AP Calculus BC Chapter 10 Part 1 AP Exam Problems
|
|
- Veronica Dixon
- 5 years ago
- Views:
Transcription
1 AP Calculus BC Chaper Par AP Eam Problems All problems are NO CALCULATOR unless oherwise indicaed Parameric Curves and Derivaives In he y plane, he graph of he parameric equaions = 5 + and y= for, is a line segmen wih slope 5 B) 5 C) D) 5 If y = and = +, hen a = is 8 B) 8 C) D) 8 A curve in he plane is defined paramerically by he equaions equaion of he line angen o he curve a = is: = + and y = + An y = C) y = y = 8 + B) y = 8 D) y = 5 Consider he curve in he y plane represened by = e and y= e for The slope of he line angen o he curve a he poin where = is (Calculaor) 86 B) C) 5 D) 5 If = + and y =, hen d y = B) C) D) 6 6 A paricle moves along he curve y = If = and =, wha is he value of? 5 B) 6 C) D) If = e and y = sin( ), hen = e sin( ) e cos( ) B) C) cos( ) e cos( ) D) e cos( ) e
2 8 For wha values of does he curve given by he parameric equaions y= + 8 have a verical angen? = and B) C), D),, and No value 9 For, an objec ravels along an ellipical pah given by he parameric equaions = cos and y = sin A he poin where =, he objec leaves he pah and ravels along he line angen o he pah a ha poin Wha is he slope of he line on which he objec ravels? B) C) an D) an an The posiion of a paricle moving in he y plane is given by he parameric equaions = and y= For wha values of is he paricle a res? B) C) D),,, A curve C is defined by he parameric equaions = + and y= Which of he following is an equaion of he line angen o he graph of C a he poin (, 8)? = B) = C) y = 8 D) 7 y= ( + ) + 8 y = ( + ) + 8 (97 BC, p 58 #) A kie flies according o he parameric equaions y= ( 8) where is measured in seconds and< 9 6 = and 8 (d) How high is he kie above he ground a ime = seconds? A wha rae is he kie rising a = seconds? A wha rae is he sring being reeled ou a = seconds? A wha ime does he kie sar o lose aliude? (97 BC5) Given he parameric equaions = ( sin) and y ( cos) Find in erms of Find an equaion of he line angen o he graph a = Find an equaion of he line angen o he graph a = =
3 (98 BC6, p 56 #9) Poin P(, y ) moves in he y plane in such a way ha = for Find he coordinaes of P in erms of if, when =, = ln and y = Wrie an equaion epressing y in erms of Find he average rae of change of y wih respec o as varies from o (d) Find he insananeous rae of change of y wih respec o when = = + and 5 (98 BC, p 58 #) The pah of a paricle is given for ime > by he parameric equaions = + and y= Find he coordinaes of each poin on he pah where he velociy of he paricle in he direcion is zero Find when = d y Find when y = 6 (989 BC, p 58 #) Consider he curve given by he parameric equaions y = = and In erms of, find Wrie an equaion for he line angen o he curve a he poin where = Find he and y coordinaes for each criical poin on he curve and idenify each poin as having a verical or horizonal angen 7 (99 BC) The posiion of a paricle a any ime is given by Find as a funcion of = and y( ) = () Lengh of Parameric Curves 8 The lengh of he curve deermined by he equaions = and y = from = o = is B) + C) + D) + + +
4 9 The lengh of he pah described by he parameric equaions, is given by = cos and y = sin, for cos + sin B) C) 9cos + 9sin D) 6 6 cos + sin cos sin+ sin cos 9cos sin + 9sin cos The lengh of he pah described by he parameric equaions, is given by = and y =, where B) + C) + D) (97 BC7) Le C be he curve defined from = o = 6 by he parameric equaions + =, y= ( 6 ) Se up bu do no evaluae an inegral epression for he lengh of C (97 BC5) Given he parameric equaions = ( sin) and y ( cos) = Se up bu do no evaluae an inegral epression represening he lengh of he curve over he inerval Epress he inegrand as a funcion of Vecors If f is a vecor valued funcion defined by f( ) ( e, cos) e sin =, hen f ( ) = + C) ( e, sin) ( e, cos) e D) ( e,cos) B) cos A paricle moves in he y plane so ha a any ime is coordinaes are y= A =, is acceleraion vecor is = and (, ) B) (, ) C) (, ) D) (, ) (, 8)
5 5 For any ime, if he posiion of a paricle in he y plane is given by ( ) y= ln +, hen he acceleraion vecor is B), +, ( + ) C), D), ( + ) ( + ), ( + ) = + and ( ln ), 6 If a paricle moves in he y plane so ha a ime > is posiion vecor is ( ) hen a ime =, is velociy vecor is +,,8 B), C),8 8 D), 8 5, 6 7 A paricle moves on a plane curve so ha a any ime > is coordinae is coordinae is ( ) The acceleraion vecor of he paricle a = is (, ) B) (, ) C) (, 6) D) (6, ) (6, ) and is y 8 A paricle moves in he y plane so ha is posiion a any ime is given by y = sin( ) Wha is he speed of he paricle when =? = and 99 B) 6 C) 688 D) (975 BC) A paricle moves on he circle by he vecor, y = so ha a ime he posiion is given Find he velociy vecor Is he paricle ever a res? Jusify your answer Give he coordinaes of he poin ha he paricle approaches as increases wihou bound (987 BC5) The posiion of a paricle moving in he y plane a any ime,, is given by he parameric equaions = sin and y = cos( ) Find he velociy vecor for he paricle a anyime, For wha values of is he paricle a res? Wrie an equaion for he pah of he paricle in erms of and y ha does no involve rigonomeric funcions 5
