the abe anwer, which wuld be hat i an unreanable anwer m a practical phical iewpint it i pintin the un almt traiht up. 7. Che dwnward t be the pitie directin. he riin i the pint where the upplie are drpped. In the ertical directin,, a 9.8 m,, and the final pitin i 6 m. he time f fliht i fund m applin Eq. -b t the ertical mtin. t a t 6 m 9.8 m t 6m t m te that the peed f the airplane de nt enter int thi calculatin. 8. he hrizntal cmpnent f the peed de nt chane durin the cure f the mtin, and. he net ertical diplacement i if the firin leel equal the landin leel. Eq. -c f then ie a. hu, and m the hrizntal. he initial f f f peed i. he final peed i. hu f f f. f 9. Che upward t be the pitie directin. he riin i pint m which the ftball i iced. he initial peed f the ftball i. m. We hae in m,, and a 9.8 m. In the hrizntal directin, c m, and 36. m. he time f fliht t reach the alpt i fund m the hrizntal mtin at cntant peed: t t 36. m 5.97 m.54. w ue thi time with the ertical mtin data and Eq. -b t find the heiht f the ftball when it reache the hrizntal lcatin f the alpt. t a t.4 m m.54.4 m 5 Pearn Educatin, Inc., Upper Saddle Rier, J. All riht reered. hi material i prtected under all cpriht law a the currentl eit. prtin f thi material ma be reprduced, in an m r b an mean, withut permiin in writin m the 49

10 Chapter 3 Kinematic in w Dimenin; Vectr Since the ball heiht i le than 3. m, the ftball de nt clear the bar. It i.76 m t lw when it reache the hrizntal lcatin f the alpt. 3. Che the riin t be where the prjectile i launched, and upward t be the pitie directin. he initial elcit f the prjectile i, the launchin anle i, a, and in. (a) he maimum heiht i fund m Eq. -c, a, with at (b) (c) (d) the maimum heiht. in in 65. m in m a 9.8m ma he ttal time in the air i fund m Eq. -b, with a ttal ertical diplacement f the ball t reach the rund. t a t in t t t 65.m in34.5 in 9.8 m 7.54 and t he time f repreent the launchin f the ball. he ttal hrizntal ditance cered i fund m the hrizntal mtin at cntant elcit. t c t 65. m c m he elcit f the prjectile.5 after firin i fund a the ectr um f the hrizntal and ertical elcitie at that time. he hrizntal elcit i a cntant c 65. m c m. he ertical elcit i fund m Eq. -a. at in t 65. m in m.5. m hu the peed f the prjectile i m. he directin abe the hrizntal i ien b. tan tan Che the riin t be at rund leel, under the place where the prjectile i launched, and upward t be the pitie directin. r the prjectile, 65. m, 37., a, 5, and in (a) he time taen t reach the rund i fund m Eq. -b, with a final heiht f. t a t 5 in t t (b) (c) t in in , Che the pitie in ince the prjectile wa launched at time t =. he hrizntal rane i fund m the hrizntal mtin at cntant elcit. t c t 65. m c m At the intant jut bee the particle reache the rund, the hrizntal cmpnent f it elcit i the cntant c 65. m c m. he ertical cmpnent i fund m Eq. -a. 5 Pearn Educatin, Inc., Upper Saddle Rier, J. All riht reered. hi material i prtected under all cpriht law a the currentl eit. prtin f thi material ma be reprduced, in an m r b an mean, withut permiin in writin m the 5

11 Giancli Phic: Principle with Applicatin, 6 th Editin (d) (e) at t in 65. m in m.4 63.m he manitude f the elcit i fund m the and cmpnent calculated in part c) abe. he directin f the elcit i 5.9 m 63.m 8.7 m 63. tan tan 5.6, and the bject i 5.9 min 5.6 belw the hrizn. (f) he maimum heiht abe the cliff tp reached b the prjectile will ccur when the - elcit i, and i fund m Eq. -c. a in ma in 65. m in 37. ma 9.8m 78. m 3. Che the riin t be the pint f releae f the ht put. Che upward t be the pitie directin. hen, 5.5in 34. m 8.67 m, a 9.8 m, and. m at the end f the mtin. Ue Eq. -b t find the time f fliht. t a t a t t t a a 9.8 Che the pitie reult ince the time mut be reater than. w calculate the hrizntal ditance traeled uin the hrizntal mtin at cntant elcit. t 5.5c 34 m m.99, Che the riin t be where the prjectile i launched, and upward t be the pitie directin. he initial elcit f the prjectile i, the launchin anle i, a, and in. in he rane f the prjectile i ien b the rane mula m Eample 3-8, R. he maimum heiht f the prjectile will ccur when it ertical peed i. Appl Eq. -c. in a in ma ma w find the anle which R. ma in in in R in ma in in c 4 c in tan 4 tan Pearn Educatin, Inc., Upper Saddle Rier, J. All riht reered. hi material i prtected under all cpriht law a the currentl eit. prtin f thi material ma be reprduced, in an m r b an mean, withut permiin in writin m the 5

