1 Motion Graphs It is said that a picture is worth a thousand words. The same can be said for a graph. Once you learn to read the graphs of the motion of objects, you can tell at a glance if the object in question was moving toward you, away from you, speeding up slowing down, or moving at constant speed.
2 Case Study #1 Imagine you are standing by the side of the road. An automobile approaches you from behind, traveling at a constant speed. Then the car passes you, disappearing off into the distance. To graph the motion of this car, you will need some data.
3 The Data Table Using a stopwatch and roadside markers, you keep track of how far the car is from you (it's displacement) at each moment in time. Negative displacements show that the car was behind you. Positive displacements show that the car was in front of you. time (s) displacement (m)
4 Graph for Case Study #1 Car is in front of you. Car is behind you.
5 Slope of Graph The slope of a displacement vs. time graph tells us the object's velocity. In this case, the line is straight, meaning that the slope (and hence, the velocity) remains constant for this automobile.
6 Calculating the Slope In mathematics, the slope of a graph is expressed as the rise over the run. rise run
7 Slope as Velocity In this particular example, we have put time ( t) on the horizontal axis and displacement ( x) on the vertical axis. The equation now becomes:
8 Finding the Velocity of Our Car Laying out a triangle on our graph helps us find the values for the rise and the run. (5.0, 20) Rise = x (1.0, -20) Run = t
9 Case Study #2 Here's a graph that describes the motion of another car you see while standing along the side of the road. Can you describe in words the motion of this car?
10 Case Study #2 If you said that this car approached you from in front, and then passed on by you, you were correct! It is also true that this car is traveling at a constant velocity (the graph line is straight.) But is it the same constant velocity the last car had?
11 Velocity from Slope It is the same velocity if the slope of the graph line is the same. To find out, we must calculate the slope. (0, 30) In the case of this graph, the rise is actually a drop. The value of the displacement drops from an original value of 30 m down to 15 meters. (5.0, -15)
12 Results for Case Study #2 And so we see that this second car is traveling a bit slower than the first one; its velocity coming in with a magnitude of 9.0 m/s instead of 10 m/s. The negative sign on the second car's velocity means that it was traveling in the opposite direction of the first car. That's easy to tell at a glance, just by looking at the two graphs. Positive slope (car #1) Negative slope (car #2)
13 Graphing Both Cars It is much easier to compare the motions of the two cars when you graph those motions on the same set of axes. Car 1 Car 2
14 Intersections and Intercepts Of particular interest is the center section of the graph. Can you figure out what's happening at each of the labelled points A, B, and C?
15 Intersections and Intercepts Point A is where the first car (that came from behind) passes you. Point B is where the two cars pass each other (in front of you). Point C is where the second car (that came from in front of you) passes you by. Where graph lines intersect the horizontal axis is where the objects pass by the observer. Where graph lines intersect each other is where the objects pass each other.
16 Moving in the Same Direction When two cars are moving in the same direction, it is much easier to compare their speeds, both in real life, and on a graph. The slopes of their graph lines are obviously different. The slope for car 1 is steeper, therefore it must be going faster. For practice, try calculating the slope of each car's graph line.
17 Check Your Work! Here are the velocities, calculated for each of the two cars, from the slopes of their graph lines. For Car 1 For Car 2
18 Case Study #4 A black car is traveling along the highway at constant velocity. A blue car follows along behind it, at the same velocity. The black car pulls to the side of the road and stops. The blue car pulls up behind it and also stops. First, we look at the data table that describes their motion, then we'll graph it. time (s) Displacement Displacement black car (m) blue car (m)
19 Case Study #4
20 Reading Information off the Graph The graph lines are parallel, so both cars have the same constant velocity. You can tell when the cars are motionless because the slope of their graph lines is zero, meaning zero velocity. While moving, the distance separating the cars remains constant at 10 meters. When parked, the distance separating the cars is now only 5 meters.
21 What's Different About This One?
22 Reading the Graph As before, the blue car followed the black car while traveling along the highway at constant velocity. Just as before, the black car pulled over and stopped. What's different is that this time, the blue car passed the black car and then pulled over to stop in front of it. We know this because, in the end, the blue car is farther out in front. In the end, the blue car's displacement is greater. The blue car passes the black car here.
