Dynamics. Option topic: Dynamics

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1 Dynamics 11 syllabusref Opion opic: Dynamics eferenceence In his cha chaper 11A Differeniaion and displacemen, velociy and acceleraion 11B Inerpreing graphs 11C Algebraic links beween displacemen, velociy and acceleraion 11D Moion under consan acceleraion

2 478 Mahs Ques Mahs C Year 11 for Queensland Inroducion The moion of sellar and earhly objecs has inrigued people for cenuries. From he movemens of he planes o he arc of a javelin o undersanding he movemens of a balle dancer, a sudy of moion is considered fundamenal o many, if no all, sciences. Kinemaics is he name given o he sudy of he moion of bodies, objecs or paricles wihou considering he forces involved. Dynamics is he sudy of moion and is causes. This moion of a body can be classified ino a number of groups, depending on he amoun of space he moving objec uses. Class A: Sraigh line moion can be seen when an ice-hockey puck is hi along he ice or when a pool ball ravels in a single direcion; ha is, in one-dimensional space. This ype of moion is called recilinear moion. Class B: Curved moion in wo-dimensional space is seen when a ball is kicked up in he air and falls o he ground or when a car rounds a bend in he road. Class C: Curved moion in hree-dimensional space is he mos comple form of moion ha occurs when a bird soars and ravels on hermals. Wihin hese hree differen ypes of moion more specific caegories also eis. Consider a ball swinging around a pole so ha i follows a circular pah. This wo-dimensional moion is referred o as circular moion for obvious reasons. Anoher ype of wo-dimensional moion is seen in a ball on a rubber band ha is pulled down and released. This is simple harmonic moion. Boh hese ypes of moion will be eamined in more deail laer in your sudy of dynamics.

3 Chaper 11 Dynamics 479 Differeniaion and displacemen, velociy and acceleraion Recilinear moion As menioned earlier, he simples form of moion occurs when an objec ravels in a sraigh line in one dimension. The funcion ha models his ype of moion is he equaion of a sraigh line: y = m + c or more specifically = m + n where is he dependen variable of displacemen of he body, (ime) is he independen variable, n is he iniial posiion of he body from he origin (a = ) and m is he slope of he line. Recilinear moion is readily illusraed if you hink of walking up and down a fenceline, wih a big pos as he origin. n Origin Recilinear moion 1 A paricular body ha is in recilinear moion can be described by he funcion = + 6. Answer he following quesions: a Complee he second row in he able below o find he displacemen a each of he given imes. Time (s), Displacemen (m), Change in displacemen b c d e f Wha is he change in displacemen during he firs second? Wha is he change in displacemen during he las second? Find he change in displacemen for each second of moion. If he change in he displacemen is he same for each ime segmen, wha can be said abou he velociy of he body hrough he scale? If he rae of displacemen is consan, he body has consan velociy. Is i eperiencing any acceleraion?

4 48 Mahs Ques Mahs C Year 11 for Queensland Anoher objec is moving wih moion ha can be modelled by he funcion = + 5 Answer he following quesions: a Complee he second row of he able below. Time (s), Displacemen (m), Change in displacemen Change in velociy b c d e f Wha is he change in displacemen during he firs second, and for he las second? Find he change in displacemen for each second of moion. If his change in displacemen is consan hroughou he moion of he body is his body under consan velociy? Wha is he change in velociy from he firs second inerval o he ne second? Complee he las row of he able. Is his body undergoing any acceleraion? Posiion The posiion of a paricle moving in a sraigh line is a measure of is locaion from a fied poin of reference, usually he origin O on he line. Posiions o he righ of O are usually aken as posiive. The poin P in his diagram has a posiion coordinae 1. O 1 P Posiive direcion Displacemen The displacemen of a moving paricle is is change in posiion relaive o a fied poin. Displacemen gives boh he disance and direcion ha a paricle is from a fied poin. For eample, a paricle which moves from S o F F via A is shown here on a displacemen ime line (or S A _ posiion ime line). 4 The disance ravelled from S o F is 1 unis. The displacemen of F from S is. How far am I from my original posiion? How far have I walked? Displacemen Disance Velociy The average velociy of a paricle is he rae of change of is posiion wih respec o ime. This can be illusraed on a posiion ime graph. For his curve, he posiion a ime is shown. The curve () is shown a righ. Posiion 1 Change in posiion δ Change in ime δ 1 Time

5 Chaper 11 Dynamics 481 Change in posiion Average velociy = Change in ime = δ = δ The insananeous velociy, v(), a ime is defined as he limiing value of he average velociy as δ (he change in ime) approaches zero. Tha is, i is he gradien of a displacemen ime graph or v() = δ δ or v() = Unis of velociy The unis of velociy are mos commonly cm/s, m/s, or km/h. Noe: 1 m/s = 3.6 km/h (verify his). Speed lim δ d d Insananeous speed is he magniude of insananeous velociy and is always posiive. disance ravelled Average speed = ime aken Noe ha disance ravelled is no necessarily he same as displacemen. WORKED Eample The posiion of a paricle, () cm, from he origin O moving in a sraigh line a any ime seconds is given by: () = 8 Find: a he iniial posiion of he paricle b he posiion afer seconds c he posiion afer 3 seconds d he average velociy during he hird second e he velociy a any ime f he iniial velociy g he velociy afer seconds h when and where he paricle is saionary. Hence, skech a displacemen ime line. THINK a 1 The iniial posiion occurs when =. Find when =. Sae is posiion. WRITE a () = 8 When =, () = () 8 = 8 The iniial posiion is 8 cm, or 8 cm lef of he origin. b 1 Find when =. b () = () 8 = 8 Sae is posiion. 1 The posiion afer seconds is 8 cm, or 8 cm lef of he origin. Coninued over page