12 Chapter 3 Kinematic in w Dimenin; Vectr 34. Che the riin t be the lcatin m which the balln i fired, and che upward a the pitie directin. Aume the b in the tree i a ditance H up m the pint at which the balln i fired, and that the tree i a ditance D hrizntall m the pint at which the balln i fired. he equatin f mtin the balln and b are a fllw, uin cntant acceleratin relatinhip. c t in t t H t Balln Balln B Ue the hrizntal mtin at cntant elcit t find the elaped time after the balln ha traeled D t the riht. D D c t t D D c Where i the balln erticall at that time? D D D in t t in Dtan Balln D D c c c Where i the b erticall at that time? te that H D tan. D H t H Dtan B D c c he b and the balln are at the ame heiht and the ame hrizntal lcatin at the ame time. hu the cllide! 35. Che the riin t be the lcatin n the rund directl belw the airplane at the time the upplie are drpped, and che upward a the pitie directin. r the upplie, 35 m,, a, and the final lcatin i m. he initial (and cntant) elcit f the upplie i 69.4 m. (a) he time the upplie t reach the rund i fund m Eq. -b. t a t at D D H (b) t a 35 m 9.8 m 6.93 hen the hrizntal ditance f trael the pacae i fund m the hrizntal mtin at cntant elcit. t 69.4 m m w the upplie hae t trael a hrizntal ditance f nl 45 m. hu the time f fliht will be le, and i fund m the hrizntal mtin at cntant elcit. t t 45 m 69.4 m 6.4. he mtin mut atif Eq. -b thi new time, but the ame ertical diplacement and acceleratin. t a t 35 m 9.8 m 6.4 a t 8.37 m t 6.4 tice that ince thi i a neatie elcit, the bject mut be prjected DOW. 5 Pearn Educatin, Inc., Upper Saddle Rier, J. All riht reered. hi material i prtected under all cpriht law a the currentl eit. prtin f thi material ma be reprduced, in an m r b an mean, withut permiin in writin m the 5

13 Giancli Phic: Principle with Applicatin, 6 th Editin (c) he hrizntal cmpnent f the peed f the upplie upn landin i the cntant hrizntal peed f 69.4 m/. he ertical peed i fund m Eq. -a. a t 8.37 m 9.8 m m hu the peed i ien b 69.4 m 68.4 m 97.4 m 36. Call the directin f the bat relatie t the water the pitie directin.. m 7.5 m (a) jer jer bat rel. rel. water rel. bat water (b) jer jer bat rel. rel. water rel. bat water 9.7 m in the directin the bat i min. m 7.5 m 5.3 m in the directin the bat i min 37. Call the directin f the flw f the rier the directin, and the directin f Huc walin relatie t the raft the directin.,.6 m.7, m Huc Huc raft rel. rel. ban rel. raft ban.7,.6 m Manitude: m Huc rel. ban.6 Directin: tan 9 relatie t rier We hae 5m car rel.. Ue the diaram, illutratin, t calculate the nw rel. nw rel. car rel. rund rund car rund ther peed. car rel. rund c 3 5 m c 3 9 m nw rel. nw rel. car car nw rel. rund tan 3 5 m tan 3 4 m nw rel. car rel. rund rund Huc rel. raft nw rel. car 3 car rel. rund Huc rel. ban raft rel. ban current nw rel. rund 39. Call the directin f the flw f the rier the directin, and the directin the bat i headed the directin. (a) m bat rel. water rel. bat rel. hre hre water. tan 7.6, relatie t hre.3 bat rel. hre water rel. hre bat rel. water 5 Pearn Educatin, Inc., Upper Saddle Rier, J. All riht reered. hi material i prtected under all cpriht law a the currentl eit. prtin f thi material ma be reprduced, in an m r b an mean, withut permiin in writin m the 53

14 Chapter 3 Kinematic in w Dimenin; Vectr (b) he pitin f the bat after 3. ecnd i ien b d t.,.3 m 3. ec bat rel. hre 3.6 m dwntream, 6.9 m acr the rier A a manitude and directin, it wuld be 7.8 m awa m the tartin pint, at an anle f 6.4 relatie t the hre. 4. If each plane ha a peed f 785 m/hr, then their relatie peed f apprach i 57 m/hr. If the plane are m apart, then the time eaie actin i fund m d. m 36 ec d t t m hr hr 4. Call eat the pitie directin and nrth the pitie directin. hen the fllwin ectr elcit relatinhip eit. (a) plane rel. plane air rel. rund rel. air rund (b) plane rel. rund, 6 m h c 45.,in 45. m h 7.7, 59 m h 7.7 m h 59 m h 54 m h 7.7 tan 7.6 eat f uth 59 he plane i awa m it intended pitin b the ditance the air ha caued it t me. he wind peed i m/h, after min (/6 h), the plane i ff cure b t m h h 7 m. 4. Call eat the pitie directin and nrth the pitie directin. hen the fllwin ectr elcit relatinhip eit. plane rel. plane air rel. rund rel. air rund 6 plane rel. air plane rel. rund air rel. rund, 6in,6 c m h plane rel. rund c 45.,in 45. m h Equate cmpnent in the abe equatin. 6in c 45. in c , wet f uth 43. rm the diaram in fiure 3-9, it i een that c.85 m c 4.4.4m. bat rel. hre bat rel. water bat rel. hre water rel. hre plane rel. air bat rel. water plane rel. rund air rel. rund 5 Pearn Educatin, Inc., Upper Saddle Rier, J. All riht reered. hi material i prtected under all cpriht law a the currentl eit. prtin f thi material ma be reprduced, in an m r b an mean, withut permiin in writin m the 54