23 The New Motorcycle Your friend just bought a new motorcycle and wants to show it off for you. Look at the graph below, and see if you can describe the motion of your friend on his new machine.
24 Describing the Motion For segment A-B, your friend is going away from you at constant velocity. For segment C-D, your friend is returning to you. For segment B-C, your friend has stopped moving.
25 Quantifying the Results Because we have grid lines and numbers on the axes, we can include numerical values in our descriptions. During segment A-B, we can see that your friend travels 30 meters out in front of you. x The slope calculation tells us how fast he was going. t
26 Quantifying the Results For segment B-C, your friend remains at a distance of 30 meters in front of you for 3 seconds. He's motionless. The slope of the graph line is quite obviously zero, but if you had to show your work, here's how you'd do it.
27 Quantifying the Results For the C-D segment, the distance between you and your friend continuously decreases as he returns to you. Again, a slope calculation tells us how fast he's going. t x
28 Curved Lines Mean Changing Slopes The slope gets steeper with time. This means the object's velocity is increasing. (As before, the increasing displacement means it's going away from you.) The slope flattens out with time. This means the object's velocity is decreasing, approaching zero. (As before, the increasing displacement means it's going away from you.)
29 Curved Lines Mean Changing Slopes The slope flattens out with time. This means the object's velocity is decreasing, approaching zero. (The decreasing displacement means the object is approaching you.) The slope gets steeper with time. This means the object's velocity is increasing. (The decreasing displacement means the object is getting closer to you.)
30 Case Study #5 An object slides down a ramp, picking up speed as it goes. We can figure out how fast the object was going by calculating the slope, but, depending upon which instant in time we pick, we will get a different answer.
31 Finding the Velocity (Slope) at a Particular Point Step 1: Draw a straight line tangent to the curved graph line at the point in time you've selected. (Here, it's at t = 3.0 seconds)
32 Finding the Velocity (Slope) at a Particular Point Step 2: Construct a right triangle, using the tangent line as its hypotenuse. (The other two sides of the triangle will become the rise and the run.) rise run
33 Finding the Velocity (Slope) at a Particular Point Step 3: By carefully reading the graph, obtain the coordinates of the end points of the tangent line. You will use these to calculate x and t. (4.0, 15) x (2.0, 3) t Warning! Do not use the coordinates of the nearby points on the curved graph line.
34 Finding the Velocity (Slope) at a Particular Point Step 4: Calculate the slope as before. (4.0, 15) x (2.0, 3) t
35 Finding the Velocity (Slope) at a Particular Point For comparison, we will now find the slope at another point on the graph. Let's use t = 5.0 s. Step 1: Draw the tangent line to the graph at t = 5.0 s.
36 Finding the Velocity (Slope) at a Particular Point Step 2: Draw the right triangle and obtain the coordinates of the tangent line's end points. (6.0, 35) x (4.0, 15) t
37 Finding the Velocity (Slope) at a Particular Point Step 3: Calculate the slope, as before. (6.0, 35) x (4.0, 15) t
38 Case Study #6 Imagine you're standing at the top of a ramp. Your friend rolls a ball up the ramp toward you. The following graph describes the motion of the ball. For additional practice, try finding the velocity of the ball at several different instants in time through slope calculation. By the way: the ramp is straight, not curved. This is a graph of the ball's motion, not a picture of the ramp!
39 Velocity vs. Time Graphs We will now remove displacement from the vertical axis of our graphs and replace it with velocity. Although slope is still rise over run, the rise part of the equation will now read change in velocity, v, instead of x. rise run
40 Slope as Acceleration Our slope equation now becomes: Since slopes can be either positive or negative, accelerations can be either positive or negative.
41 Speeding Up or Slowing Down? Warning: Many people mistakenly believe that a positive acceleration means the object is speeding up and a negative acceleration means the object is slowing down. Here's the truth: If the numerical values of an object's velocity and acceleration have the same sign (both positive or both negative), then the object is speeding up. If the velocity and acceleration have opposite signs (one of positive and the other is negative), then the object is slowing down.