6 48 Mahs Ques Mahs C Year 11 for Queensland THINK WRITE c 1 Find when = 3. c (3) = 3 (3) 8 = 5 Sae is posiion. The posiion afer 3 seconds is 5, or 5 cm o he lef of he origin. change in posiion d Average velociy = d Average velociy during he hird second change in ime ( 3) ( ) = ( 5) ( 8) cm = s = 3 cm/s d e Since velociy is he rae of change of e = d posiion, differeniae wih respec o. or v = f Iniial velociy can be found when =. Find v when =. f When =, v() = () = Iniial velociy is cm/s or cm/s o he lef. g Find v when =. g When =, v() = () = Velociy afer seconds is cm/s o he righ. h 1 The paricle is saionary when he velociy is zero. Find when v =. h The paricle is saionary when v = =. Facorise he LHS. ( 1) = 3 Solve for. = 1 4 Evaluae when = 1. (1) = 1 (1) 8 = Sae he answers. Verify using he graph of () on a graphics calculaor. For he displacemen ime line: (a) lis useful informaion: posiion a =, 1, and 3. v = when = 1 so he paricle urns a = 1 s (b) skech he displacemen ime line. The velociy is when = 1 s and is displacemen from O is 9 cm, or 9 cm o he lef of O. () = 8, (1) = 9, () = 8, (3) = 5, v(1) = = s = 1 s = = 3 s _ 9 _ 8 _ 5 (cm)

7 Chaper 11 Dynamics 483 WORKED Eample The displacemen of a paricle () moving in a sraigh line (in cm) from a fied poin O, he origin (in cm), a any ime is given by: () = Find: a he iniial displacemen b when and where he velociy of he paricle is c he average velociy of he paricle during he firs seconds of moion d he average speed in he firs seconds. THINK WRITE a Find when =. a () = () = () () () + 1 = 1 Sae he iniial posiion. The paricle is iniially a = 1 or 1 cm righ of O. b 1 Differeniae o find v. d b v() = d = 3 Se v =. When v =, 3 =. 3 4 Facorise he LHS. (3 + )( 1) = Solve for, giving soluions only where = -- or = 1 3. Therefore = 1 is he only soluion, since. Evaluae (1). (1) = (1) 3 -- (1) (1) Sae he answers. = The velociy is when = 1 s and he 1 1 posiion is -- cm or -- cm lef of O. 7 Verify using a graph of () on a graphics calculaor. c Evaluae when =. c () = () () + 1 = 3 ( ) ( ) Subsiue he values ino he average Average velociy = velociy rule. change in posiion 3 1 Average velociy = = change in ime 3 4 Evaluae. = = 1 Sae he soluion The average velociy in he firs seconds is 1 cm/s o he righ. Coninued over page

8 484 Mahs Ques Mahs C Year 11 for Queensland THINK WRITE d 1 3 Skech a displacemen ime line for he firs seconds of moion. Origin Calculae he disance ravelled in he Disance ravelled = 5 cm firs seconds. 5 Calculae he average speed. Average speed = -- =.5 Hence, he average speed for he firs seconds is.5 cm/s. d = 1 s = s = _ (cm) Noe: The disance ravelled, and also he average speed, can be calculaed wihou a line skech, providing he ime(s) when he velociy is found and he appropriae posiions found are given. Tha is (in par d above) disance = () (1) + (1) () 1 1 = = = = 5 Acceleraion The average acceleraion of a paricle a during a ime inerval 1 is he rae of change of is velociy wih respec o ime. v v 1 Tha is, average acceleraion = δv = δ The insananeous acceleraion, a, a ime is he limiing value of he average acceleraion as approaches. Tha is, i is he gradien of a velociy ime graph; or a = lim δv δ δ a = 1 1 dv d Unis of acceleraion The unis of acceleraion mach heir corresponding velociies and are usually epressed as cm/s or m/s. Noe: The acceleraion due o graviy is g = 9.8 m/s in a downward direcion. For objecs ravelling hrough he air, air resisance is ignored unless i is specified in a given problem Velociy v v 1 Change in velociy Change δv in ime δ 1 Time

9 Chaper 11 Dynamics 485 WORKED Eample 3 The posiion of a paricle (in meres) moving in a sraigh line is given by = where > and measured in seconds. Find: a he velociy a any ime b he velociy a = 5 s c he acceleraion a any ime. d Inerpre his final resul. THINK a To find he velociy, differeniae he displacemen funcion wih respec o. b To find v a = 5, subsiue for, and include unis. c To find acceleraion, differeniae v wih respec o. WRITE a = d v = = 4 4 d b A = 5 v = 4(5) 4 = 16 m/s dv c a = = 4 d d Wrie he inerpreaion of he resul. d Therefore acceleraion is consan hroughou he moion. Graphics Calculaor ip! Graphing an graph and he corresponding v graph 1. To plo he graph given () =, 4, and he corresponding v graph, ener Y 1 = X X (hink of Y 1 as and X as ), go o Y= and press MATH, selec 8:nDeriv( and ener Y 1,X,X) (so Y is v), and adjus WINDOW seings for X o [, 4]. Then press ZOOM, and selec :ZoomFi. Remember: To ener Y 1, press VARS and selec Y VARS, 1:Funcion and 1:Y1 (similarly any Y variable).. To deermine displacemen and velociy a paricular imes, press TRACE, oggle o he required graph, ener he ime (X) required and press ENTER. For eample, wih he cursor on Y1, enering 3 and pressing ENTER gives he resul in he firs screen: he displacemen is 3 a ime 3. Wih he cursor on Y, enering 3 and pressing ENTER gives he resul in he second screen: he velociy is 4 a ime 3.

10 486 Mahs Ques Mahs C Year 11 for Queensland remember remember 1. Posiion gives he locaion of a paricle relaive o a reference poin (usually he origin). The variable used is.. Displacemen is change in posiion δ or s. Displacemen is given by he magniude and direcion of he disance ravelled. d 3. Insananeous velociy, v or -----, is he ime rae of change of posiion or he rae d of change of displacemen. δ s displacemen 4. Average velociy during ime inerval δ = = ---- = δ δ ime disance ravelled 5. Speed = ime aken dv 6. Insananeous acceleraion, a or is he ime rae of change of velociy. d Mahcad Kinemaics given () Differeniaion and displacemen, velociy and acceleraion 1 Mach he skech of he displacemen ime line below o each of he following objecs moving in a sraigh line, where is he displacemen from O (he origin) a ime and is given by: a = 3 + b = 3 c = 3 d = i ii = = 3 = 4 = = 3 = 1 = = 1 = O O iii = = 3 = 4 iv = 1 = = 1 = = O _ 3 1 O For each of he following displacemen ime lines sae: i he displacemen of F from O ii he displacemen of F from S iii he disance ravelled o ge from S o F. a F b F S S O O c 11A F S _ 6 _ 5 _ 4 _ 3 1 O 1 3 In each case in quesion i akes 4 seconds o ge from S o F. For each case, find: i he average speed ii he average velociy. d _ 4 _ 3 _ 5 F S 1 O 1 3 4

