Physics Motion Math. (Read objectives on screen.)


 Kelly Shelton
 3 years ago
 Views:
Transcription
1 Physics Motion Math (Read objectives on screen.) Welcome back. When we ended the last program, your teacher gave you some motion graphs to interpret. For each section, you were to describe the motion of the object, as specifically as you could. So let s see how you did. (graph on screen) Let s look at graph one. The very first thing you must do when you are asked to interpret a motion graph is to look at the variable on the yaxis. Time will always be on the xaxis, so the other variable is very important. This graph has position d on the yaxis. And you know that as position changes, it s called delta d or displacement. The slope of each line represents displacement over time or velocity. In section ab, as time progresses, the object s position increases, so it s moving forward. But the slope changes and gets steeper, so the velocity is increasing. This is a form of accelerated motion. You can t say more than this. You can t know if the motion is uniformly accelerated without a velocity, time graph. In section bc, the object is still moving forward, but this time the graph is a straight line, which means that the slope or velocity is constant. So the motion is uniform. In cd, the object is still moving forward, but the graph is a curve, meaning that motion is accelerated. This time, the slope decreases as time progresses. If you have trouble seeing this, imagine trying to climb the hill. It gets easier as you go along on this one. Since the slope, or velocity, decreases, the motion is decelerated. You can also call it negative acceleration. The graph, de, is a flat line. This shows that the object does not change position and has no velocity. It is not in motion. From ef, as time progresses, the object s position decreases. Since the graph is a straight line, the motion is uniform. But it is going backwards. You could also say that velocity is negative. How did you do? Remember: NO SURFING. Don t skim the surface without thinking. Two graphs can look the same, but describe completely different types of motion. Watch that yaxis and decide what the slope represents before you look at each section. (graph on screen) Now we ll switch gears and look at velocity versus time graphs. Since rise over run gives us change in velocity over time, the slope of each graph will tell us what s happening to acceleration. 1
2 Let s start with section gh. Velocity increases as time progresses, so the motion is accelerated. And since the slope is a straight line, acceleration is constant, so the best description is uniformly accelerated motion. In hj, the line is flat. This does not mean that the object is motionless, because it has velocity. It just means the velocity is constant. This is uniform motion. Surfers might think that the object is moving backwards in ij. But they aren t looking at the yaxis. The only thing that is decreasing is velocity. Our object is slowing down. And since the graph is a straight line, acceleration is constant. It s also negative. So the best answer is uniformly decelerated motion. The graph for section jk looks very much like section gh. The motion is uniformly accelerated. The only difference is that the slope or acceleration is less. If you noticed that, you re not a surfer. Good for you! If you had trouble with some of the motion graphs, your local teacher may want to give you a chance to practice more before you take a quiz on interpreting these graphs. Your local teacher will stop the tape now, and we ll see you after the quiz. Good luck! I hope you all aced the quiz. It s important for you to understand what velocity and acceleration mean and how they differ. To test your understanding of these terms, how about a physics challenge? (Physics Challenge on screen) You are looking at sketches of three hills. If a ball is allowed to roll down each hill, what will happen to its velocity? Will it increase, decrease, or remain constant? And what will happen to the ball s acceleration? Will it increase, decrease, or remain constant? If you can answer these questions, you really understand the difference between velocity and acceleration. Sketch the three hills and then your teacher will pause the tape to give you time to make your decisions. Remember that your choices for each blank are increase, decrease, or remain constant. Let s see how you did, starting with the first hill. (graphic I on screen) Most of you probably figured that the ball would pick up speed as it rolls down the hill, and you re right. Velocity increases. Acceleration is a little harder. Because the hill gets steeper as the ball rolls down, it will pick up speed faster and faster. So acceleration also increases. 2
3 (table on screen) To help you understand, let s look at some sample data. You can see that velocity increases, but look at this. During the first second, the ball picked up one meter per second. During second two, it picked up two meters per second, and during the third second it picked up three meters per second. So acceleration is increasing. (graphic II on screen) On hill number two, the ball is still rolling downhill, so common sense tells us that it will pick up speed, as in hill one. Velocity increases. But this hill becomes less steep as the ball rolls down. So the rate of increase in velocity will decrease. So acceleration decreases. (table on screen) How can acceleration decrease while velocity increases? Look at sample data again. You can see that velocity increases as time passes. But look at the rate in increase. During the first second, velocity increases from zero to five meters per second, for an acceleration of five meters per second per second.. During second two, it increases from five to nine meters per second, for an acceleration of four meters per second squared. You can see that the rate of increase keeps going down as time passes. That s why acceleration decreases as velocity increases. (graphic III on screen) Finally, look at the third hill. If you said velocity increases and acceleration remains constant, you are correct. What do you think the sample data will look like? (table on screen) Let s see. As time passes the velocity increases, but notice that during each second, the ball s velocity increases by two meters per second. That makes the acceleration a constant two meters per second squared. So how did you do on the challenge question? Were you a physics surfer? Or did you dive deeper and try to understand the difference between velocity and acceleration? Our goal this year is to turn all of you into divers. And speaking of divers, it s time to dive into some linear motion math. Solving motion problems will require some basic algebra. Since the prerequisite for taking this course is two years of algebra, this review may insult some of you. But we re going to do it anyway, because it s so important. So here it goes. (cartoon on screen) In physics, we use algebraic equations. But instead of x s and y s, we use symbols which represent specific quantities or variables like distance, velocity, acceleration, and time. It is very important that you learn to rearrange equations to solve for one variable. We do this before any numbers are inserted in the equation so that when you re ready to calculate, you punch one set of buttons on the calculator and have the answer. This eliminates errors and lets you round only once. 3