15 Giancli Phic: Principle with Applicatin, 6 th Editin 44. Call the directin f the bat relatie t the water the directin, and upward the directin. Al ee the diaram. paener paener bat rel. rel. water rel. bat water.5 c 45,.5in 45 m.5, m.854,.354 m paener rel. water bat rel. water paener rel. bat paener rel. water m 45. Call the directin f the flw f the rier the directin, and the directin traiht acr the rier the directin. he bat i traelin traiht acr the rier. he bat i headed at 8.5 uptream, at a peed f.6 m. bat rel. water (a) (b) in.6 m in m water rel. bat rel. water rel. hre water hre c.6 m c m bat rel. bat rel. bat rel. hre water hre bat rel. hre water rel. hre bat rel. water 46. Call the directin f the flw f the rier the directin, and the directin traiht acr the rier the directin. rm the diaram, tan m 6 m.9. Equate the ertical cmpnent f the elcitie t find the peed f the bat relatie t the hre. bat rel. hre c in 45 bat rel. water in 45.7 m.35 m bat rel. hre c.9 Equate the hrizntal cmpnent f the elcitie. in c 45 bat rel. bat rel. water hre water rel. hre c 45 in water bat rel. bat rel. rel. hre water hre.7 m c m in.9.69 m 6 m m bat rel. hre 45 water rel. hre bat rel. water 47. Call the directin f the flw f the rier the directin, and the directin traiht acr the rier the directin. Call the lcatin f the wimmer tartin pint the riin. (a) wimmer wimmer water rel. rel. hre rel. water hre.4,.45 m,.45 m.4 m, Since the wimmer tart m the riin, the ditance cered in the and directin will be eactl prprtinal t the peed in the directin. t.4 m 67 m t 75 m.45 m wimmer rel. water water rel. hre 5 Pearn Educatin, Inc., Upper Saddle Rier, J. All riht reered. hi material i prtected under all cpriht law a the currentl eit. prtin f thi material ma be reprduced, in an m r b an mean, withut permiin in writin m the 55 wimmer rel. hre

16 Chapter 3 Kinematic in w Dimenin; Vectr (b) he time i fund m the cntant elcit relatinhip either the r directin. 75 m t t 7.45m 48. (a) Call the directin f the flw f the rier the directin, and the directin traiht acr the rier the directin. water rel..4 m.4 in in m.45 hre wimmer rel. water (b) rm the diaram her peed with repect t the hre i c.45 m c m wimmer rel. hre wimmer rel. water he time t cr the rier can be fund m the cntant elcit relatinhip. 75 m t t min.6 m water rel. hre wimmer rel. water wimmer rel. hre 49. Call eat the pitie directin and nrth the pitie directin. he fllwin i een m the diaram. Appl the law f ine t the trianle med b the three ectr. plane air rel. air rel. rel. air rund rund in in5 in5 in plane rel. air 95 in in S the plane huld head in a directin f nrth f eat. plane rel. air 35 5 plane rel. rund air rel. rund 5. ae the riin t be the lcatin at which the peeder pae the plice car, in the reference ame f the unaccelerated plice car. he peeder i traelin at 45 m h m 3.6 m h 4.8 m relatie t the rund, and the pliceman i traelin at 95 m h m 3.6 m h 6.39 m relatie t the rund. Relatie t the unaccelerated plice car, the peeder i traelin at 3.89 m, and the plice car i nt min. D all f the calculatin in the ame f reference f the unaccelerated plice car. he pitin f the peeder in the chen reference ame i ien b t. he pitin f the pliceman in the chen reference ame i ien b a t, t p p. he plice car ertae the peeder when thee tw ditance are the ame.; i.e.,. p p p t a t 3.89 m t m t t t t t t t , Pearn Educatin, Inc., Upper Saddle Rier, J. All riht reered. hi material i prtected under all cpriht law a the currentl eit. prtin f thi material ma be reprduced, in an m r b an mean, withut permiin in writin m the 56

