AP1 Kinematics (A) (C) (B) (D) Answer: C

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1 1. A ball is hrown verically upward from he ground. Which pair of graphs bes describes he moion of he ball as a funcion of ime while i is in he air? Neglec air resisance. y a v a (A) (C) y a v a (B) (D) Answer: C The velociy of he ball sars a a high iniial value (assuming he upward direcion is posiive), hen slows down a a consan rae unil i reaches i highes poin, a which poin is velociy reaches an insananeous value of zero. The ball hen acceleraes downward, increasing is speed a a consan rae unil i reurns o he same poin in space and same speed i began is journey wih. Throughou he enire ime inerval, he acceleraion remains a consan negaive value (a=-g=-9.81 m/s 2 on he surface of he Earh). EK: 3.A.1 An observer in a paricular reference frame can describe he moion of an objec using such quaniies as posiion, displacemen, disance, velociy, speed, and acceleraion. SP: 1.1 The suden can creae represenaions and models of naural or man made phenomena and sysems in he domain. LO: 3.A.1.1 The suden is able o express he moion of an objec using narraive, mahemaical, and graphical represenaions. Difficuly: 1 Page 1

2 2. The following projeciles are launched on horizonal ground wih he same iniial speed. If wo or hree siuaions have he same answer, pu he leers in he same blank space. Air resisance is negligible. A B C D (a) Rank he siuaions from leas o greaes in erms of ime he projecile is in he air. Leas Greaes Or, all of he projeciles are he same (b) Rank he siuaions from leas o greaes in erms of he horizonal disance he projecile ravels while in he air. Leas Greaes Or, all of he projeciles are he same (c) Rank he siuaions from leas o greaes in erms of he verical disance he projecile ravels while in he air. Leas Greaes Or, all of he projeciles are he same (d) Rank he siuaions from leas o greaes in erms of he acceleraion of he projecile while i is in he air. Leas Greaes Or, all of he projeciles are he same Answers: (a) A, B, C, D -- he more verical he vecor he more ime in he air since all iniial velociies are he same. (b) D, CA, B -- The 45 projecile has he greaes horizonal range. (c) A, B, C, D -- D has he larges verical componen of iniial velociy. (d) All are he same acceleraion. EK: 3.A.1 An observer in a paricular reference frame can describe he moion of an objec using such quaniies as posiion, displacemen, disance, velociy, speed, and acceleraion. quaniaively. 2.2 The suden can apply mahemaical rouines o quaniies ha describe naural phenomena. 2.3 The suden can esimae numerically quaniies ha describe naural phenomena. LO: 3.A.1.1 The suden is able o express he moion of an objec using narraive, mahemaical, and graphical represenaions. Page 2 Difficuly: 1

3 3. Two children on he playground, Bobby and Sandy, ravel down slides of idenical heigh h bu differen shapes as shown a righ. The slides are fricionless. Assuming hey sar down he slides a he same ime wih zero iniial velociy, which of he following saemens is rue? (A) Bobby reaches he boom firs wih he same average velociy as Sandy. (B) Bobby reaches he boom firs wih a larger average acceleraion han Sandy. (C) Bobby reaches he boom firs wih he same average acceleraion as Sandy. (D) They reach he boom a he same ime wih he same average acceleraion. h Sandy s Slide Bobby s Slide Answer: B Boh children begin wih graviaional poenial energy mgh a he op of he slide, which is compleely ransferred o kineic energy a he end of he slide. Bobby s poenial energy is ransferred more quickly, however, herefore he aains a higher average velociy and beas Sandy o he end of he slide. Average acceleraion is he change in velociy divided by he ime inerval. Each child has he same change in velociy, bu Bobby observes his change over a shorer period of ime, resuling in a larger average acceleraion. EK: 4.A.2 The acceleraion is equal o he rae of change of velociy wih ime, and velociy is equal o he rae of change of posiion wih ime. 5.A.2 For all sysems under all circumsances, energy, charge, linear momenum, and angular momenum are conserved. For an isolaed or a closed sysem, conserved quaniies are consan. An open sysem is one ha exchanges any conserved quaniy wih is surroundings. 5.B.4 The inernal energy of a sysem includes he kineic energy of he objecs ha make up he sysem and he poenial energy of he configuraion of he objecs ha make up he sysem. quaniaively. 7.2 The suden can connec conceps in and across domain(s) o generalize or exrapolae in and/ or across enduring undersandings and/or big ideas. 6.4 The suden can make claims and predicions abou naural phenomena based on scienific heories and models. LO: 4.A.2.1 The suden is able o make predicions abou he moion of a sysem based on he fac ha acceleraion is equal o he change in velociy per uni ime, and velociy is equal o he change in posiion per uni ime. 5.A.2.1 The suden is able o define open and closed sysems for everyday siuaions and apply conservaion conceps for energy, charge, and linear momenum o hose siuaions. 5.B.4.1 The suden is able o describe and make predicions abou he inernal energy of sysems. Difficuly: 2 Page 3