4 Let s start by having you rearrange some equations. Your teacher will pause the tape and give you a set of equations, with instructions to solve for one variable. Rearrange the equation.. Don t worry about subscripts. They are part of a symbol. When you ve finished, your teacher will check your answers. If you need some work on this skill, we ve put some examples and practice problems on the Peachstar website. Your teacher will show you how to find it. Local Teachers: Turn off tape and give students problem set number one from facilitator's guide. Number one was easy, so let s start with number two. It looks easy, but tricks many students who try to rearrange in their heads. When the unknown variable is on the bottom of a fraction, you must get it to the top. That means you start this problem by multiplying both sides by t. That gives you the equation, v times t = d. Now you can get rid of the v by dividing both sides by v. That leaves the unknown variable, t isolated. t = d divided by v. Now let s try number three. We need to isolate the a. Remember that you want to work your way into the unknown. That means getting rid of outside obstacles, like the term, v delta t before you work on the ½ and the t 2. If you started on this term before the v delta t, that would be like trying to remove the handcuffs from the princess before you got rid of the guards and dogs. It just won t work. So let s look at the v delta t. Since it is added to the term containing our unknown, we remove it by subtracting it from both sides. Now our equation reads, d v delta t = ½ a delta t 2. How do we get rid of the ½?. Since ½ times a is the same as a divided by two, all we have to do is multiply both sides by two. Now all that is left is to divide both sides by t squared, and we have our answer. Number four is the most complex. Don t worry about these subscripts. m1 is just the mass of object one. Since there is not a term added to or subtracted from the term containing our unknown, the first thing we have to do is to get the term containing the d off the bottom of the fraction. That term is d squared. Can we get rid of the square sign yet? No, that s the last thing we ll do. It s like pulling the evil twin away from the princess. First, we have to get both twins out of the basement. We do that by multiplying both sides of the equation by d squared. Now we have to get rid of the F. Since d squared is multiplied by F, we ll divide both sides by F. Now how do we get rid of the square sign? What is the opposite of squaring a number? You ve got it taking the square root. So we take the square root of both sides of the equation and we have our answer. (New set of equations on screen) If you had any trouble with this basic algebra skill, you ll need to practice. Local teachers, this is problem set 1A. (screen fades to black) 4
5 All right, we re finally ready to do some motion math. We ll start with the simplest kind of motion, which is uniform motion. There is only one basic equation that is used for uniform motion. And that s the definition of velocity. You already know that velocity equals displacement over time. Now some books use the delta signs for displacement and time, but some don t. So we re not going to be picky about it. Using the equation is the important thing. (green chalkboard on screen) Since most motion problems in physics are word problems, you ll need to know some commonly used phrases. Look at these questions. For which variable do you think each is asking you to solve? I ll wait while you write them down. OK. Here are the answers. When a problem asks how far, you are solving for the variable, delta d. How fast means that you want to calculate velocity, v. Do you know why we didn t use the delta sign here? It s because in uniform motion problems, velocity does not change. Got it? And finally, how long means time, not distance. For example, an appropriate answer to the question, How long have you known her? would be six years, not six miles. Now let s try two uniform motion problems together. Let s go over the first problem. Here it is. Anita Break and Earl E. Byrd drive 48 km east. Anita drives at a constant 88 km/h while Earl drives at a constant 92 km/h. How long will Earl have to wait on Anita at their destination? (interrupts) Before we begin, I must point out that the writer of these problems often uses strange names like I need a break and Early Bird. If you think these names are weird, just wait. I just want everyone to know that I had nothing to do with naming the characters in our problems, although I must admit that I sometimes enjoy trying to figure them out. In this problem we want to solve for time for both Anita and Earl. The displacement for each is 48 kilometers. Anita s velocity is 88 kilometers per hour and Earl s is 92 kilometers per hour. Remember that how long means length of time. We want to solve for "delta t" for each. The equation we use for uniform motion is v equals d over t. Since we re solving for t, we need to rearrange the equation. We ve already talked about this. We multiply both sides by t to get our unknown off the bottom, then divide both sides by v. So t equals d over v. Now all we have to do is plug in the numbers and chug out the answers. 5
6 Anita s time is 0.55 hours and Earl s is 0.52 hours. So Earl E. Byrd is early has to wait on Anita for 0.03 hours, which, by the way, is less than two minutes. Now for significant digits here, you ll have to stop at the hundredths because you re subtracting. Now you need to practice. When you finish, your teacher will go over them with you. Local teachers, turn off the tape and give students problem set number 2 from the facilitator s guide. Now we re ready to tackle problems involving accelerated motion. The only type of accelerated motion we will deal with is uniformly accelerated, because the value of a is a constant number. There are three basic acceleration equations that you will use. We re going to show you how these are derived from formulas you already know. Put your pencils down and watch. This is for your information only, not for your notes. (equation on screen) To derive acceleration equation number one, we start with the definition of acceleration, which is change in velocity divided by change in time. Let me give you an example before we introduce the next equation. If I asked you to calculate your change in weight for the month, how would you do it? Did you say take your weight at the end of the month and subtract your weight at the beginning? Well, in physics we use the terms, final and initial, and we use the subscripts f and i after the v s to get this equation for change in velocity. You can either read this equation in the words, change in velocity equals final velocity minus initial velocity or in symbols, delta v equals v subf minus v subi. Now we do a little substituting. We replace the delta v in the first equation with its definition from the second equation. Now all we have to do is rearrange this equation, obeying basic rules of algebra, to get our final equation. The equation we have derived is acceleration equation number one, which you will be using to solve accelerated motion problems. Deriving the second and third acceleration equations is a more complex process. But if you want to see how we get these equations, look in your physics book. (green chalkboard on screen) Here are the three acceleration equations you will use to solve problems involving uniformly accelerated motion. Put them in your notes, keeping them to the left of the paper. We ll add something on the right later. 6