11 Chaper 11 Dynamics 487 WORKED Eample 1 WORKED Eample 4 muliple choice Use he displacemen ime graph a righ. a The average speed in he firs 6 seconds is: 1 A 3 m/s B 5 -- m/s C 3 m/s D 7 m/s E 7 m/s b The average velociy in he firs 6 seconds is: 1 A 3 m/s B 5 -- m/s C 3 m/s D 7 m/s E 7 m/s 5 The displacemen ime graph for a paricle ravelling in a sraigh line is shown below. Use he graph o: a sae he displacemen when (1, 4) i = 3 ii = 1 b sae he ime when he displacemen is (1, 18) i 4 ii (7, 1) c find he average velociy from i = 3 o = 7 ii = 3 o = 1 (3, _ ) d find he average speed from i = 3 o = 7 ii = 3 o = 1. 6 The posiion of a paricle, () cm, from he origin O moving in a sraigh line a any ime seconds is given by: a () = 3 1 b () = For he above find: i he iniial posiion ii he posiion afer 3 seconds iii he posiion afer 4 seconds iv he average velociy during he fourh second v he velociy a any ime vi he iniial velociy vii he velociy afer 3 seconds viii when and where, if a all, he paricle is saionary. Give answers correc o decimal places where appropriae. 7 The displacemen of a paricle () meres moving in a sraigh line from a fied poin O, he origin, a any ime seconds is given by: () = (5 )( + 1), 5 a Find he iniial displacemen. b Find when and where he velociy is. c Find he average velociy of he paricle during he firs 4 seconds. d e 3 3 Skech a posiion ime graph for he paricle. Find he average speed in he firs 4 seconds by firs skeching a posiion ime line. 8 For each of he following posiion ime rules, where cm is he posiion a any ime seconds, find: i he posiion afer 3 seconds ii when, if a all, he velociy is iii he disance ravelled in he firs 3 seconds. a = 8 b = (Where appropriae, round off answers o decimal places.) Meres 8 (, ) Seconds (6, _ 1)

12 488 Mahs Ques Mahs C Year 11 for Queensland WORKED Eample 3 9 For each of he following, find i he velociy and ii he acceleraion a any ime. a = b = 4 + c = (5 )( + ) 1 The heigh of a projecile, h meres above he ground, seconds afer i is fired verically ino he air from a ower is given by: h = Find: a he average velociy during he firs seconds b he ime i akes o reach is greaes heigh c he greaes heigh d he acceleraion a any ime. 11 The posiion of an objec relaive o a reference poin O is given by: 1 = , [, 5], where is in meres and is in seconds. 3 a Find he iniial displacemen, velociy and acceleraion. b Find when and where he objec is a res. c Find when and where he acceleraion is. d Skech i he velociy ime graph and ii he acceleraion ime graph. 1 The displacemen from O of a paricle ravelling in a sraigh line is: = , where is in cm and is in seconds. a A wha ime does i pass hrough he origin? b Find he velociy a any ime. c When is he velociy? d Find he minimum velociy. e Skech he velociy ime graph for he firs 4 seconds. f Find he disance ravelled in he firs second. 13 Find he velociy and acceleraion a any ime for each of he following: a = b = c + 1 = Inerpreing graphs In he previous secion you have seen he relaionship beween displacemen, velociy and acceleraion of a body in moion. You should also remember ha: 1. velociy is he rae of change of displacemen. he seeper he slope of a displacemen curve (sraigh line) he greaer he velociy 3. if displacemen can be represened by a sraigh line hen i is eperiencing a consan rae of change of displacemen (velociy). Displacemen ime graphs Eamine he following graphs showing displacemen () graphed agains ime (). Remember ha ime is graphed on he horizonal ais and displacemen on he verical ais.

13 Chaper 11 Dynamics (a) Objec B sars furher away from he origin, O, han objec A. (b) Objec A reaches he origin (zero displacemen) before objec B. A B (c) Boh objecs are moving wih he same consan velociy, shown here by parallel (same) sraigh (consan) lines.. (a) Objec B remains he same disance from he origin A B a all imes; ha is, B is saionary. (Because displacemen and velociy are vecor quaniies and C involve boh magniude and direcion, he velociy of B would no be consan if B moved in a circle around he origin.) (b) Boh A and C sar a he origin and move a consan velociies, however A is moving faser han C (as shown by he seeper slope.) 3. (a) The objec sars some disance from he origin, moves a a consan velociy away from he origin. (b) I hen abruply reurns a he same consan speed a which i moved from he origin. (c) The body reurns o he same posiion from he origin as i sared. 4. (a) A and B boh move away from he origin, wih A ravelling a a faser velociy han B. A (b) A remains saionary for some ime unil B reaches B A. 5. (a) A begins some disance from he origin and moves wih a consan velociy owards he origin. A (b) B begins abou half he disance A is from he origin and remains saionary for a ime unil afer A passes B B, hen reurns o he origin a he same consan velociy as A. 6. (a) A and B sar a he same disance from he origin. (b) Iniially A moves a a greaer consan velociy han A B B. (c) A hen reduces is velociy o mach B s iniial velociy and B increases is velociy o mach A s iniial velociy. Be careful in your inerpreaion of he displacemen of a body. Displacemen is a vecor quaniy involving magniude and direcion o fully describe i. This means ha if A and B have he same displacemen from he origin hey are moving in he same direcion as well as he same disance from O. In his case shown below he displacemen of A does no equal he displacemen of B. A O B

14 49 Mahs Ques Mahs C Year 11 for Queensland From he above facs you should undersand ha if he displacemen of a body can be graphed as shown a righ, his horizonal line has a gradien of ; ha is, he velociy of he body is and i is saionary. This can be graphed as shown a righ. Logically you would appreciae ha if a body says a res (a a cerain disance from he origin) hen i has zero velociy as shown a righ. v Velociy ime graphs Eamine he following displacemen ime graphs and heir corresponding velociy ime graphs. Noe how he slope of he displacemen graph is represened in he velociy ime graph. Remember ha he slope of he line is giving essenial informaion abou he velociy because velociy is he rae of change (or derivaive) of he displacemen funcion. Displacemen increases a a consan rae; herefore, he velociy is consan. v Displacemen decreases a a consan rae; herefore, here is a negaive rae of change and a consan negaive velociy. v (a) (b) (c) Secion (a) has zero v displacemen and herefore zero velociy. Secion (b) has consan change in displacemen herefore i has consan velociy. Secion (c) has lower consan change in displacemen; herefore, i has lower velociy han secion (b). (a) (b) (c)

