1.6. Slopes of Tangents and Instantaneous Rate of Change

Size: px
Start display at page:

Download "1.6. Slopes of Tangents and Instantaneous Rate of Change"

Transcription

1 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() v c. In his equaion, is he ime, in seconds; c represens he iniial heigh of he ball, in meres; and v represens he iniial verical velociy of he ball, in meres per second. In Secion 1.5, you learned how o calculae he average rae of change of he heigh of he ball, or he average velociy of he ball, over an inerval of ime. Wha if you wan o know he velociy of he ball 1 s afer i was hi? To deermine how fas he ball is ravelling a a specific ime, ha is, o deermine he insananeous rae of change of he heigh of he ball, oher mehods are required. In his secion, you will invesigae he connecion beween average and insananeous raes of change and learn how he slope of a angen is used o deermine an insananeous rae of change. Invesigae Wha is he connecion beween he slopes of secans, he slope of a angen, and he insananeous rae of change? 1. A golf ball lying on he grass is hi so ha is iniial verical velociy is 25 m/s. The heigh, h, in meres, of he ball afer seconds can be modelled by he quadraic funcion h() Copy and complee he able for h(). Inerval h 1 2 h(2) h(1) [ 4.9(2) 2 25(2)] [ 4.9(1) 2 25(1)] Average rae _ of change, h 2. Explain how he ime inervals in he firs column are changing. 1.6 Slopes of Tangens and Insananeous Rae of Change MHR 65

2 3. Reflec Describe how he average rae of change values in he fourh column are changing in relaion o he ime inervals. 4. a) Graph he funcion. b) Reflec On he graph, skech he secans ha correspond o he average raes of change in he able. How do he secans illusrae he relaionship you described in sep 3? 5. a) Skech a angen line, a line ha ouches he graph only a he poin ha corresponds o 1 s. Describe how he values of he average rae of change in your able could be used o esimae he slope of he angen a 1 s. b) Reflec Wha does he slope of he angen represen in his siuaion? Explain. c) Wha would happen if you approached 1 from below, ha is, using.9,.99,.999, and so on? Explain. 6. Reflec Explain he relaionship beween a) he slopes of secans near a poin on a curve and he slope of he angen a ha poin b) he slope of a angen a a poin on a curve and he insananeous rae of change Relaionship Beween he Slope of Secans and he Slope of a Tangen As a poin Q becomes very close o a angen poin P, he slope of he secan line becomes closer o (approaches) he slope of he angen line. Ofen an arrow is used o denoe he word approaches. So, he above saemen may be wrien as follows: As Q P, he slope of secan PQ he slope of he angen a P. Thus, he average rae of change beween P and Q becomes closer o he value of he insananeous rae of change a P. When he graph of a funcion is given, an approximae value for he insananeous rae of change a a poin on he curve may be deermined using eiher of wo mehods: by calculaing he slope of a line passing hrough he given poin and anoher poin on he curve ha is very close o he given poin, or by skeching he angen line a he poin and calculaing he average rae of change over an inerval beween he angen poin and anoher poin on he angen line. 66 MHR Advanced Funcions Chaper 1

3 Example 1 Esimae an Insananeous Rae of Change From a Graph The graph shows he approximae disance ravelled by a parachuis in he firs 5 s afer jumping ou of a helicoper. How fas was he parachuis ravelling 2 s afer jumping ou of he helicoper? Disance (m) Disance Travelled by a Parachuis d Soluion The parachuis s velociy a 2 s corresponds o he insananeous rae of change a 2 s. Calculae an approximae value for he insananeous rae of change. Le P be he poin on he graph a 2 s. Mehod 1: Use he Slope of a Secan Deermine he slope of a secan passing hrough P and anoher poin on he curve ha is very close o P. Selec he poin Q(3, 45), which is close o P and easy o read from he graph. Disance Travelled by a Parachuis d 12 The slope of he secan PQ is m PQ _ Δd Δ Using a secan, he parachuis s velociy afer falling for 2 s is approximaely 25 m/s. Disance (m) P 2, 2 Q 3, Slopes of Tangens and Insananeous Rae of Change MHR 67