7 We derived number one, final velocity equals initial velocity plus acceleration times time. The second equation is displacement equals initial velocity times time plus one half acceleration times time squared. The third is final velocity squared equals initial velocity squared plus two times acceleration times displacement. Now for the big question. Student 1 Do we have to know these? Student 2 Are we supposed to memorize all three equations? Student 3 I was afraid it would come to this. Student 4 Surely you don t expect us to know this stuff. The answer is No, you don t have to memorize these equations. They will be given to you on every quiz and test. But you do have to use them correctly. And remember that the units must cancel out, so be consistent in the units you use. As long as the four of you are here, hang around. Local teachers are going to pause the tape and give our classroom students some more phrases from acceleration word problems that you ll need to recognize. After everyone has copied these phrases and thought about what specific information that gives us, we ll come back and the four of you can help me go over the answers. Local Teachers: Turn off tape and give students problem set number three from facilitator's guide. We already know, how far, how long, and how fast. Let s go over the other phrases. Jot down the symbols each phrase involves. Student 1 How fast was it going? asks for initial velocity and How fast will it go?, asks for final velocity. Student 2 Sometimes the object will start at rest. This tells us that initial velocity is zero. 7
8 Student 3 And sometimes the object slows down. This means that acceleration is a negative number. Student 4 Finally, when an object stops, we know that final velocity is zero and acceleration is negative. You ve got to slow down in order to stop. Good. We re about ready to try some problems. But first, let s go back to the three acceleration equations and add something. Many problems will involve objects starting from rest, and you know that this means initial velocity is zero. (table on screen) The first term on the right of each acceleration equation involves initial velocity. When initial velocity is zero, what happens to this term? Anything times zero is zero, so it goes away. And look at the other two equations. The same thing happens to their first terms when v subi equals zero. The term disappears. Let s rewrite the equations for problems where the object starts at rest. We re left with three very simple equations that are easier to rearrange and work with. Now we re ready for an example problem. Starting from rest, a ball rolls down a hill, uniformly accelerating at 3.2 m/s 2. How long does it take the ball to roll 24 meters? The first thing we see is starting from rest. This means that initial velocity is zero. So we write it down. Acceleration is 3.2 meters per second squared. And displacement is 24 meters. We want to solve for time. Look at the three acceleration equations, but use the simple ones on the right. Which one will we use? If you said number two, you re right. It s the only one that includes d, t, and a. We need to rearrange to solve for time before we do any calculating. Multiply both sides by two and divide by a. The take the square root of both sides. That gives us t equals the square root of 2 times a times d. When we plug and chug, we get the answer. It s 9.8 seconds. Don t forget to round. Skid marks at the scene of an accident show that Justin Time s car moved 64 m before it stopped. If the car decelerated at a rate of 8.0 m/s 2, how fast was Justin driving before he applied the brakes? In this problem, we know that displacement is 64 meters and that the car comes to a stop, so final velocity is zero. We also know that the car decelerated. That means that acceleration is a negative 8.0 meters per second squared. We don t know the car s initial velocity. Now, which equation will we use? We need one that includes initial and final velocities, acceleration and displacement. That s equation number three. 8
9 Now you can t use the short form, because initial velocity is not zero. Let s rearrange to solve for initial velocity. To isolate it, we need to subtract this last term from both sides. Then we ll take the square root of both sides. That gives us initial velocity equals the square root of final velocity squared minus two times acceleration times displacement. Now watch your signs here. The two negative signs will multiply out to be positive. If you re wondering if you can make two mistakes and get full credit for the right answer, the answer is no. In physics, two wrongs do not make a right. The answer is 32 meters per second. What do you think? Was Justin Time speeding? Police computers can actually use skid marks to find out, as long as the car didn t hit anything to stop it. You can calculate Justin s speed by converting from meters per second to miles per hour, using the fact that one kilometer equals 0.62 (zero point six two) miles. We ll talk about it at the beginning of our next program. Now it s time for you to try some motion problems. Your teacher will give you the problems and time to work them. Remember to watch your units, watch your signs, and pay attention to the type of motion you are dealing with. We might just slip in some uniform motion among the acceleration problems, so don t be tricked. We ll go over some of these problems in the next program, too. And we ll learn how to tell how tall a bridge is by dropping a rock from it and counting one Mississippi, two Mississippi, three Mississippi 9
Physics 303 Motion of Falling Objects
Physics 303 Motion of Falling Objects Before we start today s lesson, we need to clear up some items from our last program. First of all, did you find out if Justin Time was speeding or not? It turns out
More informationAlex s Guide to Word Problems and Linear Equations Following Glencoe Algebra 1
Alex s Guide to Word Problems and Linear Equations Following Glencoe Algebra 1 What is a linear equation? It sounds fancy, but linear equation means the same thing as a line. In other words, it s an equation
More informationPhysic 602 Conservation of Momentum. (Read objectives on screen.)
