# Motion. Acceleration. Part 2: Constant Acceleration. October Lab Phyiscs. Ms. Levine 1. Acceleration. Acceleration. Units for Acceleration.

Save this PDF as:

Size: px
Start display at page:

Download "Motion. Acceleration. Part 2: Constant Acceleration. October Lab Phyiscs. Ms. Levine 1. Acceleration. Acceleration. Units for Acceleration." ## Transcription

1 Motion ccelertion Prt : Constnt ccelertion ccelertion ccelertion ccelertion is the rte of chnge of elocity. = - o t = Δ Δt ccelertion = = - o t chnge of elocity elpsed time ccelertion is ector, lthough in one-dimensionl motion we only need the sign. Since only constnt ccelertion will be considered in this course, there is no need to differentite between erge nd instntneous ccelertion. Units for ccelertion ccelertion is the rte of chnge of. Units for ccelertion You cn derie the units by substituting the correct units into the right hnd side of these equtions. = Δ Δt m/s s = m/s C D displcement distnce speed elocity Ms. Leine

2 The unit for elocity is: 3 The metric unit for ccelertion is: m m m/s m/s C m/s D ft/s C m/s D ft/s 4 horse gllops with constnt ccelertion of 3m/s. Which sttement below is true? 5 Your elocity chnges from 60 m/s to the right to 00 m/s to the right in 0 s; wht is your erge ccelertion? C D The horse's elocity doesn't chnge. The horse moes 3m eery second. The horse's elocity increses 3m eery second. The horse's elocity increses 3m/s eery second. 6 Your elocity chnges from 60 m/s to the right to 0 m/s to the right in 0 s; wht is your erge ccelertion? 7 Your elocity chnges from 50 m/s to the left to 0 m/s to the right in 5 s; wht is your erge ccelertion? Ms. Leine

3 8 Your elocity chnges from 90 m/s to the right to 0 m/s to the right in 5.0 s; wht is your erge ccelertion? Kinemtics Eqution Return to Tble of Contents Motion t Constnt ccelertion = Δ Δt but since "Δ" mens chnge 9 Strting from rest, you ccelerte t 4.0 m/s for 6.0s. Wht is your finl elocity? Δ = - o nd = - o t t = - o - o = t Δt = t - to if we lwys let to = 0, Δt = t Soling for "" = o + t This eqution tells us how n object's elocity chnges s function of time. 0 Strting from rest, you ccelerte t 8.0 m/s for 9.0s. Wht is your finl elocity? You he n initil elocity of 5.0 m/s. You then eperience n ccelertion of -.5 m/s for 4.0s; wht is your finl elocity? Ms. Leine 3

4 You he n initil elocity of -3.0 m/s. You then eperience n ccelertion of.5 m/s for 9.0s; wht is your finl elocity? 3 How much time does it tke to ccelerte from n initil elocity of 0m/s to finl elocity of 00m/s if your ccelertion is.5 m/s? 4 How much time does it tke to come to rest if your initil elocity is 5.0 m/s nd your ccelertion is -.0 m/s? 5 n object ccelertes t rte of 3 m/s for 6 s until it reches elocity of 0 m/s. Wht ws its initil elocity? 6 n object ccelertes t rte of.5 m/s for 4 s until it reches elocity of 0 m/s. Wht ws its initil elocity? Grphing Motion t Constnt ccelertion In physics there is nother pproch in ddition to lgebric which is clled grphicl nlysis. The formul = 0 + t cn be interpreted by the grph. We just need to recll our memory from mth clsses where we lredy sw similr formul y = m + b. From these two formuls we cn some nlogies: y (dependent rible of ), 0 b (intersection with erticl is), t (independent rible), m ( slope of the grph- the rtio between rise nd run Δy/Δ). Ms. Leine 4

5 Motion t Constnt ccelertion elow we cn find the geometric eplntion to the ccelertion =Δ/Δt. If slope is equl to: m = Δy/Δ Motion t Constnt ccelertion The grph on the right hs slope of Δ/Δt, which is equl to ccelertion. Therefore, the slope of elocity s. time grph is equl to ccelertion. Then consider grph with elocity on the y-is nd time on the -is. Wht is the slope for the grph on the right? (slope) y =Δy/Δ (slope of elocity s. time) =Δ/Δt 7 The elocity s function of time is presented by the grph. Wht is the ccelertion? 8 The elocity s function of time is presented by the grph. Find the ccelertion. Motion t Constnt ccelertion The ccelertion grph s function of time cn be used to find the elocity of moing object. When the ccelertion is constnt the elocity is chnging by the sme mount ech second. This cn be shown on the grph s stright horizontl line. In order to find the chnge in elocity for certin limit of time we need to clculte the re under the ccelertion line tht is limited by the time interl. 9 The following grph shows ccelertion s function of time of moing object. Wht is the chnge in elocity during first 0 seconds? Ms. Leine 5