17 Giancli Phic: Principle with Applicatin, 6 th Editin Since the plice car den t accelerate until t., the crrect anwer i t ae the riin t be the lcatin at which the peeder pae the plice car. he peed f the peeder i. he pitin f the peeder after the 7. ecnd i t 7.. he pitin f the plice car i calculated baed n the fact that the car traeled ecnd at the riinal elcit, and then 6 ecnd under acceleratin. te that the plice car elcit mut hae the unit chaned. m 95 m h 6.39 m a. m p p 3.6 m h. 6. a 6..7 m p p p p he plice car ertae the peeder when thee tw ditance are the ame; i.e.,. p.7 m 3.6 m h 7.7 m 4 m h 7 m 5. Call eat the pitie directin and nrth the pitie directin. rm the firt diaram, thi relatie elcit relatinhip i een. carrel. carrel. carrel. carrel. treet car treet car tan Wet f rth r the ther relatie elcit relatinhip: carrel. carrel. carrel. carrel. treet car treet car tan Suth f Eat m h m h tice that the tw relatie elcitie are ppite f each ther: car rel. car rel. car car car rel. treet car rel. car car rel. treet car rel. car car rel. treet car rel. treet 53. Since the arrw will tart and end at the ame heiht, ue the rane mula deried in Eample 3-8. he rane i 7 m, and the initial peed f the arrw i 35 m/. in R 7 m 9.8 m R in.6 35 m in.6 6.,83.8 Onl the firt anwer i practical, the reult i he plumber diplacement in cmpnent ntatin i d 5 m, 5 m, m. Since thi i a 3-dimeninal prblem, it require anle t determine hi lcatin (imilar t latitude and lnitude n the urface f the Earth). In the - plane, thi fllw. d 5 tan tan 7 Suth f Eat d 5 d d d 5 Pearn Educatin, Inc., Upper Saddle Rier, J. All riht reered. hi material i prtected under all cpriht law a the currentl eit. prtin f thi material ma be reprduced, in an m r b an mean, withut permiin in writin m the 57

18 Chapter 3 Kinematic in w Dimenin; Vectr d d d m r the ertical mtin, cnider anther riht trianle, made up f d a ne le, and the ertical diplacement d a the ther le. See the ecnd fiure, z and the fllwin calculatin. d m z tan tan Belw the Hrizntal d 55.9 m d d d z d d d d d d z z m he reult i that the diplacement i 57 m, at an anle f Belw the Hrizntal. 7 Suth f Eat, and 55. Aume a cntant upward lpe, and the deceleratin i aln a traiht line. he tartin elcit aln that line i m h m 3.6 m h 33.3m. he endin elcit i m/. he acceleratin i fund m Eq. -a. 33.3m at 33.3m a 6. a 5.56m 6. he hrizntal acceleratin i he ertical acceleratin i a a c 5.56 m c3 4.7 m. hriz a ain 5.56 m in 3.8 m ert he hrizntal acceleratin i t the left in the tetb diaram, and the ertical acceleratin i dwn. 56. Manitude Directin tan 3.6 relatie t ai 75.4 See the diaram the tw pible anwer Che the directin t be the directin f train trael (the directin the paener i facin) and che the directin t be up. hi relatinhip eit amn the elcitie:. rm the diaram, find the rain rel. rain rel. train rel. rund train rund eprein the peed f the raindrp. train rel. rund tan rain rel.. rund tan rain rel. rund rain rel. rund rain rel. train train rel. rund rain rel. rund 58. Call eat the pitie directin and nrth the pitie directin. hen thi relatie elcit relatinhip fllw (ee the accmpanin diaram). plane rel. plane air rel. rund rel. air rund Equate the cmpnent f the elcit ectr. plane rel. air 45 plane rel. rund air rel. rund 5 Pearn Educatin, Inc., Upper Saddle Rier, J. All riht reered. hi material i prtected under all cpriht law a the currentl eit. prtin f thi material ma be reprduced, in an m r b an mean, withut permiin in writin m the 58