4 Mehod 2: Use Two Poins on an Approximae Tangen Line Anoher way o esimae he slope of a angen a a poin from a graph is o skech an approximae angen line hrough ha poin and hen selec a second poin on ha line. Selec he poin (3, 4). Label i S. The slope of PS is m PS _ Δd Δ Using poins on he approximae angen line, he parachuis s velociy afer falling for 2 s is approximaely 2 m/s. Disance (m) Disance Travelled by a Parachuis d P 2, 2 S 3, Noe ha boh mehods give an approximae value of he slope of he angen because he calculaions depend on eiher he accuracy of he coordinaes of he seleced poin on he curve he accuracy of he approximae angen line, as well as he accuracy of he coordinaes of he second poin on his approximae angen line In boh mehods, he closer he seleced poin is o P he beer is he approximaion of he slope of he angen line. Example 2 Esimae an Insananeous Rae of Change From a Table of Values In he able, he disance of he parachuis in Example 1 is recorded a.5-s inervals. Esimae he parachuis s velociy a 2 s. Soluion To esimae an insananeous rae of change from a able, calculae he average rae of change over a shor inerval by using poins in he able ha are closes o he angen poin. The able shows ha a 2 s he parachuis has ravelled a disance of 2 m. This corresponds o he angen poin (2, 2) on he graph shown in Example 1. A poin ha is close o (2, 2) is he poin (2.5, 31.25). (s) d (m) MHR Advanced Funcions Chaper 1

5 The average rae of change beween (2, 2) and (2.5, 31.25) is Average rae of change _ Δd Δ The parachuis s velociy a 2 s is approximaely 22.5 m/s. Noice ha we also could have chosen he poin (1.5, 11.25). Example 3 Esimae an Insananeous Rae of Change From an Equaion The funcion d() 5 2 may be used o model he approximae disance ravelled by he parachuis in Examples 1 and 2. Use he equaion o esimae he velociy of he parachuis afer 2 s. Soluion Deermine he average rae of change over shorer and shorer inervals. Inerval d 2 3 d(3) d(2) 5(3) 2 5(2) d(2.5) d(2) 5(2.5) 2 5(2) d(2.1) d(2) 5(2.1) 2 5(2) d(2.1) d(2) 5(2.1) 2 5(2) d(2.1) d(2) 5(2.1) 2 5(2) As he ime inervals decrease, he average rae of change (which corresponds o he slope of a secan line), becomes closer o (or approaches) 2. Thus, he velociy of he parachuis afer 2 s is approximaely 2 m/s. You also could have used values of less han 2. d _ _ _ _ Slopes of Tangens and Insananeous Rae of Change MHR 69

6 << >> KEY CONCEPTS An insananeous rae of change corresponds o he slope of a angen o a poin on a curve. An approximae value for an insananeous rae of change a a poin may be deermined using a graph, eiher by esimaing he slope of a secan passing hrough ha poin or by skeching he angen and esimaing he slope beween he angen poin and a second poin on he approximae angen line a able of values, by esimaing he slope beween he poin and a nearby poin in he able an equaion, by esimaing he slope using a very shor inerval beween he angen poin and a second poin found using he equaion Communicae Your Undersanding C1 C2 C3 a) Does he speedomeer of a car measure average speed or insananeous speed? Explain. b) Describe siuaions in which he insananeous speed and he average speed would be he same. Sae if each siuaion represens average rae of change or insananeous rae of change. Give reasons for your answer. a) A 3 p.m., he plane was ravelling a 85 km/h. b) The average speed ravelled by he rain during he 1-h rip was 13 km/h. c) The fire was spreading a a rae of 2 ha/h. d) 5 s afer an anisepic spray is applied, he baceria populaion is decreasing a a rae of 6 baceria per second. e) He los 4 kg per monh over a 5-monh period. f) Afer being heaed for 2 min, he waer emperaure was rising a 1 C/min. Which mehod from Examples 1, 2, and 3 is he mos accurae for finding he insananeous rae of change? Explain. 7 MHR Advanced Funcions Chaper 1

7 A Pracise For help wih quesion 1, refer o Example Consider he graph shown y a) Sae he coordinaes of he angen poin. b) Sae he coordinaes of anoher poin on he angen line. c) Use he poins you found in pars a) and b) o deermine he slope of he angen line. d) Wha does he value you found in par c) represen? x 2. a) A each of he indicaed poins on he graph, is he insananeous rae of change posiive, negaive, or zero? Explain. Heigh (m) h Heigh of a Tennis Ball A B b) Esimae he insananeous rae of change a poins A and C. c) Inerpre he values in par b) for he siuaion represened by he graph. C B Connec and Apply For help wih quesion 3, refer o Example A firework is sho ino he air such ha is heigh, h, in meres, afer seconds can be modelled by he funcion h() a) Copy and complee he able. 4. Use wo differen mehods o esimae he slope of he angen a he poin indicaed on he graph. Disance Travelled by a Bungee Jumper h Inerval h h _ b) Use he able o esimae he velociy of he firework afer 3 s. Heigh (m) Slopes of Tangens and Insananeous Rae of Change MHR 71