6 Free Fll Free Fll: ccelertion Due to Grity ll unsupported objects fll towrds Erth with the sme ccelertion. We cll this ccelertion the "ccelertion due to grity" nd it is denoted by g. g = 9.8 m/s Keep in mind, LL objects ccelerte towrds the erth t the sme rte. g is constnt! Return to Tble of Contents Click here to wtch Glileo's fmous eperiment performed on the moon It Wht stops hppens momentrily. t the top? = 0 It stops momentrily. = 0 It slows down. Wht hppens when it (negtie ccelertion) goes up? It Wht speeds hppens up when it (negtie goes down? ccelertion) It slows down. (negtie ccelertion) It speeds up. (negtie ccelertion) n object is thrown upwrd with initil elocity, o It returns Wht hppens with its when it originl lnds? elocity. n object is thrown upwrd with initil elocity, o It returns with its originl elocity. On the wy up: 0 t = 3 s t = s t = s t = 0 s On the wy down: 0 t = 0 s t = s t = s t = 3 s For ny object thrown stright up into the ir, this is wht the elocity s. time grph looks like. (m/s) n object is thrown upwrd with initil elocity, o It stops momentrily. = 0 It returns with its originl elocity but in the opposite direction. Ms. Leine 6

7 Lb Phyiscs elocity is constnt C elocity is decresing D ccelertion is decresing rock flls off cliff nd hits the ground 5 seconds lter. Wht elocity did it hit the ground with? 3 bll is thrown down off bridge with elocity of 5 m/s. Wht is its elocity seconds lter? 5 rocket is fired stright up from the ground. It returns to the ground 0 seconds lter. Wht ws its lunch speed? ccelertion is constnt n rrow is fired into the ir nd it reches its highest point 3 seconds lter. Wht ws its elocity when it ws fired? 4 Ms. Leine n corn flls from n ok tree. You note tht it tkes.5 seconds to hit the ground. How fst ws it going when it hit the ground? bll is dropped from rest nd flls (do not consider ir resistnce). Which is true bout its motion? 0 7

8 6 n object ccelertes from rest, with constnt ccelertion of 8.4 m/s, wht will its elocity be fter s? Grphing Return to Tble of Contents Velocity s. Time Grphs Similrly, the sme pproch cn be used to crete elocity s. time grph. elocity ersus time grph differs by hing the elocity on the erticl is. (m/s) Strting t the position, 0 = 4 m, you trel t constnt elocity of + m/s for 6s. b. Drw the Position ersus Time for your trel during this time. Drg nd drop the dt points on the grph in order to construct the s t pttern! elocity ersus time grph shows describes n objects elocity, it's displcement, nd ccelertion Drw line of best fit to obsere the pttern. Strting t the position, 0 = 4 m, you trel t constnt elocity of + m/s for 6s. c. Drw the Velocity ersus Time grph for your trip. Drg nd drop the dt points on the grph in order to construct the s t pttern! (m/s) Strting t the position, 0 = 0 m, you trel t constnt elocity of -m/s for 6s. b. Drw the Position ersus Time for your trel during this time. Drg nd drop the dt points on the grph in order to construct the s t pttern! Drw line of best fit to obsere the pttern. Drw line of best fit to obsere the pttern Ms. Leine 8

9 nlyzing Position s Time Grphs nlyzing Position s Time Grphs Recll erlier in this unit tht slope ws used to describe motion. positie slope is positie elocity, negtie slope is negtie elocity, nd slope of zero mens zero elocity. The slope in position s. time grph is Δ/Δt, which is equl to elocity. Δt Δ = Δ/Δt positie slope > 0 negtie slope < 0 zero slope = 0 Therefore, slope is equl to elocity on position s. time grph. positie elocity mens moing in the positie direction, negtie elocity mens moing in the negtie direction, nd zero elocity mens not moing t ll. The position ersus time grph, below, describes the motion of three different crs moing long the -is. b. Clculte the elocity of ech of the crs. Position The elocity s time grph, below, describes the motion of n object moing long the -is. 4 (m/s) Describe in words wht is hppening to the speed during the following interls. ) 0s to s b) s to 3s c) 3s to 4 sec d) 4s to 5s e) 5s to 6s 73 The elocity s time grph, below, describes the motion of n object moing long the -is. 4 (m/s) Determine the erge speed during the following interls. ) 0s to s b) s to 3s c) 3s to 4 sec d) 4s to 5s e) 5s to 6s f) 4s to 6s Summry Kinemtics is the description of how objects moe with respect to defined reference frme. Displcement is the chnge in position of n object. erge elocity is the displcement diided by the time. erge ccelertion is the chnge in elocity diided by the time. Ms. Leine 9

### Motion. Part 2: Constant Acceleration. Acceleration. October Lab Physics. Ms. Levine 1. Acceleration. Acceleration. Units for Acceleration. Moion Accelerion Pr : Consn Accelerion Accelerion Accelerion Accelerion is he re of chnge of velociy. = v - vo = Δv Δ ccelerion = = v - vo chnge of velociy elpsed ime Accelerion is vecor, lhough in one-dimensionl