19 Giancli Phic: Principle with Applicatin, 6 th Editin 5 m h c m h wind wind. rm the cmpnent f the abe equatin: 5in in m h wind- he manitude f the wind elcit i wind m h 66.6 m h m h wind wind- wind-. he directin f the wind i wind- tan tan 37. nrth f eat wind Wr in the ame f reference in which the train i at ret. hen, relatie t the train, the car i min at m/h. he car ha t trael m in that ame f reference t pa the train, and the time t pa can be fund m the cntant hrizntal elcit relatinhip. t t ame directin ame directin m 36.5 h 8 m h h he car trael m in the ame f reference f the tatinar train, but relatie t the rund, the car i traelin at 95 m/hr and relatie t the rund the car trael thi ditance: t ame directin 95 m h.5 h 4.8 m If the car and train are traelin in ppite directin, then the elcit f the car relatie t the train will be 7 m/h. hu the time t pa will be m 36 t h.. ppite directin 7 m h 7 h ppite directin he ditance traeled b the car relatie t the rund will be t 95 m h h.56 m. ppite directin 7 6. he time f fliht i fund m the cntant elcit relatinhip hrizntal mtin. t t 8. m 9.m.88 he mtin i mmetric in time it tae half the time f fliht t rie, and half t fall. hu the time the jumper t fall m hi hihet pint t the rund i.44 ec. Hi ertical peed i zer at the hihet pint. rm thi time, tartin ertical peed, and the acceleratin f rait, the maimum heiht can be fund. Call upward the pitie directin. he pint f maimum heiht i the tartin pitin, the endin pitin i, the tartin ertical peed i, and a. Ue Eq. -b t find the heiht. t a t 9.8 m m. 6. Aume that the lf ball tae ff and land at the ame heiht, that the rane mula deried in Eample 3-8 can be applied. he nl ariable i t be the acceleratin due t rait. R in R in Earth Earth Mn Mn Earth Earth Earth Mn in Mn Mn Mn Earth 8 m R in 35 m.9 R Mn.9.9 m Earth 5 Pearn Educatin, Inc., Upper Saddle Rier, J. All riht reered. hi material i prtected under all cpriht law a the currentl eit. prtin f thi material ma be reprduced, in an m r b an mean, withut permiin in writin m the 59

20 Chapter 3 Kinematic in w Dimenin; Vectr 6. he minimum peed will be that which the ball jut clear the fence; i.e., the ball ha a heiht f 7.5 m when it i 95 m hrizntall m hme plate. he riin i at hme plate, with upward a the pitie directin. r the ball,. m, 7.5 m, a, in, c, and 38. See the diaram. m (nt t cale). r the hrizntal mtin at cntant elcit, 95 m t c t, and t c. r the ertical mtin, appl Eq. -b. t a t in t t Subtitute the alue f the time f fliht the firt ccurrence nl in the abe equatin, and then le the time. tin t in t c t tan. m 7.5 m 95 m tan m 3.78 inall, ue the time with the hrizntal rane t find the initial peed. 95 m c t 3 m t c 3.78 c Che dwnward t be the pitie directin. he riin i at the pint m which the dier puh ff the cliff. In the ertical directin, the initial elcit i, the acceleratin i a 9.8 m, and the diplacement i 35 m. he time f fliht i fund m Eq. -b. 35 m t a t 35 m 9.8 m t t m he hrizntal peed (which i the initial peed) i fund m the hrizntal mtin at cntant elcit. t t 5. m.7.9 m 64. Che the riin t be the lcatin n the rund directl underneath the ball when ered, and che upward a the pitie directin. hen the ball,.5 m,, a, and the lcatin when the ball jut clear the net i.9 m. he time the ball t reach the net i calculated m Eq. -b. t a t.9 m.5 m 9.8 m t.6 m t.5743 t net 9.8 m he elcit i fund m the hrizntal mtin at cntant elcit. 5. m t m. t.5743 hi i the minimum peed required t clear the net. 7.5 m 5 Pearn Educatin, Inc., Upper Saddle Rier, J. All riht reered. hi material i prtected under all cpriht law a the currentl eit. prtin f thi material ma be reprduced, in an m r b an mean, withut permiin in writin m the 6

21 Giancli Phic: Principle with Applicatin, 6 th Editin find the full time f fliht f the ball, et the final lcatin t be =, and aain ue Eq. -b. t a t. m.5 m 9.8 m t t.5 m ttal 9.8 m he hrizntal pitin where the ball land i fund m the hrizntal mtin at cntant elcit. t 6.5 m m. Since thi i between 5. and. m, the ball land in the "d" rein. 65. Wr in the ame f reference in which the car i at ret at rund leel. In thi reference ame, the helicpter i min hrizntall with a peed f 5 m h 55 m h 6 m h m 3.6 m h 6.67 m. r the ertical mtin, che the leel f the helicpter t be the riin, and dwnward t be pitie. hen the pacae diplacement i 78. m,, and a. he time the pacae t fall i calculated m Eq. -b. 78. m t a t 78. m 9.8 m t t 3.99 ec 9.8 m he hrizntal ditance that the pacae mut me, relatie t the tatinar car, i fund m the hrizntal mtin at cntant elcit. t 6.67 m m hu the anle under the hrizntal the pacae releae will be 78. m tan tan (t inificant fiure) m 66. (a) r the uptream trip, the bat will cer a ditance f D with a net peed f u, the D D time i t u u. r the dwntream trip, the bat will cer a ditance f D D D with a net peed f u, the time i t u u D D D rund trip will be t t t u u u. hu the ttal time the. (b) r the bat t directl acr the rier, it mut be anled aaint the current in uch a wa that the net elcit i traiht acr the rier, a in the picture. hi equatin mut be atified: u. bat rel. bat rel. water rel. hre water hre hu u, and the time t a ditance D acr bat rel. hre bat rel. water water rel. hre u bat rel. hre 5 Pearn Educatin, Inc., Upper Saddle Rier, J. All riht reered. hi material i prtected under all cpriht law a the currentl eit. prtin f thi material ma be reprduced, in an m r b an mean, withut permiin in writin m the 6