15 Chaper 11 Dynamics 491 Noe ha he disjoined secions in he velociy graph coincide wih insananeous changes in velociy. As he slope of he ( ) curve increases, he velociy is increasing. Because he slope of he ( ) curve is no consan he velociy is changing. v As he slope of he ( ) curve is decreasing, he velociy is decreasing. The rae of change in displacemen is no consan. The velociy decreases a a consan rae, hence he sraigh line of he (v ) curve. v When he slope of he angen o he ( ) curve is horizonal he velociy is ; ha is, he (v ) curve cus he -ais. v Displacemen, velociy and acceleraion graphs Jus as velociy is he rae of change of displacemen, acceleraion is he rae of change of velociy, where he acceleraion can be deermined from he slope of he angen. Therefore, v a Displacemen is increasing Consan velociy Zero acceleraion a a consan rae.

16 49 Mahs Ques Mahs C Year 11 for Queensland You can hink of he hree graphs above as represening a car moving a a consan velociy away from a cerain poin, as i migh do pas any poin on he highway. Because i is moving wih consan velociy i has zero acceleraion. The acceleraor pedal is subjec o a seady, unchanging pressure. v a Displacemen is increasing Increasing velociy Consan acceleraion a an increasing rae. The hree graphs above illusrae he displacemen, velociy and acceleraion of a car perhaps as i pulls ou o overake anoher car where i has increased rae of change of displacemen, increased velociy and consan acceleraion. The pressure of he foo on he acceleraor pedal is gradually increasing. Posiive acceleraion only occurs when he velociy of a body is increasing, as i migh when overaking a car. Negaive acceleraion occurs when a body slows down or comes o res; when he velociy is decreasing. remember remember 1. Sraigh segmens in a displacemen ime graph indicae ha he moion involves consan velociy.. The seeper he slope of he graph, he greaer he velociy. 3. The equaion of a sraigh line = m + n can be used o model he displacemen, where n is he iniial disance from he origin and m is he gradien of he line. 4. Because he rae of change of displacemen represens he velociy of a body, his can be found using he derivaive of he displacemen funcion. (a) The slope of he angen o he displacemen curve holds he informaion needed abou he velociy of he body. (b) A falling or negaive slope of he displacemen curve indicaes negaive velociy. 5. Because he rae of change of velociy represens he acceleraion of a body, his can be found using he derivaive of he velociy funcion. (a) The slope of he angen o he velociy curve holds he informaion needed abou he acceleraion of he body. (b) A falling or negaive slope of he velociy curve indicaes negaive acceleraion.

17 Chaper 11 Dynamics B Inerpreing graphs 1 Describe he moion of he bodies graphed below: a b SkillSHEET 11.1 B A c d A A A B C B C B muliple choice Which descripion bes maches he graph a righ? A A sars a O and B sars away from O. When A passes B, B reurns o he origin. B B moves a a consan disance from O as A moves away from O. Afer A passes B, B reurns o he origin. C A moves away from O a a consan velociy while B wais. As A passes B, B reurns o he origin a a greaer velociy han A lef. 3 Draw a displacemen ime graph o mach he following descripions: a A sars from O and moves a a consan velociy o where B is waiing, hen A reurns o he origin wih he same velociy as i lef. b A and B se ou a he same velociy bu A sars furher away from he origin han B. A a cerain ime A sops bu when B passes A, B coninues a a greaer velociy. c A, B and C sar equally spaced away from he origin. They wai for a while hen a he same ime A and C join B, moving a he same consan speed and wai wih B. 4 Draw a velociy ime graph for he following displacemen ime represenaions: a b O B A

18 494 Mahs Ques Mahs C Year 11 for Queensland c d e f 5 muliple choice Which of he velociy ime graphs below bes maches he following descripion: wo bodies, A and B sar a he origin a he same ime wih he same consan velociy unil A moves ahead of B and B slows down. A v B v B A A B C v D v A B A A, B B 6 Draw a velociy ime graph o mach he following descripions: a An objec begins o move a a consan velociy hen gradually increases ha velociy for a ime, hen coninues o move a a consan velociy. b A body gradually increases is velociy from a resing posiion, hen abruply decreases is velociy and hen abruply increases is velociy a he same rae as i was increased earlier. c A body remains a res for some ime hen begins o move a a consan velociy; i hen abruply ceases all movemen. d A body is iniially moving wih consan velociy hen slows down a a seady rae. I abruply sars o move faser and hen abruply slows down.

19 Chaper 11 Dynamics Mach each of he following velociy ime graphs o is corresponding acceleraion ime graph. a v b v c a d v e a f a 8 Draw acceleraion ime graphs o mach he following descripions: a A body moves wih consanly increasing velociy. b A body moves wih consan velociy hen speeds up a a consan rae. c A body decreases is rae of acceleraion unil i begins o slow down. WorkSHEET11.1 Curve fiing using a graphics calculaor This graphics calculaor invesigaion allows you o plo pairs of poins and o fi a curve o hese poins. This procedure of calculaing an epression ha maches he given poins is called regression. The following daa represen he posiion of a moving body a cerain imes. Time (s), Displacemen (m), If more accurae informaion abou he posiion of he body a paricular imes is needed, a funcion ha can be used o model he pah of he body is required.

