Chapter 3 Motion in Two and Three Dimensions

Transcription

1 Chapte 3 Motion in Two and Thee Dimensions Conceptual Poblems [SSM] Can the manitude of the displacement of a paticle be less than the distance taeled b the paticle alon its path? Can its manitude be moe than the distance taeled? Eplain. Detemine the Concept The distance taeled alon a path can be epesented as a sequence of displacements. Suppose we take a tip alon some path and conside the tip as a sequence of man e small displacements. The net displacement is the ecto sum of the e small displacements, and the total distance taeled is the sum of the manitudes of the e small displacements. That is, total distance Δ + Δ + Δ + Δ,,,3... N, N + whee N is the numbe of e small displacements. (Fo this to be eactl tue we hae to take the limit as N oes to infinit and each displacement manitude oes to zeo.) Now, usin the shotest distance between two points is a staiht line, we hae Δ Δ + Δ + Δ Δ, N,,,3 N, N, whee Δ, N is the manitude of the net displacement. Hence, we hae shown that the manitude of the displacement of a paticle is less than o equal to the distance it taels alon its path. Gie an eample in which the distance taeled is a sinificant amount, et the coespondin displacement is zeo. Can the eese be tue? If so, ie an eample. 7

2 7 Chapte 3 Detemine the Concept The displacement of an object is its final position ecto minus its initial position ecto ( Δ f i ). The displacement can be less but nee moe than the distance taeled. Suppose the path is one complete tip aound Eath at the equato. Then, the displacement is but the distance taeled is πre. No, the eese cannot be tue. 3 What is the aeae elocit of a batte who hits a home un (fom when he hits the ball to when he touches home plate afte oundin the bases)? Detemine the Concept The impotant distinction hee is that aeae elocit is bein equested, as opposed to aeae speed. The aeae elocit is defined as the displacement diided b the elapsed time. Δ Δt Δt a The displacement fo an tip aound the bases is zeo. Thus we see that no matte how fast the unne taels, the aeae elocit is alwas zeo at the end of each complete cicuit of the bases. What is the coect answe if we wee asked fo aeae speed? The aeae speed is defined as the distance taeled diided b the elapsed time. a total distance Δt Fo one complete cicuit of an tack, the total distance taeled will be eate than zeo and so the aeae speed is not zeo. 4 A baseball is hit so its initial elocit upon leain the bat makes an anle of 3 aboe the hoizontal. It leaes that bat at a heiht of. m aboe the ound and lands untouched fo a sinle. Duin its fliht, fom just afte it leaes the bat to just befoe it hits the ound, descibe how the anle between its elocit and acceleation ectos chanes. Nelect an effects due to ai esistance. Detemine the Concept The anle between its elocit and acceleation ectos stats at o because the acceleation of the ball is staiht down. At the peak of the fliht of the ball the anle educes to 9 because the ball s elocit ecto is hoizontal. When the ball eaches the same eleation that it stated fom the anle is 9 3 o 6. 5 If an object is moin towad the west at some instant, in what diection is its acceleation? (a) noth, (b) east, (c) west, (d) south, (e) ma be an diection.

3 Motion in One and Two Dimensions 73 Detemine the Concept The instantaneous acceleation is the limitin alue, as Δt appoaches zeo, of Δ Δt and is in the same diection as Δ. Othe than thouh the definition of a, the instantaneous elocit and acceleation ectos ae unelated. Knowin the diection of the elocit at one instant tells one nothin about how the elocit is chanin at that instant. (e) is coect. 6 Two astonauts ae wokin on the luna suface to install a new telescope. The acceleation due to ait on the Moon is onl.64 m/s. One astonaut tosses a wench to the othe astonaut but the speed of thow is ecessie and the wench oes oe he colleaue s head. When the wench is at the hihest point of its tajecto (a) its elocit and acceleation ae both zeo, (b) its elocit is zeo but its acceleation is nonzeo, (c) its elocit is nonzeo but its acceleation is zeo, (d) its elocit and acceleation ae both nonzeo, (e) insufficient infomation is ien to choose between an of the peious choices. Detemine the Concept When the wench eaches its maimum heiht, it is still moin hoizontall but its acceleation is downwad. (d) is coect. 7 The elocit of a paticle is diected towad the east while the acceleation is diected towad the nothwest as shown in Fiue 3-7. The paticle is (a) speedin up and tunin towad the noth, (b) speedin up and tunin towad the south, (c) slowin down and tunin towad the noth, (d) slowin down and tunin towad the south, (e) maintainin constant speed and tunin towad the south. Detemine the Concept The chane in the elocit is in the same diection as the acceleation. Choose an - coodinate sstem with east bein the positie diection and noth the positie diection. Gien ou choice of coodinate sstem, the component of a is neatie and so will decease. The component of a is positie and so will incease towad the noth. (c) is coect. 8 Assume ou know the position ectos of a paticle at two points on its path, one ealie and one late. You also know the time it took the paticle to moe fom one point to the othe. Then ou can then compute the paticle s (a) aeae elocit, (b) aeae acceleation, (c) instantaneous elocit, (d) instantaneous acceleation. Detemine the Concept All ou can compute is the aeae elocit, since no instantaneous quantities can be computed and ou need two instantaneous elocities to compute the aeae acceleation. (a) is coect.

4 74 Chapte 3 9 Conside the path of a moin paticle. (a) How is the elocit ecto elated eometicall to the path of the paticle? (b) Sketch a cued path and daw the elocit ecto fo the paticle fo seeal positions alon the path. Detemine the Concept (a) The elocit ecto, as a consequence of alwas bein in the diection of motion, is tanent to the path. (b) A sketch showin two elocit ectos fo a paticle moin alon a cued path is shown to the iht. The acceleation of a ca is zeo when it is (a) tunin iht at a constant speed, (b) diin up a lon staiht incline at constant speed, (c) taelin oe the cest of a hill at constant speed, (d) bottomin out at the lowest point of a alle at constant speed, (e) speedin up as it descends a lon staiht decline. Detemine the Concept An object epeiences acceleation whenee eithe its speed chanes o it chanes diection. The acceleation of a ca moin in a staiht path at constant speed is zeo. In the othe eamples, eithe the manitude o the diection of the elocit ecto is chanin and, hence, the ca is acceleated. (b) is coect. [SSM] Gie eamples of motion in which the diections of the elocit and acceleation ectos ae (a) opposite, (b) the same, and (c) mutuall pependicula. Detemine the Concept The elocit ecto is defined b d / dt, while the acceleation ecto is defined b a d / dt. (a) A ca moin alon a staiht oad while bakin. (b) A ca moin alon a staiht oad while speedin up. (c) A paticle moin aound a cicula tack at constant speed. How is it possible fo a paticle moin at constant speed to be acceleatin? Can a paticle with constant elocit be acceleatin at the same time?

