11.7 Velocity en you are outide and notice a brik wind blowing, or you are riding in a car at 60 km/, you are imply conidering te peed of motion a calar quantity. ometime, owever, direction i alo important or even crucial. A wind blowing from te nort or from te wet (Figure 1) i ignificant for information about approacing weater ytem. It could alo affect plane a tey take off, fly, and land. Airline could not operate witout wind direction a part of teir fligt plan. A peed along wit a direction i called velocity. Velocity i a vector quantity and i communicated in a variety of format, for example, a quantity ymbol, v an actual quantity, 60 km/ [] Figure 1 a labelled vector arrow, v 1 or 60 km/ All of tee format repreent a contant velocity, wic mean tat bot te ize (peed) and te direction tay te ame. Jut a a contant peed i repreented by a traigt line on a ditance time grap, a contant velocity appear a a traigt line on a poition time grap (Figure 2). A you know, diplacement i a vector quantity. It include a ize and a direction. Velocity i alo a vector quantity becaue it i a cange in diplacement in a given time. Jut a a cange in ditance in a given time i ued to calculate peed, a cange in poition (diplacement) in a given time i ued to calculate velocity. Te direction of te velocity i alway te ame a te direction of te diplacement. ind velocitie are part of te decription of weater ytem. d 0 Contant Velocity t ample Problem 1 A train travel at a contant peed troug te countryide and a a diplacement 150 km [] in a time of 1.7. at i te velocity of te train? d = 150 km [] = 1.7 v =? v = d = 150 km [] 1.7 = 88 km/ [] Te velocity of te train i 88 km/ []. Figure 2 Te line repreenting velocity pae troug te origin. at doe ti indicate? 432 Capter 11
On a plane trip from Toronto to Vancouver, te pilot will uually announce an air peed uc a 425 km/. However, bot te pilot and te paenger know tat te direction i wet, o te velocity i 425 km/ []. Average Velocity Average velocity i defined a te overall rate of cange of poition from tart to fini. It i calculated by dividing te reultant diplacement (wic i te cange of poition) by te total time (Figure 3). Te average velocity i expreed matematically a Canging Velocity d 0 d v av = d Te diplacement, d, i a vector quantity and te direction of te average velocity, v av, will alway be te direction of te diplacement. e refer to average velocity, and ue it for calculation, in two ituation: 1. if we know te reultant diplacement and time but ave no information on te velocitie at any time during te trip; or 2. if te velocity varie (eiter in direction or ize) during te journey, but we are only intereted in te average velocity overall. Figure 3 A curved line on a poition time grap repreent a canging velocity becaue te lope of te line i canging. ample Problem 2 Monarc butterflie migrate from atern Canada to central Mexico (Figure 4), a reultant diplacement of about 3500 km [] in a time of about 91 d. at i te average velocity of te monarc butterflie in kilometre per our? d = 3500 km [] = 91 d 24 = 2.2 10 3 (or 2184 ) d v av =? v av = d = 350 0 km [ ] 2.2 103 = 1.6 km/ [] Te average velocity of te monarc butterflie, a tey fly to Mexico, i about 1.6 km/ []. (Becaue te meaurement of diplacement and time are etimate about it i bet to aume a certainty of only two ignificant digit in te anwer.) 50 120 110 100 90 80 70 60 40 30 20 o r t e r n. Madre O ccidental A. Central Flyway B. Coatal Flyway Figure 4 ocky Mountain L i m i t o f. Madre Oriental A Tran Atlantic Belt M o n a r c? B B u Monarc butterflie eat of te ocky Mountain all migrate to one valley in te central Cincua mountain of Mexico. t t e r f l y a n g e Appalacian Mountain Diplacement and Velocity 433
Te velocity equation can be rearranged to olve for eiter time or ditance. otice tat bot average velocity and diplacement are vector quantitie but time i a calar quantity. Terefore, if te time interval i te unknown variable, te anwer will not include a direction. ample Problem 3 A monarc butterfly uually flie during te day and ret at nigt on it migration. If a particular butterfly i travelling at an average velocity of 19 km/ [] for 230 km [] on one part of it journey to Mexico, ow long doe ti take? d = 230 km [] v av = 19 km/ [] =? v av = d d = v a v = 230 km [] 19 k m [] or v av = d 19 km [] = 230 k m [] = 230 km [] 19 k m [] Te butterfly would take 12 to travel 230 km [] at an average velocity of 19 km/ []. otice tat te direction divide out jut like unit. (Ti enure tat te time anwer i a calar quantity.) ometime we may need to find te reultant diplacement before we can calculate te average velocity. For example, a jogger run 52 m [] in 10.0 and ten 41 m [] for 8.0. at i te jogger average velocity? v av = d To calculate te average velocity you need to know te reultant diplacement from tart to fini. Ti diplacement can be found by adding te two diplacement vector for te two part of te run. From te vector diagram (Figure 5), te reultant diplacement mut be 11 m []. However, a vector diagram i not neceary if you aign [] a poitive (+) and [] a negative ( ). Ten you can add algebraically: d Figure 5 52 m 41 m d = (+52 m) + ( 41 m) = +11 m or 11 m [] 434 Capter 11
Te total time taken for ti diplacement i 18.0. Te average velocity i calculated from te reultant diplacement and te total time. v av = d = 11 m [ ] 18.0 = 0.61 m/ [] Te average velocity of te jogger i 0.61 m/ []. Te average velocity of te jogger i baed only on te reultant diplacement and te total time. Ti average velocity i not directly related to te velocitie during eac diplacement. For example, wen te jogger run 52 m [] in 10.0, te jogger velocity i 5.2 m/ []. For te econd part, 41 m [] in 8.0 correpond to a velocity of 5.1 m/ []. 1t diplacement v 1 = d 2nd diplacement v 2 = d = 52 m [ 10.0 ] = 41 m [ ] 8.0 = 5.2 m/ [] = 5.1 m/ [] A you can ee, tee velocitie are not obviouly related to te average velocity of 0.61 m/ []. Comparing Average peed and Average Velocity A you recall from Capter 9, average peed i calculated a te total ditance travelled during a certain lengt of time. Becaue peed i a calar quantity, ti calculation ignore te direction taken. Te jogger in te previou example travel 52 m in 10.0 and ten 41 m in 8.0 for a total ditance of 93 m in a total time of 18.0. Te average peed of te jogger i calculated a follow: average peed average velocity v av = d v av = d 93 m = = 11 m [ ] 1 8.0 18.0 = 5.2 m/ = 0.61 m/[] Compare te average peed, 5.2 m/, to te average velocity, 0.61 m/ [], for te ame jogger. ote tat te ize of te average velocity i unrelated to te average peed. Average velocity i te reultant diplacement from tart to fini, divided by te total time taken. Average velocity doe not depend on te pat taken or te peed trougout ti pat. Diplacement and Velocity 435
Undertanding Concept 1. xplain in your own word ow velocity differ from peed. 2. at i contant in contant peed and in contant velocity? Give an example of eac. 3. Tranlate eac of te following vector into a complete written vector quantity. (a) (b) (c) 40 km/ 10 m/ 5 km/ 45 (c) Calculate te tudent average peed. (d) at i er velocity wile travelling eat? (e) at i er velocity wile travelling nort? (f) Draw a vector diagram to determine te reultant diplacement. (g) Calculate te tudent average velocity. 7. A fier et out to ceck i fiing net and ead 15 km/ [] for 0.20. After topping for 0.50, e travel at 12 km/ [] for 0.10 to get to te next et of fiing net. (a) at i te fier diplacement to get to te firt et of net? (b) at i te diplacement wen going from te firt to te next et of net? (c) at i te fier reultant diplacement from te dock to te econd et of net? (d) at i average velocity from te dock to te econd et of net? (e) at i te average peed from te dock to te econd et of net? (f) Te fier return to te dock. ow wat are i reultant diplacement and average velocity? 4. Grey wale migrate from te Cukci ea to te Aleutian Iland in an average time of 25 d before eading eat to travel along te coat. at i te grey wale average velocity, in kilometre per day, between point A and B on te map own in Figure 6? (ote tat te ditance between latitude line 10 apart i equal to 1100 km.) 5. ile delivering flier, Brad walk 1.00 km [], 0.50 km [], 1.00 km [], and finally 0.50 km [] in a total time of 1.5. (a) at i te total ditance travelled? (b) at i Brad average peed? (c) at i te reultant diplacement? (d) at i Brad average velocity? (e) y are you able to calculate te average velocity witout knowing te time for eac part of te trip? 6. A tudent travel 6.0 m [] in 3.0 and ten 10.0 m [] in 4.0. (a) at i er peed wile travelling eat? (b) at i er peed wile travelling nort? 70 60 50 40 30 0 Bering ea Figure 6 Cuckci ea A B Pacific Ocean Alaka wale migration route 10 latitude = 1100 km CAADA UITD TAT MXI 436 Capter 11