6 (99 BC) A ime,, he posiion of a paricle moving along a pah in he y plane is given by he parameric equaions = e sin and y= e cos Find he slope of he pah of he paricle a ime = Find he speed of he paricle when = Find he disance raveled by he paricle along he pah from = o = (99 BC) The posiion of a paricle a any ime is given by Find he magniude of he velociy vecor a = 5 Find he oal disance raveled by he paricle from = o = 5 = and y( ) = () (99 BC) A paricle moves along he graph of y = cos so ha componen of acceleraion is always A ime =, he paricle is a he poin (, ) and he velociy vecor of he paricle is (, ) Find he and y coordinaes of he posiion of he paricle in erms of Find he speed of he paricle when is posiion is(, cos ) (995 BC) Two paricles move in he y plane For ime, he posiion of paricle A is given by = and y= ( ), and he posiion of paricle B is given by = and y = Find he velociy vecor for each paricle a ime = Se up an inegral epression ha gives he disance raveled by paricle A from = o = Do no evaluae Deermine he eac ime a which he paricles collide; ha is, when he paricles are a he same poin a he same ime Jusify your answer 5 (997 BC) During he ime period from = o = 6 seconds, a paricle moves along he pah () cos y () = 5sin given by = ( ) and ( ) Find he posiion of he paricle when = 5 On a se of and y aes, skech he graph of he pah of he paricle from = o = 6 Indicae he direcion of he paricle along his pah How many imes does he paricle pass hrough he poin found in par? (d) Find he velociy vecor for he paricle a any ime (e) Wrie and evaluae an inegral epression, in erms of sine and cosine, ha gives he disance he paricle ravels from ime = 5 o = 75 6
7 6 (999 BC) A paricle moves in he y plane so ha is posiion a any ime,, is given by ( ) = ln( + ) and y( ) = sin Skech he pah of he paricle on a se of and y aes Indicae he direcion of moion along he pah A wha ime,, does ( ) aain is minimum value? Wha is he posiion ( (), y() ) of he paricle a his ime? A wha ime,, is he paricle on he y ais? Find he speed and he acceleraion vecor of he paricle a his ime 7 ( BC) A moving paricle has posiion ( (), y() ) a ime The posiion of he paricle a ime = is (, 6), and he velociy vecor a any ime > is given by,+ Find he acceleraion vecor a ime = Find he posiion of he paricle a ime = For wha ime > does he line angen o he pah of he paricle a ( (), ()) (d) y have a slope of 8? The paricle approaches a line as Find he slope of his line Show he work ha leads o your conclusion 8 ( BC) An objec moving along a curve in he y plane has posiion ( (), y() ) a ime = = for A ime =, he objec is a posiion (, wih cos( ) and sin( ) 5) Wrie an equaion for he line angen o he curve a (, 5) Find he speed of he objec a ime = Find he oal disance raveled by he objec over he ime inerval (d) Find he posiion of he objec a ime = 7
8 9 ( BC) The figure above shows he pah raveled by a roller coaser car over he ime inerval 8seconds The posiion of he car a ime seconds can be modeled paramerically by = + sin, y= ( )( cos ), where and y are measured in meers The derivaives of hese funcions are given by = + cos, y = ( )sin + cos (d) Find he slope of he pah a ime = Show he compuaions ha lead o your answer Find he acceleraion vecor of he car a he ime when he car s horizonal posiion is = Find he ime a which he car is a is maimum heigh, and find he speed, in m/sec, of he car a his ime For <<8, here are wo imes a which he car is a ground level (y = ) Find hese wo imes and wrie an epression ha gives he average speed, in m/sec, of he car beween hese wo imes Do no evaluae he epression (B BC) A paricle moves in he y plane so ha is posiion a any ime,, is given by ( ) = sin( ) and y ( ) = Skech he pah of he paricle on a se of and y aes Indicae he direcion of moion along he pah Find he range of ( ) and he range of y( ) Find he smalles posiive value of for which he coordinae of he paricle is a local maimum Wha is he speed of he paricle a his ime? (d) Is he disance raveled by he paricle in greaer han5? Jusify your answer 8
9 ( BC) A paricle sars a poin A on he posiive ais a ime = and ravels along he curve from A o B o C o D, as shown above The coordinaes of he paricle s posiion ( ( ), y ( )) are differeniable funcions of, where 9cos sin + = = 6 and y = is no eplicily given A ime = 9, he paricle reaches is final posiion a poin D on he posiive ais A poin C, is posiive? A poin C, is posiive? Give a reason for each answer The slope of he curve is undefined a poin B A wha ime is he paricle a B? 5 The line angen o he curve a he poin ( (8), y (8)) has equaion y = Find he 9 velociy vecor and he speed of he paricle a his poin (d) How far apar are poins A and D, he iniial and final posiions, respecively, of he paricle? (B BC) A paricle moves in he y plane so ha he posiion of he paricle a any ime 7 is given by () = e + e and y () = e e (d) Find he velociy vecor for he paricle in erms of, and find he speed of he paricle a ime = Find in erms of, and find lim Find each value a which he line angen o he pah of he paricle is horizonal, or eplain why none eiss Find each value a which he line angen o he pah of he paricle is verical, or eplain why none eiss 9
10 ( BC) An objec moving along a curve in he y plane has posiion ( (), y() ) a ime wih = + cos( ) The derivaive is no eplicily given A ime =, he objec is a posiion (, 8) Find he coordinae of he posiion of he objec a ime = A ime =, he value of = 7 Wrie an equaion for he line angen o he curve a he poin( (), y ()) Find he speed of he objec a ime = (d) For, he line angen o he curve a ( (), y() ) has a slope of + Find he acceleraion vecor of he objec a ime = (B BC) A paricle moving along a curve in he plane has posiion ( ( ), y ( )) a ime, where 9 = + and 5 = e + e for all real values of A ime =, he paricle is a he poin (, ) Find he speed of he paricle and is acceleraion vecor a ime = Find an equaion of he line angen o he pah of he paricle a ime = Find he oal disance raveled by he paricle over he ime inerval (d) Find he coordinae of he posiion of he paricle a ime = 5 (5B BC) A paricle moving along a curve in he y plane has posiion ( ( ), y ( )) a ime, where = and = ln( + ( ) ) A ime =, he paricle is a he poin (, 5) A =, he objec is a poin P wih coordinae Find he acceleraion vecor a ime = and he speed a ime = Find he y coordinae of P Wrie an equaion for he line angen o he curve a P (d) For wha value of, if any, is he objec a res? Eplain your reasoning 6 (6 BC) An objec moving along a curve in he y plane is a posiion ((), y()) a ime, where = sin ( e ) and = for > A ime =, he objec is a he poin + (6, ) Find he acceleraion vecor and he speed of he objec a ime = The curve has a verical angen line a one poin A wha ime is he objec a his poin? Le m() denoe he slope of he line angen o he curve a he poin ((), y()) Wrie an epression for m() in erms of and use i o evaluae lim m ( ) (d) The graph of he curve has a horizonal asympoe y = c Wrie, bu do no evaluae, and epression involving an improper inegral ha represens his value c
