Kinematics in Two-Dimensions

Transcription

1 Slide 1 / 92 Slide 2 / 92 Kinemtics in Two-imensions Slide 3 / 92 How to Use this File ch topic is composed of brief direct instruction There re formtie ssessment questions fter eer topic denoted b blck tet nd number in the upper left. > Students work in groups to sole these problems but use student responders to enter their own nswers. > esigned for SMRT Response P student response sstems. Vector Nottion Projectile Motion Uniform irculr Motion Reltie Motion Slide 4 / 92 Tble of ontents lick on the topic to go to tht section > Use onl s mn questions s necessr for sufficient number of students to lern topic. Full informtion on how to tech with NJTL courses cn be found t njctl.org/courses/teching methods Slide 5 / 92 Slide 6 / 92 Position nd Velocit Vectors Vector Nottion Motion problems in one dimension re interesting, but frequentl, objects re moing in two, nd een three dimensions (four, when ou count time s dimension in specil nd generl reltiit). This is where the ector nottion lerned erlier comes in er hnd, nd we will strt b defining position ector,. Return to Tble of ontents

2 Slide 7 / 92 Slide 8 / 92 Slide 9 / 92 erge Velocit Slide 10 / 92 Instntneous Velocit s n object moes from one point in spce to nother, the erge elocit of its motion cn be described s the displcement of the object diided b the time it tkes to moe. To find the instntneous elocit (the elocit t specific point in time) requires the time interl to be so smll tht it cn effectiel be reduced to 0, which is represented s limit. (erge elocit ector) (instntneous elocit ector) Nottion note: it will be ssumed tht ll the motion ectors re time dependent, so fter this slide (t), (t) nd z(t) will be shown s, nd z (sme conention for elocit nd ccelertion). Slide 11 / 92 Instntneous Velocit omponents The instntneous elocit hs three different components:,, nd z (n of which cn equl zero). ch component is shown below: Slide 12 / 92 erge ccelertion ccelertion is the rte t which the elocit is chnging, nd the erge ccelertion cn be found b tking the difference of the finl nd initil elocit nd diiding it b the time it tkes for tht eent to occur. Vector representtion:

3 Slide 13 / 92 Instntneous ccelertion Just s we cn find the elocit t specific point in time, we cn lso find the instntneous ccelertion using limit. Slide 14 / 92 Instntneous ccelertion The instntneous ccelertion hs three different components:,, nd z (n of which cn equl zero). ch component is shown below: Vector representtion: Slide 15 / 92 1 The ector,, describes the position of prticle s function of time. Find the epression for the elocit nd ccelertion ectors epressed s function of time. Slide 15 () / 92 1 The ector,, describes the position of prticle s function of time. Find the epression for the elocit nd ccelertion ectors epressed s function of time. [This object is pull tb] Slide 16 / 92 2 The ector,, describes the position of prticle s function of time. Find the epression for the elocit nd ccelertion ectors epressed s function of time. Slide 16 () / 92 2 The ector,, describes the position of prticle s function of time. Find the epression for the elocit nd ccelertion ectors epressed s function of time. [This object is pull tb]

4 Slide 17 / 92 Integrtion The unit on One imension Kinemtics showed how to obtin position from elocit, nd elocit from ccelertion through integrtion techniques. The sme method works for two nd three dimensions. Slide 18 / 92 Integrtion Here is it wht it looks like from ector point of iew, where we strt with ccelertion nd integrte twice to get to position: ch component is shown below, nd since we re onl looking for instntneous lues, we will lee out the limits of integrtion: Slide 19 / 92 3 The ector,, describes the ccelertion of prticle s function of time. Find the epression for the elocit nd position ectors epressed s function of time. Slide 19 () / 92 3 The ector,, describes the ccelertion of prticle s function of time. Find the epression for the elocit nd position ectors epressed s function of time. [This object is pull tb] Slide 20 / 92 Slide 20 () / 92