42 Here's How to Read that Information from Velocity vs. Time Graphs Object speeding up Velocity is + v (first quadrant) Object slowing down v Velocity is + (first quadrant) Acceleration is + Acceleration is (positive slope) (negative slope) t t Object slowing down Object speeding up t t Velocity is v (fourth quadrant) Acceleration is (negative slope) Velocity is v (fourth quadrant) Acceleration is + (positive slope)
43 Case Study #7 A bus has finished loading passengers, the door closes, and it accelerates away from the curb. Since the graph line is straight, the bus has a constant acceleration. The value of the bus's acceleration can be calculated from the slope of the graph.
44 Obtaining the Acceleration from the Slope Draw a right triangle and obtain the slope by finding the rise over the run. (5.0, 25) v (1.0, 5) t
45 Obtaining the Acceleration from the Slope (5.0, 25) v (1.0, 5) t
46 Obtaining the Acceleration at a Glance An astute observer will notice that the velocity increases in step-like fashion by 5 m/s for every second of elapsed time. This allows us to know at a glance that the acceleration must be 5 m/s2.
47 Displacement from Area The area between the graph line and the horizontal axis stands for the displacement of the object from its point of origin. In this case, the area is shaped like a triangle. Area of a triangle = ½ base height
48 Displacement from Area We use the scale on the vertical axis to determine the value of the height of the triangle. height A=½b h For the base we use 6 s. For the height we use 30 m/s. The scale on the horizontal axis is used to determine the value of the base.
49 Displacement from Area A=½b h A = ½ (6 s)(30 m/s) A = 90 m The usual units for area are meters squared, but in this case, the area actually stands for displacement, which is measured in meters. This tells us that the bus has traveled 90 meters during the first 6 seconds since it left the curb.
50 Relationship Between Displacement and Time Here is the equation we use to calculate the displacement of an object accelerating uniformly from rest: x = ½ a t 2 Expressed as a proportionality, it says that displacement is directly proportional to the square of the time. x t 2 This means that, in twice the amount of time, the object will be displaced 22 or four times as much.
51 Displacement Time Squared The area of the red triangle is 10 meters. It represents the displacement of the bus during the first 2 seconds. The area of the large triangle outlined in yellow is 40 meters. It represents the displacement of the bus during the first 4 seconds One can tell at a glance that the area of the yellow triangle is four times the area of the red triangle.
52 Case Study #8 A policeman waits by the side of the road with a radar unit. At the instant a speeding car passes by, the police car begins to accelerate from rest in an effort to catch up to the speeder and pull him over. Here's the data: time (s) speeder's cops's velocity (m/s) velocity (m/s) Question: At what time does the policeman catch up with the speeder?
53 The Cop and the Speeder This is not the point where the policeman catches up with the speeder. It is only the instant when they both happen to be going at the same velocity!
54 The Cop and the Speeder The true point where the cop catches up with the speeder is the point where they have both traveled the same distance forward from the point of origin (where the speeder first passed the cop when parked on the side of the road.) You must find the moment in time when the area under the graph line for cop is the same as the area under the graph line for the speeder!
55 The Cop and the Speeder The area for the speeder is the red rectangle. The area for the cop is the cross-hatched triangle. Clearly, the areas (and the displacements they represent) are not yet equal at t= 3.0 s.
56 The Cop and the Speeder The area for the speeder is the red rectangle. The area for the cop is the cross-hatched triangle. At t = 4 s, the areas are still not equal. How much farther do we have to go?
57 The Cop and the Speeder The area for the cop is the sum of the areas of the crosshatched triangle and the cross-hatched rectangle. At t= 8 s, the area for the speeder is the same as the area for the cop. The cop has finally caught him.
58 The Cop and the Speeder If it isn't obvious from the graph that the areas are the same, then you should do the calculation. For the speeder: A=b h A = (8.0 s)(30 m/s) A = 240 meters Total displacement = 240 meters. For the cop: A = ½ b h [for the triangle] A = ½ (4.0 s)(40 m/s) A = 80 meters A = b h [for the rectangle] A = (4.0 s)(40 m/s) A = 160 meters Total displacement = 80 m +160 m = 240 m.
