Position Time Graphs 12.1

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12.1 Positio Time Graphs Figure 3 Motio with fairly costat speed Chapter 12 Distace (m) A Crae Flyig Figure 1 Distace time graph showig motio with costat speed A Crae Flyig Positio (m [E] of pod) We

1 12.1 Positio Time Graphs Figure 3 Motio with fairly costat speed Chapter 12 Distace (m) A Crae Flyig Figure 1 Distace time graph showig motio with costat speed A Crae Flyig Positio (m [E] of pod) We kow that positio ad displacemet are vectors, ad iclude directio. Is it possible to represet vector motio o a graph? Absolutely. Positio is represeted o a positio time graph, which looks very much like a distace time graph. Look at Figures 1 ad 2. Oe has a scalar quatity o the y-axis, while the other has a vector quatity. What differeces ca you detect? If a bird is flyig at costat speed i a straight lie (Figure 3), it is coverig equal distaces i equal itervals of time. This is illustrated by the straight lie o a distace time (d-t) graph (Figure 1). A positio time (d -t) graph for this motio (Figure 2) looks idetical to the distace time graph: the shape ad uits are the same, but the positio time graph icludes the directio of motio i the label for the y-axis. Figures ad 5 represet the same trip to the video store. They show that ot all distace time ad correspodig positio time graphs are the same (as i Figures 1 ad 2). How may ways are Figure ad Figure 5 differet? Firstly, the y-axis is labelled differetly. Secodly, the last segmet of the trip is differet oly i that the lie is sloped upward (Figure ) rather tha dowward (Figure 5). All lie segmets i both graphs are the same legth ad slope (except for the egative versus positive slope for the last segmet). This illustrates how positio (a vector quatity) is differet from distace (a scalar quatitiy). More iformatio is icluded i a positio time graph. We kow, for example, that the video store is orth of the referece poit (assume home). We also kow from the positio time graph that the shopper has retured home. Figure 2 Positio time graph showig motio eastward with costat velocity

2 Trip to the Video Store Trip to the Video Store Distace (m) 12 8 Positio (m [N] of home) Figure Distace time graph, showig a retur trip to the video store 3 5 Figure 5 Positio time graph, showig the same retur trip to the video store as Figure Let us ow look at a graph that is a little differet (Figure ). How would you walk-the-graph? You would 1. start from the referece poit (home, ) ad move east at a costat velocity; 2. stop (for whatever reaso) for about a equal period of time; 3. start walkig back home at about the same speed (approximately the same size of slope) but i the opposite directio (velocity is egative);. walk right past your home ad cotiue west for a while (a egative displacemet); 5. slow dow gradually before stoppig ad turig east, toward home, agai; the. retur to your origial positio, home. Goig for a Walk Positio (m [E] of home) Time (mi.) 5 Figure A positio time graph of a short walk from home Displacemet, Velocity, ad Acceleratio 7

3 As you ca see, positio time graphs ca have a egative slope (a decreasig value of d ), whereas distace time graphs caot. This is because positio ca decrease while distace always icreases. Recall that the slope of a distace time graph is over ru, which is the same as the speed of the motio. The slope of a positio time graph is, likewise, the /ru or the velocity of the motio. Distace Time Graph slope = di stace time v = d t Positio Time Graph slope = displ acemet time v = d t Positio (m [N] of fiish lie) d 1 d 2 Fiishig a Race at the Start/Fiish Lie d () t (ru) egative slope t 1 t 2 back through the referece poit Note that i Figure 7 the displacemet d (d 2 d 1 ), is egative, therefore the slope is egative, ad so the velocity is egative. What does this mea? We ca see that positive o the graph is defied as orth, so egative is therefore south. The ruer is headig south. A egative slope o a positio time graph idicates motio i the opposite directio. A egative slope above the x-axis idicates motio toward the referece poit, but the same slope below the x-axis idicates motio away from the referece poit. Positio ad displacemet are vector quatities: they have both size ad directio. This meas that they ca be either positive or egative. Notice that Figure 7 shows the ruer goig a short distace beyod the referece poit (the fiish lie). Sample Problem 1 Caadias Doova Bailey, Bruy Suri, Gleroy Gilbert, ad Robert Esmie wo the 199 Olympic gold medal i the m relay race. Whe Bailey fiished the m relay race the team s time was 37.9 s, as described graphically i Figure 8. From the graph, determie Bailey s velocity i the fial stretch of his -m leg of the race. Assume that orth is positive ad south, egative. d 1 = +. m d 2 = +5. m t 1 = 35.2 s t 2 = 37.1 s v = d t = ( 5.. ) m ( ) s = 11 m/s From the slope of the lie o the positio time graph, Bailey s velocity while fiishig the race was 11 m/s [S]. 8 Chapter 12 Positio (m [N] of fiish lie) Figure 7 I which directio is the ruer movig? What is happeig to the ruer as the lie crosses the x-axis? Bailey Fiishes the Race Figure 8 The last few secods of the m relay of the wiig Caadia team