5 Slide 21 / 92 Slide 22 / 92 Instntneous lues Once the ector for position, elocit or ccelertion is found, either b differentition or integrtion, the instntneous lue cn be found b substituting the lue of time in for t. Nottion note: When ou find the lue of the position, elocit or ector, just lee it in ector nottion - don't worr bout the units - t this point in our phsics eduction, its ssumed ou know them! Slide 22 () / 92 Slide 23 / 92 6 Wht is the elocit of n object t t = 3 s if its ccelertion is described b? Slide 23 () / 92 Slide 24 / 92 6 Wht is the elocit of n object t t = 3 s if its ccelertion is described b? Projectile Motion [This object is pull tb] Return to Tble of ontents

6 Slide 25 / 92 Projectile Motion He ou eer thrown n object in the ir or kicked soccer bll to friend nd wtched the pth in spce it followed? The pth is described b mthemtics nd phsics - it is prbolic pth - nother reson wh ou studied prbols in mthemtics. Slide 26 / 92 Projectile Motion The ectors re cting s studied erlier - is mimum t the lunch point, decreses under the influence of the grittionl field, reches zero t the pe, nd then increses until it reches the negtie of the initil elocit right before it strikes the ground. The boe is n - plot tht shows the pth of the object - nd shows t rious points, the elocit ectors. Tke minute nd discuss the behior of the ectors. Now tht the behior hs been reiewed, wht else do ou notice bout this picture? Slide 27 / 92 Projectile Motion Slide 28 / 92 Projectile Velocit Just s in mthemtics where ector is resoled into two perpendiculr ectors ( nd ), in rel life, the motion is independent of the motion nd cn be delt with seprtel. Vector nlsis for the elocit gies us: totl The ectors chnge becuse fter lunch, the onl force cting on the bll in the direction is grit. ut, neglecting friction, there re NO forces cting in the direction. θ So is constnt throughout the motion. Slide 29 / 92 Velocit of Projectile Slide 30 / 92 ccelertion of Projectile = -g = -g = -g = -g = -g In 1 Kinemtics, ou re used to the elocit of the object t its pe being zero. For 2 Kinemtics, the elocit is zero, but it hs totl elocit becuse it still hs elocit component in the direction. Wht is the direction of the ccelertion ector t ech point? Ner the surfce of the plnet rth, there is zero ccelertion in the direction, nd constnt ccelertion, with mgnitude, g, in the negtie direction. This is true, regrdless of the direction of the elocit or displcement of the projectile. = 0 = -g

7 Slide 31 / 92 Motion of Projectile Slide 32 / 92 Motion of Projectile You know from eperience tht this motion is prbol. Let's see if this cn be deried mthemticll, b emining the position equtions in the nd direction. In the bsence of gien initil point, we re free to set 0 = 0 = 0. The ccelertion in the direction is zero, nd the ccelertion in the direction is "-g." We're using prmetric equtions here, where t is the prmeter, nd we're free to mnipulte the nd equtions simultneousl, since the both re true for n gien t. For more info - see our mth techer! The constnts re combined, represented b nd, nd is now epressed in terms of. Plug = 2.8 nd = 0.18 nd see wht our grphing clcultor or other electronic deice plots. Slide 33 / 92 Motion of Projectile Slide 34 / 92 Projectile isplcement Here's the plot - looks like the sketch boe. Tht's good. We now he group of equtions nd ectors to work with to predict the future motion of projectile, gien initil conditions. totl θ Slide 35 / 92 7 Which of the following sttements re true regrding projectile motion? Slide 35 () / 92 7 Which of the following sttements re true regrding projectile motion? is constnt. is constnt. Totl ccelertion is +g when the object is rising nd -g when it is flling. In the bsence of friction, the trjector depends upon the mss of the object. The elocit of the object is zero t the pe of the motion. The horizontl motion is independent of the erticl motion. Totl ccelertion is +g when the object is rising nd -g when it is flling. In the bsence of friction, the trjector depends upon the mss of the object. [This object is pull tb] The elocit of the object is zero t the pe of the motion. The horizontl motion is independent of the erticl motion.