22 Chapter 3 Kinematic in w Dimenin; Vectr D D the rier i t. he ame relatinhip wuld be in effect crin u u bac, the time t cme bac i ien b t t and the ttal time i t t t D u he peed mut be reater than the peed u. he elcit f the bat relatie t the hre when in uptream i u. If u, the bat will nt me uptream at all, and the firt part f the trip wuld be impible. Al, in part b, we ee that i lner than u in the trianle, ince i the hptenue. 67. Che the riin t be the pint m which the prjectile i launched, and che upward a the pitie directin. he diplacement f the prjectile i 55 m, and the hrizntal rane f the prjectile i 95 m. he acceleratin in the directin i a, and the time f fliht i 7.6. he hrizntal elcit i fund m the hrizntal mtin at cntant elcit. 95 m t 5.7 m t 7.6 Calculate the initial elcit m the ien data and Eq. -b. t a t 55 m m m hu the initial elcit and directin f the prjectile are: 57.6 m tan tan m 5.7 m 57.6 m 63m 68. Che dwnward t be the pitie directin thi prblem. (a) he ertical cmpnent f her acceleratin i directed dwnward, and it manitude will be (b) ien b a a in.8 m in 3..9 m. he time t reach the bttm f the hill i calculated m Eq. -b, with a diplacement f 335 m,, and a.9 m. t a t 335 m.9 m t t 335 m.9 m he prper initial peed will be the which the ball ha traeled a hrizntal ditance mewhere between.78 m and. m while it chane heiht m. m t.6 m with a htin anle f 38.. Che the riin t be at the htin lcatin f the baetball, with upward a the pitie directin. hen the ertical diplacement i.5 m, a 9.8 m, in, and the (cntant) elcit i c. See the diaram (nt t cale). r the hrizntal mtin at cntant elcit, t c t and t c..78 m. m.5 m 5 Pearn Educatin, Inc., Upper Saddle Rier, J. All riht reered. hi material i prtected under all cpriht law a the currentl eit. prtin f thi material ma be reprduced, in an m r b an mean, withut permiin in writin m the 6

23 Giancli Phic: Principle with Applicatin, 6 th Editin r the ertical mtin, applin Eq. -b. t a t in t t Subtitute the eprein the time f fliht and le the initial elcit. in t t in tan c c c r r c tan.78 m, the hrtet ht: 9.8 m.78 m.8 m. c m.78 m tan 38.. m, the lnet ht: 9.8 m. m. m. c m. m tan Che the riin t be the lcatin at water leel directl underneath the dier when he left the bard. Che upward a the pitie directin. r the dier, 5. m, the final pitin i. m (water leel), a, the time f fliht i t.3, and the hrizntal diplacement i (a) (b) 3. m. he hrizntal elcit i determined m the hrizntal mtin at cntant elcit. 3. m t.3m t.3 he initial elcit i fund uin Eq. -b. t a t m 5. m m.3.5 m he manitude and directin f the initial elcit i.3m.5 m 3.4 m.5 m tan tan 48 abe the hrizntal.3m he maimum heiht will be reached when the elcit i zer. Ue Eq. -c. ma a.5 m 9.8 m 5. m (c) ma 5.3 m find the elcit when he enter the water, the hrizntal elcit i the (cntant) alue f.3m. he ertical elcit i fund m Eq. -a. at.5 m 9.8 m.3. m. he manitude and directin f thi elcit i ien b 5 Pearn Educatin, Inc., Upper Saddle Rier, J. All riht reered. hi material i prtected under all cpriht law a the currentl eit. prtin f thi material ma be reprduced, in an m r b an mean, withut permiin in writin m the 63

24 Chapter 3 Kinematic in w Dimenin; Vectr.3m. m.458 m m. m tan tan 77 belw the hrizntal.3m 7. (a) Che the riin t be the lcatin where the car leae the ramp, and che upward t be the pitie directin. At the end f it fliht er the 8 car, the car mut be at.5 m. Al the car,, a,, and m. he time f fliht i fund m the hrizntal mtin at cntant elcit: t t. hat eprein the time i ued in Eq. -b the ertical mtin. t a t (b) 9.8 m m.5 m 36 m Aain che the riin t be the lcatin where the car leae the ramp, and che upward t be the pitie directin. he diplacement f the car at the end f it fliht er the 8 car mut aain be.5 m. r the car, in, a, c, and m. he launch anle i. he time f fliht i fund m the hrizntal mtin at cntant elcit. t t c hat eprein the time i ued in Eq. -b the ertical mtin. t a t in c c 9.8 m m tan c m tan.5 m c m 7. Che the riin t be the pint at rund leel directl belw where the ball wa hit. Call upward the pitie directin. r the ball, we hae 8 m, 6, a,.9 m, and. m. (a) find the hrizntal diplacement f the ball, the hrizntal elcit and the time f fliht are needed. he (cntant) hrizntal elcit i ien b c. he time f fliht i fund m Eq. -b. t a t in t t t in in 4 5 Pearn Educatin, Inc., Upper Saddle Rier, J. All riht reered. hi material i prtected under all cpriht law a the currentl eit. prtin f thi material ma be reprduced, in an m r b an mean, withut permiin in writin m the 64