11 7 (6B BC) An objec moving along a curve in he y plane is a posiion ((), y()) a ime, where = an( e ) and = sec( e ) for A ime =, he objec is a he poin (, ) Wrie an equaion for he line angen o he curve a posiion (, ) Find he acceleraion vecor and he speed of he objec a ime = Find he oal disance raveled by he objec over he ime inerval (d) Is here a ime a which he objec is on he y ais? Eplain why or why no 8 (7B BC) An objec moving along a curve in he y plane is a posiion ( ( ), y( ) ) a ime wih = arcan + (,) = + for A ime =, he objec is a posiion and ln( ) Find he speed of he objec a ime = Find he oal disance raveled by he objec over he ime inerval Find () (d) For >, here is a poin on he curve where he line angen o he curve has slope A wha ime is he objec a his poin? Find he acceleraion vecor a his poin 9 (8B BC) A paricle moving along a curve in he y plane has posiion ( ( ), y( ) ) a ime wih = and = cos The paricle is a posiion (, 5) a ime = Find he acceleraion vecor a ime = Find he y coordinae of he posiion of he paricle a ime = On he inerval, a wha ime does he speed of he paricle firs reach 5? (d) Find he oal disance raveled by he paricle over he ime inerval Answers Parameric Curves and Derivaives A 998 BC # 9% B 985 BC # 7% C 988 BC # 68% D 99 BC #5 5% 5 A 99 BC #6 9% 6 B 99 BC # 75% 7 E 997 BC # 9% 8 C 997 BC #8 6% 9 D BC # 79% C BC #7 9% A BC #7 6% Lengh of Parameric Curves 8 D 99 BC # 8% 9 D 997 BC #5 66% C 998 BC # 7% Vecors E 998 BC #77 87% D 985 BC # 69% 5 E 988 BC #5 75% 6 A 99 BC #8 86% 7 E 998 BC # 75% 8 C BC #8 6%
AP Calculus BC - Parametric equations and vectors Chapter 9- AP Exam Problems solutions
AP Calculus BC - Parameric equaions and vecors Chaper 9- AP Exam Problems soluions. A 5 and 5. B A, 4 + 8. C A, 4 + 4 8 ; he poin a is (,). y + ( x ) x + 4 4. e + e D A, slope.5 6 e e e 5. A d hus d d
More informationParametrics and Vectors (BC Only)
Paramerics and Vecors (BC Only) The following relaionships should be learned and memorized. The paricle s posiion vecor is r() x(), y(). The velociy vecor is v(),. The speed is he magniude of he velociy
More information!!"#"$%&#'()!"#&'(*%)+,&',-)./0)1-*23)
"#"$%&#'()"#&'(*%)+,&',-)./)1-*) #$%&'()*+,&',-.%,/)*+,-&1*#$)()5*6$+$%*,7&*-'-&1*(,-&*6&,7.$%$+*&%'(*8$&',-,%'-&1*(,-&*6&,79*(&,%: ;..,*&1$&$.$%&'()*1$$.,'&',-9*(&,%)?%*,('&5
More informationAP CALCULUS BC 2016 SCORING GUIDELINES
6 SCORING GUIDELINES Quesion A ime, he posiion of a paricle moving in he xy-plane is given by he parameric funcions ( x ( ), y ( )), where = + sin ( ). The graph of y, consising of hree line segmens, is
More informationAnswers to 1 Homework
Answers o Homework. x + and y x 5 y To eliminae he parameer, solve for x. Subsiue ino y s equaion o ge y x.. x and y, x y x To eliminae he parameer, solve for. Subsiue ino y s equaion o ge x y, x. (Noe:
More information2.1: What is physics? Ch02: Motion along a straight line. 2.2: Motion. 2.3: Position, Displacement, Distance
Ch: Moion along a sraigh line Moion Posiion and Displacemen Average Velociy and Average Speed Insananeous Velociy and Speed Acceleraion Consan Acceleraion: A Special Case Anoher Look a Consan Acceleraion
More information15. Vector Valued Functions
1. Vecor Valued Funcions Up o his poin, we have presened vecors wih consan componens, for example, 1, and,,4. However, we can allow he componens of a vecor o be funcions of a common variable. For example,
More informationCALCULUS EXPLORATION OF THE SECOND FUNDAMENTAL THEOREM OF CALCULUS. Second Fundamental Theorem of Calculus (Chain Rule Version):
CALCULUS EXPLORATION OF THE SECOND FUNDAMENTAL THEOREM OF CALCULUS 6 cos Secon Funamenal Theorem of Calculus: f a 4 a f 6 cos Secon Funamenal Theorem of Calculus (Chain Rule Version): g f a E. Use he Secon
More informationMEI Mechanics 1 General motion. Section 1: Using calculus
Soluions o Exercise MEI Mechanics General moion Secion : Using calculus. s 4 v a 6 4 4 When =, v 4 a 6 4 6. (i) When = 0, s = -, so he iniial displacemen = - m. s v 4 When = 0, v = so he iniial velociy
More informationAP CALCULUS AB 2003 SCORING GUIDELINES (Form B)
SCORING GUIDELINES (Form B) Quesion A blood vessel is 6 millimeers (mm) long Disance wih circular cross secions of varying diameer. x (mm) 6 8 4 6 Diameer The able above gives he measuremens of he B(x)
More informationMidterm Exam Review Questions Free Response Non Calculator
Name: Dae: Block: Miderm Eam Review Quesions Free Response Non Calculaor Direcions: Solve each of he following problems. Choose he BEST answer choice from hose given. A calculaor may no be used. Do no
More informationKINEMATICS IN ONE DIMENSION
KINEMATICS IN ONE DIMENSION PREVIEW Kinemaics is he sudy of how hings move how far (disance and displacemen), how fas (speed and velociy), and how fas ha how fas changes (acceleraion). We say ha an objec
More informationIB Physics Kinematics Worksheet
IB Physics Kinemaics Workshee Wrie full soluions and noes for muliple choice answers. Do no use a calculaor for muliple choice answers. 1. Which of he following is a correc definiion of average acceleraion?