### PHYSICS 211 MIDTERM I 21 April 2004 PHYSICS MIDERM I April 004 Exm is closed book, closed notes. Use only your formul sheet. Write ll work nd nswers in exm booklets. he bcks of pges will not be grded unless you so request on the front of

### Linear Motion. Kinematics Quantities Liner Motion Physics 101 Eyres Kinemtics Quntities Time Instnt t Fundmentl Time Interl Defined Position x Fundmentl Displcement Defined Aerge Velocity g Defined Aerge Accelertion g Defined 1 Kinemtics

### Introduction to Mechanics Practice using the Kinematics Equations Introduction to Mechnics Prctice using the Kinemtics Equtions Ln Sheridn De Anz College Jn 24, 2018 Lst time finished deriing the kinemtics equtions some problem soling prctice Oeriew using kinemtics equtions

### 2/20/ :21 AM. Chapter 11. Kinematics of Particles. Mohammad Suliman Abuhaiba,Ph.D., P.E. //15 11:1 M Chpter 11 Kinemtics of Prticles 1 //15 11:1 M Introduction Mechnics Mechnics = science which describes nd predicts the conditions of rest or motion of bodies under the ction of forces It is

### PHYS Summer Professor Caillault Homework Solutions. Chapter 2 PHYS 1111 - Summer 2007 - Professor Cillult Homework Solutions Chpter 2 5. Picture the Problem: The runner moves long the ovl trck. Strtegy: The distnce is the totl length of trvel, nd the displcement

### (3.2.3) r x x x y y y. 2. Average Velocity and Instantaneous Velocity 2 1, (3.2.2) Lecture 3- Kinemtics in Two Dimensions Durin our preious discussions we he been tlkin bout objects moin lon the striht line. In relity, howeer, it rrely hppens when somethin moes lon the striht pth. For

### 2/2/ :36 AM. Chapter 11. Kinematics of Particles. Mohammad Suliman Abuhaiba,Ph.D., P.E. //16 1:36 AM Chpter 11 Kinemtics of Prticles 1 //16 1:36 AM First Em Wednesdy 4//16 3 //16 1:36 AM Introduction Mechnics Mechnics = science which describes nd predicts the conditions of rest or motion

### 1/31/ :33 PM. Chapter 11. Kinematics of Particles. Mohammad Suliman Abuhaiba,Ph.D., P.E. 1/31/18 1:33 PM Chpter 11 Kinemtics of Prticles 1 1/31/18 1:33 PM First Em Sturdy 1//18 3 1/31/18 1:33 PM Introduction Mechnics Mechnics = science which describes nd predicts conditions of rest or motion

### PhET INTRODUCTION TO MOTION IB PHYS-1 Nme: Period: Dte: Preprtion: DEVIL PHYSICS BADDEST CLASS ON CAMPUS PhET INTRODUCTION TO MOTION 1. Log on to computer using your student usernme nd pssword. 2. Go to https://phet.colordo.edu/en/simultion/moing-mn.

### Physics 207 Lecture 5 Phsics 07 Lecture 5 Agend Phsics 07, Lecture 5, Sept. 0 Chpter 4 Kinemtics in or 3 dimensions Independence of, nd/or z components Circulr motion Cured pths nd projectile motion Frmes of reference dil nd

### 4-6 ROTATIONAL MOTION Chpter 4 Motions in Spce 51 Reinforce the ide tht net force is needed for orbitl motion Content We discuss the trnsition from projectile motion to orbitl motion when bll is thrown horizontlly with eer

### _3-----"/- ~StudI_G u_id_e_-..,...-~~_~ e- / Dte Period Nme CHAPTR 3-----"/- StudIG uide-..,...- [-------------------- Accelerted Motion Vocbulry Review Write the term tht correctly completes the sttement. Use ech term once. ccelertion verge

### Prep Session Topic: Particle Motion Student Notes Prep Session Topic: Prticle Motion Number Line for AB Prticle motion nd similr problems re on the AP Clculus exms lmost every yer. The prticle my be prticle, person, cr, etc. The position,

### Distance And Velocity Unit #8 - The Integrl Some problems nd solutions selected or dpted from Hughes-Hllett Clculus. Distnce And Velocity. The grph below shows the velocity, v, of n object (in meters/sec). Estimte the totl

### Goals: Determine how to calculate the area described by a function. Define the definite integral. Explore the relationship between the definite Unit #8 : The Integrl Gols: Determine how to clculte the re described by function. Define the definite integrl. Eplore the reltionship between the definite integrl nd re. Eplore wys to estimte the definite

### Goals: Determine how to calculate the area described by a function. Define the definite integral. Explore the relationship between the definite Unit #8 : The Integrl Gols: Determine how to clculte the re described by function. Define the definite integrl. Eplore the reltionship between the definite integrl nd re. Eplore wys to estimte the definite