Physic 602 Conservation of Momentum (Read objectives on screen.) Good. You re back. We re just about ready to start this lab on conservation of momentum during collisions and explosions. In the lab, we
More informationChapter 1 Review of Equations and Inequalities
Chapter 1 Review of Equations and Inequalities Part I Review of Basic Equations Recall that an equation is an expression with an equal sign in the middle. Also recall that, if a question asks you to solve
More information( )( b + c) = ab + ac, but it can also be ( )( a) = ba + ca. Let s use the distributive property on a couple of
Factoring Review for Algebra II The saddest thing about not doing well in Algebra II is that almost any math teacher can tell you going into it what s going to trip you up. One of the first things they
More informationTHE MOVING MAN: DISTANCE, DISPLACEMENT, SPEED & VELOCITY
THE MOVING MAN: DISTANCE, DISPLACEMENT, SPEED & VELOCITY Background Remember graphs are not just an evil thing your teacher makes you create, they are a means of communication. Graphs are a way of communicating
More informationPhysics 30S Unit 1 Kinematics
Physics 30S Unit 1 Kinematics Mrs. Kornelsen Teulon Collegiate Institute 1 P a g e Grade 11 Physics Math Basics Answer the following questions. Round all final answers to 2 decimal places. Algebra 1. Rearrange
More informationPHYSICS Kinematics in One Dimension
PHYSICS Kinematics in One Dimension August 13, 2012 www.njctl.org 1 Motion in One Dimension Return to Table of Contents 2 Distance We all know what the distance between two objects is... So what is it?
More informationMath 138: Introduction to solving systems of equations with matrices. The Concept of Balance for Systems of Equations
Math 138: Introduction to solving systems of equations with matrices. Pedagogy focus: Concept of equation balance, integer arithmetic, quadratic equations. The Concept of Balance for Systems of Equations
More informationQuadratic Equations Part I
Quadratic Equations Part I Before proceeding with this section we should note that the topic of solving quadratic equations will be covered in two sections. This is done for the benefit of those viewing
More informationSolving Quadratic & Higher Degree Equations
Chapter 9 Solving Quadratic & Higher Degree Equations Sec 1. Zero Product Property Back in the third grade students were taught when they multiplied a number by zero, the product would be zero. In algebra,
More informationGravity: How fast do objects fall? Teacher Advanced Version (Grade Level: 8 12)
Gravity: How fast do objects fall? Teacher Advanced Version (Grade Level: 8 12) *** Experiment with Audacity and Excel to be sure you know how to do what s needed for the lab*** Kinematics is the study
More informationMath101, Sections 2 and 3, Spring 2008 Review Sheet for Exam #2:
Math101, Sections 2 and 3, Spring 2008 Review Sheet for Exam #2: 03 17 08 3 All about lines 3.1 The Rectangular Coordinate System Know how to plot points in the rectangular coordinate system. Know the
More informationSolving Quadratic & Higher Degree Equations
Chapter 7 Solving Quadratic & Higher Degree Equations Sec 1. Zero Product Property Back in the third grade students were taught when they multiplied a number by zero, the product would be zero. In algebra,
More informationSquaring and Unsquaring
PROBLEM STRINGS LESSON 8.1 Squaring and Unsquaring At a Glance (6 6)* ( 6 6)* (1 1)* ( 1 1)* = 64 17 = 64 + 15 = 64 ( + 3) = 49 ( 7) = 5 ( + ) + 1= 8 *optional problems Objectives The goal of this string
More information161 Sp18 T1 grades (out of 40, max 100)
Grades for test Graded out of 40 (scores over 00% not possible) o Three perfect scores based on this grading scale!!! o Avg = 57 o Stdev = 3 Scores below 40% are in trouble. Scores 4060% are on the bubble
More informationAstronomy 102 Math Review
Astronomy 102 Math Review 2003August06 Prof. Robert Knop r.knop@vanderbilt.edu) For Astronomy 102, you will not need to do any math beyond the highschool alegbra that is part of the admissions requirements
More informationAlgebra & Trig Review
Algebra & Trig Review 1 Algebra & Trig Review This review was originally written for my Calculus I class, but it should be accessible to anyone needing a review in some basic algebra and trig topics. The
More informationAP Physics I Summer Assignment. Mrs. Verdi
AP Physics I Summer Assignment Mrs. Verdi 20152016 ALL STUDENTS ENROLLED IN AP PHYSICS: Please email me at averdi@leepublicschools.net before the start of school. Leave your first and last name so I know
More informationPlease bring the task to your first physics lesson and hand it to the teacher.