5 Motion in One and Two Dimensions 75 Detemine the Concept A paticle epeiences acceleated motion when eithe its speed o diection of motion chanes. A paticle moin at constant speed in a cicula path is acceleatin because the diection of its elocit ecto is chanin. If a paticle is moin at constant elocit, it is not acceleatin. 3 [SSM] Imaine thowin a dat staiht upwad so that it sticks into the ceilin. Afte it leaes ou hand, it steadil slows down as it ises befoe it sticks. (a) Daw the dat s elocit ecto at times t and t, whee t and t occu afte it leaes ou hand but befoe it impacts the ceilin, and Δt t t is small. Fom ou dawin find the diection of the chane in elocit Δ, and thus the diection of the acceleation ecto. (b) Afte it has stuck in the ceilin fo a few seconds, the dat falls down to the floo. As it falls it speeds up, of couse, until it hits the floo. Repeat Pat (a) to find the diection of its acceleation ecto as it falls. (c) Now imaine tossin the dat hoizontall. What is the diection of its acceleation ecto afte it leaes ou hand, but befoe it stikes the floo? Detemine the Concept The acceleation ecto is in the same diection as the chane in elocit ecto, Δ. (a) The sketch fo the dat thown upwad is shown below. The acceleation ecto is in the diection of the chane in the elocit ecto Δ. (b) The sketch fo the fallin dat is shown below. Aain, the acceleation ecto is in the diection of the chane in the elocit ecto Δ. (c) The acceleation ecto is in the diection of the chane in the elocit ecto and hence is downwad as shown below: Δ Δ Δ 4 As a bunee jumpe appoaches the lowest point in he descent, the ubbe cod holdin he stetches and she loses speed as she continues to moe downwad. Assumin that she is doppin staiht down, make a motion diaam to find the diection of he acceleation ecto as she slows down b dawin he elocit ectos at times t and t, whee t and t ae two instants duin the potion of he descent that she is losin speed and t t is small. Fom ou

6 76 Chapte 3 dawin find the diection of the chane in elocit Δ, and thus the diection of the acceleation ecto. Detemine the Concept The acceleation ecto is in the same diection as the chane in elocit ecto, Δ. The dawin is shown to the iht. Δ 5 Afte eachin the lowest point in he jump at time t low, a bunee jumpe moes upwad, ainin speed fo a shot time until ait aain dominates he motion. Daw he elocit ectos at times t and t, whee Δt t t is small and t < t low < t. Fom ou dawin find the diection of the chane in elocit Δ, and thus the diection of the acceleation ecto at time t low. Detemine the Concept The acceleation ecto is in the same diection as the chane in the elocit ecto, Δ. The dawin is shown to the iht. Δ 6 A ie is.76 km wide. The banks ae staiht and paallel (Fiue 3-8). The cuent is 4. km/h and is paallel to the banks. A boat has a maimum speed of 4. km/h in still wate. The pilot of the boat wishes to o on a staiht line fom A to B, whee the line AB is pependicula to the banks. The pilot should (a) head diectl acoss the ie, (b) head 53º upsteam fom the line AB, (c) head 37º upsteam fom the line AB, (d) ie up the tip fom A to B is not possible with a boat of this limited speed, (e) do none of the aboe. Detemine the Concept We can decide what the pilot should do b considein the speeds of the boat and of the cuent. The speed of the steam is equal to the maimum speed of the boat in still wate. The best the boat can do is, while facin diectl upsteam, maintain its position elatie to the bank. So, the pilot should ie up. (d) is coect.

7 Motion in One and Two Dimensions 77 7 [SSM] Duin a hea ain, the dops ae fallin at a constant elocit and at an anle of west of the etical. You ae walkin in the ain and notice that onl the top sufaces of ou clothes ae ettin wet. In what diection ae ou walkin? Eplain. Detemine the Concept You must be walkin west to make it appea to ou that the ain is eactl etical. 8 In Poblem 7, what is ou walkin speed if the speed of the dops elatie to the ound is 5. m/s? Detemine the Concept Let YG epesent ou elocit elatie to the ound and RG epesent the elocit of the ain elatie to the ound. Then ou speed elatie to the ound is ien b RG ( 5. m/s) sin.9 m/s YG YG 9 Tue o false (Inoe an effects due to ai esistance): (a) (b) (c) When a pojectile is fied hoizontall, it takes the same amount of time to each the ound as an identical pojectile dopped fom est fom the same heiht. When a pojectile is fied fom a cetain heiht at an upwad anle, it takes lone to each the ound than does an identical pojectile dopped fom est fom the same heiht. When a pojectile is fied hoizontall fom a cetain heiht, it has a hihe speed upon eachin the ound than does an identical pojectile dopped fom est fom the same heiht. (a) Tue. In the absence of ai esistance, both pojectiles epeience the same downwad acceleation. Because both pojectiles hae initial etical elocities of zeo, thei etical motions must be identical. (b) Tue. When a pojectile is fied fom a cetain heiht at an upwad anle, its time in the ai is lae because it fist ises to a eate heiht befoe beinnin to fall with zeo initial etical elocit.

8 78 Chapte 3 (c) Tue. When a pojectile is fied hoizontall, its elocit upon eachin the ound has a hoizontal component in addition to the etical component it has when it is dopped fom est. The manitude of this elocit is elated to its hoizontal and etical components thouh the Pthaoean Theoem. A pojectile is fied at 35º aboe the hoizontal. An effects due to ai esistance ae neliible. At the hihest point in its tajecto, its speed is m/s. The initial elocit had a hoizontal component of (a), (b) ( m/s) cos 35º, (c) ( m/s) sin 35º, (d) ( m/s)/cos 35º, (e) m/s. Detemine the Concept In the absence of ai esistance, the hoizontal component of the pojectile s elocit is constant fo the duation of its fliht. At the hihest point, the speed is the hoizontal component of the initial elocit. The etical component is zeo at the hihest point. (e) is coect. [SSM] A pojectile is fied at 35º aboe the hoizontal. An effects due to ai esistance ae neliible. The initial elocit of the pojectile in Poblem has a etical component that is (a) less than m/s, (b) eate than m/s, (c) equal to m/s, (d) cannot be detemined fom the data ien. Detemine the Concept (a) is coect. Because the initial hoizontal elocit is m/s and the launch anle is less than 45 deees, the initial etical elocit must be less than m/s. A pojectile is fied at 35º aboe the hoizontal. An effects due to ai esistance ae neliible. The pojectile lands at the same eleation of launch, so the etical component of the impact elocit is (a) the same as the etical component of its initial elocit in both manitude and diection, (b) the same as the etical component of its initial elocit, (c) less than the etical component of its initial elocit, (d) less than the etical component of its initial elocit. Detemine the Concept (b) is coect. The landin speed is the same as the launch speed. Because the hoizontal component of its initial elocit does not chane, the etical component of the elocit at landin must be the same manitude but oppositel diected its etical component at launch. 3 Fiue 3-9 epesents the tajecto of a pojectile oin fom A to E. Ai esistance is neliible. What is the diection of the acceleation at point B? (a) up and to the iht, (b) down and to the left, (c) staiht up, (d) staiht down, (e) The acceleation of the ball is zeo.