4 4. You are asked o experimenally deermine he acceleraion of a skier raveling down a snow-covered hill of uniform slope as accuraely as possible. Which combinaion of equipmen and equaion would be mos useful in your endeavor? equipmen equaion (A) ape measure, sopwach x = x 0 + v x a x 2 (B) phoo gaes, sopwach v 2 x = v 2 x0 + 2a x (x x 0 ) (C) radar gun, ape measure v x = v x0 + a x (D) phoo gaes, radar gun v x = v 0 x + v x 2 Answer: A Measure he ime i akes he skier o go a se disance using he sopwach, and use he ape measure o deermine he disance. You may hen use he equaion x = x 0 + v x a x 2 o solve for he acceleraion, recognizing ha he iniial velociy and posiion of he skier are zero. EK: 3.A.1 An observer in a paricular reference frame can describe he moion of an objec using such quaniies as posiion, displacemen, disance, velociy, speed, and acceleraion. 4.A.2 The acceleraion is equal o he rae of change of velociy wih ime, and velociy is equal o he rae of change of posiion wih ime. quaniaively. 2.1 The suden can jusify he selecion of a mahemaical rouine o solve problems. 4.2 The suden can design a plan for collecing daa o answer a paricular scienific quesion. LO: 3.A.1.2 The suden is able o design an experimenal invesigaion of he moion of an objec. 4.A.2.3 The suden is able o creae mahemaical models and analyze graphical relaionships for acceleraion, velociy, and posiion of he cener of mass of a sysem and use hem o calculae properies of he moion of he cener of mass of a sysem. Page 4 Difficuly: 2

5 5. An eagle flies a consan velociy horizonally across he sky, carrying a urle in is alons. The eagle releases he urle while in fligh. From he eagle s perspecive, he urle falls verically wih speed v 1. From an observer on he ground s perspecive, a a paricular insan he urle falls a an angle wih speed v 2. Wha is he speed of he eagle wih respec o an observer on he ground? (A) v 1 + v 2 (B) v 1 v 2 (C) v 1 2 v 2 2 (D) v 2 2 v 1 2 Answer: D Call he velociy of he urle wih respec o he eagle v TE, also known as v 1. Call he velociy of he urle wih respec o he ground v TG, also known as v 2. You are asked o find he velociy of he eagle wih respec o he ground, v EG. Analyzing he righ riangle, you can use he Pyhagorean Theorem o solve for he magniude of v EG.! v TG = v! EG + v! TE! v TG = v! EG + v!! vtg = v!! 2 TE v! 2 = v! EG + v! 1 v = v EG v TE =! v 1 + v 1 2 v EG = v 2 2 v 1 2 v EG vtg = v 2 v TE = v 1 EK: 3.A.1 An observer in a paricular reference frame can describe he moion of an objec using such quaniies as posiion, displacemen, disance, velociy, speed, and acceleraion. 4.A.2 The acceleraion is equal o he rae of change of velociy wih ime, and velociy is equal o he rae of change of posiion wih ime. quaniaively. 2.2 The suden can apply mahemaical rouines o quaniies ha describe naural phenomena. LO: 3.A.1.1 The suden is able o express he moion of an objec using narraive, mahemaical, and graphical represenaions. 4.A.2.3 The suden is able o creae mahemaical models and analyze graphical relaionships for acceleraion, velociy, and posiion of he cener of mass of a sysem and use hem o calculae properies of he moion of he cener of mass of a sysem. Difficuly: 3 Page 5