Preenrolment task for 2014 entry Physics Why do I need to complete a preenrolment task? This bridging pack serves a number of purposes. It gives you practice in some of the important skills you will
More informationSECTION 2. Objectives. Describe motion in terms changing velocity. Compare graphical representations of accelerated and nonaccelerated motions.
SECTION Plan and Prepare Preview Vocabulary Academic Vocabulary Remind students that rate describes how something changes compared to something else. In physics, a rate usually refers to a change over
More informationAP Physics 1 Kinematics 1D
AP Physics 1 Kinematics 1D 1 Algebra Based Physics Kinematics in One Dimension 2015 08 25 www.njctl.org 2 Table of Contents: Kinematics Motion in One Dimension Position and Reference Frame Displacement
More informationMITOCW big_picture_derivatives_512kbmp4
MITOCW big_picture_derivatives_512kbmp4 PROFESSOR: OK, hi. This is the second in my videos about the main ideas, the big picture of calculus. And this is an important one, because I want to introduce
More informationKEY CONCEPTS AND PROCESS SKILLS
Measuring 74 40 to 23 50minute sessions ACTIVITY OVERVIEW L A B O R AT O R Y Students use a cart, ramp, and track to measure the time it takes for a cart to roll 100 centimeters. They then calculate
More informationPHYSICS LAB: CONSTANT MOTION
PHYSICS LAB: CONSTANT MOTION Introduction Experimentation is fundamental to physics (and all science, for that matter) because it allows us to prove or disprove our hypotheses about how the physical world
More informationof 8 28/11/ :25
Paul's Online Math Notes Home Content Chapter/Section Downloads Misc Links Site Help Contact Me Differential Equations (Notes) / First Order DE`s / Modeling with First Order DE's [Notes] Differential Equations
More informationSolving Quadratic & Higher Degree Equations
Chapter 9 Solving Quadratic & Higher Degree Equations Sec 1. Zero Product Property Back in the third grade students were taught when they multiplied a number by zero, the product would be zero. In algebra,
More informationMotion and Forces. Describing Motion
CHAPTER Motion and Forces LESSON 1 Describing Motion What do you think? Read the two statements below and decide whether you agree or disagree with them. Place an A in the Before column if you agree with
More informationSCIENCE 1206 Unit 3. Physical Science Motion
SCIENCE 1206 Unit 3 Physical Science Motion Section 1: Units, Measurements and Error What is Physics? Physics is the study of motion, matter, energy, and force. Qualitative and Quantitative Descriptions
More informationTake the Anxiety Out of Word Problems
Take the Anxiety Out of Word Problems I find that students fear any problem that has words in it. This does not have to be the case. In this chapter, we will practice a strategy for approaching word problems
More informationVelocity, Speed, and Acceleration. Unit 1: Kinematics
Velocity, Speed, and Acceleration Unit 1: Kinematics Speed vs Velocity Speed is a precise measurement of how fast you are going. It is your distance traveled over time. Speed is a scalar quantity. To measure
More informationIn this lesson about Displacement, Velocity and Time, you will:
Slide 1 Module 3, Lesson 2  Objectives & Standards In this lesson about Displacement, Velocity and Time, you will: Pb: Demonstrate an understanding of the principles of force and motionand relationships
More informationRecitation Questions 1D Motion (part 1)
Recitation Questions 1D Motion (part 1) 18 January Question 1: Two runners (This problem is simple, but it has the same template as most of the problems that you ll be doing for this unit. Take note of
More informationDIFFERENTIAL EQUATIONS
DIFFERENTIAL EQUATIONS Basic Concepts Paul Dawkins Table of Contents Preface... Basic Concepts... 1 Introduction... 1 Definitions... Direction Fields... 8 Final Thoughts...19 007 Paul Dawkins i http://tutorial.math.lamar.edu/terms.aspx
More informationPH 2213 : Chapter 05 Homework Solutions
PH 2213 : Chapter 05 Homework Solutions Problem 5.4 : The coefficient of static friction between hard rubber and normal street pavement is about 0.90. On how steep a hill (maximum angle) can you leave
More informationAlgebra Exam. Solutions and Grading Guide
Algebra Exam Solutions and Grading Guide You should use this grading guide to carefully grade your own exam, trying to be as objective as possible about what score the TAs would give your responses. Full
More informationIntroduction to Algebra: The First Week
Introduction to Algebra: The First Week Background: According to the thermostat on the wall, the temperature in the classroom right now is 72 degrees Fahrenheit. I want to write to my friend in Europe,
More informationMITOCW ocw18_02f07lec02_220k
MITOCW ocw18_02f07lec02_220k The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high quality educational resources for free.