### Plane curvilinear motion is the motion of a particle along a curved path which lies in a single plane. Plne curiliner motion is the motion of prticle long cured pth which lies in single plne. Before the description of plne curiliner motion in n specific set of coordintes, we will use ector nlsis to describe

### Lecture 5. Today: Motion in many dimensions: Circular motion. Uniform Circular Motion Lecture 5 Physics 2A Olg Dudko UCSD Physics Tody: Motion in mny dimensions: Circulr motion. Newton s Lws of Motion. Lws tht nswer why questions bout motion. Forces. Inerti. Momentum. Uniform Circulr Motion

### Kinematics in Two-Dimensions Slide 1 / 92 Slide 2 / 92 Kinemtics in Two-imensions www.njctl.org Slide 3 / 92 How to Use this File ch topic is composed of brief direct instruction There re formtie ssessment questions fter eer topic

### Plane curvilinear motion is the motion of a particle along a curved path which lies in a single plane. Plne curiliner motion is the motion of prticle long cured pth which lies in single plne. Before the description of plne curiliner motion in n specific set of coordintes, we will use ector nlsis to describe

### Properties of Integrals, Indefinite Integrals. Goals: Definition of the Definite Integral Integral Calculations using Antiderivatives Block #6: Properties of Integrls, Indefinite Integrls Gols: Definition of the Definite Integrl Integrl Clcultions using Antiderivtives Properties of Integrls The Indefinite Integrl 1 Riemnn Sums - 1 Riemnn

### SECTION B Circular Motion SECTION B Circulr Motion 1. When person stnds on rotting merry-go-round, the frictionl force exerted on the person by the merry-go-round is (A) greter in mgnitude thn the frictionl force exerted on the

### 1. Find the derivative of the following functions. a) f(x) = 2 + 3x b) f(x) = (5 2x) 8 c) f(x) = e2x I. Dierentition. ) Rules. *product rule, quotient rule, chin rule MATH 34B FINAL REVIEW. Find the derivtive of the following functions. ) f(x) = 2 + 3x x 3 b) f(x) = (5 2x) 8 c) f(x) = e2x 4x 7 +x+2 d)

### In-Class Problems 2 and 3: Projectile Motion Solutions. In-Class Problem 2: Throwing a Stone Down a Hill MASSACHUSETTS INSTITUTE OF TECHNOLOGY Deprtment of Physics Physics 8T Fll Term 4 In-Clss Problems nd 3: Projectile Motion Solutions We would like ech group to pply the problem solving strtegy with the

### Student Session Topic: Particle Motion Student Session Topic: Prticle Motion Prticle motion nd similr problems re on the AP Clculus exms lmost every yer. The prticle my be prticle, person, cr, etc. The position, velocity or ccelertion my be

### What determines where a batted baseball lands? How do you describe MTIN IN TW R THREE DIMENIN 3 LEARNING GAL studing this chpter, ou will lern:?if cr is going round cure t constnt speed, is it ccelerting? If so, in wht direction is it ccelerting? Wht determines where

### SECTION B Circular Motion SECTION B Circulr Motion 1. When person stnds on rotting merry-go-round, the frictionl force exerted on the person by the merry-go-round is (A) greter in mgnitude thn the frictionl force exerted on the

### JURONG JUNIOR COLLEGE JURONG JUNIOR COLLEGE 2010 JC1 H1 8866 hysics utoril : Dynmics Lerning Outcomes Sub-topic utoril Questions Newton's lws of motion 1 1 st Lw, b, e f 2 nd Lw, including drwing FBDs nd solving problems by

### FULL MECHANICS SOLUTION FULL MECHANICS SOLUION. m 3 3 3 f For long the tngentil direction m 3g cos 3 sin 3 f N m 3g sin 3 cos3 from soling 3. ( N 4) ( N 8) N gsin 3. = ut + t = ut g sin cos t u t = gsin cos = 4 5 5 = s] 3 4 o

### DESCRIBING MOTION: KINEMATICS IN ONE DIMENSION DESCRIBING MOTION: KINEMATICS IN ONE DIMENSION Responses to Questions. A cr speedometer mesures only speed. It does not give ny informtion bout the direction, so it does not mesure velocity.. If the velocity

### ( dg. ) 2 dt. + dt. dt j + dh. + dt. r(t) dt. Comparing this equation with the one listed above for the length of see that Arc Length of Curves in Three Dimensionl Spce If the vector function r(t) f(t) i + g(t) j + h(t) k trces out the curve C s t vries, we cn mesure distnces long C using formul nerly identicl to one tht we

### 20 MATHEMATICS POLYNOMIALS 0 MATHEMATICS POLYNOMIALS.1 Introduction In Clss IX, you hve studied polynomils in one vrible nd their degrees. Recll tht if p(x) is polynomil in x, the highest power of x in p(x) is clled the degree of

### 7.1 Integral as Net Change and 7.2 Areas in the Plane Calculus 7.1 Integrl s Net Chnge nd 7. Ares in the Plne Clculus 7.1 INTEGRAL AS NET CHANGE Notecrds from 7.1: Displcement vs Totl Distnce, Integrl s Net Chnge We hve lredy seen how the position of n oject cn e