8 Slide 36 / 92 8 mrble is shot b mrble luncher nd follows prbolic pth s shown. ir resistnce is negligible. Which of the following shows the direction of the elocit t point Y, the highest point on the pth? Slide 36 () / 92 8 mrble is shot b mrble luncher nd follows prbolic pth s shown. ir resistnce is negligible. Which of the following shows the direction of the elocit t point Y, the highest point on the pth? None. None. [This object is pull tb] Slide 37 / 92 9 mrble is shot b mrble luncher nd follows prbolic pth s shown. ir resistnce is negligible. Which of the following shows the direction of the net force t point X? Slide 37 () / 92 9 mrble is shot b mrble luncher nd follows prbolic pth s shown. ir resistnce is negligible. Which of the following shows the direction of the net force t point X? None. None. [This object is pull tb] Slide 38 / 92 Soling Projectile Motion Problems Seerl different problems will now be soled. The criticl point in soling these is relizing tht in the bsence of friction, the onl force cting on the projectile in flight is the grittionl force. Without this force, the projectile would moe in stright line nd neer hit the ground - nd lee the plnet. With grit, the projectile gets pulled to the ground (unless it reches sufficient elocit to orbit the plnet - but this will be coered in the Uniersl Grittion unit of this course). Slide 39 / 92 Time to fll You sole for the time to fll - using the kinemtics equtions for the is. Nothing er interesting is going on in the direction - the elocit sts constnt (no ccelertion). ut, the displcement depends on how long the bll sts in the ir. Strt with this problem - bll is pushed off height, H, boe the ground, with n initil elocit 0 in the horizontl direction. 0 sed on this, in most cses, do ou begin with the kinemtics equtions in the or in the direction? H

9 Slide 40 / 92 Slide 41 / 92 Slide 42 / Two cnnon blls, with different msses nd initil elocities, re lunched horizontll off cliff t the sme time. Which will strike the ground first? Slide 42 () / Two cnnon blls, with different msses nd initil elocities, re lunched horizontll off cliff t the sme time. Which will strike the ground first? The cnnon bll with the gretest mss. The cnnon bll with the smllest mss. The cnnon bll with the gretest initil elocit. The cnnon bll with the smllest initil elocit. The both strike the ground t the sme time. The cnnon bll with the gretest mss. The cnnon bll with the smllest mss. The cnnon bll with the gretest initil elocit. The cnnon bll with the smllest initil elocit. [This object is pull tb] The both strike the ground t the sme time. Slide 43 / Two cnnon blls, with different msses nd initil elocities, re lunched horizontll off cliff t the sme time. Which will go further in the direction? Slide 43 () / Two cnnon blls, with different msses nd initil elocities, re lunched horizontll off cliff t the sme time. Which will go further in the direction? The cnnon bll with the gretest mss. The cnnon bll with the smllest mss. The cnnon bll with the gretest initil elocit. The cnnon bll with the smllest initil elocit. The both strike the ground t the sme time. The cnnon bll with the gretest mss. The cnnon bll with the smllest mss. The cnnon bll with the gretest initil elocit. The cnnon bll with the smllest initil elocit. [This object is pull tb] The both strike the ground t the sme time.

10 Slide 44 / 92 Finding Mimum Height (pe) nd Time to pe Slide 45 / 92 Finding Time to pe n object is propelled with n initil elocit, 0, t n ngle θ with the ground. Find the time it tkes to rech point (pe) nd its mimum height. θ 0 0 The initil elocit in the direction is 0sinθ nd t the pe, the elocit in the direction is zero. This is the time to pe. There re two pproches to finding the mimum height. We'll tke the more complicted one first (which inoles finding the time to pe)! Slide 46 / 92 Slide 47 / 92 Finding Mimum Height (pe) id ou notice how we used the first two Kinemtics equtions to find the mimum height? If the problem did not sk for the time to pe, but just for the height, the third Kinemtics eqution could he been used, s it ws deried from combining the first two Kinemtics equtions: Δ = h nd = 0 for mimum height Mtches the nswer from using the first two Kinemtics equtions. Slide 48 / 92 Finding Mimum isplcement Slide 49 / 92 Finding Mimum isplcement The lst problem to be worked is to find the mimum displcement in the direction - point (, 0). Use the second kinemtics eqution, relizing tht = 0 = 0. 0 θ 0 (, 0) θ 0 0 (, 0) The first step is to find out how long the projectile is in the ir. Since the projectile tkes s long to rech point from the ground s it does to go from point to, we cn just double the pe time. ut tht's using smmetr rgument, nd we cn be more forml. How? There re two solutions for t. Mn times, the mth gies two solutions nd onl one is phsicll relent. In this cse, both solutions re lid - the show the initil nd finl times t which the projectile is on the ground. One more step...