More informationGround Rules. PC1221 Fundamentals of Physics I. Kinematics. Position. Lectures 3 and 4 Motion in One Dimension. A/Prof Tay Seng Chuan
Ground Rules PC11 Fundamenals of Physics I Lecures 3 and 4 Moion in One Dimension A/Prof Tay Seng Chuan 1 Swich off your handphone and pager Swich off your lapop compuer and keep i No alking while lecure
More informationPracticing Problem Solving and Graphing
Pracicing Problem Solving and Graphing Tes 1: Jan 30, 7pm, Ming Hsieh G20 The Bes Ways To Pracice for Tes Bes If need more, ry suggesed problems from each new opic: Suden Response Examples A pas opic ha
More informationPhysics 180A Fall 2008 Test points. Provide the best answer to the following questions and problems. Watch your sig figs.
Physics 180A Fall 2008 Tes 1-120 poins Name Provide he bes answer o he following quesions and problems. Wach your sig figs. 1) The number of meaningful digis in a number is called he number of. When numbers
More informationAP Calculus BC 2004 Free-Response Questions Form B
AP Calculus BC 200 Free-Response Quesions Form B The maerials included in hese files are inended for noncommercial use by AP eachers for course and exam preparaion; permission for any oher use mus be sough
More information3.6 Derivatives as Rates of Change
3.6 Derivaives as Raes of Change Problem 1 John is walking along a sraigh pah. His posiion a he ime >0 is given by s = f(). He sars a =0from his house (f(0) = 0) and he graph of f is given below. (a) Describe
More informationMotion along a Straight Line
chaper 2 Moion along a Sraigh Line verage speed and average velociy (Secion 2.2) 1. Velociy versus speed Cone in he ebook: fer Eample 2. Insananeous velociy and insananeous acceleraion (Secions 2.3, 2.4)
More informationKinematics Vocabulary. Kinematics and One Dimensional Motion. Position. Coordinate System in One Dimension. Kinema means movement 8.
Kinemaics Vocabulary Kinemaics and One Dimensional Moion 8.1 WD1 Kinema means movemen Mahemaical descripion of moion Posiion Time Inerval Displacemen Velociy; absolue value: speed Acceleraion Averages
More informationSolution: b All the terms must have the dimension of acceleration. We see that, indeed, each term has the units of acceleration
PHYS 54 Tes Pracice Soluions Spring 8 Q: [4] Knowing ha in he ne epression a is acceleraion, v is speed, is posiion and is ime, from a dimensional v poin of view, he equaion a is a) incorrec b) correc
More information4.5 Constant Acceleration
4.5 Consan Acceleraion v() v() = v 0 + a a() a a() = a v 0 Area = a (a) (b) Figure 4.8 Consan acceleraion: (a) velociy, (b) acceleraion When he x -componen of he velociy is a linear funcion (Figure 4.8(a)),
More information10.1 EXERCISES. y 2 t 2. y 1 t y t 3. y e
66 CHAPTER PARAMETRIC EQUATINS AND PLAR CRDINATES SLUTIN We use a graphing device o produce he graphs for he cases a,,.5,.,,.5,, and shown in Figure 7. Noice ha all of hese curves (ecep he case a ) have
More informationPHYSICS 220 Lecture 02 Motion, Forces, and Newton s Laws Textbook Sections
PHYSICS 220 Lecure 02 Moion, Forces, and Newon s Laws Texbook Secions 2.2-2.4 Lecure 2 Purdue Universiy, Physics 220 1 Overview Las Lecure Unis Scienific Noaion Significan Figures Moion Displacemen: Δx
More informationBe able to sketch a function defined parametrically. (by hand and by calculator)
Pre Calculus Uni : Parameric and Polar Equaions (7) Te References: Pre Calculus wih Limis; Larson, Hoseler, Edwards. B he end of he uni, ou should be able o complee he problems below. The eacher ma provide
More informationPhys 221 Fall Chapter 2. Motion in One Dimension. 2014, 2005 A. Dzyubenko Brooks/Cole
Phys 221 Fall 2014 Chaper 2 Moion in One Dimension 2014, 2005 A. Dzyubenko 2004 Brooks/Cole 1 Kinemaics Kinemaics, a par of classical mechanics: Describes moion in erms of space and ime Ignores he agen
More informationAP CALCULUS AB/CALCULUS BC 2016 SCORING GUIDELINES. Question 1. 1 : estimate = = 120 liters/hr
AP CALCULUS AB/CALCULUS BC 16 SCORING GUIDELINES Quesion 1 (hours) R ( ) (liers / hour) 1 3 6 8 134 119 95 74 7 Waer is pumped ino a ank a a rae modeled by W( ) = e liers per hour for 8, where is measured
More information10.6 Parametric Equations
0_006.qd /8/05 9:05 AM Page 77 Secion 0.6 77 Parameric Equaions 0.6 Parameric Equaions Wha ou should learn Evaluae ses of parameric equaions for given values of he parameer. Skech curves ha are represened
More informationMultiple Choice Solutions 1. E (2003 AB25) () xt t t t 2. A (2008 AB21/BC21) 3. B (2008 AB7) Using Fundamental Theorem of Calculus: 1
Paricle Moion Soluions We have inenionally included more maerial han can be covered in mos Suden Sudy Sessions o accoun for groups ha are able o answer he quesions a a faser rae. Use your own judgmen,
More informationAP CALCULUS AB 2003 SCORING GUIDELINES (Form B)
SCING GUIDELINES (Form B) Quesion 4 A paricle moves along he x-axis wih velociy a ime given by v( ) = 1 + e1. (a) Find he acceleraion of he paricle a ime =. (b) Is he speed of he paricle increasing a ime
More informationChapter Let. 1) k be a vector-valued function. (a) Evaluate f (0). (b) What is the domain of f () t? (c) Is f () t continuous at t = 1?