6 6. A car ravels hrough a rainsorm a consan speed v C as shown in he diagram a righ. Rain is falling verically a a consan speed v R wih respec o he ground. If he back windshield of he car, highlighed in he diagram, is se a an angle of θ wih he verical, wha is he maximum speed he car can ravel and sill have rain hi he back windshield? v R v C (A) v R cosθ (C) v R sinθ (B) v R anθ (D) v R (1 sinθ) back windshield Answer: (B) v R anθ In order for he rain o jus hi he windshield, he angle of he velociy vecor for he rain wih respec o he car mus mach he angle of he back windshield. The velociy of he rain wih respec o he car (v RC ) can be found as he vecor sum of he velociy of he rain wih respec o he ground (v R ) and he velociy of he ground wih respec o he car (-v C ) as shown in he diagram a righ. From his diagram, i is a sraighforward applicaion of rigonomery o find he speed of he car a which his condiion occurs. v RC = v RG + v GC v GC = v C v RG = v R anθ = v GC v RG = v C v R v C = v R anθ EK: 3.A.1 An observer in a paricular reference frame can describe he moion of an objec using such quaniies as posiion, displacemen, disance, velociy, speed, and acceleraion. 4.A.2 The acceleraion is equal o he rae of change of velociy wih ime, and velociy is equal o he rae of change of posiion wih ime. quaniaively. 2.2 The suden can apply mahemaical rouines o quaniies ha describe naural phenomena. LO: 3.A.1.1 The suden is able o express he moion of an objec using narraive, mahemaical, and graphical represenaions. 4.A.2.3 The suden is able o creae mahemaical models and analyze graphical relaionships for acceleraion, velociy, and posiion of he cener of mass of a sysem and use hem o calculae properies of he moion of he cener of mass of a sysem. Page 6 Difficuly: 3

7 7. A cargo plane flies horizonally a a speed of 140 m/s a a heigh of 50 m above he ground. A supply package is dropped ou of he boom of he plane a ime =0. Two seconds laer, a second package is dropped ou of he boom of he plane. Air resisance is negligible. Wha happens o he separaion beween he packages as hey fall hrough he air? (A) The separaion beween packages decreases. (B) The separaion beween packages increases. (C) The separaion beween packages remains he same. (D) Canno answer wihou knowing he mass of he packages. Answer: (B) The separaion beween he packages increases. The firs package ou of he plane has more ime o accelerae while i is in he air, so i is moving a a faser speed han he second package dropped, herefore he separaion beween he packages increases while hey fall. EK: 4.A.2 The acceleraion is equal o he rae of change of velociy wih ime, and velociy is equal o he rae of change of posiion wih ime. quaniaively. LO: 4.A.2.1 The suden is able o make predicions abou he moion of a sysem based on he fac ha acceleraion is equal o he change in velociy per uni ime, and velociy is equal o he change in posiion per uni ime. Difficuly: 2 Page 7

8 8. A cargo plane flies horizonally a a speed of 140 m/s a a heigh of 50 m above he ground. A supply package is dropped ou of he boom of he plane a ime =0. Two seconds laer, a second package is dropped ou of he boom of he plane. Air resisance is negligible. How far apar will he packages land on he ground? (A) 70 m (B) 140 m (C) 280 m (D) 420 m Answer: (C) 280m. The packages will land on he ground a he same separaion as he separaion during heir release, which is 140 m/s 2s = 280m. EK: 4.A.2 The acceleraion is equal o he rae of change of velociy wih ime, and velociy is equal o he rae of change of posiion wih ime. quaniaively. 2.2 The suden can apply mahemaical rouines o quaniies ha describe naural phenomena. LO: 4.A.2.3 The suden is able o creae mahemaical models and analyze graphical relaionships for acceleraion, velociy, and posiion of he cener of mass of a sysem and use hem o calculae properies of he moion of he cener of mass of a sysem. Page 8 Difficuly: 1