11 Slide 50 / 92 Finding Mimum isplcement The mimum displcement is frequentl lbeled R (rnge). Time in the ir 0 θ 0 (, 0) Slide 51 / t wht ngle will projectile he the gretest erticl displcement? ouble ngle formul (trigonometr) Mimum displcement (rnge) Slide 51 () / t wht ngle will projectile he the gretest erticl displcement? Slide 52 / t wht ngle will projectile he the gretest horizontl displcement? [This object is pull tb] 90 0 Slide 52 () / t wht ngle will projectile he the gretest horizontl displcement? 0 0 Slide 53 / mrble luncher fires two mrbles with the sme initil elocit but t different ngles to the horizontl. Which pir of ngles will result in the sme mimum displcement in the direction? nd nd nd [This object is pull tb] 20 0 nd 80 0 None of the boe.

12 Slide 53 () / mrble luncher fires two mrbles with the sme initil elocit but t different ngles to the horizontl. Which pir of ngles will result in the sme mimum displcement in the direction? 0 0 nd 30 0 Slide 54 / 92 Uniform irculr Motion 30 0 nd nd nd 80 0 None of the boe. [This object is pull tb] Return to Tble of ontents Slide 55 / 92 Uniform irculr Motion Slide 56 / 92 Uniform irculr Motion constnt speeddecresing speedincresing speed Uniform irculr Motion occurs when n object moes in circle with constnt speed, nd its ccelertion is perpendiculr to its elocit. The elocit nd displcement ectors re tngent to the circle. In the boe picture, UM is represented on the left. Wh is it necessr for the ccelertion to be perpendiculr to the elocit for UM? Slide 57 / 92 Uniform irculr Motion constnt speeddecresing speedincresing speed This perpendiculr ccelertion is clled centripetl (centerseeking) ccelertion. The ccelertion is cused b the centripetl force - nd when force is perpendiculr to n object's displcement, then no Work is done on the prticle. With zero work, the Work-nerg eqution sttes tht K f = K 0, so the speed remins constnt. Slide 58 / cr is driing with decresing speed on cured pth. Which digrm shows the correct direction for the elocit nd the ccelertion? constnt speeddecresing speedincresing speed Use the Work-nerg eqution to eplin wht is going on in the middle nd right digrms. These two digrms will be eplined in more detil in the nmics unit of this course s the inole free bod digrms.

13 Slide 58 () / cr is driing with decresing speed on cured pth. Which digrm shows the correct direction for the elocit nd the ccelertion? Slide 59 / cr is driing with constnt speed on cured pth. Which digrm shows the correct direction for the elocit nd the ccelertion? [This object is pull tb] Slide 59 () / cr is driing with constnt speed on cured pth. Which digrm shows the correct direction for the elocit nd the ccelertion? Slide 60 / cr is driing with incresing speed on cured pth. Which digrm shows the correct direction for the elocit nd the ccelertion? [This object is pull tb] Slide 60 () / cr is driing with incresing speed on cured pth. Which digrm shows the correct direction for the elocit nd the ccelertion? Slide 61 / Is the elocit for n object in Uniform irculr Motion constnt? plin our nswer. Students tpe their nswers here [This object is pull tb]

14 Slide 61 () / Is the elocit for n object in Uniform irculr Motion constnt? plin our nswer. Students tpe their nswers here Slide 62 / 92 entripetl ccelertion erition You'e lred lerned tht for objects in uniform circulr motion (the speed is constnt); now it will be deried. No, the speed of the object is constnt, but since the object is continull chnging the direction of its motion, its elocit is chnging. The mgnitude of the elocit sts constnt. P1 P2 ompre these two tringles. This is position sketch of UM. s n object moes from point P 1 to P 2, it undergoes liner displcement of Δs nd sweeps out n ngle Φ. [This object is pull tb] This is sketch of the elocities t the two points. The mgnitudes re equl, but their differences in direction gie rise to Δ. Slide 63 / 92 Slide 64 / 92 entripetl ccelertion erition entripetl ccelertion erition P1 P2 Since 1 is perpendiculr to r 1 nd 2 is perpendiculr to r 2, the position nd the elocit tringles both subtend the sme ngle, Δθ. P1 P2 The elocit tringle hs been rotted nd superimposed on the position tringle so ou cn how the ngles re the sme. Knowing tht the tringles re similr, the rtios of their corresponding sides re equl (geometr): oth of these re isosceles tringles, nd he the sme erte ngle. geometric proof shows tht these re similr tringles. nd since 1 = 2 =, nd r 1 = r 2 = r, this simplifies s: or Slide 65 / 92 entripetl ccelertion erition Slide 66 / 92 entripetl ccelertion erition P2 P1 To find the instntneous ccelertion, we first he to come up with representtion for the erge ccelertion s before: The instntneous ccelertion is the lue of the erge ccelertion s Δt 0. The instntneous ccelertion t eer point on the circle is the sme - nd is clled centripetl ccelertion, c.