### Operations with Polynomials 38 Chpter P Prerequisites P.4 Opertions with Polynomils Wht you should lern: How to identify the leding coefficients nd degrees of polynomils How to dd nd subtrct polynomils How to multiply polynomils

### ARITHMETIC OPERATIONS. The real numbers have the following properties: a b c ab ac REVIEW OF ALGEBRA Here we review the bsic rules nd procedures of lgebr tht you need to know in order to be successful in clculus. ARITHMETIC OPERATIONS The rel numbers hve the following properties: b b

### Ch AP Problems Ch. 7.-7. AP Prolems. Willy nd his friends decided to rce ech other one fternoon. Willy volunteered to rce first. His position is descried y the function f(t). Joe, his friend from school, rced ginst him,

### Calculus AB Section I Part A A CALCULATOR MAY NOT BE USED ON THIS PART OF THE EXAMINATION lculus Section I Prt LULTOR MY NOT US ON THIS PRT OF TH XMINTION In this test: Unless otherwise specified, the domin of function f is ssumed to e the set of ll rel numers for which f () is rel numer..

### Chapters Five Notes SN AA U1C5 Chpters Five Notes SN AA U1C5 Nme Period Section 5-: Fctoring Qudrtic Epressions When you took lger, you lerned tht the first thing involved in fctoring is to mke sure to fctor out ny numers or vriles

### Definition of Continuity: The function f(x) is continuous at x = a if f(a) exists and lim Mth 9 Course Summry/Study Guide Fll, 2005  Limits Definition of Limit: We sy tht L is the limit of f(x) s x pproches if f(x) gets closer nd closer to L s x gets closer nd closer to. We write lim f(x)

### Indefinite Integral. Chapter Integration - reverse of differentiation Chpter Indefinite Integrl Most of the mthemticl opertions hve inverse opertions. The inverse opertion of differentition is clled integrtion. For exmple, describing process t the given moment knowing the

### THREE-DIMENSIONAL KINEMATICS OF RIGID BODIES THREE-DIMENSIONAL KINEMATICS OF RIGID BODIES 1. TRANSLATION Figure shows rigid body trnslting in three-dimensionl spce. Any two points in the body, such s A nd B, will move long prllel stright lines if

### MA123, Chapter 10: Formulas for integrals: integrals, antiderivatives, and the Fundamental Theorem of Calculus (pp. MA123, Chpter 1: Formuls for integrls: integrls, ntiderivtives, nd the Fundmentl Theorem of Clculus (pp. 27-233, Gootmn) Chpter Gols: Assignments: Understnd the sttement of the Fundmentl Theorem of Clculus.

### Interpreting Integrals and the Fundamental Theorem Interpreting Integrls nd the Fundmentl Theorem Tody, we go further in interpreting the mening of the definite integrl. Using Units to Aid Interprettion We lredy know tht if f(t) is the rte of chnge of

### DO NOT OPEN THIS EXAM BOOKLET UNTIL INSTRUCTED TO DO SO. PHYSICS 1 Fll 017 EXAM 1: October 3rd, 017 8:15pm 10:15pm Nme (printed): Recittion Instructor: Section #: DO NOT OPEN THIS EXAM BOOKLET UNTIL INSTRUCTED TO DO SO. This exm contins 5 multiple-choice questions,

### Review of basic calculus Review of bsic clculus This brief review reclls some of the most importnt concepts, definitions, nd theorems from bsic clculus. It is not intended to tech bsic clculus from scrtch. If ny of the items below

### Section 14.3 Arc Length and Curvature Section 4.3 Arc Length nd Curvture Clculus on Curves in Spce In this section, we ly the foundtions for describing the movement of n object in spce.. Vector Function Bsics In Clc, formul for rc length in

### 3. Vectors. Vectors: quantities which indicate both magnitude and direction. Examples: displacemement, velocity, acceleration Rutgers University Deprtment of Physics & Astronomy 01:750:271 Honors Physics I Lecture 3 Pge 1 of 57 3. Vectors Vectors: quntities which indicte both mgnitude nd direction. Exmples: displcemement, velocity,

### Kinematics equations, some numbers Kinemtics equtions, some numbers Kinemtics equtions: x = x 0 + v 0 t + 1 2 t2, v = v 0 + t. They describe motion with constnt ccelertion. Brking exmple, = 1m/s. Initil: x 0 = 10m, v 0 = 10m/s. x(t=1s)

### AP Physics 1. Slide 1 / 71. Slide 2 / 71. Slide 3 / 71. Circular Motion. Topics of Uniform Circular Motion (UCM) Slide 1 / 71 Slide 2 / 71 P Physics 1 irculr Motion 2015-12-02 www.njctl.org Topics of Uniform irculr Motion (UM) Slide 3 / 71 Kinemtics of UM lick on the topic to go to tht section Period, Frequency,