8 5. The daa show he percen of households ha play games over he Inerne. Year % Households Source: Saisics Canada, Canada a a Glance 26, page 9, Household Inerne use a home by Inerne aciviy. a) Deermine he average rae of change, in percen, of households ha played games over he Inerne from 1999 o 23. b) Esimae he insananeous rae of change in percen of households ha played games over he Inerne i) in 2 ii) in 22 c) Compare he values found in pars a) and b). Explain any similariies and differences. 6. The able shows he consumer price index (CPI) every 5 years from 1955 o 25. Year CPI Source: Saisics Canada, CANSIM Table a) Deermine he average rae of change in he CPI from 1955 o 25. b) Esimae he insananeous rae of change in he CPI for i) 1965 ii) 1985 iii) 2 c) Compare he values found in pars a) and b). Explain any similariies and differences. 7. A soccer ball is kicked Reasoning and Proving Represening Selecing Tools ino he air such ha is heigh, h, in meres, Problem Solving afer seconds can be Connecing Reflecing Communicaing modelled by he funcion h() a) Deermine he average rae of change of he heigh of he ball from 1 s o 2 s. b) Esimae he insananeous rae of change of he heigh of he ball afer 1 s. c) Skech he curve and he angen. d) Inerpre he average rae of change and he insananeous rae of change for his siuaion. 8. On Earh, he heigh, h, in meres, of a free-falling objec afer seconds can be modelled by he funcion h() k, while on Venus, he heigh can be modelled by h() k, where and k is he heigh, in meres, from which he objec is dropped. Suppose a rock is dropped from a heigh of 6 m on each plane. a) Deermine he average rae of change of he heigh of he rock in he firs 3 s afer i is dropped. b) Esimae he insananeous rae of change of he heigh of he rock 3 s afer i is dropped. c) Inerpre he values in pars a) and b) for his siuaion. 9. A manufacurer esimaes ha he cos, C, in dollars, of producing x MP3 players can be modelled by C(x). 15x 3 1x. a) Deermine he average rae of change of he cos of producing from 1 o 2 MP3 players. b) Esimae he insananeous rae of change of he cos of producing 2 MP3 players. c) Inerpre he values found in pars a) and b) for his siuaion. d) Does he cos ever decrease? Explain. CONNECTIONS The CPI measures he average price of consumer goods and services purchased by households. The percen change in he CPI is one measure of inflaion. 72 MHR Advanced Funcions Chaper 1

9 1. Suppose he revenue, Reasoning and Proving Represening Selecing Tools R, in dollars, from he sales of x MP3 players Problem Solving described in quesion 9 Connecing Reflecing Communicaing is given by R(x) x( x 2 ). a) Deermine he average rae of change of revenue from selling from 1 o 2 MP3 players. b) Esimae he insananeous rae of change of revenue from he sale of 2 MP3 players. c) Inerpre he values found in pars a) and b) for his siuaion. d) The profi, P, in dollars, from he sale of x MP3 players is given by he profi funcion P(x) R(x) C(x). Deermine an equaion for he profi funcion. e) Deermine he average rae of change of profi on he sale of from 1 o 2 MP3 players. f) Esimae he insananeous rae of change of profi on he sale of 2 MP3 players. g) Inerpre he values found in pars e) and f) for his siuaion. 11. A worldwide disribuor of baskeballs deermines ha he yearly profi, P, in housands of dollars, earned on he sale of x housand baskeballs can be modelled by he funcion P(x).9x x 2 9x, where x [, 25]. a) Deermine he average rae of change of profi earned on he sale of from i) 2 o 6 baskeballs ii) 16 o 2 baskeballs b) Wha conclusions can you make by comparing he values in par b)? Explain your reasoning. c) Esimae he insananeous rae of change of profi earned on he sale of i) 5 baskeballs ii) 18 baskeballs d) Wha conclusions can you make from he values found in pars c)? Explain your reasoning. e) Use Technology Graph he funcion. How does he graph suppor your answers in pars a) and c)? C Exend and Challenge 12. The populaion, P, of a small own afer years can be modelled by he funcion P() , where represens he beginning of his year. a) Wrie an expression for he average rae of change of he populaion from 8 o 8 h. b) Use he expression in par a) o deermine he average rae of change of he populaion when i) h 2 ii) h 4 iii) h 5 c) Wha do he values you found in par b) represen? d) Describe how he expression in par a) could be used o esimae he insananeous rae of change of he populaion afer 8 years. e) Use he mehod you described in par d) o esimae he insananeous rae of change of he populaion afer 8 years. 13. Mah Cones Deermine he exac value of Mah Cones Find m given 3 m m Slopes of Tangens and Insananeous Rae of Change MHR 73