Chaper. Le f() = sin i+ ( 3+ ) j ln( + ) k be a vecor-valued funcion. (a) Evaluae f (). (b) Wha is he domain of f ()? (c) Is f () coninuous a =? Chaper. Le f() = sin i+ ( 3+ ) j ln( + ) k be a vecor-valued
More informationChapters 6 & 7: Trigonometric Functions of Angles and Real Numbers. Divide both Sides by 180
Algebra Chapers & : Trigonomeric Funcions of Angles and Real Numbers Chapers & : Trigonomeric Funcions of Angles and Real Numbers - Angle Measures Radians: - a uni (rad o measure he size of an angle. rad
More informationMechanics Acceleration The Kinematics Equations
Mechanics Acceleraion The Kinemaics Equaions Lana Sheridan De Anza College Sep 27, 2018 Las ime kinemaic quaniies graphs of kinemaic quaniies Overview acceleraion he kinemaics equaions (consan acceleraion)
More informationPhysics 221 Fall 2008 Homework #2 Solutions Ch. 2 Due Tues, Sept 9, 2008
Physics 221 Fall 28 Homework #2 Soluions Ch. 2 Due Tues, Sep 9, 28 2.1 A paricle moving along he x-axis moves direcly from posiion x =. m a ime =. s o posiion x = 1. m by ime = 1. s, and hen moves direcly
More informationDisplacement ( x) x x x
Kinemaics Kinemaics is he branch of mechanics ha describes he moion of objecs wihou necessarily discussing wha causes he moion. 1-Dimensional Kinemaics (or 1- Dimensional moion) refers o moion in a sraigh
More informationSpeed and Velocity. Overview. Velocity & Speed. Speed & Velocity. Instantaneous Velocity. Instantaneous and Average
Overview Kinemaics: Descripion of Moion Posiion and displacemen velociy»insananeous acceleraion»insananeous Speed Velociy Speed and Velociy Speed & Velociy Velociy & Speed A physics eacher walks 4 meers
More informationPhysics 20 Lesson 5 Graphical Analysis Acceleration
Physics 2 Lesson 5 Graphical Analysis Acceleraion I. Insananeous Velociy From our previous work wih consan speed and consan velociy, we know ha he slope of a posiion-ime graph is equal o he velociy of
More informationa. Show that these lines intersect by finding the point of intersection. b. Find an equation for the plane containing these lines.
Mah A Final Eam Problems for onsideraion. Show all work for credi. Be sure o show wha you know. Given poins A(,,, B(,,, (,, 4 and (,,, find he volume of he parallelepiped wih adjacen edges AB, A, and A.
More information2. What is the displacement of the bug between t = 0.00 s and t = 20.0 s? A) cm B) 39.9 cm C) cm D) 16.1 cm E) +16.
1. For which one of he following siuaions will he pah lengh equal he magniude of he displacemen? A) A jogger is running around a circular pah. B) A ball is rolling down an inclined plane. C) A rain ravels
More informationx i v x t a dx dt t x
Physics 3A: Basic Physics I Shoup - Miderm Useful Equaions A y A sin A A A y an A y A A = A i + A y j + A z k A * B = A B cos(θ) A B = A B sin(θ) A * B = A B + A y B y + A z B z A B = (A y B z A z B y
More information3, so θ = arccos
Mahemaics 210 Professor Alan H Sein Monday, Ocober 1, 2007 SOLUTIONS This problem se is worh 50 poins 1 Find he angle beween he vecors (2, 7, 3) and (5, 2, 4) Soluion: Le θ be he angle (2, 7, 3) (5, 2,
More informationRoller-Coaster Coordinate System
Winer 200 MECH 220: Mechanics 2 Roller-Coaser Coordinae Sysem Imagine you are riding on a roller-coaer in which he rack goes up and down, wiss and urns. Your velociy and acceleraion will change (quie abruply),
More information(π 3)k. f(t) = 1 π 3 sin(t)
Mah 6 Fall 6 Dr. Lil Yen Tes Show all our work Name: Score: /6 No Calculaor permied in his par. Read he quesions carefull. Show all our work and clearl indicae our final answer. Use proper noaion. Problem
More information1. VELOCITY AND ACCELERATION
1. VELOCITY AND ACCELERATION 1.1 Kinemaics Equaions s = u + 1 a and s = v 1 a s = 1 (u + v) v = u + as 1. Displacemen-Time Graph Gradien = speed 1.3 Velociy-Time Graph Gradien = acceleraion Area under
More informationKinematics Motion in 1 Dimension and Graphs
Kinemaics Moion in 1 Dimension and Graphs Lana Sheridan De Anza College Sep 27, 2017 Las ime moion in 1-dimension some kinemaic quaniies graphs Overview velociy and speed acceleraion more graphs Kinemaics
More informationChapter 2. Motion in One-Dimension I
Chaper 2. Moion in One-Dimension I Level : AP Physics Insrucor : Kim 1. Average Rae of Change and Insananeous Velociy To find he average velociy(v ) of a paricle, we need o find he paricle s displacemen
More informationToday: Graphing. Note: I hope this joke will be funnier (or at least make you roll your eyes and say ugh ) after class. v (miles per hour ) Time