4 The costat slope i Figure 8 shows that the velocity of the ruer is costat. This meas that the istataeous velocity has the same value throughout. Recall what a distace time graph for costat acceleratio looks like. Cosider ow the case of the costat acceleratio of a boat from rest (Figure 9). You ca see from the graph that the slope of the lie is steadily icreasig. What does this mea? Sice the slope of the lie o the positio time graph is icreasig, the velocity must be icreasig. It is o loger costat. Now we ca do somethig ew: we ca determie the istataeous velocity at, say,.7 s. We do this by drawig a taget to the curve at that poit. The slope of the taget at a poit o a positio-time graph yields the istataeous velocity. Istataeous velocity is the chage i positio over a extremely short period of time. Istataeous velocity is like istataeous speed plus a directio. Both quatities are calculated from a slope, but istataeous velocity must iclude a directio because it is a vector quatity. Sample Problem 2 A boat accelerates uiformly for seve secods (Figure 9). What is the istataeous velocity at.9 s? Assume that east is positive. d 1 = + m d 2 = +5 m t 1 = 3. s t 2 = 7. s v = d t = ( 5) m ( 7. 3.)s = 9.8 m/s The istataeous velocity of the boat at.9 s is 9.8 m/s [E]. Positio (m [E] of referece poit) 5 3 A Acceleratig Boat m 5 m Figure 9 A taget to the curve at ay poit yields the istataeous velocity at that poit. Displacemet, Velocity, ad Acceleratio 9

5 Uderstadig Cocepts 1. How ca the velocity of a object be determied from a positio time graph? 2. Usig the labels o the graphs i Figure, write brief descriptios about the motio that the graphs might describe. Iclude the directio ad relative size of the differet velocities. (a) (b) Positio (km [N] of home) Positio (km [N] of school) 3. Figure 11 presets a positio time graph for a ball rollig dow a slight iclie. Positio (m [dow iclie] from top) Figure Figure 11 A Ball Rollig o a Iclie (a) What is happeig to the velocity as the ball rolls dow the iclie? How does the graph show this iformatio? (b) What kid of motio does the positio time graph represet? (c) What is the displacemet after 2. s,. s ad. s? Use Figure 12 to aswer questios ad 5. Positio (m [E] of home) A Figure 12. Describe the velocity for graph segmet A by idicatig (a) whether the motio represets costat velocity; (b) how the average velocity is foud; (c) whether the istataeous velocity chages; (d) how the size ad directio of the average ad istataeous velocities compare. 5. What are the similarities ad differeces betwee segmets A ad B?. (a) Draw a positio time graph for the motio of the air puck V2 (Table 1). Table 1 The Travels of a Air Puck Dot umber Total time (ms) Positio (mm [N]) (b) Determie the istataeous velocity at 3. ms, 5. ms, V3 ad 9. ms. (c) Describe the velocity of the air puck from start to fiish. (d) Calculate the average velocity for the total time period. (e) How do the istataeous velocities, calculated i (b), compare with the average velocity from (d)? B 5 Chapter 12 SKILLS HANDBOOK: V2 Costructig a Lie Graph V3 Slopes ad Itercepts

6 7. A wid-up toy is attached to a ticker tape ad released. Table 2 shows the positios immediately after release. Table 2 A Ticker Tape Experimet Dot umber Total time (ms) Positio (mm [E] of start) Which method of commuicatio i questio 8 provides the best iformatio? Explai your aswer.. Cosider Caadia moutai-bike champio, Aliso Sydor s false start i a race. The dots i Figure 13 represet Sydor s positio every.5 s from the start (positio = ). (a) Sketch a positio time graph for Sydor s false start. W -2m S 18m N E (a) Draw a positio time graph for the toy. (b) What is the displacemet betwee (i). ms ad 3. ms? (ii) 3. ms ad. ms? (c) What do your aswers to (b) tell you about the motio of the toy? (d) How would you reach the same coclusio as you did i (c), just by lookig at the graph? (e) Determie the istataeous velocity at 25. ms ad 35. ms. (f) Calculate the average velocity for the total time period. Explorig 8. Look agai at the graph i Figure (page 7). Show the same motio as (a) a displacemet vector diagram (without a scale) (b) a distace time graph (c) a velocity time graph Figure 13 (b) Draw a scale vector diagram to represet Sydor s displacemet. Reflectig 11. Whe ew cocepts are beig created, scietists go through a stage of cofusio before everythig seems to fit ito place agai. I this regard, how do you feel right ow about the itroductio of positio time graphs? Do you have cofidece i your uderstadig of this ew cocept? Challege 1,2 Aristotle ad Galileo had trouble visualizig ad measurig acceleratio due to gravity. Why? Displacemet, Velocity, ad Acceleratio 51

We will conclude the chapter with the study a few methods and techniques which are useful

We will conclude the chapter with the study a few methods and techniques which are useful Chapter : Coordiate geometry: I this chapter we will lear about the mai priciples of graphig i a dimesioal (D) Cartesia system of coordiates. We will focus o drawig lies ad the characteristics of the graphs