KINEMATICS IN ONE DIMENSION

KINEMATICS 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 information

5.2. The Natural Logarithm. Solution

5.2. The Natural Logarithm. Solution 5.2 The Naural Logarihm The number e is an irraional number, similar in naure o π. Is non-erminaing, non-repeaing value is e 2.718 281 828 59. Like π, e also occurs frequenly in naural phenomena. In fac,

More information

PROBLEMS 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 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 information

Math 115 Final Exam December 14, 2017

Math 115 Final Exam December 14, 2017 On my honor, as a suden, I have neiher given nor received unauhorized aid on his academic work. Your Iniials Only: Iniials: Do no wrie in his area Mah 5 Final Exam December, 07 Your U-M ID # (no uniqname):

More information

Physics 20 Lesson 5 Graphical Analysis Acceleration

Physics 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 information

AP Calculus BC Chapter 10 Part 1 AP Exam Problems

AP Calculus BC Chapter 10 Part 1 AP Exam Problems 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

More information

Displacement ( x) x x x

Displacement ( 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 information

5.1 - Logarithms and Their Properties

5.1 - Logarithms and Their Properties Chaper 5 Logarihmic Funcions 5.1 - Logarihms and Their Properies Suppose ha a populaion grows according o he formula P 10, where P is he colony size a ime, in hours. When will he populaion be 2500? We

More information

2.1: What is physics? Ch02: Motion along a straight line. 2.2: Motion. 2.3: Position, Displacement, Distance

2.1: 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 information

Math 111 Midterm I, Lecture A, version 1 -- Solutions January 30 th, 2007

Math 111 Midterm I, Lecture A, version 1 -- Solutions January 30 th, 2007 NAME: Suden ID #: QUIZ SECTION: Mah 111 Miderm I, Lecure A, version 1 -- Soluions January 30 h, 2007 Problem 1 4 Problem 2 6 Problem 3 20 Problem 4 20 Toal: 50 You are allowed o use a calculaor, a ruler,

More information

4.1 - Logarithms and Their Properties

4.1 - Logarithms and Their Properties Chaper 4 Logarihmic Funcions 4.1 - Logarihms and Their Properies Wha is a Logarihm? We define he common logarihm funcion, simply he log funcion, wrien log 10 x log x, as follows: If x is a posiive number,

More information

IB Physics Kinematics Worksheet

IB 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 information

3.6 Derivatives as Rates of Change

3.6 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 information

Lecture 2-1 Kinematics in One Dimension Displacement, Velocity and Acceleration Everything in the world is moving. Nothing stays still.

Lecture 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 information

Lab #2: Kinematics in 1-Dimension

Lab #2: Kinematics in 1-Dimension Reading Assignmen: Chaper 2, Secions 2-1 hrough 2-8 Lab #2: Kinemaics in 1-Dimension Inroducion: The sudy of moion is broken ino wo main areas of sudy kinemaics and dynamics. Kinemaics is he descripion

More information

SPH3U1 Lesson 03 Kinematics

SPH3U1 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 information

1. Kinematics I: Position and Velocity

1. 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 information

Solutionbank Edexcel AS and A Level Modular Mathematics

Solutionbank Edexcel AS and A Level Modular Mathematics Page of 4 Soluionbank Edexcel AS and A Level Modular Mahemaics Exercise A, Quesion Quesion: Skech he graphs of (a) y = e x + (b) y = 4e x (c) y = e x 3 (d) y = 4 e x (e) y = 6 + 0e x (f) y = 00e x + 0

More information

72 Calculus and Structures

72 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 information

!!"#"$%&#'()!"#&'(*%)+,&',-)./0)1-*23)

!!#$%&#'()!#&'(*%)+,&',-)./0)1-*23) "#"$%&#'()"#&'(*%)+,&',-)./)1-*) #$%&'()*+,&',-.%,/)*+,-&1*#$)()5*6$+$%*,7&*-'-&1*(,-&*6&,7.$%$+*&%'(*8$&',-,%'-&1*(,-&*6&,79*(&,%: ;..,*&1$&$.$%&'()*1$$.,'&',-9*(&,%)?%*,('&5