9 Motion in One and Two Dimensions 79 Detemine the Concept (d) is coect. In the absence of ai esistance, the acceleation of the ball depends onl on the chane in its elocit and is independent of its elocit. As the ball moes alon its tajecto between points A and C, the etical component of its elocit deceases and the chane in its elocit is a downwad pointin ecto. Between points C and E, the etical component of its elocit inceases and the chane in its elocit is also a downwad pointin ecto. Thee is no chane in the hoizontal component of the elocit. 4 Fiue 3-9 epesents the tajecto of a pojectile oin fom A to E. Ai esistance is neliible. (a) At which point(s) is the speed the eatest? (b) At which point(s) is the speed the least? (c) At which two points is the speed the same? Is the elocit also the same at these points? Detemine the Concept In the absence of ai esistance, the hoizontal component of the elocit emains constant thouhout the fliht. The etical component has its maimum alues at launch and impact. (a) The speed is eatest at A and E. (b) The speed is least at point C. (c) The speed is the same at A and E. No. The hoizontal components ae equal at these points but the etical components ae oppositel diected, so the elocit is not the same at A and E. 5 [SSM] Tue o false: (a) (b) (c) (d) (e) If an object's speed is constant, then its acceleation must be zeo. If an object's acceleation is zeo, then its speed must be constant. If an object's acceleation is zeo, its elocit must be constant. If an object's speed is constant, then its elocit must be constant. If an object's elocit is constant, then its speed must be constant. Detemine the Concept Speed is a scala quantit, wheeas acceleation, equal to the ate of chane of elocit, is a ecto quantit. (a) False. Conside a ball moin in a hoizontal cicle on the end of a stin. The ball can moe with constant speed (a scala) een thouh its acceleation (a ecto) is alwas chanin diection. (b) Tue. Fom its definition, if the acceleation is zeo, the elocit must be constant and so, theefoe, must be the speed. (c) Tue. An object s elocit must chane in ode fo the object to hae acceleation othe than zeo.

10 8 Chapte 3 (d) False. Conside an object moin at constant speed alon a cicula path. Its elocit chanes continuousl alon such a path. (e) Tue. If the elocit of an object is constant, then both its diection and manitude (speed) must be constant. 6 The initial and final elocities of a paticle ae as shown in Fiue 3-3. Find the diection of the aeae acceleation. Detemine the Concept The aeae acceleation ecto is defined b aa Δ / Δt. The diection of a a is that of Δ f i, as shown to the iht. Δ f i f i 7 The automobile path shown in Fiue 3-3 is made up of staiht lines and acs of cicles. The automobile stats fom est at point A. Afte it eaches point B, it taels at constant speed until it eaches point E. It comes to est at point F. (a) At the middle of each sement (AB, BC, CD, DE, and EF), what is the diection of the elocit ecto? (b) At which of these points does the automobile hae a nonzeo acceleation? In those cases, what is the diection of the acceleation? (c) How do the manitudes of the acceleation compae fo sements BC and DE? Detemine the Concept The elocit ecto is in the same diection as the chane in the position ecto while the acceleation ecto is in the same diection as the chane in the elocit ecto. Choose a coodinate sstem in which the diection is noth and the diection is east. (a) Path AB BC CD DE EF Diection of elocit ecto noth notheast east southeast south (b) Path Diection of acceleation ecto AB noth BC southeast CD DE southwest EF noth (c) The manitudes ae compaable, but lae fo DE because the adius of the path is smalle thee.

11 Motion in One and Two Dimensions 8 8 Two cannons ae pointed diectl towad each othe as shown in Fiue 3-3. When fied, the cannonballs will follow the tajectoies shown P is the point whee the tajectoies coss each othe. If we want the cannonballs to hit each othe, should the un cews fie cannon A fist, cannon B fist, o should the fie simultaneousl? Inoe an effects due to ai esistance. Detemine the Concept We ll assume that the cannons ae identical and use a constant-acceleation equation to epess the displacement of each cannonball as a function of time. Hain done so, we can then establish the condition unde which the will hae the same etical position at a ien time and, hence, collide. The modified diaam shown below shows the displacements of both cannonballs. Epess the displacement of the cannonball fom cannon A at an time t afte bein fied and befoe an collision: Epess the displacement of the cannonball fom cannon A at an time t afte bein fied and befoe an collision: Δ t + t Δ t + t If the uns ae fied simultaneousl, t t and the balls ae the same distance t below the line of siht at all times. Also, because the cannons ae identical, the cannonballs hae the same hoizontal component of elocit and will each the hoizontal midpoint at the same time. Theefoe, the should fie the uns simultaneousl. Remaks: This is the monke and hunte poblem in disuise. If ou imaine a monke in the position shown below, and the two uns ae fied simultaneousl, and the monke beins to fall when the uns ae fied, then the monke and the two cannonballs will all each point P at the same time.

12 8 Chapte 3 9 Galileo wote the followin in his Dialoue concenin the two wold sstems: Shut ouself up... in the main cabin below decks on some lae ship, and... han up a bottle that empties dop b dop into a wide essel beneath it. When ou hae obseed [this] caefull... hae the ship poceed with an speed ou like, so lon as the motion is unifom and not fluctuatin this wa and that. The doplets will fall as befoe into the essel beneath without doppin towads the sten, althouh while the dops ae in the ai the ship uns man spans. Eplain this quotation. Detemine the Concept The doplet leain the bottle has the same hoizontal elocit as the ship. Duin the time the doplet is in the ai, it is also moin hoizontall with the same elocit as the est of the ship. Because of this, it falls into the essel, which has the same hoizontal elocit. Because ou hae the same hoizontal elocit as the ship does, ou see the same thin as if the ship wee standin still. 3 A man swins a stone attached to a ope in a hoizontal cicle at constant speed. Fiue 3-33 epesents the path of the ock lookin down fom aboe. (a) Which of the ectos could epesent the elocit of the stone? (b) Which could epesent the acceleation? Detemine the Concept (a) Because A and D ae tanent to the path of the stone, eithe of them could epesent the elocit of the stone. (b) Let the ectos A () t and A ( t + Δt) be of equal lenth but point in slihtl diffeent diections as the stone moes aound the cicle. These two ectos and ΔA ae shown in the diaam aboe. Note that ΔA is neal pependicula to A () t. Fo e small time inteals, ΔA and A ( t) ae pependicula to one anothe. Theefoe, d A / dt is pependicula to A and onl the ecto E could epesent the acceleation of the stone.