59 Case Study #9 You throw a ball straight up in the air; it comes back down and you catch it. Here are the graphs that describe the motion of the ball. Both graphs describe the same motion!
60 Characteristics of the v vs. t graph Velocities are positive when the ball is rising. The ball is slowing down as it rises through the air. The ball reaches its highest point at t = 3.0 s, where it is motionless for an instant. Velocities are negative when the ball is falling. The ball speeds up (larger negative numbers) as it falls back to earth.
61 Characteristics of the v vs. t graph t v The slope of the graph is the ball's acceleration.
62 Characteristics of the v vs. t graph This area represents the displacement of the ball from when it was thrown on up to its highest point. It is a positive area. The displacement is in the positive or upward direction. This area represents the displacement of the ball from its highest point back down to its original point of origin. It is a negative area. This displacement is in the negative or downward direction.
63 Characteristics of the v vs. t graph Here's the upward displacement which occurs during the first 3 seconds. A=½b h A = ½ (3.0 s)(29.4 m/s) A = meters Here's the downward displacement which occurs during the last 3 seconds. A=½b h A = ½ (3.0 s)( 29.4 m/s) A = 44.1 meters The total displacement is zero. The ball ends up back where it started.
64 Characteristics of the x vs. t graph At t = 1 s, the slope of the graph (ball's velocity is m/s. The ball is slowing down as it rises. x t At t = 3 s, the slope of the graph is zero, because the ball is motionless at its highest point. t At t = 5 s, the slope of the graph (ball's velocity) is 19.6 m/s. The ball is on its way back down. x
In the previous section, you have come across many examples of motion. You have learnt that to describe the motion of an object we must know its position at different points of time. The position of an
2-1 Position, Displacement, and Distance In describing an object s motion, we should first talk about position where is the object? A position is a vector because it has both a magnitude and a direction:
1 of 7 9/5/2009 6:12 PM Chapter 2 Homework Due: 9:00am on Tuesday, September 8, 2009 Note: To understand how points are awarded, read your instructor's Grading Policy. [Return to Standard Assignment View]
Physics 2A, Sec B00: Mechanics -- Winter 2011 Instructor: B. Grinstein Final Exam INSTRUCTIONS: Use a pencil #2 to fill your scantron. Write your code number and bubble it in under "EXAM NUMBER;" an entry
1. What is the average speed of an object that travels 6.00 meters north in 2.00 seconds and then travels 3.00 meters east in 1.00 second? 9.00 m/s 3.00 m/s 0.333 m/s 4.24 m/s 2. What is the distance traveled
Revised Pages PART ONE Mechanics CHAPTER Motion Along a Line 2 Despite its enormous mass (425 to 9 kg), the Cape buffalo is capable of running at a top speed of about 55 km/h (34 mi/h). Since the top speed
After completing this chapter you should be able to solve problems involving motion in a straight line with constant acceleration model an object moving vertically under gravity understand distance time
At the skate park on the ramp 1 On the ramp When a cart rolls down a ramp, it begins at rest, but starts moving downward upon release covers more distance each second When a cart rolls up a ramp, it rises
Candidates should be able to : Derive the equations of motion for constant acceleration in a straight line from a velocity-time graph. Select and use the equations of motion for constant acceleration in
Chapter 3: Falling Objects and Projectile Motion 1. Neglecting friction, if a Cadillac and Volkswagen start rolling down a hill together, the heavier Cadillac will get to the bottom A. before the Volkswagen.