15 Slide 67 / 92 entripetl ccelertion Slide 68 / Wht is the centripetl ccelertion of bll tht is swung in circle of rdius, 1.0 m, with elocit of 5.0 m/s? One more hnd epression: m/s m/s m/s 2 10 m/s 2 25 m/s 2 Slide 68 () / Wht is the centripetl ccelertion of bll tht is swung in circle of rdius, 1.0 m, with elocit of 5.0 m/s? Slide 69 / bll is swung in circle of 9.0 m nd its centripetl ccelertion is 1 m/s 2. Wht is its elocit? m/s m/s m/s 3.0 m/s 5.0 m/s 2 10 m/s 2 9 m/s 18 m/s 25 m/s 2 [This object is pull tb] 81 m/s Slide 69 () / bll is swung in circle of 9.0 m nd its centripetl ccelertion is 1 m/s 2. Wht is its elocit? 3.0 m/s 3.0 m/s 9 m/s 18 m/s 81 m/s [This object is pull tb] Slide 70 / Wht is the rdius of the orbit of n object moing in circle with speed of 15 m/s nd centripetl ccelertion of 45 m/s 2? 0.33 m 3.0 m 5.0 m 10 m 15 m

16 Slide 70 () / Wht is the rdius of the orbit of n object moing in circle with speed of 15 m/s nd centripetl ccelertion of 45 m/s 2? Slide 71 / bll spins in horizontl orbit with period of 5.0 s. The orbit's rdius is 0.98 m. Wht is the centripetl ccelertion of the bll? 0.33 m 1.3 m/s m 5.0 m 2.5 m/s m/s 2 10 m 7.7 m/s 2 15 m [This object is pull tb] 15 m/s 2 Slide 71 () / bll spins in horizontl orbit with period of 5.0 s. The orbit's rdius is 0.98 m. Wht is the centripetl ccelertion of the bll? 1.3 m/s 2 Slide 72 / 92 Reltie Motion 2.5 m/s m/s m/s 2 15 m/s 2 [This object is pull tb] Return to Tble of ontents Slide 73 / 92 Reference Frmes Slide 74 / 92 Reference Frmes In order to describe where something is, ou need to relte it to something else. Tht's the purpose of reference frme - ou choose coordinte sstem in spce nd mke mesurements reltie to the sstem. There is no one preferred reference frme - nd frequentl, there is need to trnslte mesurements mde in one frme to nother. One reference frme cn een be moing compred to the other (or is the other w round?). s long s the frmes moe with constnt elocit reltie to ech other, we cn use Glilen trnsformtion. ccelerting reference frmes re the proince of Specil Reltiit nd won't be discussed now. efore the mth is done, think bout different reference frmes. If ou're sitting in bus tht is going 25 m/s (56 mph), the person sitting net to ou, reltie to the reference frme tht hs ou t the origin, is not chnging his position, nd his speed reltie to ou is 0 m/s. If someone is stnding t bus stop, reltie to reference sstem tht hs her t the origin, the person in the bus is moing t 25 m/s, nd is getting further w from her. nd if bus is moing prllel to our bus with the sme elocit, person in tht bus would gree with ou. Your set mte looks sttionr. Who is correct? Wh, ou ll re!