13 Motion in One and Two Dimensions 83 3 Tue o false: (a) (b) (c) (d) An object cannot moe in a cicle unless it has centipetal acceleation. An object cannot moe in a cicle unless it has tanential acceleation. An object moin in a cicle cannot hae a aiable speed. An object moin in a cicle cannot hae a constant elocit. (a) Tue. An object acceleates when its elocit chanes; that is, when eithe its speed o its diection chanes. When an object moes in a cicle the diection of its motion is continuall chanin. (b) False. An object moin in a cicula path at constant speed has a tanential acceleation of zeo. (c) False. A ood eample is a ock weded in the tead of an automobile tie. Its speed chanes whenee the ca s speed chanes. (d) Tue. The elocit ecto of an object moin in a cicle is continuall chanin diection. 3 Usin a motion diaam, find the diection of the acceleation of the bob of a pendulum when the bob is at a point whee it is just eesin its diection. Pictue the Poblem In the diaam, (a) shows the pendulum just befoe it eeses diection and (b) shows the pendulum just afte it has eesed its diection. The acceleation of the bob is in the diection of the chane in the elocit Δ f i and is tanent to the pendulum tajecto at the point of eesal of diection. This makes sense because, at an etemum of motion,, so thee is no centipetal acceleation. Howee, because the elocit is eesin diection, the tanential acceleation is nonzeo. (a) (b) i i f f Δ f i 33 [SSM] Duin ou ookie bunee jump, ou fiend ecods ou fall usin a camcode. B analzin it fame b fame, he finds that the - component of ou elocit is (ecoded ee / th of a second) as follows:

14 84 Chapte 3 t (s) (m/s) (a) Daw a motion diaam. Use it to find the diection and elatie manitude of ou aeae acceleation fo each of the eiht successie.5 s time inteals in the table. (b) Comment on how the component of ou acceleation does o does not a in sin and manitude as ou eese ou diection of motion. Detemine the Concept (a) The motion diaam shown below was constucted usin the data in the table shown below the motion diaam. The column fo Δ in the table was calculated usin Δ i i and the column fo a was calculated usin a ( i i ) Δt. + a 9 9 a 89 a a 78 a a 67 a 45 5 a i Δ a ae (m/s) (m/s) (m/s )

15 Motion in One and Two Dimensions (b) The acceleation ecto alwas points upwad and so the sin of its component does not chane. The manitude of the acceleation ecto is eatest when the bunee cod has its maimum etension (ou speed, the manitude of ou elocit, is least at this time and times nea it) and is less than this maimum alue when the bunee cod has less etension. Estimation and Appoimation 34 Estimate the speed in mph with which wate comes out of a aden hose usin ou past obseations of wate comin out of aden hoses and ou knowlede of pojectile motion. Pictue the Poblem Based on ou epeience with aden hoses, ou pobabl know that the maimum ane of the wate is achieed when the hose is inclined at about 45 with the etical. A easonable estimate of the ane of such a steam is about 4. m when the initial heiht of the steam is. m. Use constantacceleation equations to obtain epessions fo the and coodinates of a doplet of wate in the steam and then eliminate time between these equations to obtain an equation that ou can sole fo. θ (,) R Use constant-acceleation equations to epess the and components of a molecule of wate in the steam: Because, R, a, and a : and + t + at + t + a t t () and + t t ()

16 86 Chapte 3 Epess and in tems of and θ : Substitute in equations () and () to obtain: cosθ and sinθ whee θ is the anle the steam makes with the hoizontal. ( cosθ )t (3) and + ( sinθ ) t t (4) Eliminatin t fom equations (3) + tanθ cos θ and (4) ields: ( ) When the steam of wate hits the + tanθ R R cos θ ound, and R: ( ) Solin this equation fo ies: R [ R tanθ + ] cos θ Substitute numeical alues and ealuate : cos m/s ( 4. m) [( 4. m) tan m] 5.63 m/s Use a conesion facto, found in Appendi A, to conet m/s to mi/h: 5.63 m/s 3 mi/h mi/h.447 m/s 35 [SSM] You won a contest to spend a da with all team duin thei spin tainin camp. You ae allowed to t to hit some balls thown b a pitche. Estimate the acceleation duin the hit of a fastball thown b a majo leaue pitche when ou hit the ball squael-staiht back at the pitche. You will need to make easonable choices fo ball speeds, both just befoe and just afte the ball is hit, and of the contact time the ball has with the bat. Detemine the Concept The manitude of the acceleation of the ball is ien b Δ a a. Let afte epesent the elocit of the ball just afte its collision with Δt the bat and befoe its elocit just befoe this collision. Most majo leaue pitches can thow a fastball at least 9 mi/h and some occasionall thow as fast as mi/h. Let s assume that the pitche thows ou an 8 mph fastball.

17 Motion in One and Two Dimensions 87 The manitude of the acceleation afte befoe of the ball is: a. Δt Assumin that afte and befoe ae both 8 mi/h and that the ball is in contact with the bat fo ms: Conetin a to m/s ields: 8 mi/h a ms a 6 mi/h ms 7 4 m/s ( 8 mi/h).447 m/s mi/h 6 mi/h ms 36 Estimate how fa ou can thow a ball if ou thow it (a) hoizontall while standin on leel ound, (b) at θ 45º aboe hoizontal while standin on leel ound, (c) hoizontall fom the top of a buildin m hih, (d) at θ 45º aboe hoizontal fom the top of a buildin m hih. Inoe an effects due to ai esistance. Pictue the Poblem Duin the fliht of the ball the acceleation is constant and equal to 9.8 m/s diected downwad. We can find the fliht time fom the etical pat of the motion, and then use the hoizontal pat of the motion to find the hoizontal distance. We ll assume that the elease point of the ball is. m aboe ou feet. A sketch of the motion that includes coodinate aes, the initial and final positions of the ball, the launch anle, and the initial elocit follows. θ (,) R Obiousl, how fa ou thow the ball will depend on how fast ou can thow it. A majo leaue baseball pitche can thow a fastball at 9 mi/h o so. Assume that ou can thow a ball at two-thids that speed to obtain:.447 m/s 6 mi/h mi/h 7m/s