### Practice Final. Name: Problem 1. Show all of your work, label your answers clearly, and do not use a calculator. Nme: MATH 2250 Clculus Eric Perkerson Dte: December 11, 2015 Prctice Finl Show ll of your work, lbel your nswers clerly, nd do not use clcultor. Problem 1 Compute the following limits, showing pproprite

### Infinite Geometric Series Infinite Geometric Series Finite Geometric Series ( finite SUM) Let 0 < r < 1, nd let n be positive integer. Consider the finite sum It turns out there is simple lgebric expression tht is equivlent to

### Mathematics of Motion II Projectiles Chmp+ Fll 2001 Dn Stump 1 Mthemtics of Motion II Projectiles Tble of vribles t time v velocity, v 0 initil velocity ccelertion D distnce x position coordinte, x 0 initil position x horizontl coordinte

### Section 5.1 #7, 10, 16, 21, 25; Section 5.2 #8, 9, 15, 20, 27, 30; Section 5.3 #4, 6, 9, 13, 16, 28, 31; Section 5.4 #7, 18, 21, 23, 25, 29, 40 Mth B Prof. Audrey Terrs HW # Solutions by Alex Eustis Due Tuesdy, Oct. 9 Section 5. #7,, 6,, 5; Section 5. #8, 9, 5,, 7, 3; Section 5.3 #4, 6, 9, 3, 6, 8, 3; Section 5.4 #7, 8,, 3, 5, 9, 4 5..7 Since

### Math 1102: Calculus I (Math/Sci majors) MWF 3pm, Fulton Hall 230 Homework 2 solutions Mth 1102: Clculus I (Mth/Sci mjors) MWF 3pm, Fulton Hll 230 Homework 2 solutions Plese write netly, nd show ll work. Cution: An nswer with no work is wrong! Do the following problems from Chpter III: 6,

### a) mass inversely proportional b) force directly proportional 1. Wht produces ccelertion? A orce 2. Wht is the reltionship between ccelertion nd ) mss inersely proportionl b) orce directly proportionl 3. I you he orce o riction, 30N, on n object, how much orce is

### HIGHER SCHOOL CERTIFICATE EXAMINATION MATHEMATICS 3 UNIT (ADDITIONAL) AND 3/4 UNIT (COMMON) Time allowed Two hours (Plus 5 minutes reading time) HIGHER SCHOOL CERTIFICATE EXAMINATION 998 MATHEMATICS 3 UNIT (ADDITIONAL) AND 3/4 UNIT (COMMON) Time llowed Two hours (Plus 5 minutes reding time) DIRECTIONS TO CANDIDATES Attempt ALL questions ALL questions

### The momentum of a body of constant mass m moving with velocity u is, by definition, equal to the product of mass and velocity, that is Newtons Lws 1 Newton s Lws There re three lws which ber Newton s nme nd they re the fundmentls lws upon which the study of dynmics is bsed. The lws re set of sttements tht we believe to be true in most

### AB Calculus Review Sheet AB Clculus Review Sheet Legend: A Preclculus, B Limits, C Differentil Clculus, D Applictions of Differentil Clculus, E Integrl Clculus, F Applictions of Integrl Clculus, G Prticle Motion nd Rtes This is

### Section 6.1 INTRO to LAPLACE TRANSFORMS Section 6. INTRO to LAPLACE TRANSFORMS Key terms: Improper Integrl; diverge, converge A A f(t)dt lim f(t)dt Piecewise Continuous Function; jump discontinuity Function of Exponentil Order Lplce Trnsform

### y = f(x) This means that there must be a point, c, where the Figure 1 Clculus Investigtion A Men Slope TEACHER S Prt 1: Understnding the Men Vlue Theorem The Men Vlue Theorem for differentition sttes tht if f() is defined nd continuous over the intervl [, ], nd differentile

### Chapter 5 1. = on [ 1, 2] 1. Let gx ( ) e x. . The derivative of g is g ( x) e 1 Chpter 5. Let g ( e. on [, ]. The derivtive of g is g ( e ( Write the slope intercept form of the eqution of the tngent line to the grph of g t. (b Determine the -coordinte of ech criticl vlue of g. Show

### ( ) where f ( x ) is a. AB Calculus Exam Review Sheet. A. Precalculus Type problems. Find the zeros of f ( x). AB Clculus Exm Review Sheet A. Preclculus Type prolems A1 Find the zeros of f ( x). This is wht you think of doing A2 A3 Find the intersection of f ( x) nd g( x). Show tht f ( x) is even. A4 Show tht f

### Sample Problems for the Final of Math 121, Fall, 2005 Smple Problems for the Finl of Mth, Fll, 5 The following is collection of vrious types of smple problems covering sections.8,.,.5, nd.8 6.5 of the text which constitute only prt of the common Mth Finl.