20 496 Mahs Ques Mahs C Year 11 for Queensland Even before hese poins are ploed i can be readily seen ha hey do no follow a sraigh line. We will endeavour o fi a quadraic epression o his daa. Tha is, find values of a, b and c such ha y = a + b + c 1 Follow he keysrokes and seps given below on your graphics calculaor. Keysrokes on graphics calculaor Commens {,, 6, 9, 11,13} Ener he daa for ime. Use curly brackes and commas beween each number. STO nd ALPHA T I M E ENTER Sores his as a lis named TIME. Use a maimum of 4 leers in he lis name. {, 5, 9, 9, 8, 4} Ener he daa for displacemen. STO nd ALPHA ENTER D I S P Sores his as a lis named DISP. nd LIST Shows DISP sored as L7 and TIME sored as L8. nd STAT PLOT selec 1 (plo on) ENTER. Selec scaergram hen XLIST nd LIST (choose TIME) Ses up he STAT PLOTS menu as a scaergram and ses he independen () lis as TIME. Selec YLIST DISP) nd LIST (choose Ses he dependen (y) lis as DISP. GRAPH ZOOM or check WINDOW seings for hese daa. nd CATALOG (scroll down) DiagnosicOn ENTER STAT selec CALC menu, hen choose 5: QuadReg ENTER nd LIST selec TIME, nd LIST selec DISP, VARS selec Y-VARS FUNCTION Y 1 ENTER Graphs daa as a scaergram and adjuss scale and view. Displays Done when se. Chooses a quadraic regression procedure ha will sore he resul in Y 1. Ses he informaion as [ independen variable, dependen variable, sored in Y 1 ]. Values of a, b and c are displayed. GRAPH Graphs he curve in Y 1 over he poins. The values of a, b and c obained can be used in he quadraic equaion ha models he moion of he body.

21 Chaper 11 Dynamics 497 a Differeniae Y 1 as indicaed in he previous aciviy. This is he epression for he velociy of he body during is moion. b Ener his epression in Y and graph his equaion. c Wha does his indicae abou he velociy of he body? d Differeniae he epression in Y and ener his in Y 3. This is he epression for he acceleraion of he body during is moion. e Wha does his indicae abou he acceleraion of he body? f Wrie a comprehensive saemen ha links he iniial informaion you have used o perform his regression procedure, all he way hrough o he epression for he acceleraion eperienced by he body during is moion. Algebraic links beween displacemen, velociy and acceleraion You have already considered he graphical links beween displacemen, velociy and acceleraion and have used he concep ha he slope of he angen o he curve will give informaion abou he rae of change of displacemen (velociy) and he rae of change of velociy (acceleraion). Algebraic epressions reinforce wha common sense and graphical informaion have shown us. The simples displacemen funcion is = n, where n is any real number. This means ha he displacemen from he origin is consan (n unis) and remains here. n v Therefore when = n d v = = d dv a = = d As we have seen, recilinear moion is modelled by a linear equaion, = m + n where he objec sars n unis from he origin (a = ). Therefore when = m + n v d v = = m d dv a = = d n m where m and n are consan hroughou.

22 498 Mahs Ques Mahs C Year 11 for Queensland When a quadraic equaion = l + m + n can be used o model he moion of a body, you would epec is moion o follow he shape of a parabolic curve, where he gradien changes wih every value of ; l, m and n are consan hroughou. Therefore when = l n + m + n d v = = l + m d dv a = = l d Hence he acceleraion is consan. If a cubic funcion models he moion of a body hen = k 3 + l + m + n d v = = 3k + l + m d dv a = = 6k + l d In his case he acceleraion is no consan because i depends on (ime). In his course, our sudy is resriced o moion under consan acceleraion, herefore moion modelled by a cubic funcion will no be eamined. Ecluding he case when here is no moion, where = n, here are only wo ypes of displacemen funcion ha we will eamine linear and quadraic. Linear = m + n Quadraic = l + m + n = l + m + n = m + n where consan velociy = m and acceleraion = where velociy = + m and consan acceleraion = l As you would appreciae from your earlier algebra sudies, he sign of he consans l, m and n is highly significan in deermining he shape and posiion of he graph produced and herefore he differen syles of moion ha he graph of displacemen represens. remember remember 1. Consan displacemen is represened algebraically by = n.. A linear equaion = m + n represens moion under consan velociy. 3. A quadraic equaion = l + m + n represens moion under consan acceleraion.

23 Chaper 11 Dynamics C Algebraic links beween displacemen, velociy and acceleraion 1 Skech he syles of displacemen graphs ha occur when each of he following condiions apply. Use your graphics calculaor o help you. a n < and m > in a linear funcion b n < and m < in a linear funcion c m > and l and n < in a quadraic funcion d m < and l and n > in a quadraic funcion e n < and l and m > in a quadraic funcion. Describe he moion involved in each of he cases graphed in quesion 1 above. Moion under consan acceleraion Even hough our sudy in his course is resriced o moion involving consan acceleraion, his sudy does include many siuaions of pracical significance. The common eample of bodies in free fall owards he Earh s surface, regardless of heir size, mass or composiion, involves moion under consan acceleraion ha of acceleraion due o graviy. Galileo illusraed his propery by dropping various sized cannon balls from he op of he Leaning Tower of Pisa. Provided he effecs of air resisance are minimised (ha is, he objec should no be oo large in area or fall oo far) all bodies will fall wih consan acceleraion. Oher variaions will be considered laer in your sudies. Since he acceleraion due o graviy depends on he disance from he cenre of he Earh, i varies slighly a differen places on he Earh s surface being greaer a he poles han a he equaor and less a high aliudes. I is usual o sae he consan of acceleraion due o graviy as 9.8 m/s unless you are old oherwise. Since acceleraion is consan, he displacemen of a falling objec can be modelled by he general quadraic funcion = l + m + n

24 5 Mahs Ques Mahs C Year 11 for Queensland However, when moion under he influence of graviy is involved, he dependen variable is changed o y (insead of ) o indicae verical movemen. This is wrien as y = l + m + n This funcion can be used o model he moion of a body in free fall or one ossed upwards. The imporan difference is ha he velociy and acceleraion are in differen direcions. When vecor noaion is used he verical displacemen vecor is wrien as ỹ = (l + m + n) Because j does no depend on he derivaive of his funcion is as follows: ṽ = (l + m) and ã = l The following analysis eamines he moion of a ball as i ress a he op of a cliff. As soil falls away he ball begins o move. (a) A =, find v. A = he ball is a res, herefore v =. (b) If he boom of he cliff is he origin and he cliff is h meres high, find y when =. A =, ỹ = h (c) Given he consan acceleraion owards he origin equals 9.8 m/s (a negaive value is used o indicae ha he direcion of he acceleraion is downwards), find m and l. Use he original equaion for displacemen and differeniae: ỹ = (l + m + n) (1) ṽ = (l + m) () ã = l (3) From par (a) above when = ṽ = [l() + m] in () bu ṽ = (iniially a res) Therefore, m = From par (b) above, if a =, ỹ = h ỹ = (l + m + n) (1) ỹ = [l() + m() + n] = h j (as given a = ) Therefore, n = h From (c) g = 9.8 m/s and ã = l = = l j Therefore, l = 4.9 Therefore he general equaion for his moion can be wrien: ỹ = ( h) Do no aemp o memorise hese equaions. Equaions (1), () and (3) are he basic equaions and all ohers are specific rearrangemens of hese. In general, 1. l can be readily found if he acceleraion is given: a = l.. m can be readily found if he iniial velociy is given: v = l() + m. 3. n can be readily found if he iniial displacemen is given: y = l() + m() + n