18 88 Chapte 3 Thee is no acceleation in the diection, so the hoizontal motion is one of constant elocit. Epess the hoizontal position of the ball as a function of time: t ( cosθ )t () Assumin that the elease point of the ball is a distance aboe the ound, epess the etical position of the ball as a function of time: + + ( sinθ ) t + a t t + a t () Eliminatin t between equations () a + tanθ + cos θ and () ields: ( ) (a) If ou thow the ball hoizontall ou equation becomes: Substitute numeical alues to obtain: At impact, and R: Solin fo R ields: a m/s. m + ( 7 m/s) 9.8 m/s. m + R ( 7 m/s) R 7 m (b) Fo θ 45 we hae: At impact, and R: 9.8 m/s. m + ( tan 45 ) + ( 7 m/s) cos m/s. m + ( tan 45 ) R + R ( 7 m/s) cos 45 Use the quadatic fomula o ou aphin calculato to obtain: R 75m (c) If ou thow the ball hoizontall fom the top of a buildin that is m hih ou equation becomes: 9.8 m/s 4. m + ( 7 m/s)

19 Motion in One and Two Dimensions 89 At impact, and R: 9.8 m/s 4. m + R ( 7 m/s) Sole fo R to obtain: R 45m (d) If ou thow the ball at an anle of 45 fom the top of a buildin that is m hih ou equation becomes: At impact, and R: 9.8 m/s 4 m + ( tan 45 ) + ( 7 m/s) cos m/s 4 m + ( tan 45 ) R + R ( 7 m/s) cos 45 Use the quadatic fomula o ou aphin calculato to obtain: R 85m 37 In 978, Geoff Capes of Geat Bitain thew a hea bick a hoizontal distance of 44.5 m. Find the appoimate speed of the bick at the hihest point of its fliht, nelectin an effects due to ai esistance. Assume the bick landed at the same heiht it was launched. Pictue the Poblem We ll inoe the heiht of Geoff s elease point aboe the ound and assume that he launched the bick at an anle of 45. Because the elocit of the bick at the hihest point of its fliht is equal to the hoizontal component of its initial elocit, we can use constant-acceleation equations to elate this elocit to the bick s and coodinates at impact. The diaam shows an appopiate coodinate sstem and the bick when it is at point P with coodinates (, ). P (, ) θ R 44.5 m Usin a constant-acceleation equation, epess the and coodinates of the bick as a function of time: and + t + at + t + at

20 9 Chapte 3 Because,, a, and a : t and t t Eliminate t between these equations tanθ to obtain: ( ) When the bick stikes the ound whee we hae used and R: ( ) θ. tan tanθ R R whee R is the ane of the bick. Sole fo to obtain: R tanθ Substitute numeical alues and ealuate : ( 9.8m/s )( 44.5m) 5m/s tan 45 Note that, at the bick s hihest point,. Position, Displacement, Velocit and Acceleation Vectos 38 A wall clock has a minute hand with a lenth of.5 m and an hou hand with a lenth of.5 m. Take the cente of the clock as the oiin, and use a Catesian coodinate sstem with the positie ais pointin to 3 o'clock and the positie ais pointin to o clock. Usin unit ectosî and ĵ, epess the position ectos of the tip of the hou hand ( A ) and the tip of the minute hand ( B ) when the clock eads (a) :, (b) 3:, (c) 6:, (d) 9:. Pictue the Poblem Let the + diection be staiht up and the + diection be to the iht. (a) At :, both hands ae positioned alon the + ais. The position ecto fo the tip of the hou hand at: is: A (.5m)j ˆ The position ecto fo the tip of the minute hand at : is: B (.5m)j ˆ

21 Motion in One and Two Dimensions 9 (b) At 3:, the minute hand is positioned alon the + ais, while the hou hand is positioned alon the + ais. The position ecto fo the tip of the hou hand at 3: is: The position ecto fo the tip of the minute hand at 3: is: A B (.5m)iˆ (.5m)j ˆ (c) At 6:, the minute hand is positioned alon the ais, while the hou hand is positioned alon the + ais. The position ecto fo the tip of the hou hand at 6: is: The position ecto fo the tip of the minute hand at 6: is: A B (.5m)j ˆ (.5m)j ˆ (d) At 9:, the minute hand is positioned alon the + ais, while the hou hand is positioned alon the ais. The position ecto fo the tip of the hou hand at 9: is: The position ecto fo the tip of the minute hand at 9: is: A B (.5m)iˆ (.5m)j ˆ 39 [SSM] In Poblem 38, find the displacements of the tip of each hand (that is, Δ A and Δ B ) when the time adances fom 3: P.M. to 6: P.M. Pictue the Poblem Let the + diection be staiht up, the + diection be to the iht, and use the ectos descibin the ends of the hou and minute hands in Poblem 38 to find the displacements ΔA and ΔB. The displacement of the minute hand ΔB B6 B3 as time adances fom 3: P.M. to 6: P.M. is ien b: Fom Poblem 38: B (.5m)j ˆ and B (.5m)j ˆ Substitute and simplif to obtain: Δ (.5m) ˆj (.5m) ˆj 6 B 3

22 9 Chapte 3 The displacement of the hou hand as time adances fom 3: P.M. to 6: P.M. is ien b: ΔA A 6 A Fom Poblem 38: A (.5m)j ˆ and A (.5m)iˆ 6 Substitute and simplif to obtain: ΔA (.5m) ˆj (.5m)iˆ In Poblem 38, wite the ecto that descibes the displacement of a fl if it quickl oes fom the tip of the minute hand to the tip of the hou hand at 3: P.M. Pictue the Poblem Let the positie diection be staiht up, the positie diection be to the iht, and use the ectos descibin the ends of the hou and minute hands in Poblem 38 to find the displacement D of the fl as it oes fom the tip of the minute hand to the tip of the hou hand at 3: P.M. The displacement of the fl is ien b: D A 3 B 3 Fom Poblem 38: A (.5m)iˆ and B (.5m)j ˆ Substitute fo A 3 and 3 3 ˆ 3 B to obtain: D (.5m) i (.5m)j 4 A bea, awakenin fom winte hibenation, staes diectl notheast fo m and then due east fo m. Show each displacement aphicall and aphicall detemine the sinle displacement that will take the bea back to he cae, to continue he hibenation. Pictue the Poblem The sinle displacement fo the bea to make it back to its cae is the ecto D. Its manitude D and diection θ can be detemined b dawin the bea s displacement ectos to scale. Ν 45 m θ ˆ m D Ε