More information

The average rate of change between two points on a function is d t

The average rate of change between two points on a function is d t SM Dae: Secion: Objecive: The average rae of change beween wo poins on a funcion is d. For example, if he funcion ( ) represens he disance in miles ha a car has raveled afer hours, hen finding he slope

More information

Motion along a Straight Line

Motion 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 information

SPH3U: Projectiles. Recorder: Manager: Speaker:

SPH3U: Projectiles. Recorder: Manager: Speaker: SPH3U: Projeciles Now i s ime o use our new skills o analyze he moion of a golf ball ha was ossed hrough he air. Le s find ou wha is special abou he moion of a projecile. Recorder: Manager: Speaker: 0

More information

In this chapter the model of free motion under gravity is extended to objects projected at an angle. When you have completed it, you should

In 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 information

Welcome Back to Physics 215!

Welcome 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 information

Practicing Problem Solving and Graphing

Practicing 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 information

6. Solve by applying the quadratic formula.

6. Solve by applying the quadratic formula. Dae: Chaper 7 Prerequisie Skills BLM 7.. Apply he Eponen Laws. Simplify. Idenify he eponen law ha you used. a) ( c) ( c) ( c) ( y)( y ) c) ( m)( n ). Simplify. Idenify he eponen law ha you used. 8 w a)

More information

4.5 Constant Acceleration

4.5 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 information

MEI Mechanics 1 General motion. Section 1: Using calculus

MEI 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 information

EXERCISES FOR SECTION 1.5

EXERCISES FOR SECTION 1.5 1.5 Exisence and Uniqueness of Soluions 43 20. 1 v c 21. 1 v c 1 2 4 6 8 10 1 2 2 4 6 8 10 Graph of approximae soluion obained using Euler s mehod wih = 0.1. Graph of approximae soluion obained using Euler

More information

WEEK-3 Recitation PHYS 131. of the projectile s velocity remains constant throughout the motion, since the acceleration a x

WEEK-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 information

Answers to 1 Homework

Answers 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 information

Parametrics and Vectors (BC Only)

Parametrics 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

University Physics with Modern Physics 14th Edition Young TEST BANK

University 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 information

Kinematics Vocabulary. Kinematics and One Dimensional Motion. Position. Coordinate System in One Dimension. Kinema means movement 8.

Kinematics 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 information

Math 116 Second Midterm March 21, 2016

Math 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 information

Chapter 15: Phenomena. Chapter 15 Chemical Kinetics. Reaction Rates. Reaction Rates R P. Reaction Rates. Rate Laws

Chapter 15: Phenomena. Chapter 15 Chemical Kinetics. Reaction Rates. Reaction Rates R P. Reaction Rates. Rate Laws Chaper 5: Phenomena Phenomena: The reacion (aq) + B(aq) C(aq) was sudied a wo differen emperaures (98 K and 35 K). For each emperaure he reacion was sared by puing differen concenraions of he 3 species

More information

1998 Calculus AB Scoring Guidelines

1998 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 information

Dynamics. Option topic: Dynamics

Dynamics. 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 information

Physics 180A Fall 2008 Test points. Provide the best answer to the following questions and problems. Watch your sig figs.

Physics 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 information

Simulation-Solving Dynamic Models ABE 5646 Week 2, Spring 2010

Simulation-Solving Dynamic Models ABE 5646 Week 2, Spring 2010 Simulaion-Solving Dynamic Models ABE 5646 Week 2, Spring 2010 Week Descripion Reading Maerial 2 Compuer Simulaion of Dynamic Models Finie Difference, coninuous saes, discree ime Simple Mehods Euler Trapezoid

More information

Ground Rules. PC1221 Fundamentals of Physics I. Kinematics. Position. Lectures 3 and 4 Motion in One Dimension. A/Prof Tay Seng Chuan

Ground 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 information

Chapter 7: Solving Trig Equations

Chapter 7: Solving Trig Equations Haberman MTH Secion I: The Trigonomeric Funcions Chaper 7: Solving Trig Equaions Le s sar by solving a couple of equaions ha involve he sine funcion EXAMPLE a: Solve he equaion sin( ) The inverse funcions

More information

and v y . The changes occur, respectively, because of the acceleration components a x and a y

and 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 information

Chapters 6 & 7: Trigonometric Functions of Angles and Real Numbers. Divide both Sides by 180

Chapters 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 information

The equation to any straight line can be expressed in the form:

The equation to any straight line can be expressed in the form: Sring Graphs Par 1 Answers 1 TI-Nspire Invesigaion Suden min Aims Deermine a series of equaions of sraigh lines o form a paern similar o ha formed by he cables on he Jerusalem Chords Bridge. Deermine he

More information

Physics 221 Fall 2008 Homework #2 Solutions Ch. 2 Due Tues, Sept 9, 2008

Physics 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 information

1. VELOCITY AND ACCELERATION

1. 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 information

Physics 101 Fall 2006: Exam #1- PROBLEM #1

Physics 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 information

Math 2214 Solution Test 1B Fall 2017

Math 2214 Solution Test 1B Fall 2017 Mah 14 Soluion Tes 1B Fall 017 Problem 1: A ank has a capaci for 500 gallons and conains 0 gallons of waer wih lbs of sal iniiall. A soluion conaining of 8 lbsgal of sal is pumped ino he ank a 10 galsmin.

More information

Physics for Scientists and Engineers I

Physics 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 information

AP CALCULUS AB/CALCULUS BC 2016 SCORING GUIDELINES. Question 1. 1 : estimate = = 120 liters/hr

AP 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 information

not to be republished NCERT MATHEMATICAL MODELLING Appendix 2 A.2.1 Introduction A.2.2 Why Mathematical Modelling?

not to be republished NCERT MATHEMATICAL MODELLING Appendix 2 A.2.1 Introduction A.2.2 Why Mathematical Modelling? 256 MATHEMATICS A.2.1 Inroducion In class XI, we have learn abou mahemaical modelling as an aemp o sudy some par (or form) of some real-life problems in mahemaical erms, i.e., he conversion of a physical

More information

AP CALCULUS AB 2003 SCORING GUIDELINES (Form B)

AP 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 information

Math 116 Practice for Exam 2

Math 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 information

Math 2214 Solution Test 1A Spring 2016

Math 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 information

MA Study Guide #1

MA Study Guide #1 MA 66 Su Guide #1 (1) Special Tpes of Firs Order Equaions I. Firs Order Linear Equaion (FOL): + p() = g() Soluion : = 1 µ() [ ] µ()g() + C, where µ() = e p() II. Separable Equaion (SEP): dx = h(x) g()

More information

Math 105 Second Midterm March 16, 2017

Math 105 Second Midterm March 16, 2017 Mah 105 Second Miderm March 16, 2017 UMID: Insrucor: Iniials: Secion: 1. Do no open his exam unil you are old o do so. 2. Do no wrie your name anywhere on his exam. 3. This exam has 9 pages including his

More information

PHYSICS 149: Lecture 9

PHYSICS 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 information

EQUATIONS REVIEW I Lesson Notes. Example 1. Example 2. Equations Review. 5 2 x = 1 6. Simple Equations

EQUATIONS REVIEW I Lesson Notes. Example 1. Example 2.  Equations Review. 5 2 x = 1 6. Simple Equations Equaions Review x + 3 = 6 EQUATIONS REVIEW I Example Simple Equaions a) a - 7 = b) m - 9 = -7 c) 6r = 4 d) 7 = -9x Example Simple Equaions a) 6p + = 4 b) 4 = 3k + 6 c) 9 + k = + 3k d) 8-3n = -8n + 3 EQUATIONS

More information

ACCUMULATION. Section 7.5 Calculus AP/Dual, Revised /26/2018 7:27 PM 7.5A: Accumulation 1

ACCUMULATION. 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 information

Topic 1: Linear motion and forces

Topic 1: Linear motion and forces TOPIC 1 Topic 1: Linear moion and forces 1.1 Moion under consan acceleraion Science undersanding 1. Linear moion wih consan elociy is described in erms of relaionships beween measureable scalar and ecor

More information

Phys1112: DC and RC circuits

Phys1112: DC and RC circuits Name: Group Members: Dae: TA s Name: Phys1112: DC and RC circuis Objecives: 1. To undersand curren and volage characerisics of a DC RC discharging circui. 2. To undersand he effec of he RC ime consan.

More information

Of all of the intellectual hurdles which the human mind has confronted and has overcome in the last fifteen hundred years, the one which seems to me

Of all of the intellectual hurdles which the human mind has confronted and has overcome in the last fifteen hundred years, the one which seems to me Of all of he inellecual hurdles which he human mind has confroned and has overcome in he las fifeen hundred years, he one which seems o me o have been he mos amazing in characer and he mos supendous in

More information

F.LE.A.4: Exponential Growth

F.LE.A.4: Exponential Growth Regens Exam Quesions F.LE.A.4: Exponenial Growh www.jmap.org Name: F.LE.A.4: Exponenial Growh 1 A populaion of rabbis doubles every days according o he formula P = 10(2), where P is he populaion of rabbis

More information

x i v x t a dx dt t x

x 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 information

15. Vector Valued Functions

15. 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 information

Q.1 Define work and its unit?