25 Chaper 11 Dynamics 51 WORKED Eample 4 Eamine he moion of a sone hrown upward wih a velociy of 6 m/s from he op of a cliff 6 m high. Find he ime aken for i o reach he ground and is velociy on impac. THINK WRITE 1 As he acceleraion is consan, he ỹ = (l + m + n) verical displacemen is quadraic. ṽ = (l + m) Wrie he hree equaions of moion. ã = l The posiion of he origin needs o be Le he origin be he op of he cliff. saed. Generally where he moion begins or finishes can be hough of as he origin. 3 Use he given informaion o find each A =, ỹ = value of n, m and l. ỹ = [l() + m() + n] = herefore n = Rewrie he equaion for ỹ wih he ỹ = (l + m) new informaion. 4 Consan downwards acceleraion away ã = l from he origin so he acceleraion is = 9.8 negaive. 9.8 = l l = ỹ = ( m) Iniial velociy a = is 6 m/s up A =, v = 6 or ṽ = 6 (posiive sign). 6 j = [l() + m] j m = 6 Rewrie ỹ wih all curren informaion. ỹ = ( ) Find he ime when he sone reaches A ỹ = 6 he ground a ỹ = 6. 6 j = ( ) j Drop he vecor noaion. = Solve for using he quadraic formula where a = 4.9, b = 6 and c = 6. b ± b 4ac = a 6 ± 36 4( 4.9) ( 6) = ( 4.9) = 4.16 s or.94 s Rejec he negaive soluion. Sae he soluion. Therefore he sone reaches he ground 4.16 s afer i was hrown upwards. 7 Find he velociy on impac by finding v ṽ = ( ) when = A = 4.16 s Noe ha he negaive sign indicaes a ṽ = [ 9.8(4.16) + 6] vecor moving downwards, in he same = j direcion as he negaive acceleraion. Therefore, he sone reaches he ground wih a velociy of m/s downwards 4.16 s afer launching.

26 5 Mahs Ques Mahs C Year 11 for Queensland WORKED Eample 5 A sone (A) is hrown upwards from a cliff wih a velociy of 3 m/s. Afer (A) has been in moion for 4 s anoher sone (B) is dropped from he same poin. Find when and where he wo sones will mee. THINK 1 Draw a diagram and sae he origin posiion and direcion of posiive moion. Wrie he 3 equaions of moion. WRITE/DRAW 3 m/s A O ỹ = (l + m + n) ṽ = (l + m) ã = l The origin is a he op of he cliff and here is posiive moion up. Sone A: Sone A: Subsiue in he given informaion o A =, ỹ = find ỹ. ỹ = [l() + m() + n] = herefore, n = ỹ = (l + m) 3 Use informaion abou v o find m. A =, ṽ = 3 ṽ = [l() + m] = j m = 3 4 Rewrie ỹ. ỹ = (l + 3) Use informaion abou a o find l. A =, ã = 9.8 m/s ã = l 9.8 = l l = 4.9 ỹ = ( ) 5 For sone B: Repea he mehod used Sone B: for sone A. A =, a = 9.8 and l = 4.9 For many of your problems l = 4.9. A =, v = ṽ = [l() + m] Therefore m = A =, ỹ = = [l() + m() + n] j Rewrie he equaion for ỹ using all Therefore n = curren informaion. ỹ = Find where he wo sones mee: The wo sones mee where ỹ A = ỹ B when ỹ A = ỹ B; bu B = A 4. ( 4.9 A + 3 A ) j = 4.9 B j Epress all informaion in erms of A. ( 4.9 A + 3 A ) = 4.9( A 4) = 4.9( A 8 A + 16) = 39. A 3 A 78.4 = 9. A 78.4 A = 8.5 s. B

27 Chaper 11 Dynamics 53 THINK 7 Find he displacemen of B a A = 8.5 s and B = A 4 = 4.5 s. The informaion for sone B is easier o use o find he displacemen because A has had upwards and downwards moion (boh unknown) whereas B has had downwards moion only. WRITE y B = 4.9 (4.5) = 1.1 m The sones mee 1 m below he op of he cliff 8.5 seconds afer sone A is hrown. In general his procedure can be shorened. If an objec sars a he origin, n = and m has he value of he iniial velociy (upwards is posiive, downwards is negaive). If he body is influenced by graviy hen l = 4.9. If an objec is iniially a res hen m =, bu if an objec is par of a sysem ha is iself moving, and he objec is released from ha moving sysem, hen he iniial velociy (m) of he objec will be he velociy of he whole sysem a he insan prior o release. remember remember 1. If he origin is he saring poin, hen n =.. If he body is iniially a res, hen m =. 3. In verical moion, displacemen up is posiive; displacemen down is negaive. 4. Velociy up is posiive; velociy down is negaive. 5. Acceleraion up is posiive; acceleraion down is negaive. 11D Moion under consan acceleraion 1 A paricle moving from res wih consan acceleraion reaches a speed of 16 m/s in 4 seconds. Find: a he acceleraion b he disance ravelled. An objec ravelling a 8 m/s acceleraes uniformly over a disance of meres unil i reaches a speed of 18 m/s. Find: a he acceleraion b he ime aken. 3 a A racing car acceleraes consanly from res and covers a disance of 4 meres in 1 seconds. Find is velociy a he end of he 4 meres. b Anoher car ravels he 4 meres wih a consan acceleraion of 1 m/s. Find is ime for he 4 meres.