More informationPHYSICS LAB: CONSTANT MOTION
PHYSICS LAB: CONSTANT MOTION Introduction Experimentation is fundamental to physics (and all science, for that matter) because it allows us to prove or disprove our hypotheses about how the physical world
More informationchapter 1 function notebook September 04, 2015 Foundational Skills Algebra 1 Sep 8 8:34 AM
Foundational Skills of Algebra 1 Sep 8 8:34 AM 1 In this unit we will see how key vocabulary words are connected equation variable expression solving evaluating simplifying Order of operation Sep 8 8:40
More informationPHY2048 Physics with Calculus I
PHY2048 Physics with Calculus I Section 584761 Prof. Douglas H. Laurence Exam 1 (Chapters 2 6) February 14, 2018 Name: Solutions 1 Instructions: This exam is composed of 10 multiple choice questions and
More informationLAB 2  ONE DIMENSIONAL MOTION
Name Date Partners L021 LAB 2  ONE DIMENSIONAL MOTION OBJECTIVES Slow and steady wins the race. Aesop s fable: The Hare and the Tortoise To learn how to use a motion detector and gain more familiarity
More informationMath 119 Main Points of Discussion
Math 119 Main Points of Discussion 1. Solving equations: When you have an equation like y = 3 + 5, you should see a relationship between two variables, and y. The graph of y = 3 + 5 is the picture of this
More informationConceptual Explanations: Simultaneous Equations Distance, rate, and time
Conceptual Explanations: Simultaneous Equations Distance, rate, and time If you travel 30 miles per hour for 4 hours, how far do you go? A little common sense will tell you that the answer is 120 miles.
More informationALevel Notes CORE 1
ALevel Notes CORE 1 Basic algebra Glossary Coefficient For example, in the expression x³ 3x² x + 4, the coefficient of x³ is, the coefficient of x² is 3, and the coefficient of x is 1. (The final 4 is
More informationLesson 39: Kinetic Energy & Potential Energy
Lesson 39: Kinetic Energy & Potential Energy Kinetic Energy You ve probably heard of kinetic energy in previous courses using the following definition and formula Any object that is moving has kinetic
More informationChapter 2: Motion a Straight Line
Formula Memorization: Displacement What is a vector? Average Velocity Average Speed Instanteous Velocity Average Acceleration Instantaneous Acceleration Constant Acceleration Equation (List all five of
More informationChapter 2 Motion in One Dimension. Slide 21
Chapter 2 Motion in One Dimension Slide 21 MasteringPhysics, PackBack Answers You should be on both by now. MasteringPhysics first reading quiz Wednesday PackBack should have email & be signed up 2014
More information3.3 Acceleration An example of acceleration Definition of acceleration Acceleration Figure 3.16: Steeper hills
3.3 Acceleration Constant speed is easy to understand. However, almost nothing moves with constant speed for long. When the driver steps on the gas pedal, the speed of the car increases. When the driver
More informationCalculus II. Calculus II tends to be a very difficult course for many students. There are many reasons for this.
Preface Here are my online notes for my Calculus II course that I teach here at Lamar University. Despite the fact that these are my class notes they should be accessible to anyone wanting to learn Calculus
More informationBasic Math Problems Unit 1
Basic Math Problems Unit 1 Name Period Using fractions: When you are using fractions in science, we need to convert them into decimals. You can do this by dividing the top number by the bottom number.
More informationLesson 8: Velocity. Displacement & Time
Lesson 8: Velocity Two branches in physics examine the motion of objects: Kinematics: describes the motion of objects, without looking at the cause of the motion (kinematics is the first unit of Physics
More informationSUMMER MATH PACKET ADVANCED ALGEBRA A COURSE 215
SUMMER MATH PACKET ADVANCED ALGEBRA A COURSE 5 Updated May 0 MATH SUMMER PACKET INSTRUCTIONS Attached you will find a packet of exciting math problems for your enjoyment over the summer. The purpose of
More informationChapter 2 Section 2: Acceleration
Chapter 2 Section 2: Acceleration Motion Review Speed is the rate that an object s distance changes Distance is how far an object has travelled Speed = distance/time Velocity is rate that an object s displacement
More informationBuilding Concepts: Solving Systems of Equations Algebraically
Lesson Overview In this TINspire lesson, students will investigate pathways for solving systems of linear equations algebraically. There are many effective solution pathways for a system of two linear
More informationFinding Limits Graphically and Numerically
Finding Limits Graphically and Numerically 1. Welcome to finding limits graphically and numerically. My name is Tuesday Johnson and I m a lecturer at the University of Texas El Paso. 2. With each lecture
More informationLesson Plan by: Stephanie Miller
Lesson: Pythagorean Theorem and Distance Formula Length: 45 minutes Grade: Geometry Academic Standards: MA.G.1.1 2000 Find the lengths and midpoints of line segments in one or twodimensional coordinate
More informationTo factor an expression means to write it as a product of factors instead of a sum of terms. The expression 3x
Factoring trinomials In general, we are factoring ax + bx + c where a, b, and c are real numbers. To factor an expression means to write it as a product of factors instead of a sum of terms. The expression
More informationv a =, to calculate acceleration. t You had also rearranged it as v = a t using basic Algebra (or cross multiplying) Velocity
1 GRADE 1 SCIENCE ACCELERATION DISTANCE VS VELOCITY **MORE IMPORTANTLY: VELOCITY VS BRAKING DISTANCE** Name: Date: 1. You have studied the formula v a =, to calculate acceleration. t You had also rearranged
More informationPHY 123 Lab 1  Error and Uncertainty and the Simple Pendulum
To print higherresolution math symbols, click the HiRes Fonts for Printing button on the jsmath control panel. PHY 13 Lab 1  Error and Uncertainty and the Simple Pendulum Important: You need to print
More informationUnit 2  Linear Motion and Graphical Analysis
Unit 2  Linear Motion and Graphical Analysis Motion in one dimension is particularly easy to deal with because all the information about it can be encapsulated in two variables: x, the position of the
More informationMITOCW MIT8_01F16_W01PS05_360p
MITOCW MIT8_01F16_W01PS05_360p You're standing at a traffic intersection. And you start to accelerate when the light turns green. Suppose that your acceleration as a function of time is a constant for
More informationIntroduction. So, why did I even bother to write this?