25 Giancli Phic: Principle with Applicatin, 6 th Editin (b) 9.8m 8 m in 6 8 m in m.9 m 5.34,.365 Che the pitie time, ince the ball wa hit at t. he hrizntal diplacement f the ball will be fund b the cntant elcit relatinhip hrizntal mtin. t c t 8 m c m 68 m he center fielder catche the ball riht at rund leel. He ran 5 m m = m t catch the ball, hi aerae runnin peed wuld be d m a 7.8 m 7.3m t Since the ball i bein cauht at the ame heiht m which it wa truc, ue the rane mula t find the hrizntal ditance the ball trael. in 3 m in 55 R m 9.8 m hen a een m abe, the lcatin f hme plate, the pint where the ball mut be cauht, and the initial lcatin f the utfielder are hwn in the diaram. he dar arrw hw the directin in which the utfielder mut run. he lenth f that ditance i fund m the law f cine a applied t the trianle. a b ab c Lcatin f catchin ball m 85 m Hme plate Initial lcatin f utfielder he anle c 37.7 m at which the utfielder huld run i fund m the law f ine. in in in in 8 r m 37.7 Since , the anle mut be btue, we che 97. Aume that the utfielder time runnin i the ame a the time f fliht f the ball. he time f fliht f the ball i fund m the hrizntal mtin f the ball at cntant elcit. R m R t c t t 5.35 c 3 m c 55 hu the aerae elcit f the utfielder mut be 97 relatie t the utfielder' line f iht t hme plate. a t d 37.7 m 7. m 5.35 at an anle f 74. Che the riin t be the pint at the tp f the buildin m which the ball i ht, and call upward the pitie directin. he initial elcit i 8 m at an anle f 4. he acceleratin due t rait i a. (a) c 8 m c m in 8 m in 4.4 m 5 Pearn Educatin, Inc., Upper Saddle Rier, J. All riht reered. hi material i prtected under all cpriht law a the currentl eit. prtin f thi material ma be reprduced, in an m r b an mean, withut permiin in writin m the 65

26 Chapter 3 Kinematic in w Dimenin; Vectr (b) Since the hrizntal elcit i nwn and the hrizntal ditance i nwn, the time f fliht can be fund m the cntant elcit equatin hrizntal mtin. 55 m t t m With that time f fliht, calculate the ertical pitin f the ball uin Eq. -b. t a t.4 m m m S the ball will trie 33 m belw the tp f the buildin. 75. When htin the un erticall, half the time f fliht i pent min upward. hu the upward fliht tae tw ecnd. Che upward a the pitie directin. Since at the tp f the fliht, the ertical elcit i zer, find the launchin elcit m Eq. -a. at at 9.8 m. 9.6 m Uin thi initial elcit and an anle f 45 in the rane mula will ie the maimum rane the un. in 9.6 m in 45 R 39 m 9.8 m 5 Pearn Educatin, Inc., Upper Saddle Rier, J. All riht reered. hi material i prtected under all cpriht law a the currentl eit. prtin f thi material ma be reprduced, in an m r b an mean, withut permiin in writin m the 66