28 54 Mahs Ques Mahs C Year 11 for Queensland Eample 4 SkillSHEET 11. WORKED 4 A rain ravelling a a consan speed deceleraes uniformly for 3 seconds over a disance of 7 meres, coming o a sop. Find: a he iniial speed b he acceleraion. 5 A parachuis free-falls from an aircraf for 6 seconds. If he acceleraion due o graviy is 9.8 m/s, find: a he speed of he parachuis afer 6 seconds b he disance ravelled afer 6 seconds. 6 A ball is hrown up from he ground wih an iniial velociy of 19.6 m/s. The acceleraion due o graviy is 9.8 m/s downwards, ha is, 9.8 m/s. Find: a he maimum heigh aained by he ball b he oal ime aken for he ball o reurn o he ground. 7 A sone is dropped from a bridge which is 39. meres above a river. a How long does i ake he sone o reach he waer? b Wha is is speed on impac? 8 A ball is dropped from a ower and reaches he ground in 4 seconds. Find: a he heigh of he ower b he velociy of he ball when i his he ground. 9 A paricle is projeced verically up from he op of a building ha is 5 meres above he ground. If he iniial speed of he paricle is 8 m/s, find: a he maimum heigh, above he ground, ha i reaches b oal ime aken o reach he ground c he speed of he paricle when i reaches he ground. 1 A rain ravels a disance of 18 meres in 9 seconds while acceleraing uniformly from res. Wha is is velociy a he end of 5 meres? 11 A car acceleraes uniformly from res, increasing is speed from 5 m/s o 5 m/s in 1 seconds. Find: a he acceleraion b he disance ravelled, from res, in 1 seconds c he ime aken o increase is speed from 15 m/s o 3 m/s. 1 A spriner acceleraes uniformly o his op speed afer running 3 meres of a 1-mere race. He mainains his speed for he remainder of he race and akes 1.4 seconds o complee i. Find: a he op speed of he ahlee b he ime aken o reach he op speed. 13 A ram is ravelling a 16 m/s when he brakes are applied, reducing he speed o 6 m/s in seconds. Assuming he reardaion is consan, find: a he acceleraion b he disance ravelled seconds afer he brakes are applied c how long afer applying he brakes he ram comes o a sop d he braking disance of he ram. 14 A car moving from res wih uniform acceleraion akes 1 seconds o ravel 144 meres. Wha is is speed afer 6 seconds?

29 Chaper 11 Dynamics 55 WORKED Eample 5 15 A bus ravels 6 meres in 1 seconds and he ne 6 meres in 15 seconds. If he acceleraion is consan, find: a how much furher i will ravel before coming o res b how many more seconds i akes before coming o res. 16 A juggler hrows balls verically ino he air so ha hey rise o a heigh of 4.4 meres above he ground. He fails o cach one and i his he ground wih a speed of imes ha of is iniial speed. Find: a he speed of projecion of he ball b he heigh from which he ball is hrown c he oal ime he ball is in he air. 17 An objec is projeced verically up from a 14-mere ower and reaches he ground 4 seconds laer. a Wha is he projecion speed of he objec? b Wha is he maimum heigh above he ground ha is aained by he objec? 18 a An objec is dropped from he op of a building 39. m high. Calculae: i is velociy when i is halfway o he ground ii is velociy on sriking he ground iii he ime aken o each he ground. b Repea he above quesions for he case when he objec is hrown o he ground wih a velociy of 4.9 m/s. 19 a A sone rolls off he op of a cliff and is found o ake 4 seconds o reach he sea below. Wha is he heigh of he cliff? b Wha is he difference in ime o reach he boom beween par a and if he sone were launched verically upwards from his cliff wih a velociy of m/s? A verical sli 1.5 m long is posiioned in a sone wall 9.8 m below he op of he wall. A small objec is dropped from he op of he wall so ha i falls in line wih he sli. For wha lengh of ime is he falling objec visible hrough he sli? 1 A body is dropped from he op of a 1-m high ower a he same ime as a body is launched verically upwards from he boom of he ower wih a velociy of 5 m/s. Find when and where he wo bodies are a he same heigh above he ground. A body is rising wih a velociy of m/s and i releases a small paricle when i is 5 m above he ground. How long will i ake for he small paricle o reach he ground? 3 A cage is descending ino a well a a consan velociy of m/s when an objec falls hrough he wire of he cage. If he objec reaches he waer a he boom of he well 1 seconds before he cage find he heigh above he waer level a which he objec fell ou of he cage. 4 A fireworks rocke is fired verically upwards wih uniform acceleraion of 19.6 m/s. Afer seconds a small paricle is released from he rocke. How long afer release will he paricle fall o he ground? WorkSHEET11.

30 56 Mahs Ques Mahs C Year 11 for Queensland summary Differeniaion and displacemen, velociy and acceleraion Posiion gives he locaion of a paricle relaive o a reference poin (usually he origin). The variable used is. Displacemen is change in posiion δ or s. Displacemen and posiion have he same value if a body is iniially a he origin and so he wo erms are ofen used o mean he same hing. Hence, he same variable is used, namely. d Insananeously velociy, v or -----, is he rae of change of posiion or he d rae of change of displacemen wih respec o ime. δ s displacemen Average velociy during ime inerval δ = = ---- = δ δ ime disance ravelled Speed = ime aken dv Insananeous acceleraion, a or is he ime rae of change of velociy. d Inerpreing graphs Sraigh segmens in a displacemen ime graph indicae ha he moion involves consan velociy. The seeper he slope of he graph, he greaer he velociy. The equaion of a sraigh line = m + n can be used o model he displacemen, where n is he iniial disance from he origin and m is he gradien of he line. Because he rae of change of displacemen represens he velociy of a body, his can be found using he derivaive of he displacemen funcion. (a) The slope of he angen o he displacemen curve holds he informaion needed abou he velociy of he body. (b) A falling or negaive slope of he displacemen curve indicaes negaive velociy. Because he rae of change of velociy represens he acceleraion of a body, his can be found using he derivaive of he velociy funcion. (a) The slope of he angen o he velociy curve holds he informaion needed abou he acceleraion of he body. (b) A falling or negaive slope of he velociy curve indicaes negaive acceleraion. Algebraic links beween displacemen, velociy and acceleraion Consan displacemen is represened algebraically by = n. A linear equaion = m + n represens moion under consan velociy. A quadraic equaion = l + m + n represens moion under consan acceleraion. Moion under consan acceleraion If he origin is he saring poin, hen n =. If he body is iniially a res, hen m =. In verical moion, displacemen up is posiive; displacemen down is negaive. Velociy up is posiive; velociy down is negaive. Acceleraion up is posiive; acceleraion down is negaive.