17 Slide 75 / 92 Glilen Trnsformtion hoose reference frme bsed on rtesin coordintes - nd for clrit, we'll just work in two dimensions - the sme principles ppl in three dimensions. This will be the frme. Then, choose the sstem nd put it in motion with constnt elocit, (elocit of the sstem with respect to the sstem), in the direction. ' In the bus emple, reference frme would be the womn t the bus stop, nd ou nd our set mte would be using the sstem. ' t ' Slide 76 / 92 Glilen Trnsformtion rp rp P ' The set mte is t point P. You re t the origin of the coordinte sstem. r P is the position of the set mte reltie to ou. The womn t the bus stop is t the origin of the sstem. r P is the position of the set mte reltie to her. Strt the problem with the two reference sstems coincident with ech other t t = 0. fter time t, the distnce treled b the bus is = 0 + t (ssuming = 0). Using ector nlsis, derie the eqution for the reltionship of the positions mesured b both obserers. Slide 77 / 92 Glilen Trnsformtion Slide 78 / 92 Glilen Trnsformtion ' P The ector nlsis gies us the First Glilen trnsformtion: ' P rp rp t rp ' Tke the time deritie of both sides: V is constnt t rp ' There is no trnsformtion in the direction perpendiculr to the motion. The component of point P sts constnt (s does the z); onl the component chnges. This is the Second Glilen trnsformtion. hnd w to keep trck of the subscripts is to note tht the "internl" subscript () on the right side of the equtions re the sme. The "eternl" subscripts on the right side mtch the subscripts on the left side of the eqution (P). Slide 79 / 92 Glilen Trnsformtion - ccelertion ' P rp Slide 80 / n irplne is fling south (ssume south is in the negtie direction) t speed of 500 km/h t constnt ltitude (positie direction). You re t rest on the ground. Wht is the elocit of plne pssenger in the nd directions reltie to our position? t rp ' Tke the time deritie of the second trnsformtion: = 500 km/h = -500 km/h = -500 km/h = 500 km/h V is constnt = 500 km/h = 0 km/h The ccelertion of n object mesured in two reference frmes moing with constnt elocit reltie to ech other is the sme. = -500 km/h = -500 km/h = -500 km/h = 0 km/h

18 Slide 80 () / n irplne is fling south (ssume south is in the negtie direction) t speed of 500 km/h t constnt ltitude (positie direction). You re t rest on the ground. Wht is the elocit of plne pssenger in the nd directions reltie to our position? = 500 km/h = -500 km/h = -500 km/h = 500 km/h = 500 km/h = 0 km/h = -500 km/h = -500 km/h = -500 km/h = 0 km/h [This object is pull tb] Slide 81 () / You re in bus, moing t constnt speed of 30 m/s. The person net to ou gets up nd strts wlking up the isle with n ccelertion of 0.42 m/s 2 from our ntge point. person on the side of the rod simultneousl mesures the the person's ccelertion. Wht mesurement does she obtin for his ccelertion? Slide 81 / You re in bus, moing t constnt speed of 30 m/s. The person net to ou gets up nd strts wlking up the isle with n ccelertion of 0.42 m/s 2 from our ntge point. person on the side of the rod simultneousl mesures the the person's ccelertion. Wht mesurement does she obtin for his ccelertion? 0.21 m/s m/s m/s m/s m/s 2 Slide 82 / Which of the following is required in order to use Glilen trnsformtion between two reference frmes? oth frmes must be moing in the sme direction. The frmes must be moing in opposite directions m/s m/s m/s 2 The frmes must be moing perpendiculr to ech other. The frmes must moe with constnt elocit reltie to ech other m/s m/s 2 [This object is pull tb] The frmes must moe with constnt ccelertion reltie to ech other. Slide 82 () / 92 Slide 83 / Which of the following is required in order to use Glilen trnsformtion between two reference frmes? ot problem oth frmes must be moing in the sme direction. Finish point ' WG The frmes must be moing in opposite directions. The frmes must be moing perpendiculr to ech other. The frmes must moe with constnt elocit reltie to ech other. [This object is pull tb] The frmes must moe with constnt ccelertion reltie to ech other. Strt point Two kids re on bot cpble of mimum speed of 4.2 m/s, nd wish to cross rier 1200 m to point directl cross from their strting point. If the speed of the wter in the rier is 2.8 m/s, how much time is required for the crossing (ssume their engine is operting t mimum speed)? '