Q.1 Define work and its unit? CHP # 6 ORK AND ENERGY Q.1 Define work and is uni? A. ORK I can be define as when we applied a force on a body and he body covers a disance in he direcion of force, hen we say ha work is done. I is a scalar

More information

AP Calculus BC - Parametric equations and vectors Chapter 9- AP Exam Problems solutions

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 information

Math 10C: Relations and Functions PRACTICE EXAM

Math 10C: Relations and Functions PRACTICE EXAM Mah C: Relaions and Funcions PRACTICE EXAM. Cailin rides her bike o school every day. The able of values shows her disance from home as ime passes. An equaion ha describes he daa is: ime (minues) disance

More information

AP CALCULUS AB 2003 SCORING GUIDELINES (Form B)

AP 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 information

2.7. Some common engineering functions. Introduction. Prerequisites. Learning Outcomes

2.7. Some common engineering functions. Introduction. Prerequisites. Learning Outcomes Some common engineering funcions 2.7 Inroducion This secion provides a caalogue of some common funcions ofen used in Science and Engineering. These include polynomials, raional funcions, he modulus funcion

More information

Q2.1 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 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 information

Unit 1 Test Review Physics Basics, Movement, and Vectors Chapters 1-3

Unit 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 information

Math 333 Problem Set #2 Solution 14 February 2003

Math 333 Problem Set #2 Solution 14 February 2003 Mah 333 Problem Se #2 Soluion 14 February 2003 A1. Solve he iniial value problem dy dx = x2 + e 3x ; 2y 4 y(0) = 1. Soluion: This is separable; we wrie 2y 4 dy = x 2 + e x dx and inegrae o ge The iniial

More information

Solution: b All the terms must have the dimension of acceleration. We see that, indeed, each term has the units of acceleration

Solution: 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 information

Chapter 2. Motion in One-Dimension I

Chapter 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 information

2) Of the following questions, which ones are thermodynamic, rather than kinetic concepts?

2) Of the following questions, which ones are thermodynamic, rather than kinetic concepts? AP Chemisry Tes (Chaper 12) Muliple Choice (40%) 1) Which of he following is a kineic quaniy? A) Enhalpy B) Inernal Energy C) Gibb s free energy D) Enropy E) Rae of reacion 2) Of he following quesions,

More information

, where P is the number of bears at time t in years. dt (a) If 0 100, lim Pt. Is the solution curve increasing or decreasing?

, where P is the number of bears at time t in years. dt (a) If 0 100, lim Pt. Is the solution curve increasing or decreasing? CALCULUS BC WORKSHEET 1 ON LOGISTIC GROWTH Work he following on noebook paper. Use your calculaor on 4(b) and 4(c) only. 1. Suppose he populaion of bears in a naional park grows according o he logisic

More information

Kinematics Motion in 1 Dimension and Graphs

Kinematics 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 information

3 at MAC 1140 TEST 3 NOTES. 5.1 and 5.2. Exponential Functions. Form I: P is the y-intercept. (0, P) When a > 1: a = growth factor = 1 + growth rate

3 at MAC 1140 TEST 3 NOTES. 5.1 and 5.2. Exponential Functions. Form I: P is the y-intercept. (0, P) When a > 1: a = growth factor = 1 + growth rate 1 5.1 and 5. Eponenial Funcions Form I: Y Pa, a 1, a > 0 P is he y-inercep. (0, P) When a > 1: a = growh facor = 1 + growh rae The equaion can be wrien as The larger a is, he seeper he graph is. Y P( 1

More information

Check in: 1 If m = 2(x + 1) and n = find y when. b y = 2m n 2

Check 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 information

4.6 One Dimensional Kinematics and Integration

4.6 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 information

Final Exam. Tuesday, December hours

Final Exam. Tuesday, December hours San Francisco Sae Universiy Michael Bar ECON 560 Fall 03 Final Exam Tuesday, December 7 hours Name: Insrucions. This is closed book, closed noes exam.. No calculaors of any kind are allowed. 3. Show all