31 Chaper 11 Dynamics 57 CHAPTER review (3, 1) Quesions 1 o 3 refer o he graph a righ, which shows he posiion of a paricle moving in a sraigh line,, as a funcion of. Displacemen in he direcion righ is aken o be posiive. (cm) muliple choice The iniial posiion is a: A 8 B C 13 D 5 E (1, _ 4) (s) 11A muliple choice The paricle is ravelling o he lef when: A 3 < < 1 B < < 3 C 8 < < 1 D 1 < < 13 E > 1 11A 3 muliple choice The average velociy beween = 3 and = 13, in cm/s, is: A 1.8 B 1 C 1.8 D 1 E A Quesions 4 o 6 refer o a paricle moving in a sraigh line, which has a displacemen, cm, from he origin a any ime, seconds, given by: () = muliple choice The iniial velociy in cm/s is: A 4 B C D 5 E 6 muliple choice The disance ravelled in he firs 4 seconds, in cm, is: A 8 B 9 C 1 D 3 E 7 muliple choice The number of imes he paricle passes hrough he origin is: A 1 B C D 4 E 3 muliple choice A paricle moves in a sraigh line so ha is posiion a any ime is given by: = 3 The acceleraion of he paricle when = 1 is: A 1 B 4 C D 6 E 8 11A 11A 11A 11A

32 58 Mahs Ques Mahs C Year 11 for Queensland 11A 8 A paricle moves in a sraigh line so ha is displacemen from a fied origin is given by: = a Find when and where he paricle firs comes o res. b When is he velociy negaive? c Illusrae he moion using a displacemen ime line. Quesions 9 o 11 refer o he following informaion. A car acceleraes uniformly from res, reaching a speed of m/s afer 5 seconds. I mainains his speed for 13 seconds before deceleraing uniformly o res in 4 seconds. 11B 9 muliple choice The velociy ime graph represening he moion of he car is: A B v (m/s) C v (m/s) v (m/s) (s) 5 13 (s) 5 5 (s) 9 D v (m/s) E v (m/s) (s) 5 18 (s) 11B 11B 1 11 muliple choice The oal disance ravelled by he car is: A 44 m B 4 m C 35 m D 41 m E 4 m muliple choice The deceleraion of he car, in m/s, is equal o: A 5 B 4 C 4 D 5 E Quesions 1 and 13 refer o he velociy ime graph, a righ, of a paricle moving in a sraigh line compleing a disance of 15 meres. v (m/s) 1 11B 1 muliple choice A wha ime is he paricle firs a res? A 4 s B 5 s C 6 s D 3 s E 5 s _ (s) 11B 13 muliple choice The oal ime aken o ravel he 15 meres is: A 5 s B 3 s C s D 4 s E 8 s

33 Chaper 11 Dynamics muliple choice Which of he following equaions for bes maches he graph? A = 3 + B = + 3 C = + 3 D = E = C 15 Skech he graph of he displacemen,, versus ime,, if acceleraion is consan and a = m/s. Use he condiions given below. When = s, velociy, v = m/s and displacemen, = 4 m. 11C 16 A sone is projeced verically up from he ground wih an iniial velociy of 4.5 m/s. Taking he acceleraion due o graviy o be 9.8 m/s, find: a he maimum heigh reached by he sone b he imes a which is heigh is meres above he ground. 11D muliple choice An objec is dropped from he op of a -mere high building. If he acceleraion due o graviy is 9.8 m/s, wha will be he heigh of he objec afer 5 seconds? A 151 m B 49 m C 1.5 m D m E 77.5 m muliple choice A paricle iniially moving a 6 m/s is subjec o a consan reardaion of m/s. The disance, in meres, ravelled before coming o res is: A 7 B 8 C 9 D 1 E 1 muliple choice A ram ravels 5 meres in 5 seconds when acceleraing uniformly from res. The acceleraion, in m/s, is: A.4 B 1.6 C 1.5 D 1. E.65 11D 11D 11D A parachuis drops from an aeroplane so ha he consan acceleraion during free fall due o graviy and air resisance is 8 m/s. The parachue is released afer 6 seconds, uniformly rearding he parachuis in 8 seconds o a consan speed of.5 m/s. This speed is mainained unil he parachuis reaches he ground which is 111 meres below he poin of release. a How long is he parachuis in he air? b Afer how long has he parachuis fallen half he disance (answer o he neares enh of a second)? 11D

34 51 Mahs Ques Mahs C Year 11 for Queensland 11D 11D 1 Jogger A is running in a sraigh line a a consan speed of 4 m/s when passing jogger B who has sopped o ie a lace. Jogger B heads off in he same direcion as jogger A 6 seconds laer, acceleraing uniformly a m/s unil reaching a speed of 5 m/s. a Skech a velociy ime graph showing he moion of boh joggers. b How long is i afer jogger A firs passes jogger B unil B caches up o A? c How far has jogger B ravelled o cach jogger A? d How far ahead will jogger B be afer jogger B has ravelled 5 meres? The velociy of a paricle moving in a sraigh line is given by: v = 4, > Find he acceleraion when v = 3. Modelling and applicaion es yourself CHAPTER 11 Car A is 6 meres from he cenre of he inersecion when i sars from res and acceleraes uniformly a 4 m/s, reaching a speed of 4 m/s which i mainains. A he insan car A akes off, car B is 78 meres from he cenre of he inersecion and ravelling a a consan speed of 8 m/s. When car B is 5 meres from he cenre of he inersecion i deceleraes uniformly a 5 m/s. a Which car ges o he cenre of he inersecion firs? b How far pas he cenre of he inersecion is he firs car when he second car reaches i? c If all oher condiions remain he same, wha consan acceleraion would: ii he second car need o have for a collision o occur? ii he firs car need o have for a collision o occur? d Car A 6 m If all oher condiions remain he same, a wha consan speed would: ii he firs car need o ravel for a collision o occur? (Use a graphics calculaor or a numerical mehod o assis.) ii he second car need o ravel for a collision o occur? (Give answers correc o decimal places where appropriae.) Car B 78 m Cenre of inersecion