19 Slide 84 / 92 ot problem Slide 85 / 92 ot problem Finish point Strt point ' WG ' This is clssic Glilen trnsformtion problem. The, es, represent coordinte sstem fied to the erth. The bot will strt t point nd go to point. Finish point Strt point ' WG ' The wter current will push the bot to the right. So, if the bot is heded directl for point, it will wind up somewhere to the right of it on the opposite shore. The '' es represent the flowing wter reference frme - it is moing t elocit WG = 2.8 m/s to the right ( WG is the nottion for the elocit of the wter, reltie to the fied ground). From our own eperience, if ou were driing the bot, would ou point the bow t point nd just strt the engine? You wnt to im the bot to the left of point. The bot's mimum speed is 4.2 m/s. Tr drwing ector digrm tht shows the bot's elocit, nd wht elocit the bot hs becuse of the current pushing it to the right. dd the WG to the two ectors nd mke tringle. Hint: ou re free to moe WG from where it is shown boe. Slide 86 / 92 ot problem Slide 87 / 92 ot problem W WG G ' WG ' The current ( WG) pushes the bot to the right - so lthough it is pointing to the left, the bot is ctull going stright cross the rier - G, the elocit of the bot in the ground fied sstem. The mimum speed of the bot is W - the elocit of the bot oer the wter reference frme (''). It represents the direction tht the bot's bow is pointing. fied obserer in the '' frme ( duck floting with the current) would see the bot under power moe opposite the current flow (to the left) nd w from the duck. W WG G How long does it tke the bot to rech the other side? ' WG ' We're now red to sole the problem b using ector nlsis. Note how the inner subscripts "cncel" nd onl the outer subscripts remin on the right. The three elocit ectors form right tringle, so we cn use Pthgors to find the mgnitude of G: Slide 88 / 92 ot problem Slide 89 / 92 ot problem W WG G ' WG The distnce to be treled from point to is 1200 m, so the time it tkes to cross, gien G = 3.13 m/s is: W WG G ' WG ' Wht ngle, θ, with the erticl, should the bot be pointing t to get to point? W WG G ' ow Stern The ol to the left represents the bot. fling duck looking down on the bot would see it moe from point to, but it would be pointing to the left s it worked ginst the current - s if it ws sliding cross the rier.

20 Slide 90 / Two silors re on motor whlebot treling t speed of 10.0 km/h, nd wish to cross rier 2.0 km wide to point directl cross from their strting point. If the speed of the rier is 9.0 km/h prllel to the shore, how much time is required for the crossing? Slide 90 () / Two silors re on motor whlebot treling t speed of 10.0 km/h, nd wish to cross rier 2.0 km wide to point directl cross from their strting point. If the speed of the rier is 9.0 km/h prllel to the shore, how much time is required for the crossing? 0.45 h 0.45 h 0.90 h 1.0 h 0.90 h 1.0 h 10.0 h The bot will neer rech the other shore h [This object is pull tb] The bot will neer rech the other shore. Slide 91 / Two silors re on motor whlebot tht trels t speed of 10.0 km/h nd wish to cross rier 2.0 km wide in the quickest time. Wht direction should the point the bow? rw nd lbel the elocit ector digrm. The speed of the rier is 9.0 km/h prllel to the shore. How fr down rier do the trel before the rech the other shore? Slide 91 () / Two silors re on motor whlebot tht trels t speed of 10.0 km/h nd wish to cross rier 2.0 km wide in the quickest time. Wht direction should the point the bow? rw nd lbel the elocit ector digrm. The speed of the rier is 9.0 km/h prllel to the shore. How fr down rier do the trel before the rech the other wg shore? Point the bow perpendiculr to bw bg the shore. [This object is pull tb] Slide 92 / n irplne is fling with constnt speed of 1200 km/h through the ir, while eperiencing cross wind with speed of 500 km/h reltie to the ground. Wht is the irplne's speed reltie to ground? Slide 92 () / n irplne is fling with constnt speed of 1200 km/h through the ir, while eperiencing cross wind with speed of 500 km/h reltie to the ground. Wht is the irplne's speed reltie to ground? 700 km/h 1200 km/h 700 km/h 1200 km/h 1300 km/h 1600 km/h 500 km/h 1300 km/h 1600 km/h 500 km/h 1700 km/h 1700 km/h 2500 km/h 2500 km/h [This object is pull tb]