Introduction This review was originally written for my Calculus I class, but it should be accessible to anyone needing a review in some basic algebra and trig topics. The review contains the occasional
More informationSTEP 1: Ask Do I know the SLOPE of the line? (Notice how it s needed for both!) YES! NO! But, I have two NO! But, my line is
EQUATIONS OF LINES 1. Writing Equations of Lines There are many ways to define a line, but for today, let s think of a LINE as a collection of points such that the slope between any two of those points
More informationChapter 1. Foundations of GMAT Math. Arithmetic
Chapter of Foundations of GMAT Math In This Chapter QuickStart Definitions Basic Numbers Greater Than and Less Than Adding and Subtracting Positives and Negatives Multiplying and Dividing Distributing
More informationSystems of Equations. Red Company. Blue Company. cost. 30 minutes. Copyright 2003 Hanlonmath 1
Chapter 6 Systems of Equations Sec. 1 Systems of Equations How many times have you watched a commercial on television touting a product or services as not only the best, but the cheapest? Let s say you
More informationPROFESSOR: WELCOME BACK TO THE LAST LECTURE OF THE SEMESTER. PLANNING TO DO TODAY WAS FINISH THE BOOK. FINISH SECTION 6.5
1 MATH 16A LECTURE. DECEMBER 9, 2008. PROFESSOR: WELCOME BACK TO THE LAST LECTURE OF THE SEMESTER. I HOPE YOU ALL WILL MISS IT AS MUCH AS I DO. SO WHAT I WAS PLANNING TO DO TODAY WAS FINISH THE BOOK. FINISH
More informationBridging the gap between GCSE and A level mathematics
Bridging the gap between GCSE and A level mathematics This booklet is designed to help you revise important algebra topics from GCSE and make the transition from GCSE to A level a smooth one. You are advised
More informationLesson 21 Not So Dramatic Quadratics
STUDENT MANUAL ALGEBRA II / LESSON 21 Lesson 21 Not So Dramatic Quadratics Quadratic equations are probably one of the most popular types of equations that you ll see in algebra. A quadratic equation has
More informationLesson 39: Kinetic Energy & Potential Energy
Lesson 39: Kinetic Energy & Potential Energy Kinetic Energy WorkEnergy Theorem Potential Energy Total Mechanical Energy We sometimes call the total energy of an object (potential and kinetic) the total
More informationAlgebra. Here are a couple of warnings to my students who may be here to get a copy of what happened on a day that you missed.
This document was written and copyrighted by Paul Dawkins. Use of this document and its online version is governed by the Terms and Conditions of Use located at. The online version of this document is
More informationConservation of Momentum
Learning Goals Conservation of Momentum After you finish this lab, you will be able to: 1. Use Logger Pro to analyze video and calculate position, velocity, and acceleration. 2. Use the equations for 2dimensional
More informationAP Physics 1 Summer Assignment
AP Physics 1 Summer Assignment Gaithersburg High School Mr. Schultz Welcome to AP Physics 1! This is a college level physics course that is fun, interesting and challenging on a level you ve not yet experienced.
More informationPrealgebra. Edition 5
Prealgebra Edition 5 Prealgebra, Edition 5 2009, 2007, 2005, 2004, 2003 Michelle A. Wyatt (M A Wyatt) 2009, Edition 5 Michelle A. Wyatt, author Special thanks to Garry Knight for many suggestions for the
More informationNumerical and Algebraic Expressions and Equations
Numerical and Algebraic Expressions and Equations Sometimes it's hard to tell how a person is feeling when you're not talking to them face to face. People use emoticons in emails and chat messages to show
More informationacceleration versus time. LO Determine a particle s change in position by graphical integration on a graph of velocity versus time.