9 9. An objec slides one meer down a fricionless ramp of consan slope as shown a righ (no o scale). A suden measures he ime i akes for he objec o ravel various displacemens using a sopwach. Three consecuive rials are measured, and he daa is recorded as shown below. Displacemen (m) Avg. Time (s) θ Deermine he acceleraion of he objec. Answer: m/s 2 There are several mehods of arriving a he answer, including, bu no limied o, aking he slope of he v- graph (calculaing velociy for he various ime inervals) or he use of he kinemaic equaions. EK: 3.A.1 An observer in a paricular reference frame can describe he moion of an objec using such quaniies as posiion, displacemen, disance, velociy, speed, and acceleraion. SP: 1.1 The suden can creae represenaions and models of naural or man made phenomena and sysems in he domain. 2.3 The suden can esimae numerically quaniies ha describe naural phenomena. 5.1 The suden can analyze daa o idenify paerns or relaionships. LO: 3.A.1.3 The suden is able o analyze experimenal daa describing he moion of an objec and is able o express he resuls of he analysis using narraive, mahemaical, and graphical represenaions. Difficuly: 1 Page 9

10 10. An objec slides one meer down a fricionless ramp of consan slope as shown a righ (no o scale). A suden measures he ime i akes for he objec o ravel various displacemens using a sopwach. Three consecuive rials are measured, and he daa is recorded as shown below. Displacemen (m) Avg. Time (s) θ Deermine he angle θ of he ramp. Answer: 5 Afer finding he acceleraion of he objec, recognize ha he acceleraion of he objec is a=gsin(θ). Solving for hea, hen, gives: a a = g sinθ θ = sin 1 g = m 2 sin 1 s 9.81 m s = 5 2 EK: 3.A.1 An observer in a paricular reference frame can describe he moion of an objec using such quaniies as posiion, displacemen, disance, velociy, speed, and acceleraion. 3.B.2 Free-body diagrams are useful ools for visualizing forces being exered on a single objec and wriing he equaions ha represen a physical siuaion. SP: 1.1 The suden can creae represenaions and models of naural or man made phenomena and sysems in he domain. 2.3 The suden can esimae numerically quaniies ha describe naural phenomena. 5.1 The suden can analyze daa o idenify paerns or relaionships. LO: 3.A.1.3 The suden is able o analyze experimenal daa describing he moion of an objec and is able o express he resuls of he analysis using narraive, mahemaical, and graphical represenaions. 3.B.2.1 The suden is able o creae and use free-body diagrams o analyze physical siuaions o solve problems wih moion qualiaively and quaniaively. Page 10 Difficuly: 3

11 11. A pirae capain in her ship spies her firs mae in a dinghy five kilomeers away. The pirae capain sails her ship oward he dinghy a a rae of eigh kilomeers per hour. The firs mae rows his dinghy oward he pirae ship a a rae of wo kilomeers per hour. When he capain iniially spies he firs mae a a disance of five kilomeers, her parro, Polly, begins flying back and forh beween he wo a a rae of 40 kilomeers per hour. How far does Polly fly in oal if she coninues her back-and-forh journey unil he pirae ship mees he dinghy? Answer: 20 kilomeers The ship and dinghy approach each oher a a combined 10 km per hour, herefore i akes hem 0.5 hours o mee. During his enire ime period, he parro flies a 40 km/hr, herefore he oal disance raveled by he parro is 20 kilomeers. EK: 3.A.1 An observer in a paricular reference frame can describe he moion of an objec using such quaniies as posiion, displacemen, disance, velociy, speed, and acceleraion. 4.A.2 The acceleraion is equal o he rae of change of velociy wih ime, and velociy is equal o he rae of change of posiion wih ime. quaniaively. 2.2 The suden can apply mahemaical rouines o quaniies ha describe naural phenomena. LO: 3.A.1.1 The suden is able o express he moion of an objec using narraive, mahemaical, and graphical represenaions. 4.A.2.1 The suden is able o make predicions abou he moion of a sysem based on he fac ha acceleraion is equal o he change in velociy per uni ime, and velociy is equal o he change in posiion per uni ime. Difficuly: 2 Page 11