27 CHAPER 4: Dnamic: ewtn Law f Mtin Anwer t Quetin. he child tend t remain at ret (ewtn t Law), unle a ce act n her. he ce i applied t the wan, nt the child, and the wan accelerate ut m under the child, main it l lie the child fall bacward relatie t the wan. If the child i tandin in the wan, the ce f ictin between the child and the bttm f the wan will prduce an acceleratin f the feet, pullin the feet ut m under the child, al main the child fall bacward.. (a) Mar ee the b ta tatinar with repect t the rund. here i n hrizntal ce n the b ince the truc bed i mth, and the b cannt accelerate. hu Mar wuld decribe the mtin f the b in term f ewtn t law there i n ce n the b, it de nt accelerate. (b) Chri, m hi nn-inertial reference ame, wuld a methin abut the b bein thrwn bacward in the truc, and perhap ue ewtn nd law t decribe the effect f that ce. But the urce f that ce wuld be impible t pecif. 3. If the acceleratin f an bject i zer, then b ewtn ecnd law, the net ce mut be zer. here can be ce actin n the bject a ln a the ectr um f the ce i zer. 4. If nl nce ce act n the bject, then the net ce cannt be zer. hu the bject cannt hae zer acceleratin, b ewtn ecnd law. he bject can hae zer elcit an intant. r eample, an bject thrwn traiht up under the influence f rait ha a elcit f zer at the tp f it path, but ha a nn-zer net ce and nn-zer acceleratin thruhut the entire fliht. 5. (a) A ce i needed t bunce the ball bac up, becaue the ball chane directin, and accelerate. If the ball accelerate, there mut be a ce. (b) he paement eert the ce n the lf ball. 6. When u tr t wal eat, u puh n the rund (r n the l in thi cae) with a wetward ce. When u puh wetward n the maie Earth, the Earth me imperceptibl, but b ewtn 3 rd law there i an eatward ce n u, which prpel u ward. When walin n the l, the relatiel liht and unretricted l i ee t me, and when u puh it wetward, it me wetward a u me eatward. 7. B ewtn 3 rd law, the de r wall eert a ce n ur ft equal in manitude t the ce with which u hit the de r wall. If u hit the de r wall with a lare ce, then there will be a lare ce n ur ft, cauin pain. Onl a ce n ur ft caue pain. 8. (a) When u are runnin, the tppin ce i a ce f ictin between ur feet and the rund. Yu puh ward with ur feet n the rund, and thu the rund puhe bacward n u, lwin ur peed. (b) A fat pern can run abut meter per ecnd, perhap tae a ditance f 5 meter er which t tp. he 5 meter wuld be abut 5 tride, f meter each. he acceleratin can be fund m Eq. -c. m a a m m 5 Pearn Educatin, Inc., Upper Saddle Rier, J. All riht reered. hi material i prtected under all cpriht law a the currentl eit. prtin f thi material ma be reprduced, in an m r b an mean, withut permiin in writin m the 67

28 Chapter 4 Dnamic: ewtn Law f Mtin 9. When iin a harp pull, the e i the uddenne f the applicatin f the ce. When a lare, udden ce i applied t the bttm trin, the bttm trin will hae a lare tenin in it. Becaue f the tne inertia, the upper trin de nt immediatel eperience the lare ce. he bttm trin mut hae mre tenin in it, and will brea firt. If a lw and tead pull i applied, the tenin in the bttm trin increae. We apprimate that cnditin a cniderin the tne t be in equilibrium until the trin brea. he ee-bd diaram the tne wuld l lie thi diaram. While the tne i in equilibrium, ewtn nd law tate that up dwn m. hu the tenin in the upper trin i in t be larer than the tenin in the lwer trin becaue f the weiht f the tne, and the upper trin will brea firt.. he acceleratin f bth rc i fund b diidin their weiht (the ce f rait n them) b their ma. he - rc ha a ce f rait n it that i twice a reat a the ce f rait n the - rc, but al twice a reat a ma a the - rc, the acceleratin i the ame bth.. Onl the pund readin wuld be crrect. he prin cale wr n the fact that a certain ce (the weiht f the bject bein weihed) will tretch the prin a certain ditance. hat ditance i prprtinal t the prduct f the ma and the acceleratin due t rait. Since the acceleratin due t rait i maller b a factr f 6 n the mn, the weiht f the bject i maller b a factr f 6, and the prin will be pulled t nl ne-ith f the ditance that it wa pulled n the Earth. he ma itelf den t chane when min t the Mn, and a ma readin n the Mn wuld be incrrect.. When u pull the rpe at an anle, nl the hrizntal cmpnent f the pullin ce will be acceleratin the b acr the table. hi i a maller hrizntal ce than riinall ued, and the hrizntal acceleratin f the b will decreae. 3. Let u find the acceleratin f the Earth, aumin the ma f the eel fallin bject i m =. If the ma f the Earth i M, then the acceleratin f the Earth wuld be fund uin ewtn 3 rd law and ewtn nd law. Ma m a m M Earth bject Earth Earth Since the Earth ha a ma that i n the rder f, then the acceleratin f the Earth i n the 5 4 rder f, r abut m. hi tin acceleratin i undetectable. 4. (a) lift the bject n the Earth require a ce the ame ize a it weiht n Earth, m 98. lift the bject n the Mn require a ce the ame ize a it (b) Earth Earth 5 dwn tne weiht n the Mn, m m 6 6. Mn Mn Mn he hrizntal acceleratin ce wuld be the ame in each cae, becaue the ma f the bject i the ame n bth the Earth and the Mn, and bth bject wuld hae the ame acceleratin t thrw them with the ame peed. S b ewtn ecnd law, the ce wuld hae t be the ame. 5. In a tu f war, the team that puhe hardet aaint the rund win. It i true that bth team hae the ame ce n them due t the tenin in the rpe. But the winnin team puhe harder aaint the rund and thu the rund puhe harder n the winnin team, main a net unbalanced ce. up m 5 Pearn Educatin, Inc., Upper Saddle Rier, J. All riht reered. hi material i prtected under all cpriht law a the currentl eit. prtin f thi material ma be reprduced, in an m r b an mean, withut permiin in writin m the 68