VELOCITY, ACCELERATION, FORCE velocity Velocity v is a vector, with units of meters per second ( m s ). Velocity indicates the rate of change of the object s position ( r ); i.e., velocity tells you how
CT16-1 Which of the following is necessary to make an object oscillate? i. a stable equilibrium ii. little or no friction iii. a disturbance A: i only B: ii only C: iii only D: i and iii E: All three Answer:
CHAPTER 13 SECTION 13-1 Geometry and Algebra The Distance Formula COORDINATE PLANE consists of two perpendicular number lines, dividing the plane into four regions called quadrants X-AXIS - the horizontal
Physics Midterm Review Packet January 2010 This Packet is a Study Guide, not a replacement for studying from your notes, tests, quizzes, and textbook. Midterm Date: Thursday, January 28 th 8:15-10:15 Room:
More Chapter 3 Projectile motion simulator http://www.walter-fendt.de/ph11e/projectile.htm The equations of motion for constant acceleration from chapter 2 are valid separately for both motion in the x
Free Fall: Observing and Analyzing the Free Fall Motion of a Bouncing Ping-Pong Ball and Calculating the Free Fall Acceleration (Teacher s Guide) 2012 WARD S Science v.11/12 OVERVIEW Students will measure
4277(a) Semester 2, 2011 Page 1 of 9 THE UNIVERSITY OF SYDNEY EDUH 1017 - SPORTS MECHANICS NOVEMBER 2011 Time allowed: TWO Hours Total marks: 90 MARKS INSTRUCTIONS All questions are to be answered. Use
Chapter 3.8 & 6 Solutions P3.37. Prepare: We are asked to find period, speed and acceleration. Period and frequency are inverses according to Equation 3.26. To find speed we need to know the distance traveled
Solving Simultaneous Equations and Matrices The following represents a systematic investigation for the steps used to solve two simultaneous linear equations in two unknowns. The motivation for considering
Practice Midterm 1 1) When a parachutist jumps from an airplane, he eventually reaches a constant speed, called the terminal velocity. This means that A) the acceleration is equal to g. B) the force of
Week 7 homework IMPORTANT NOTE ABOUT WEBASSIGN: In the WebAssign versions of these problems, various details have been changed, so that the answers will come out differently. The method to find the solution
Chapters and 6 Introduction to Vectors A vector quantity has direction and magnitude. There are many examples of vector quantities in the natural world, such as force, velocity, and acceleration. A vector
Graphing Equations with Color Activity Students must re-write equations into slope intercept form and then graph them on a coordinate plane. 2011 Lindsay Perro Name Date Between The Lines Re-write each
Chapter 7: Momentum and Impulse 1. When a baseball bat hits the ball, the impulse delivered to the ball is increased by A. follow through on the swing. B. rapidly stopping the bat after impact. C. letting
SOLID MECHANICS TUTORIAL MECHANISMS KINEMATICS - VELOCITY AND ACCELERATION DIAGRAMS This work covers elements of the syllabus for the Engineering Council exams C105 Mechanical and Structural Engineering
Exam Three Momentum Concept Questions Isolated Systems 4. A car accelerates from rest. In doing so the absolute value of the car's momentum changes by a certain amount and that of the Earth changes by:
Name: Earth 110 Exploration of the Solar System Assignment 1: Celestial Motions and Forces Due in class Tuesday, Jan. 20, 2015 Why are celestial motions and forces important? They explain the world around
Physics: Principles and Applications, 6e Giancoli Chapter 4 Dynamics: Newton's Laws of Motion Conceptual Questions 1) Which of Newton's laws best explains why motorists should buckle-up? A) the first law
. Slope of a Line (-) 67. 600 68. 00. SLOPE OF A LINE In this section In Section. we saw some equations whose graphs were straight lines. In this section we look at graphs of straight lines in more detail
Chapter 4: Newton s Laws: Explaining Motion 1. All except one of the following require the application of a net force. Which one is the exception? A. to change an object from a state of rest to a state
Chapter 1 INVESTIGATIONS AND FUNCTIONS 1.1.1 1.1.4 This opening section introduces the students to man of the big ideas of Algebra 2, as well as different was of thinking and various problem solving strategies.