More information

4 Two movies, together, run for 3 hours. One movie runs 12 minutes longer than the other. How long is each movie?

4 Two movies, together, run for 3 hours. One movie runs 12 minutes longer than the other. How long is each movie? Algebra Problems 1 A number is increased by 12. The resul is 28. A) Wrie an equaion o find he number. B) Solve your equaion o find he number. 2 A number is decreased by 6. The resul is 15. A) Wrie an equaion

More information

Exponential and Logarithmic Functions

Exponential and Logarithmic Functions Chaper 5 Eponenial and Logarihmic Funcions Chaper 5 Prerequisie Skills Chaper 5 Prerequisie Skills Quesion 1 Page 50 a) b) c) Answers may vary. For eample: The equaion of he inverse is y = log since log

More information

PHYS 1401 General Physics I Test 3 Review Questions

PHYS 1401 General Physics I Test 3 Review Questions PHYS 1401 General Physics I Tes 3 Review Quesions Ch. 7 1. A 6500 kg railroad car moving a 4.0 m/s couples wih a second 7500 kg car iniially a res. a) Skech before and afer picures of he siuaion. b) Wha

More information

Some Basic Information about M-S-D Systems

Some Basic Information about M-S-D Systems Some Basic Informaion abou M-S-D Sysems 1 Inroducion We wan o give some summary of he facs concerning unforced (homogeneous) and forced (non-homogeneous) models for linear oscillaors governed by second-order,

More information

a 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)

a 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 information

Constant Acceleration

Constant Acceleration Objecive Consan Acceleraion To deermine he acceleraion of objecs moving along a sraigh line wih consan acceleraion. Inroducion The posiion y of a paricle moving along a sraigh line wih a consan acceleraion

More information

Chapter 7 Response of First-order RL and RC Circuits

Chapter 7 Response of First-order RL and RC Circuits Chaper 7 Response of Firs-order RL and RC Circuis 7.- The Naural Response of RL and RC Circuis 7.3 The Sep Response of RL and RC Circuis 7.4 A General Soluion for Sep and Naural Responses 7.5 Sequenial

More information

Note: For all questions, answer (E) NOTA means none of the above answers is correct.

Note: For all questions, answer (E) NOTA means none of the above answers is correct. Thea Logarihms & Eponens 0 ΜΑΘ Naional Convenion Noe: For all quesions, answer means none of he above answers is correc.. The elemen C 4 has a half life of 70 ears. There is grams of C 4 in a paricular

More information

CHEAPEST PMT ONLINE TEST SERIES AIIMS/NEET TOPPER PREPARE QUESTIONS

CHEAPEST PMT ONLINE TEST SERIES AIIMS/NEET TOPPER PREPARE QUESTIONS CHEAPEST PMT ONLINE TEST SERIES AIIMS/NEET TOPPER PREPARE QUESTIONS For more deails see las page or conac @aimaiims.in Physics Mock Tes Paper AIIMS/NEET 07 Physics 06 Saurday Augus 0 Uni es : Moion in

More information

13.1 Accelerating Objects

13.1 Accelerating Objects 13.1 Acceleraing Objec A you learned in Chaper 12, when you are ravelling a a conan peed in a raigh line, you have uniform moion. However, mo objec do no ravel a conan peed in a raigh line o hey do no

More information

Section A: Forces and Motion

Section A: Forces and Motion I is very useful o be able o make predicions abou he way moving objecs behave. In his chaper you will learn abou some equaions of moion ha can be used o calculae he speed and acceleraion of objecs, and

More information

RC, RL and RLC circuits

RC, RL and RLC circuits Name Dae Time o Complee h m Parner Course/ Secion / Grade RC, RL and RLC circuis Inroducion In his experimen we will invesigae he behavior of circuis conaining combinaions of resisors, capaciors, and inducors.

More information

PHYSICS 220 Lecture 02 Motion, Forces, and Newton s Laws Textbook Sections

PHYSICS 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 information

Chapter 3 Kinematics in Two Dimensions

Chapter 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 information

Decimal moved after first digit = 4.6 x Decimal moves five places left SCIENTIFIC > POSITIONAL. a) g) 5.31 x b) 0.

Decimal moved after first digit = 4.6 x Decimal moves five places left SCIENTIFIC > POSITIONAL. a) g) 5.31 x b) 0. PHYSICS 20 UNIT 1 SCIENCE MATH WORKSHEET NAME: A. Sandard Noaion Very large and very small numbers are easily wrien using scienific (or sandard) noaion, raher han decimal (or posiional) noaion. Sandard

More information