23 Motion in One and Two Dimensions 93 D m and θ 3 Remaks: The diection of D is 8 + θ 3 4 A scout walks.4 km due East fom camp, then tuns left and walks.4 km alon the ac of a cicle centeed at the campsite, and finall walks.5 km diectl towad the camp. (a) How fa is the scout fom camp at the end of his walk? (b) In what diection is the scout s position elatie to the campsite? (c) What is the atio of the final manitude of the displacement to the total distance walked? Pictue the Poblem The fiue shows the paths walked b the Scout. The lenth of path A is.4 km; the lenth of path B is.4 km; and the lenth of path C is.5 km: Noth C B θ A East (a) Epess the distance fom the campsite to the end of path C: (b) Detemine the anle θ subtended b the ac at the oiin (campsite): (c) Epess the total distance as the sum of the thee pats of his walk: Substitute the ien distances to find the total:.4 km.5 km.9km θ adians aclenth adius ad 57.3 His diection fom camp is ad Noth of East. d d + d + d tot east ac.4km.4km towad camp d tot.4 km +.4 km +.5 km 6.3 km Epess the atio of the manitude of his displacement to the total distance he walked and substitute to obtain a numeical alue fo this atio: Manitude of his displacement Total distance walked.9km 6.3km 7

24 94 Chapte 3 43 [SSM] The faces of a cubical stoae cabinet in ou aae has 3.-m-lon edes that ae paallel to the z coodinate planes. The cube has one cone at the oiin. A cockoach, on the hunt fo cumbs of food, beins at that cone and walks alon thee edes until it is at the fa cone. (a) Wite the oach's displacement usin the set of ˆ i, j ˆ, and k ˆ unit ectos, and (b) find the manitude of its displacement. Pictue the Poblem While thee ae seeal walkin outes the cockoach could take to et fom the oiin to point C, its displacement will be the same fo all of them. One possible oute is shown in the fiue. (a) The oach s displacement D duin its tip fom the oiin to D A + B + C ˆ point C is: ( 3.m) i + ( 3.m) j + ( 3.m)k ˆ ˆ (b) The manitude of the oach s displacement is ien b: D D + D + D z Substitute numeical alues fo D, D, and D z and ealuate D to obtain: D ( 3.m) + ( 3.m) + ( 3.m) 5.m 44 You ae the naiato of a ship at sea. You eceie adio sinals fom two tansmittes A and B, which ae km apat, one due south of the othe. The diection finde shows ou that tansmitte A is at a headin of 3º south of east fom the ship, while tansmitte B is due east. Calculate the distance between ou ship and tansmitte B. Pictue the Poblem The diaam shows the locations of the tansmittes elatie to the ship and defines the distances sepaatin the tansmittes fom each othe and fom the ship. We can find the distance between the ship and tansmitte B usin tionomet.

25 Motion in One and Two Dimensions 95 S θ D SB A D SA D AB B Relate the distance between A and B to the distance fom the ship to A and the anle θ : tanθ D D AB SB DAB D SB tanθ Substitute numeical alues and ealuate D SB : D km 5.7 m tan3 SB Velocit and Acceleation Vectos 45 A stationa ada opeato detemines that a ship is km due south of him. An hou late the same ship is km due southeast. If the ship moed at constant speed and alwas in the same diection, what was its elocit duin this time? Pictue the Poblem Fo constant speed and diection, the instantaneous elocit is identical to the aeae elocit. Take the oiin to be the location of the stationa ada and let the + diection be to the East and the + diection be to the Noth. () N Δ E () Epess the aeae elocit: Detemine the position ectos and : Δ a () Δt and ( km) ˆj ( 4.km) iˆ + ( 4.km)j ˆ

26 96 Chapte 3 Find the displacement ecto Δ : Substitute fo Δ and Δt in equation () to find the aeae elocit. Δ a ( 4.km) iˆ + ( 4.km)j ˆ ( 4.km) iˆ + ( 4.km).h ˆj ( 4 km/h) iˆ + ( 4.km/h)j ˆ 46 A paticle s position coodinates (, ) ae (. m, 3. m) at t ; (6. m, 7. m) at t. s; and (3 m, 4 m) at t 5. s. (a) Find the manitude of the aeae elocit fom t to t. s. (b) Find the manitude of the aeae elocit fom t to t 5. s. Pictue the Poblem The aeae elocit is the chane in position diided b the elapsed time. (a) The manitude of the aeae elocit is ien b: a Δ Δt Find the position ectos and the displacement ecto: Find the manitude of the displacement ecto fo the inteal between t and t. s: (. m) iˆ ( 3. m)j ˆ + ( 6. m) iˆ ( 7. m)j ˆ + and Δ + Δ ( 4. m) iˆ ( 4. m)j ˆ ( 4. m) + ( 4. m) 5.66 m Substitute to detemine a : 5.66 m. s a.8 m/s θ is ien b: θ tan Δ Δ,, tan 4. m 4. m 45 measued fom the positie ais.

27 Motion in One and Two Dimensions 97 (b) Repeat (a), this time usin the displacement between t and t 5. s to obtain: ( 3 m) iˆ ( 4 m)j ˆ 5 + Δ Δ m + m ( m) iˆ ( m)j ˆ ( ) ( ) 5.6 m 5 5.6m a 3.m/s 5.s and m θ tan 45 m measued fom the + ais. 47 [SSM] A paticle moin at a elocit of 4. m/s in the + diection is ien an acceleation of 3. m/s in the + diection fo. s. Find the final speed of the paticle. Pictue the Poblem The manitude of the elocit ecto at the end of the s of acceleation will ie us its speed at that instant. This is a constant-acceleation poblem. Find the final elocit ecto of the paticle: iˆ + ˆj iˆ + a tˆj The manitude of is: + ( ) iˆ 4.m/s + ( 3. m/s )(.s) ( 4.m/s) iˆ + ( 6. m/s)j ˆ ˆj Substitute fo and and ealuate : ( 4. m/s) + ( 6. m/s) 7. m/s 48 Initiall, a swift-moin hawk is moin due west with a speed of 3 m/s; 5. s late it is moin due noth with a speed of m/s. (a) What ae the manitude and diection of Δ a duin this 5.-s inteal? (b) What ae the manitude and diection of a a duin this 5.-s inteal? Pictue the Poblem Choose a coodinate sstem in which noth coincides with the + diection and east with the + diection. Epessin the hawk s elocit ectos is the fist step in deteminin Δ and a a.