A cannon shoots a clown directly upward with a speed of 20 m/s. What height will the clown reach? How much time will the clown spend in the air? Projectile Motion 1:Horizontally Launched Projectiles Two
KINEMTICS OF PRTICLES RELTIVE MOTION WITH RESPECT TO TRNSLTING XES In the previous articles, we have described particle motion using coordinates with respect to fixed reference axes. The displacements,
Solutions to old Exam 1 problems Hi students! I am putting this old version of my review for the first midterm review, place and time to be announced. Check for updates on the web site as to which sections
1. A mixed martial artist kicks his opponent in the nose with a force of 200 newtons. Identify the action-reaction force pairs in this interchange. (A) foot applies 200 newton force to nose; nose applies
Summer Math Eercises For students who are entering Pre-Calculus It has been discovered that idle students lose learning over the summer months. To help you succeed net fall and perhaps to help you learn
Supplemental Questions The fastest of all fishes is the sailfish. If a sailfish accelerates at a rate of 14 (km/hr)/sec [fwd] for 4.7 s from its initial velocity of 42 km/h [fwd], what is its final velocity?
Derivatives as Rates of Change One-Dimensional Motion An object moving in a straight line For an object moving in more complicated ways, consider the motion of the object in just one of the three dimensions
CONDENSED LESSON 7.1 Polynomial Degree and Finite Differences In this lesson you will learn the terminology associated with polynomials use the finite differences method to determine the degree of a polynomial
Module 8 Lesson 4: Applications of Vectors So now that you have learned the basic skills necessary to understand and operate with vectors, in this lesson, we will look at how to solve real world problems
EXPERIMENT 3 Analysis of a freely falling body Dependence of speed and position on time Objectives to verify how the distance of a freely-falling body varies with time to investigate whether the velocity
Big Bend Community College Beginning Algebra MPC 095 Lab Notebook Beginning Algebra Lab Notebook by Tyler Wallace is licensed under a Creative Commons Attribution 3.0 Unported License. Permissions beyond
Class: Date: Chapter 07 Test A Multiple Choice Identify the choice that best completes the statement or answers the question. 1. An example of a vector quantity is: a. temperature. b. length. c. velocity.
Name Period Date Show all work neatly on separate paper. (You may use both sides of your paper.) Problems should be labeled clearly. If I can t find a problem, I ll assume it s not there, so USE THE TEMPLATE
1. When driving your car Into traffic from a parked position, you should: A. Sound your horn and pull Into the other lane. B. Signal and proceed when safe. C. Signal other traffic and pull directly into
8. Potential Energy and Conservation of Energy Potential Energy: When an object has potential to have work done on it, it is said to have potential energy, e.g. a ball in your hand has more potential energy
I. A mechanical device shakes a ball-spring system vertically at its natural frequency. The ball is attached to a string, sending a harmonic wave in the positive x-direction. +x a) The ball, of mass M,
Get to Know Golf! John Dunigan Get to Know Golf is an initiative designed to promote the understanding the laws that govern ball flight. This information will help golfers develop the most important skill
The Physics and Math of Ping-pong and How It Affects Game Play 1 The Physics and Math of Ping-pong and How It Affects Game Play By: Connor Thompson & Andrew Johnson The Practical Applications of Advanced
Mathematics 31 Pre-calculus and Limits Overview After completing this section, students will be epected to have acquired reliability and fluency in the algebraic skills of factoring, operations with radicals
Chapter 2 Solutions 4. We find the aerage elocity from = (x 2 x 1 )/(t 2 t 1 ) = ( 4.2 cm 3.4 cm)/(6.1 s 3.0 s) = 2.5 cm/s (toward x). 6. (a) We find the elapsed time before the speed change from speed
Individuals: The objects that are described by a set of data. They may be people, animals, things, etc. (Also referred to as Cases or Records) Variables: The characteristics recorded about each individual.