### Identify graphs of linear inequalities on a number line. COMPETENCY 1.0 KNOWLEDGE OF ALGEBRA SKILL 1.1 Identify grphs of liner inequlities on number line. - When grphing first-degree eqution, solve for the vrible. The grph of this solution will be single point

### ( ) as a fraction. Determine location of the highest AB Clculus Exm Review Sheet - Solutions A. Preclculus Type prolems A1 A2 A3 A4 A5 A6 A7 This is wht you think of doing Find the zeros of f ( x). Set function equl to 0. Fctor or use qudrtic eqution if

### Improper Integrals. Type I Improper Integrals How do we evaluate an integral such as Improper Integrls Two different types of integrls cn qulify s improper. The first type of improper integrl (which we will refer to s Type I) involves evluting n integrl over n infinite region. In the grph

### Answers to the Conceptual Questions Chpter 3 Explining Motion 41 Physics on Your Own If the clss is not too lrge, tke them into freight elevtor to perform this exercise. This simple exercise is importnt if you re going to cover inertil forces

### Week 10: Line Integrals Week 10: Line Integrls Introduction In this finl week we return to prmetrised curves nd consider integrtion long such curves. We lredy sw this in Week 2 when we integrted long curve to find its length.

### 13.4 Work done by Constant Forces 13.4 Work done by Constnt Forces We will begin our discussion of the concept of work by nlyzing the motion of n object in one dimension cted on by constnt forces. Let s consider the following exmple: push

### Test , 8.2, 8.4 (density only), 8.5 (work only), 9.1, 9.2 and 9.3 related test 1 material and material from prior classes Test 2 8., 8.2, 8.4 (density only), 8.5 (work only), 9., 9.2 nd 9.3 relted test mteril nd mteril from prior clsses Locl to Globl Perspectives Anlyze smll pieces to understnd the big picture. Exmples: numericl Lesson 1: Qudrtic Equtions Qudrtic Eqution: The qudrtic eqution in form is. In this section, we will review 4 methods of qudrtic equtions, nd when it is most to use ech method. 1. 3.. 4. Method 1: Fctoring

### Section 6: Area, Volume, and Average Value Chpter The Integrl Applied Clculus Section 6: Are, Volume, nd Averge Vlue Are We hve lredy used integrls to find the re etween the grph of function nd the horizontl xis. Integrls cn lso e used to find

### ( ) where f ( x ) is a. AB Calculus Exam Review Sheet. A. Precalculus Type problems. Find the zeros of f ( x). AB Clculus Exm Review Sheet A. Preclculus Type prolems A1 Find the zeros of f ( x). This is wht you think of doing A2 A3 Find the intersection of f ( x) nd g( x). Show tht f ( x) is even. A4 Show tht f

### Chapter 4 Contravariance, Covariance, and Spacetime Diagrams Chpter 4 Contrvrince, Covrince, nd Spcetime Digrms 4. The Components of Vector in Skewed Coordintes We hve seen in Chpter 3; figure 3.9, tht in order to show inertil motion tht is consistent with the Lorentz

### Mathematics Extension 1 04 Bored of Studies Tril Emintions Mthemtics Etension Written by Crrotsticks & Trebl. Generl Instructions Totl Mrks 70 Reding time 5 minutes. Working time hours. Write using blck or blue pen. Blck pen

### HIGHER SCHOOL CERTIFICATE EXAMINATION MATHEMATICS 4 UNIT (ADDITIONAL) Time allowed Three hours (Plus 5 minutes reading time) HIGHER SCHOOL CERTIFICATE EXAMINATION 999 MATHEMATICS UNIT (ADDITIONAL) Time llowed Three hours (Plus 5 minutes reding time) DIRECTIONS TO CANDIDATES Attempt ALL questions ALL questions re of equl vlue

### BIFURCATIONS IN ONE-DIMENSIONAL DISCRETE SYSTEMS BIFRCATIONS IN ONE-DIMENSIONAL DISCRETE SYSTEMS FRANCESCA AICARDI In this lesson we will study the simplest dynmicl systems. We will see, however, tht even in this cse the scenrio of different possible

### Physics 207 Lecture 7 Phsics 07 Lecture 7 Agend: Phsics 07, Lecture 7, Sept. 6 hpter 6: Motion in (nd 3) dimensions, Dnmics II Recll instntneous velocit nd ccelertion hpter 6 (Dnmics II) Motion in two (or three dimensions)

### MATH 144: Business Calculus Final Review MATH 144: Business Clculus Finl Review 1 Skills 1. Clculte severl limits. 2. Find verticl nd horizontl symptotes for given rtionl function. 3. Clculte derivtive by definition. 4. Clculte severl derivtives

### 4.4 Areas, Integrals and Antiderivatives . res, integrls nd ntiderivtives 333. Ares, Integrls nd Antiderivtives This section explores properties of functions defined s res nd exmines some connections mong res, integrls nd ntiderivtives. In order