+v Today: Graphing v (miles per hour ) 9 8 7 6 5 4 - - Time Noe: I hope his joke will be funnier (or a leas make you roll your eyes and say ugh ) afer class. Do yourself a favor! Prof Sarah s fail-safe
More informationQ2.1 This is the x t graph of the motion of a particle. Of the four points P, Q, R, and S, the velocity v x is greatest (most positive) at
Q2.1 This is he x graph of he moion of a paricle. Of he four poins P, Q, R, and S, he velociy is greaes (mos posiive) a A. poin P. B. poin Q. C. poin R. D. poin S. E. no enough informaion in he graph o
More information1998 Calculus AB Scoring Guidelines
AB{ / BC{ 1999. The rae a which waer ows ou of a pipe, in gallons per hour, is given by a diereniable funcion R of ime. The able above shows he rae as measured every hours for a {hour period. (a) Use a
More informationWEEK-3 Recitation PHYS 131. of the projectile s velocity remains constant throughout the motion, since the acceleration a x
WEEK-3 Reciaion PHYS 131 Ch. 3: FOC 1, 3, 4, 6, 14. Problems 9, 37, 41 & 71 and Ch. 4: FOC 1, 3, 5, 8. Problems 3, 5 & 16. Feb 8, 018 Ch. 3: FOC 1, 3, 4, 6, 14. 1. (a) The horizonal componen of he projecile
More information4.6 One Dimensional Kinematics and Integration
4.6 One Dimensional Kinemaics and Inegraion When he acceleraion a( of an objec is a non-consan funcion of ime, we would like o deermine he ime dependence of he posiion funcion x( and he x -componen of
More informationAP CALCULUS AB 2004 SCORING GUIDELINES (Form B)
4 SCORING GUIDELINES (Form B) Quesion A es plane flies in a sraigh line wih (min) 5 1 15 5 5 4 posiive velociy v (), in miles per v ()(mpm) 7. 9. 9.5 7. 4.5.4.4 4. 7. minue a ime minues, where v is a differeniable
More informationMath 116 Second Midterm March 21, 2016
Mah 6 Second Miderm March, 06 UMID: EXAM SOLUTIONS Iniials: Insrucor: Secion:. Do no open his exam unil you are old o do so.. Do no wrie your name anywhere on his exam. 3. This exam has pages including
More informationAP CALCULUS AB 2017 SCORING GUIDELINES
AP CALCULUS AB 17 SCORING GUIDELINES 16 SCORING GUIDELINES Quesion For, a paricle moves along he x-axis. The velociy of he paricle a ime is given by v ( ) = 1 + sin. The paricle is a posiion x = a ime.
More informationACCUMULATION. Section 7.5 Calculus AP/Dual, Revised /26/2018 7:27 PM 7.5A: Accumulation 1
ACCUMULATION Secion 7.5 Calculus AP/Dual, Revised 2019 vie.dang@humbleisd.ne 12/26/2018 7:27 PM 7.5A: Accumulaion 1 APPLICATION PROBLEMS A. Undersand he quesion. I is ofen no necessary o as much compuaion
More informationSMT 2014 Calculus Test Solutions February 15, 2014 = 3 5 = 15.
SMT Calculus Tes Soluions February 5,. Le f() = and le g() =. Compue f ()g (). Answer: 5 Soluion: We noe ha f () = and g () = 6. Then f ()g () =. Plugging in = we ge f ()g () = 6 = 3 5 = 5.. There is a
More informationPhysics 3A: Basic Physics I Shoup Sample Midterm. Useful Equations. x f. x i v x. a x. x i. v xi v xf. 2a x f x i. y f. a r.
Physics 3A: Basic Physics I Shoup Sample Miderm Useful Equaions A y Asin A A x A y an A y A x A = A x i + A y j + A z k A * B = A B cos(θ) A x B = A B sin(θ) A * B = A x B x + A y B y + A z B z A x B =
More informationMEI STRUCTURED MATHEMATICS 4758
OXFORD CAMBRIDGE AND RSA EXAMINATIONS Advanced Subsidiary General Cerificae of Educaion Advanced General Cerificae of Educaion MEI STRUCTURED MATHEMATICS 4758 Differenial Equaions Thursday 5 JUNE 006 Afernoon
More information02. MOTION. Questions and Answers
CLASS-09 02. MOTION Quesions and Answers PHYSICAL SCIENCE 1. Se moves a a consan speed in a consan direcion.. Reprase e same senence in fewer words using conceps relaed o moion. Se moves wi uniform velociy.
More informationPROBLEMS FOR MATH 162 If a problem is starred, all subproblems are due. If only subproblems are starred, only those are due. SLOPES OF TANGENT LINES
PROBLEMS FOR MATH 6 If a problem is sarred, all subproblems are due. If onl subproblems are sarred, onl hose are due. 00. Shor answer quesions. SLOPES OF TANGENT LINES (a) A ball is hrown ino he air. Is
More informationPhysics for Scientists and Engineers I
Physics for Scieniss and Engineers I PHY 48, Secion 4 Dr. Beariz Roldán Cuenya Universiy of Cenral Florida, Physics Deparmen, Orlando, FL Chaper - Inroducion I. General II. Inernaional Sysem of Unis III.
More informationAP CALCULUS AB 2003 SCORING GUIDELINES (Form B)
SCORING GUIDELINES (Form B) Quesion A ank conains 15 gallons of heaing oil a ime =. During he ime inerval 1 hours, heaing oil is pumped ino he ank a he rae 1 H ( ) = + ( 1 + ln( + 1) ) gallons per hour.