476 CHAPTER 7 Graphs, Equations, and Inequalities 7. Quadratic Equations Now Work the Are You Prepared? problems on page 48. OBJECTIVES 1 Solve Quadratic Equations by Factoring (p. 476) Solve Quadratic
Science teaching unit Disclaimer The Department for Children, Schools and Families wishes to make it clear that the Department and its agents accept no responsibility for the actual content of any materials
Revised form 081695R Force Concept Inventory Originally published in The Physics Teacher, March 1992 by David Hestenes, Malcolm Wells, and Gregg Swackhamer Revised August 1995 by Ibrahim Halloun, Richard
2 ONE- DIMENSIONAL MOTION Chapter 2 One-Dimensional Motion Objectives After studying this chapter you should be able to derive and use formulae involving constant acceleration; be able to understand the
CHAPTER 2 1* What is the approximate average velocity of the race cars during the Indianapolis 500? Since the cars go around a closed circuit and return nearly to the starting point, the displacement is
MASSACHUSETTS INSTITUTE OF TECHNOLOGY Physics Department 8.01L: Physics I November 7, 2015 Prof. Alan Guth Problem Set #8 Solutions Due by 11:00 am on Friday, November 6 in the bins at the intersection
Mathematics on the Soccer Field Katie Purdy Abstract: This paper takes the everyday activity of soccer and uncovers the mathematics that can be used to help optimize goal scoring. The four situations that
R. Polnomials In this section we want to review all that we know about polnomials. We start with the basic operations on polnomials, that is adding, subtracting, and multipling. Recall, to add subtract
[ Assignment View ] [ Pri Eðlisfræði 2, vor 2007 28. Sources of Magnetic Field Assignment is due at 2:00am on Wednesday, March 7, 2007 Credit for problems submitted late will decrease to 0% after the deadline
Student Name: Teacher: District: Date: Miami-Dade County Public Schools Assessment: 9_12 Mathematics Algebra II Interim 2 Description: Mid-Year 2014 - Algebra II Form: 201 1. During a physics experiment,
G.1 EE1.el3 (EEE1023): Electronics III Mechanics lecture 7 Moment of a force, torque, equilibrium of a body Dr Philip Jackson http://www.ee.surrey.ac.uk/teaching/courses/ee1.el3/ G.2 Moments, torque and
Instructor s Solutions Manual, Section 5.1 Exercise 1 Solutions to Exercises, Section 5.1 1. Find all numbers t such that ( 1 3,t) is a point on the unit circle. For ( 1 3,t)to be a point on the unit circle
Pre Calculus Worksheet. 1. Which of the 1 parent functions we know from chapter 1 are power functions? List their equations and names.. Analyze each power function using the terminology from lesson 1-.
EXPERIMENTAL ERROR AND DATA ANALYSIS 1. INTRODUCTION: Laboratory experiments involve taking measurements of physical quantities. No measurement of any physical quantity is ever perfectly accurate, except
Section 4: The Basics of Satellite Orbits MOTION IN SPACE VS. MOTION IN THE ATMOSPHERE The motion of objects in the atmosphere differs in three important ways from the motion of objects in space. First,
This booklet will discuss some of the principles involved in the design of a roller coaster. It is intended for the middle or high school teacher. Physics students may find the information helpful as well.
Lecture 11: Chapter 5, Section 3 Relationships between Two Quantitative Variables; Correlation Display and Summarize Correlation for Direction and Strength Properties of Correlation Regression Line Cengage
Notes on indifference curve analysis of the choice between leisure and labor, and the deadweight loss of taxation Jon Bakija This example shows how to use a budget constraint and indifference curve diagram
Jump Shot Mathematics Howard Penn Abstract In this paper we exae variations of standard calculus problems in the context of shooting a basketball jump shot. We believe that many students will find this
MFF 3a: Charged Particle and a Straight Current-Carrying Wire... 2 MFF3a RT1: Charged Particle and a Straight Current-Carrying Wire... 3 MFF3a RT2: Charged Particle and a Straight Current-Carrying Wire...
Experiment 0 Introduction to Data Analysis Using an Excel Spreadsheet I. Purpose The purpose of this introductory lab is to teach you a few basic things about how to use an EXCEL 2010 spreadsheet to do
Unit 7 Quadratic Relations of the Form y = ax 2 + bx + c Lesson Outline BIG PICTURE Students will: manipulate algebraic expressions, as needed to understand quadratic relations; identify characteristics
Physics Lab Report Guidelines Summary The following is an outline of the requirements for a physics lab report. A. Experimental Description 1. Provide a statement of the physical theory or principle observed