Chapter: Chapter 2 Learning Objectives LO 2.1.0 Solve problems related to position, displacement, and average velocity to solve problems. LO 2.1.1 Identify that if all parts of an object move in the same
More informationChapter 5 Simplifying Formulas and Solving Equations
Chapter 5 Simplifying Formulas and Solving Equations Look at the geometry formula for Perimeter of a rectangle P = L W L W. Can this formula be written in a simpler way? If it is true, that we can simplify
More informationAP Physics 1 Summer Assignment
Name: Email address (write legibly): AP Physics 1 Summer Assignment Packet 3 The assignments included here are to be brought to the first day of class to be submitted. They are: Problems from Conceptual
More informationMITOCW free_body_diagrams
MITOCW free_body_diagrams This is a bungee jumper at the bottom of his trajectory. This is a pack of dogs pulling a sled. And this is a golf ball about to be struck. All of these scenarios can be represented
More information3 Acceleration. positive and one is negative. When a car changes direction, it is also accelerating. In the figure to the
What You ll Learn how acceleration, time, and velocity are related the different ways an object can accelerate how to calculate acceleration the similarities and differences between straight line motion,
More informationThe total time traveled divided by the total time taken to travel it. Average speed =
Unit 3: Motion V = d t Average speed The total time traveled divided by the total time taken to travel it Mathematically: Average speed = Total Distance Travelled Total Time Traveled So just how fast were
More informationVectors. Vector Practice Problems: Oddnumbered problems from
Vectors Vector Practice Problems: Oddnumbered problems from 3.13.21 After today, you should be able to: Understand vector notation Use basic trigonometry in order to find the x and y components of a
More informationExperimenting with Forces
A mother hears a loud crash in the living room. She walks into the room to see her sevenyearold son looking at a broken vase on the floor. How did that happen? she asks. I don t know. The vase just fell
More informationSUMMER MATH PACKET. Geometry A COURSE 227
SUMMER MATH PACKET Geometry A COURSE 7 MATH SUMMER PACKET INSTRUCTIONS Attached you will find a packet of exciting math problems for your enjoyment over the summer. The purpose of the summer packet is
More informationAccelerated CP Geometry Summer Packet
Accelerated CP Geometry Summer Packet The math teachers at Cherry Creek High School want each student to be successful, no matter which level or math course the student is in. We know that it is critical
More information2015 SUMMER MATH PACKET
Name: Date: 05 SUMMER MATH PACKET College Algebra Trig.  I understand that the purpose of the summer packet is for my child to review the topics they have already mastered in previous math classes and
More informationMITOCW MITRES18_005S10_DerivOfSinXCosX_300k_512kbmp4
MITOCW MITRES18_005S10_DerivOfSinXCosX_300k_512kbmp4 PROFESSOR: OK, this lecture is about the slopes, the derivatives, of two of the great functions of mathematics: sine x and cosine x. Why do I say great
More informationModule 3 Study Guide. GCF Method: Notice that a polynomial like 2x 2 8 xy+9 y 2 can't be factored by this method.
Module 3 Study Guide The second module covers the following sections of the textbook: 5.45.8 and 6.16.5. Most people would consider this the hardest module of the semester. Really, it boils down to your
More informationPosition, Speed and Velocity Position is a variable that gives your location relative to an origin. The origin is the place where position equals 0.
Position, Speed and Velocity Position is a variable that gives your location relative to an origin. The origin is the place where position equals 0. The position of this car at 50 cm describes where the
More informationSolving with Absolute Value
Solving with Absolute Value Who knew two little lines could cause so much trouble? Ask someone to solve the equation 3x 2 = 7 and they ll say No problem! Add just two little lines, and ask them to solve
More informationPOLYNOMIAL EXPRESSIONS PART 1
POLYNOMIAL EXPRESSIONS PART 1 A polynomial is an expression that is a sum of one or more terms. Each term consists of one or more variables multiplied by a coefficient. Coefficients can be negative, so
More informationFinal Review Sheet. B = (1, 1 + 3x, 1 + x 2 ) then 2 + 3x + 6x 2
Final Review Sheet The final will cover Sections Chapters 1,2,3 and 4, as well as sections 5.15.4, 6.16.2 and 7.17.3 from chapters 5,6 and 7. This is essentially all material covered this term. Watch
More informationChapter 3 ALGEBRA. Overview. Algebra. 3.1 Linear Equations and Applications 3.2 More Linear Equations 3.3 Equations with Exponents. Section 3.
4 Chapter 3 ALGEBRA Overview Algebra 3.1 Linear Equations and Applications 3.2 More Linear Equations 3.3 Equations with Exponents 5 LinearEquations 3+ what = 7? If you have come through arithmetic, the
More informationPHYSICS 107. Lecture 5 Newton s Laws of Motion
PHYSICS 107 Lecture 5 Newton s Laws of Motion First Law We saw that the type of motion which was most difficult for Aristotle to explain was horizontal motion of nonliving objects, particularly after they've
More informationUnit 5. Linear equations and inequalities OUTLINE. Topic 13: Solving linear equations. Topic 14: Problem solving with slope triangles
Unit 5 Linear equations and inequalities In this unit, you will build your understanding of the connection between linear functions and linear equations and inequalities that can be used to represent and
More informationAP Physics 1 Summer Assignment
AP Physics 1 Summer Assignment Welcome to AP Physics 1 It is a college level physics course that is fun, interesting and challenging on a level you ve not yet experienced. This summer assignment will review
More informationEdexcel AS and A Level Mathematics Year 1/AS  Pure Mathematics
Year Maths A Level Year  Tet Book Purchase In order to study A Level Maths students are epected to purchase from the school, at a reduced cost, the following tetbooks that will be used throughout their
More informationUsing Scientific Measurements
Section 3 Main Ideas Accuracy is different from precision. Significant figures are those measured precisely, plus one estimated digit. Scientific notation is used to express very large or very small numbers.
More informationLecture 2. When we studied dimensional analysis in the last lecture, I defined speed. The average speed for a traveling object is quite simply
Lecture 2 Speed Displacement Average velocity Instantaneous velocity Cutnell+Johnson: chapter 2.12.2 Most physics classes start by studying the laws describing how things move around. This study goes
More informationThe Celsius temperature scale is based on the freezing point and the boiling point of water. 12 degrees Celsius below zero would be written as
Prealgebra, Chapter 2  Integers, Introductory Algebra 2.1 Integers In the real world, numbers are used to represent real things, such as the height of a building, the cost of a car, the temperature of
More information