28 98 Chapte 3 (a) The chane in the hawk s elocit duin this inteal is: W Δ a N and N ae ien b: ( 3 m/s)iˆ and ( m/s)j ˆ Substitute fo W ealuate Δ : and N and W W Δ a + N [ iˆ ] ( m/s) ˆj ( 3 m/s) ( 3 m/s) iˆ ( m/s)j ˆ The manitude of b: Δ a is ien Δ a Δ + Δ Substitute numeical alues and ealuate Δ: Δ a ( 3 m/s) + ( m/s) 36 m/s The diection of Δ is ien b: W θ N Δ a Δ θ tan Δ whee θ is measued fom the + ais. θ Substitute numeical alues and ealuate θ : (b) The hawk s aeae acceleation duin this inteal is: Substitute fo m/s θ tan 3 m/s a a Δt Δ a 34 Δ and Δt to obtain: ( 3 m/s) iˆ + ( m/s) The manitude of a a is ien b: a a + a a a a 5. s ˆj ( ) iˆ 6. m/s + ( 4. m/s )j ˆ Substitute numeical alues and ealuate a : a a a ( 6. m/s ) + ( 4. m/s ) 7. m/s

29 The diection of a a is ien b: Motion in One and Two Dimensions 99 θ tan a a Substitute numeical alues and 4. m/s ealuate θ : θ tan m/s whee θ is measued fom the + ais. Remaks: Because a a is in the same diection as Δ a, the calculation of θ in Pat (b) was not necessa. 49 At t, a paticle located at the oiin has a elocit of 4 m/s at θ 45º. At t 3. s, the paticle is at m and 8 m and has a elocit of 3 m/s at θ 5º. Calculate (a) the aeae elocit and (b) the aeae acceleation of the paticle duin this 3.-s inteal. Pictue the Poblem The initial and final positions and elocities of the paticle ae ien. We can find the aeae elocit and aeae acceleation usin thei definitions b fist calculatin the ien displacement and elocities usin unit ectos i ˆ and ˆj. (a) The aeae elocit of the paticle is the atio of its displacement to the elapsed time: The displacement of the paticle duin this inteal of time is: a Δ Δt Δ + ( m) iˆ ( 8m)j ˆ Substitute to find the aeae elocit: a ( m) iˆ + ( 8m) 3.s ˆj ( 33.3m/s) iˆ + ( 6.7 m/s) ( 33m/s) iˆ + ( 7 m/s)j ˆ ˆj (b) The aeae acceleation is: and a a Δ Δt Δt ae ien b: [( ) ] iˆ 4m/s cos45 + [( 4m/s) sin 45 ] ( 8.8m/s) iˆ + ( 8.8m/s)j ˆ and [( 3m/s) cos5 ] iˆ + [( 3m/s) sin 5 ] ( 9.8m/s) iˆ + (.98m/s)j ˆ ˆj ˆj

30 Chapte 3 Substitute fo and a a and ealuate a a : [ ˆj ] ( 9.8 m/s) iˆ + (.98m/s) ˆj ( 8.8 m/s) iˆ + ( 8.8 m/s) ( 9. m/s) iˆ + ( 5.3 m/s) ˆj ( 3. m/s ) iˆ + (.8 m/s )j ˆ 3. s 3. s 5 At time zeo, a paticle is at 4. m and 3. m and has elocit (. m/s) ˆ i + ( 9. m/s) j ˆ. The acceleation of the paticle is constant and is ien b a (4. m/s ) ˆ i + (3. m/s ) j ˆ. (a) Find the elocit at t. s. (b) Epess the position ecto at t 4. s in tems of î and ĵ. In addition, ie the manitude and diection of the position ecto at this time. Pictue the Poblem The acceleation is constant so we can use the constantacceleation equations in ecto fom to find the elocit at t. s and the position ecto at t 4. s. (a) The elocit of the paticle, as a function of time, is ien b: Substitute fo and a to find (. s): + at ( ) iˆ 9. m/s) ˆ j [ iˆ. s (. m/s) + ( + (4. m/s ) + (3. m/s ) ˆj ](. s) ( m/s) iˆ + ( 3. m/s) ˆj (b) Epess the position ecto as a function of time: + + a t t Substitute numeical alues and ealuate : (4. m) iˆ + (3. m) ˆ + j + [(. m/s) iˆ + ( 9. m/s) ˆj ]( 4. s) [(4. m/s ) iˆ (3. m/s ) ˆ + j]( 4. s) (44. m) iˆ + ( 9. m) ˆj (44 m) iˆ + ( 9. m) ˆj The manitude of is ien b. Substitute numeical + alues and ealuate (4. s): (4. s) ( 44.m) + ( 9.m) 45 m

31 The diection of is ien b θ tan. Because is in the 4 th quadant, its diection measued fom the + ais is: Motion in One and Two Dimensions 9.m θ tan.6 44.m 5 [SSM] A paticle has a position ecto ien b (3t) ˆ i + (4t 5t ) j ˆ, whee is in metes and t is in seconds. Find the instantaneous-elocit and instantaneous-acceleation ectos as functions of time t. Pictue the Poblem The elocit ecto is the time-deiatie of the position ecto and the acceleation ecto is the time-deiatie of the elocit ecto. Diffeentiate with espect to time: d d dt dt 3ˆ i + ( 4 t)j ˆ [( ) iˆ 3t + ( 4t 5t ) ˆj ] Diffeentiate with espect to time: whee has units of m/s if t is in seconds. d d a [ 3ˆ i + ( 4 t) ˆj ] dt dt ( m/s )j ˆ 5 A paticle has a constant acceleation of a (6. m/s ) ˆ i + (4. m/s ) j ˆ. At time t, the elocit is zeo and the position ecto is ( m) ˆ i. (a) Find the elocit and position ectos as functions of time t. (b) Find the equation of the paticle s path in the plane and sketch the path. Pictue the Poblem We can use constant-acceleation equations in ecto fom to find the elocit and position ectos as functions of time t. In (b), we can eliminate t fom the equations iin the and components of the paticle to find an epession fo as a function of. (a) Use + at with to find : [ ˆj ]t ( ) ( ) 6. m/s ˆ i + 4. m/s Use + t + a with ( m)iˆ t to find : [( ) ( ) t ] iˆ m + 3. m/s + (. m/s ) [ t ]j ˆ

32 Chapte 3 (b) Obtain the and components of the path fom the ecto equation in (a): Eliminate t fom these equations and sole fo to obtain: ( 3. m/s ) m + t and ( ). m/s t 3 3 m The aph of staiht line m is shown below. Note that the path in the plane is a, m , m 53 Statin fom est at a dock, a moto boat on a lake heads noth while ainin speed at a constant 3. m/s fo s. The boat then heads west and continues fo s at the speed that it had at s. (a) What is the aeae elocit of the boat duin the 3-s tip? (b) What is the aeae acceleation of the boat duin the 3-s tip? (c) What is the displacement of the boat duin the 3-s tip? Pictue the Poblem The displacements of the boat ae shown in the followin fiue. Let the + diection be to the east and the + diection be to the noth. We need to detemine each of the displacements in ode to calculate the aeae elocit of the boat duin the 3-s tip.