12 12. A rain ravels eas oward Chicago a 80 km/hr. A man on he rain runs from he fron of he rain oward he rear of he rain a 10 km/hr. As he runs, he carries a plae of frui wih him. He noices a gian spider on he plae and hrows he plae away from him (oward he rear of he rain) a 20 km/hr. The sarled spider jumps oward he man a 5 km/hr. The insan afer he spider jumps oward he man, how fas is he spider approaching Chicago? Answer: 55 km/hr The spider s velociy wih respec o he ground is he vecor sum of he four given velociies, 80 km/hr eas, -10 km/ hr eas, -20 km/hr eas, and 5 km/hr eas, for a resulan of 55 km/hr eas. EK: 3.A.1c An observer in a paricular reference frame can describe he moion of an objec using such quaniies as posiion, displacemen, disance, velociy, speed, and acceleraion. A choice of reference frame deermines he direcion and he magniude of each of hese quaniies. 4.A.2 The acceleraion is equal o he rae of change of velociy wih ime, and velociy is equal o he rae of change of posiion wih ime. quaniaively. 2.2 The suden can apply mahemaical rouines o quaniies ha describe naural phenomena. LO: 3.A.1.1 The suden is able o express he moion of an objec using narraive, mahemaical, and graphical represenaions. 4.A.2.1 The suden is able o make predicions abou he moion of a sysem based on he fac ha acceleraion is equal o he change in velociy per uni ime, and velociy is equal o he change in posiion per uni ime. Page 12 Difficuly: 2

13 13. Three penguins are arranged in he cener of a pach of fricionless ice floaing across he ocean wih a velociy of 2 m/s wes. The mass of penguin A is 38 kg, he mass of penguin B is 30 kg, and he mass of penguin C is 23 kg. A ime =0, he penguins push off each oher, each wih a force of 20 newons, such ha hey all slide away from he cener of he floaing ice pach a an angle of 120 from each oher as shown in he diagram a righ. Describe he moion of he cener of mass of he hree-penguin sysem a ime =3s. 120 B A C Answer: The cener of mass coninues in is curren sae of moion, raveling wes a 2 m/s. Because here are no exernal forces on he sysem, he velociy of he cener of mass of he sysem canno be changed. EK: 4.A.1 The linear moion of a sysem can be described by he displacemen, velociy, and acceleraion of is cener of mass. 4.A.3 Forces ha sysems exer on each oher are due o ineracions beween objecs in he sysems. If he ineracing objecs are pars of he same sysem, here will be no change in he cener-of-mass velociy of ha sysem. 5.D.3a The velociy of he cener of mass of he sysem canno be changed by an ineracion wihin he sysem. In an isolaed sysem (a sysem wih no exernal forces), he velociy of he cener of mass does no change. quaniaively. 7.2 The suden can connec conceps in and across domain(s) o generalize or exrapolae in and/or across enduring undersandings and/or big ideas. LO: 4.A.1.1 The suden is able o use represenaions of he cener of mass of an isolaed wo-objec sysem o analyze he moion of he sysem qualiaively and semiquaniaively. 4.A.3.2 The suden is able o use visual or mahemaical represenaions of he forces beween objecs in a sysem o predic wheher or no here will be a change in he cener-of-mass velociy of ha sysem. 5.D.3.1 The suden is able o predic he velociy of he cener of mass of a sysem when here is no ineracion ouside of he sysem bu here is an ineracion wihin he sysem (i.e., he suden simply recognizes ha ineracions wihin a sysem do no affec he cener of mass moion of he sysem and is able o deermine ha here is no exernal force). Difficuly: 2 Page 13

14 14. A fisherman in a small fishing boa a res in a lake hooks a gian log floaing in he lake 30 meers away. The fisherman reels he log in. During his process, he boa moves 12 meers in he direcion of he log. If he mass of he boa and fisherman is 400 kg, wha is he mass of he log? Assume fricionless. Answer: m log = 267 kg Because here are no exernal forces on he sysem, and he cener of mass is iniially a res, he cener of mass mus remain a res. Se x=0 as he cener of mass of he fisherman / log sysem: x cm = x fisherman m fisherman + x log m log m fisherman + m log = 0 x fisherman m fisherman + x log m log = 0 m log = x fisherman m fisherman x log = (12m)(400kg) (18m) = 267kg EK: 4.A.1 The linear moion of a sysem can be described by he displacemen, velociy, and acceleraion of is cener of mass. 5.D.3 The velociy of he cener of mass of he sysem canno be changed by an ineracion wihin he sysem. quaniaively. 2.2 The suden can apply mahemaical rouines o quaniies ha describe naural phenomena. LO: 4.A.1.1 The suden is able o use represenaions of he cener of mass of an isolaed wo-objec sysem o analyze he moion of he sysem qualiaively and semiquaniaively. 5.D.3.1 The suden is able o predic he velociy of he cener of mass of a sysem when here is no ineracion ouside of he sysem bu here is an ineracion wihin he sysem (i.e., he suden simply recognizes ha ineracions wihin a sysem do no affec he cener of mass moion of he sysem and is able o deermine ha here is no exernal force). Page 14 Difficuly: 2