More informationSuggested Practice Problems (set #2) for the Physics Placement Test
Deparmen of Physics College of Ars and Sciences American Universiy of Sharjah (AUS) Fall 014 Suggesed Pracice Problems (se #) for he Physics Placemen Tes This documen conains 5 suggesed problems ha are
More informationt A. 3. Which vector has the largest component in the y-direction, as defined by the axes to the right?
Ke Name Insrucor Phsics 1210 Exam 1 Sepember 26, 2013 Please wrie direcl on he exam and aach oher shees of work if necessar. Calculaors are allowed. No noes or books ma be used. Muliple-choice problems
More informationChapter 15 Oscillatory Motion I
Chaper 15 Oscillaory Moion I Level : AP Physics Insrucor : Kim Inroducion A very special kind of moion occurs when he force acing on a body is proporional o he displacemen of he body from some equilibrium
More informationLecture 2-1 Kinematics in One Dimension Displacement, Velocity and Acceleration Everything in the world is moving. Nothing stays still.
Lecure - Kinemaics in One Dimension Displacemen, Velociy and Acceleraion Everyhing in he world is moving. Nohing says sill. Moion occurs a all scales of he universe, saring from he moion of elecrons in
More informationand v y . The changes occur, respectively, because of the acceleration components a x and a y
Week 3 Reciaion: Chaper3 : Problems: 1, 16, 9, 37, 41, 71. 1. A spacecraf is raveling wih a veloci of v0 = 5480 m/s along he + direcion. Two engines are urned on for a ime of 84 s. One engine gives he
More informationx(m) t(sec ) Homework #2. Ph 231 Introductory Physics, Sp-03 Page 1 of 4
Homework #2. Ph 231 Inroducory Physics, Sp-03 Page 1 of 4 2-1A. A person walks 2 miles Eas (E) in 40 minues and hen back 1 mile Wes (W) in 20 minues. Wha are her average speed and average velociy (in ha
More informationA. Using Newton s second law in one dimension, F net. , write down the differential equation that governs the motion of the block.
Simple SIMPLE harmonic HARMONIC moion MOTION I. Differenial equaion of moion A block is conneced o a spring, one end of which is aached o a wall. (Neglec he mass of he spring, and assume he surface is
More informationUniversity Physics with Modern Physics 14th Edition Young TEST BANK
Universi Phsics wih Modern Phsics 14h Ediion Young SOLUTIONS MANUAL Full clear download (no formaing errors) a: hps://esbankreal.com/download/universi-phsics-modern-phsics- 14h-ediion-oung-soluions-manual-/
More informationSOLUTIONS TO CONCEPTS CHAPTER 3
SOLUTIONS TO ONEPTS HPTER 3. a) Disance ravelled = 50 + 40 + 0 = 0 m b) F = F = D = 50 0 = 30 M His displacemen is D D = F DF 30 40 50m In ED an = DE/E = 30/40 = 3/4 = an (3/4) His displacemen from his
More informationNon-uniform circular motion *
OpenSax-CNX module: m14020 1 Non-uniform circular moion * Sunil Kumar Singh This work is produced by OpenSax-CNX and licensed under he Creaive Commons Aribuion License 2.0 Wha do we mean by non-uniform
More informationApplications of the Basic Equations Chapter 3. Paul A. Ullrich
Applicaions of he Basic Equaions Chaper 3 Paul A. Ullrich paullrich@ucdavis.edu Par 1: Naural Coordinaes Naural Coordinaes Quesion: Why do we need anoher coordinae sysem? Our goal is o simplify he equaions
More informationTopics covered in tutorial 01: 1. Review of definite integrals 2. Physical Application 3. Area between curves. 1. Review of definite integrals
MATH4 Calculus II (8 Spring) MATH 4 Tuorial Noes Tuorial Noes (Phyllis LIANG) IA: Phyllis LIANG Email: masliang@us.hk Homepage: hps://masliang.people.us.hk Office: Room 3 (Lif/Lif 3) Phone number: 3587453
More informationWelcome Back to Physics 215!
Welcome Back o Physics 215! (General Physics I) Thurs. Jan 19 h, 2017 Lecure01-2 1 Las ime: Syllabus Unis and dimensional analysis Today: Displacemen, velociy, acceleraion graphs Nex ime: More acceleraion
More information1.6. Slopes of Tangents and Instantaneous Rate of Change
1.6 Slopes of Tangens and Insananeous Rae of Change When you hi or kick a ball, he heigh, h, in meres, of he ball can be modelled by he equaion h() 4.9 2 v c. In his equaion, is he ime, in seconds; c represens
More informationPhysics for Scientists and Engineers. Chapter 2 Kinematics in One Dimension
Physics for Scieniss and Engineers Chaper Kinemaics in One Dimension Spring, 8 Ho Jung Paik Kinemaics Describes moion while ignoring he agens (forces) ha caused he moion For now, will consider moion in
More informationSPH3U1 Lesson 03 Kinematics
SPH3U1 Lesson 03 Kinemaics GRAPHICAL ANALYSIS LEARNING GOALS Sudens will Learn how o read values, find slopes and calculae areas on graphs. Learn wha hese values mean on boh posiion-ime and velociy-ime
More informationMath 116 Practice for Exam 2
Mah 6 Pracice for Exam Generaed Ocober 3, 7 Name: SOLUTIONS Insrucor: Secion Number:. This exam has 5 quesions. Noe ha he problems are no of equal difficuly, so you may wan o skip over and reurn o a problem
More informationPhysics 101 Fall 2006: Exam #1- PROBLEM #1
Physics 101 Fall 2006: Exam #1- PROBLEM #1 1. Problem 1. (+20 ps) (a) (+10 ps) i. +5 ps graph for x of he rain vs. ime. The graph needs o be parabolic and concave upward. ii. +3 ps graph for x of he person
More information1 1 + x 2 dx. tan 1 (2) = ] ] x 3. Solution: Recall that the given integral is improper because. x 3. 1 x 3. dx = lim dx.