33 Motion in One and Two Dimensions 3 (a) The aeae elocit of the boat is ien b: Δ a () Δt The total displacement of the boat is ien b: Δ Δ a N N + Δ ( Δt ) ˆj + Δt ( iˆ ) N W W W () To calculate the displacement we fist hae to find the speed afte the fist s: W N, f a N Δt N Substitute numeical alues and ealuate W : ( 3. m/s )( s) 6 m/s W Substitute numeical alues in equation () and ealuate Δ ( 3 s): Δ ( 3 s) ( 3. m/s )( s) ˆj ( 6 m/s)( s) iˆ ( 6 m) ˆj ( 6 m)iˆ Substitute numeical alues in equation () to find the boat s aeae elocit: a Δ Δ t ( 6 m)( iˆ + ˆj ) 3 s ( m/s)( iˆ + ˆj ) (b) The aeae acceleation of the boat is ien b: a a Δ f i Δt Δt Substitute numeical alues and ealuate a : 6m/s iˆ 3s a ( ) aa (. m/s )i ˆ (c) The displacement of the boat fom the dock at the end of the 3-s tip was one of the intemediate esults we obtained in Pat (a). Δ ( 6 m) ˆj + ( 6 m) ( 6 m)( iˆ + ˆj ) iˆ 54 Statin fom est at point A, ou ide ou motoccle noth to point B 75. m awa, inceasin speed at stead ate of. m/s. You then aduall tun towad the east alon a cicula path of adius 5. m at constant speed fom B to point C until ou diection of motion is due east at C. You then continue eastwad, slowin at a stead ate of. m/s until ou come to est at point D. (a) What is ou aeae elocit and acceleation fo the tip fom A to D? (b) What is ou displacement duin ou tip fom A to C? (c) What distance did ou tael fo the entie tip fom A to D?

34 4 Chapte 3 Pictue the Poblem The followin diaam summaizes the infomation ien in the poblem statement. Let the + diection be to the east and the + diection be to the noth. To find ou aeae elocit ou ll need to find ou displacement Δ and the total time equied fo ou to make this tip. You can epess Δ as the sum of ou displacements Δ AB, Δ BC, and Δ CD. The total time fo ou tip is the sum of the times equied fo each of the sements. Because the acceleation is constant (but diffeent) alon each of the sements of ou tip, ou can use constant-acceleation equations to find each of these quantities. Noth C D 7 5 m 5 m B 5 m Δ A (a) The aeae elocit fo ou tip is ien b: The total displacement fo ou tip is the sum of the displacements alon the thee sements: The total time fo ou tip is the sum of the times fo the thee sements: The displacements of the thee sements of the tip ae: East Δ a () Δt Δ Δ + Δ + Δ () AB AB BC BC CD Δt Δt + Δt + Δt (3) CD Δ ˆ i ( 75. m)j ˆ AB +, Δ 5. m iˆ BC + 5. m and Δ Δ iˆ ˆj ( ) ( )j ˆ CD CD +,

35 Motion in One and Two Dimensions 5 In ode to find Δ CD, ou need to find the time fo the C to D sement of the tip. Use a constantacceleation equation to epess Δ : CD Because C B : CD C CD CD ( Δt ) Δ Δt + a (4) CD C B A + aabδtab o, because A, a Δt (5) C AB AB Use a constant-acceleation equation to elate Δt AB to Δ AB : Substitute numeical alues and ealuate Δt AB : Δ AB AΔtAB + a o, because A, AB AB ( Δt ) AB AB ( Δt ) AB Δ a Δt (75. m). m/s Δt AB AB 8.66 s Δ a AB AB Substitute fo Δt AB in equation (5) to obtain: Δ AB C aab aab a AB Δ AB Substitute numeical alues and ealuate C : C (. m/s )( 75. m) 7.3 m/s The time equied fo the cicula sement BC is ien b: Δt BC π B Substitute numeical alues and ealuate Δt BC : The time to tael fom C to D is ien b: Substitute numeical alues and ealuate Δt : CD (5. m) π 7.3 m/s Δt BC ΔCD D ΔtCD acd acd o, because D, C Δt CD a CD 7.3 m/s. m/s Δt CD C s 7.3 s

36 6 Chapte 3 Now we can use equation (4) to ealuate Δ Δt CD : ( 7.3 m/s)( 7.3 s) + (. m/s )( 7.3 s) 5 m CD The thee displacements that ou need to add in ode to et Δ ae: Δ ˆ i ( 75. m)j ˆ AB +, Δ 5. m iˆ BC + 5. m and Δ 5 m iˆ ˆ CD + ( ) ( )j ˆ ( ) j, Substitute in equation () to obtain: Δ ˆ i + ( 75. m) ˆj + ( 5. m) iˆ + ( 5. m) ˆj + ( 5 m) ( m) iˆ + ( 5 m)j ˆ iˆ + ˆj Substitutin in equation (3) fo Δt AB, Δt BC, and ΔtCD ields: Δ t 8.66 s s s 3.5s Use equation () to find a : ( m) iˆ + ( 5 m) a 3.5s ˆj ( 6.56 m/s) iˆ + ( 4. m/s)j ˆ Because the final and initial elocities ae zeo, the aeae acceleation is zeo. (b) Epess ou displacement fom A to C as the sum of the displacements fom A to B and fom B to C: Δ AC Δ AB + Δ Fom (a) ou hae: Δ ˆ i ( 75. m)j ˆ AB + and Δ 5. m iˆ BC + 5. m BC ( ) ( )j ˆ Substitute and simplif to obtain: Δ i ( ) ˆ AC ˆ m j + ( 5. m) + ( 5. m) ˆj ( 5. m) iˆ + ( 5 m)j ˆ iˆ (c) Epess the total distance ou taeled as the sum of the distances taeled alon the thee sements: d d + d + d tot AB BC CD whee d BC π and is the adius of the cicula ac.

37 Motion in One and Two Dimensions 7 Substitutin fo d BC ields: d tot d AB + π + dcd Substitute numeical alues and ealuate d tot : d tot 75. m + π 34 m ( 5. m) + 5 m Relatie Velocit 55 A plane flies at an aispeed of 5 km/h. A wind is blowin at 8 km/h towad the diection 6º east of noth. (a) In what diection should the plane head in ode to fl due noth elatie to the ound? (b) What is the speed of the plane elatie to the ound? Pictue the Poblem Choose a coodinate sstem in which noth is the + diection and east is the + diection. Let θ be the anle between noth and the diection of the plane s headin. The elocit of the plane elatie to the ound, PG, is the sum of the elocit of the plane elatie to the ai, PA, and the elocit of the ai elatie to the ound, AG. That is, PG PA + AG. The pilot must head in such a diection that the east-west component of PG is zeo in ode to make the plane fl due noth. W AG 3 PA θ N S PG E (a) Fom the diaam one can see that: Solin fo θ ields: Substitute numeical alues and ealuate θ : cos3 AG θ sin PA sinθ AG cos3 PA (8 km/h)cos3 sin θ 5 km/h 6. 6 west of noth (b) Because the plane is headed due noth, add the noth components of PA and AG to detemine the plane s ound speed: PG (5 km/h)cos6. + (8 km/h)sin3 8 km/s