### FORM FIVE ADDITIONAL MATHEMATIC NOTE. ar 3 = (1) ar 5 = = (2) (2) (1) a = T 8 = 81 FORM FIVE ADDITIONAL MATHEMATIC NOTE CHAPTER : PROGRESSION Arithmetic Progression T n = + (n ) d S n = n [ + (n )d] = n [ + Tn ] S = T = T = S S Emple : The th term of n A.P. is 86 nd the sum of the first

### Math 1431 Section M TH 4:00 PM 6:00 PM Susan Wheeler Office Hours: Wed 6:00 7:00 PM Online ***NOTE LABS ARE MON AND WED Mth 43 Section 4839 M TH 4: PM 6: PM Susn Wheeler swheeler@mth.uh.edu Office Hours: Wed 6: 7: PM Online ***NOTE LABS ARE MON AND WED t :3 PM to 3: pm ONLINE Approimting the re under curve given the type

### Big idea in Calculus: approximation Big ide in Clculus: pproximtion Derivtive: f (x) = df dx f f(x +h) f(x) =, x h rte of chnge is pproximtely the rtio of chnges in the function vlue nd in the vrible in very short time Liner pproximtion:

### Physics 105 Exam 2 10/31/2008 Name A Physics 105 Exm 2 10/31/2008 Nme_ A As student t NJIT I will conduct myself in professionl mnner nd will comply with the proisions of the NJIT Acdemic Honor Code. I lso understnd tht I must subscribe to

### Improper Integrals. The First Fundamental Theorem of Calculus, as we ve discussed in class, goes as follows: Improper Integrls The First Fundmentl Theorem of Clculus, s we ve discussed in clss, goes s follows: If f is continuous on the intervl [, ] nd F is function for which F t = ft, then ftdt = F F. An integrl

### Stuff You Need to Know From Calculus Stuff You Need to Know From Clculus For the first time in the semester, the stuff we re doing is finlly going to look like clculus (with vector slnt, of course). This mens tht in order to succeed, you

### Adding and Subtracting Rational Expressions 6.4 Adding nd Subtrcting Rtionl Epressions Essentil Question How cn you determine the domin of the sum or difference of two rtionl epressions? You cn dd nd subtrct rtionl epressions in much the sme wy

### Math Lecture 23 Mth 8 - Lecture 3 Dyln Zwick Fll 3 In our lst lecture we delt with solutions to the system: x = Ax where A is n n n mtrix with n distinct eigenvlues. As promised, tody we will del with the question of

### W. We shall do so one by one, starting with I 1, and we shall do it greedily, trying Vitli covers 1 Definition. A Vitli cover of set E R is set V of closed intervls with positive length so tht, for every δ > 0 nd every x E, there is some I V with λ(i ) < δ nd x I. 2 Lemm (Vitli covering)

### Section 7.2 Velocity. Solution Section 7.2 Velocity In the previous chpter, we showed tht velocity is vector becuse it hd both mgnitude (speed) nd direction. In this section, we will demonstrte how two velocities cn be combined to determine

### 3.1 Exponential Functions and Their Graphs . Eponentil Functions nd Their Grphs Sllbus Objective: 9. The student will sketch the grph of eponentil, logistic, or logrithmic function. 9. The student will evlute eponentil or logrithmic epressions.

### Section 4: Integration ECO4112F 2011 Reding: Ching Chpter Section : Integrtion ECOF Note: These notes do not fully cover the mteril in Ching, ut re ment to supplement your reding in Ching. Thus fr the optimistion you hve covered hs een sttic

### Equations, expressions and formulae Get strted 2 Equtions, epressions nd formule This unit will help you to work with equtions, epressions nd formule. AO1 Fluency check 1 Work out 2 b 2 c 7 2 d 7 2 2 Simplify by collecting like terms. b

### 5.1 How do we Measure Distance Traveled given Velocity? Student Notes . How do we Mesure Distnce Trveled given Velocity? Student Notes EX ) The tle contins velocities of moving cr in ft/sec for time t in seconds: time (sec) 3 velocity (ft/sec) 3 A) Lel the x-xis & y-xis

### SOLUTIONS FOR ADMISSIONS TEST IN MATHEMATICS, COMPUTER SCIENCE AND JOINT SCHOOLS WEDNESDAY 5 NOVEMBER 2014 SOLUTIONS FOR ADMISSIONS TEST IN MATHEMATICS, COMPUTER SCIENCE AND JOINT SCHOOLS WEDNESDAY 5 NOVEMBER 014 Mrk Scheme: Ech prt of Question 1 is worth four mrks which re wrded solely for the correct nswer. Answers to Een Numbered Problems Chpter 4. 0.96 s 4..7 cm to 7 m 6. 8.5 m 8. 36 K 0. () -7 5.0 0 W (b) -5 5.0 0 W. 3.0 db 4. () 6. () -4.3 0 W m (b) 8. db - 7.96 0 W m (b) 09 db (c).8 m IA 8. () I (b) Answers to selected problems from Essentil Physics, Chpter 3 1. FBD 1 is the correct free-body dirm in ll five cses. As fr s forces re concerned, t rest nd constnt velocity situtions re equivlent. 3. ()