. Use Simpson s rule wih n 4 o esimae an () +. Soluion: Since we are using 4 seps, 4 Thus we have [ ( ) f() + 4f + f() + 4f 3 [ + 4 4 6 5 + + 4 4 3 + ] 5 [ + 6 6 5 + + 6 3 + ]. 5. Our funcion is f() +.
More informationUnit 1 Test Review Physics Basics, Movement, and Vectors Chapters 1-3
A.P. Physics B Uni 1 Tes Reiew Physics Basics, Moemen, and Vecors Chapers 1-3 * In sudying for your es, make sure o sudy his reiew shee along wih your quizzes and homework assignmens. Muliple Choice Reiew:
More informationAP CALCULUS AB 2017 SCORING GUIDELINES
AP CALCULUS AB 17 SCORING GUIDELINES /CALCULUS BC 15 SCORING GUIDELINES Quesion (minues) v ( ) (meers per minue) 1 4 4 4 15 Johanna jogs along a sraigh pah. For 4, Johanna s velociy is given by a differeniable
More informationPHYSICS 149: Lecture 9
PHYSICS 149: Lecure 9 Chaper 3 3.2 Velociy and Acceleraion 3.3 Newon s Second Law of Moion 3.4 Applying Newon s Second Law 3.5 Relaive Velociy Lecure 9 Purdue Universiy, Physics 149 1 Velociy (m/s) The
More informationGuest Lecturer Friday! Symbolic reasoning. Symbolic reasoning. Practice Problem day A. 2 B. 3 C. 4 D. 8 E. 16 Q25. Will Armentrout.
Pracice Problem day Gues Lecurer Friday! Will Armenrou. He d welcome your feedback! Anonymously: wrie somehing and pu i in my mailbox a 111 Whie Hall. Email me: sarah.spolaor@mail.wvu.edu Symbolic reasoning
More informationChapter 3 Kinematics in Two Dimensions
Chaper 3 KINEMATICS IN TWO DIMENSIONS PREVIEW Two-dimensional moion includes objecs which are moing in wo direcions a he same ime, such as a projecile, which has boh horizonal and erical moion. These wo
More informationOne-Dimensional Kinematics
One-Dimensional Kinemaics One dimensional kinemaics refers o moion along a sraigh line. Een hough we lie in a 3-dimension world, moion can ofen be absraced o a single dimension. We can also describe moion
More information1. Kinematics I: Position and Velocity
1. Kinemaics I: Posiion and Velociy Inroducion The purpose of his eperimen is o undersand and describe moion. We describe he moion of an objec by specifying is posiion, velociy, and acceleraion. In his
More informationIn this chapter the model of free motion under gravity is extended to objects projected at an angle. When you have completed it, you should
Cambridge Universiy Press 978--36-60033-7 Cambridge Inernaional AS and A Level Mahemaics: Mechanics Coursebook Excerp More Informaion Chaper The moion of projeciles In his chaper he model of free moion
More informationCheck in: 1 If m = 2(x + 1) and n = find y when. b y = 2m n 2
7 Parameric equaions This chaer will show ou how o skech curves using heir arameric equaions conver arameric equaions o Caresian equaions find oins of inersecion of curves and lines using arameric equaions
More informationa 10.0 (m/s 2 ) 5.0 Name: Date: 1. The graph below describes the motion of a fly that starts out going right V(m/s)
Name: Dae: Kinemaics Review (Honors. Physics) Complee he following on a separae shee of paper o be urned in on he day of he es. ALL WORK MUST BE SHOWN TO RECEIVE CREDIT. 1. The graph below describes he
More informationDynamics. Option topic: Dynamics
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
More informationt = x v = 18.4m 44.4m/s =0.414 s.
1 Assuming he horizonal velociy of he ball is consan, he horizonal displacemen is x = v where x is he horizonal disance raveled, is he ime, and v is he (horizonal) velociy Convering v o meers per second,
More informationMath 2214 Solution Test 1A Spring 2016
Mah 14 Soluion Tes 1A Spring 016 sec Problem 1: Wha is he larges -inerval for which ( 4) = has a guaraneed + unique soluion for iniial value (-1) = 3 according o he Exisence Uniqueness Theorem? Soluion
More informationCourse II. Lesson 7 Applications to Physics. 7A Velocity and Acceleration of a Particle
Course II Lesson 7 Applicaions o Physics 7A Velociy and Acceleraion of a Paricle Moion in a Sraigh Line : Velociy O Aerage elociy Moion in he -ais + Δ + Δ 0 0 Δ Δ Insananeous elociy d d Δ Δ Δ 0 lim [ m/s
More information72 Calculus and Structures
72 Calculus and Srucures CHAPTER 5 DISTANCE AND ACCUMULATED CHANGE Calculus and Srucures 73 Copyrigh Chaper 5 DISTANCE AND ACCUMULATED CHANGE 5. DISTANCE a. Consan velociy Le s ake anoher look a Mary s
More informationChapter 1 Limits, Derivatives, Integrals, and Integrals
Chaper 1 Limis, Derivaives, Inegrals, and Inegrals Problem Se 1-1 1. a. 9 cm b. From o.1: average rae 6. 34 cm/s From o.01: average rae 7. 1 cm/s From o.001: average rae 7. 0 cm/s So he insananeous rae
More informationLecture 23 Damped Motion
Differenial Equaions (MTH40) Lecure Daped Moion In he previous lecure, we discussed he free haronic oion ha assues no rearding forces acing on he oving ass. However No rearding forces acing on he oving
More information- Graphing: Position Velocity. Acceleration
Tes Wednesday, Jan 31 in 101 Clark Hall a 7PM Main Ideas in Class Today - Graphing: Posiion Velociy v avg = x f f x i i a avg = v f f v i i Acceleraion Pracice ess & key online. Tes over maerial up o secion
More information