15 15. Two balls are launched off he edge a cliff of heigh h wih an iniial velociy v 0. The red ball is launched horizonally. The green ball is launched a an angle of θ above he horizonal. Neglec air resisance. (a) Derive an expression for he ime he red ball is in he air. (b) Derive an expression for he horizonal disance raveled by he red ball while i is in he air. (c) Derive an expression for he ime he green ball is in he air. (d) Derive an expression for he horizonal disance raveled by he green ball while while i is in he air. (e) If he iniial launch velociy v 0 of he balls is 100 m/s, he green ball is launched a an angle θ=30, and he balls land 600 meers apar from each oher, wha is he heigh of he cliff? (Noe: calculaor use srongly encouraged for his sep). Noe: This is a challenge problem ha is algebra-inensive and ouside he scope of wha a suden would see on an AP-1 exam. Difficuly: 3 Page 15

16 Answers: (a) red = 2h g AP1 Kinemaics (b) Δx red = v 0 2h g (c) green = v sinθ + v 2 sin 2 θ + 2gh 0 0 g (d) Δx green = v 0 cosθ (e) H=81m ( ) v 0 sinθ + v 0 2 sin 2 θ + 2gh g EK: 3.A.1 An observer in a paricular reference frame can describe he moion of an objec using such quaniies as posiion, displacemen, disance, velociy, speed, and acceleraion. SP: 1.1 The suden can creae represenaions and models of naural or man made phenomena and sysems in he domain. 1.4 The suden can use represenaions and models o analyze siuaions or solve problems qualiaively and quaniaively. 2.2 The suden can apply mahemaical rouines o quaniies ha describe naural phenomena. 2.3 The suden can esimae numerically quaniies ha describe naural phenomena. LO: 3.A.1.1 The suden is able o express he moion of an objec using narraive, mahemaical, and graphical represenaions. Page 16 Difficuly: 3

17 16. Using he following hree scenarios, answer he quesions ha follow. All hree cannons launch an idenical mass wih he same launch speed. The wo angled projeciles are launched a he same angle above he horizonal. The wo projeciles launched off a cliff are launched from he same heigh above he ground. Neglec air resisance. Illusraions are no drawn o scale (a) Rank he order of he resulan velociy a he projecile s maximum heigh from leas o greaes. Explain your reasoning. (b) Rank he order of he final resulan velociy jus before he projecile his he ground from leas o greaes. Explain your reasoning. Answers: (a) 2 = 3 < 1 A maximum heigh, cannon 1 has a velociy of v. A heir maximum heighs, boh cannons 2 and 3 have a velociy of vcos θ. (b) 2 < 1 = 3 Cannon wo has he lowes resulan velociy jus before hiing he ground as is speed will be v. Cannons 1 and 3, however, boh have speed v while hey are a heigh h. They also hen boh ravel he same verical disance h o he ground, gaining he same amoun of speed, resuling in he same final velociy. This can also be approached from an analyical perspecive, solving for he final speeds of cannons 1 and 3 righ before hey hi he ground, obaining v = v gh for boh cannons. EK: 3.A.1 An observer in a paricular reference frame can describe he moion of an objec using such quaniies as posiion, displacemen, disance, velociy, speed, and acceleraion. SP: 1.1 The suden can creae represenaions and models of naural or man made phenomena and sysems in he domain. 1.4 The suden can use represenaions and models o analyze siuaions or solve problems qualiaively and quaniaively. 2.2 The suden can apply mahemaical rouines o quaniies ha describe naural phenomena. LO: 3.A.1.1 The suden is able o express he moion of an objec using narraive, mahemaical, and graphical represenaions. Difficuly: 2 Page 17

1. The graph shows the variation with time t of the velocity v of an object.

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