RATE OF CHANGE AND INSTANTANEOUS VELOCITY
|
|
- Doreen Wells
- 5 years ago
- Views:
Transcription
1 RATE OF CHANGE AND INSTANTANEOUS VELOCITY Section 2.2A Calculus AP/Dual, Revised /30/2018 1:34 AM 2.2A: Rates of Change 1
2 AVERAGE VELOCITY A. Rates of change play a role whenever we study the relationship between two changing quantities. A familiar example is velocity, which is the rate of change of position with respect to time. B. If an object is traveling in a straight line, the average velocity over a given time interval from point A to point B C. We cannot define instantaneous velocity as a ratio because we would have to divide by the length of the time interval, which is zero. However we can estimate the instantaneous velocity by computing the average velocity over successively smaller intervals. 7/30/2018 1:34 AM 2.2A: Rates of Change 2
3 AVERAGE RATE OF CHANGE A. The average rate of change describes the constant rate at which a function would have to change over an interval if the function were to achieve the same vertical displacement over the length of the interval investigated. B. Average Velocity Formula: ഥx s t f b f a b a Change in Position Time Elapsed 7/30/2018 1:34 AM 2.2A: Rates of Change 3
4 STEPS IN AVERAGE RATE OF CHANGE A. Plug your a and b in to get the corresponding f a and f b. B. Plug into the formula and reduce. C. Average Rate of Change uses only endpoints of the interval. It does not reflect any fluctuations within the interval. D. Label. 7/30/2018 1:34 AM 2.2A: Rates of Change 4
5 EXAMPLE 1 A car travels along a straight road and the driver observes the mile markers along the trip as shown in the table. What are the units for the average speed over the 24 minutes? Explain your answer. Time (in min.) Mile Marker The average speed is Miles f b f a y b a x Minute f b f a b a min miles 3 2 miles per minute over 24 minutes 7/30/2018 1:34 AM 2.2A: Rates of Change 5
6 EXAMPLE 2 Water is pumped into a tank at a rate modeled by W t 2000e t2 /20 liters per hour for 0 t 8, where t is measured in hours. Using the table below, estimate R 2 and show all work that leads to the answer. Explain your answer. t (hours) R t (Liters/Hr) R ' f ( a) f b b R '2 a 7/30/2018 1:34 AM 2.2A: Rates of Change 6 R '2 liters liters hour hour 3 hours 1 hour f ( 3) f ( 1) 3 1
7 EXAMPLE 2 Water is pumped into a tank at a rate modeled by W t 2000e t2 /20 liters per hour for 0 t 8, where t is measured in hours. Using the table below, estimate R 2 and show all work that leads to the answer. Explain your answer. R '2 t (hours) R t (Liters/Hr) liters liters hour hour 3 hours 1 hour liters 240 hour liters R ' hours hour 2 liters The rate of change is 120 ( 120 liters per hour per hour ) at t 2. 2 hour 7/30/2018 1:34 AM 2.2A: Rates of Change 7
8 YOUR TURN Jo Ann jogs along a straight path. For 0 t 40, Jo s speed is given by a differential function v t where t is measured in minutes and v t is measured in meters per minute. Using the table below, estimate v 16 and show all work that leads to the answer. Explain your answer and label accordingly. t (Min) v t (Met/Min) meters The rate of change is 5 or 5 meters per min per min at t min 7/30/2018 1:34 AM 2.2A: Rates of Change 8
9 EXAMPLE 3 A tennis ball is dropped from a height of 100 feet, its height s at time t is given by the position function, d t 16t where d is measured in feet and t is measured in seconds. Find the average velocity over the time interval of 1, 2. f ( a) f b d b a 2 d ft./ sec 7/30/2018 1:34 AM 2.2A: Rates of Change 9
10 EXAMPLE 3 A tennis ball is dropped from a height of 100 feet, its height s at time t is given by the position function, d t 16t where d is measured in feet and t is measured in seconds. Find the average velocity over the time interval of 1, 2. 7/30/2018 1:34 AM 2.2A: Rates of Change 10
11 EXAMPLE 4 If an object is dropped from a tall building, then the distance it has fallen after t seconds is given by the function d t 16t 2. Find its average speed (average rate of change) between a and a + h seconds. 16 7/30/2018 1:34 AM 2.2A: Rates of Change 11
12 YOUR TURN Suppose you are driving to a friend s house. After 30 minutes, you are 15 miles from your house. After 3 hours, you are 60 miles from your house. What is your average rate of change between these times (in hours)? Label accordingly. s ' t 18 miles / hour 7/30/2018 1:34 AM 2.2A: Rates of Change 12
13 INSTANTANEOUS VELOCITY A. If we look at a graph of the position of the object, a secant line can be drawn through the points t 1, f a and t 2, f b. The average velocity of the object would be the slope of the secant line through t 1, f a and t 2, f b. B. If the secant line from t 2 is moved closer to t 1, the secant line gets closer and closer to becoming a tangent line to the graph at t 1, f a. The slope of the tangent line is the instantaneous velocity of the object when the time is t 1 C. The instantaneous rate of change at is the limit of the average rates of change as gets closer and closer to We must consider values of on both the left side of and on the right side of as we take the limit. D. An expression such as s quotient. t f b f a b a is called a difference 7/30/2018 1:34 AM 2.2A: Rates of Change 13
14 INSTANTANEOUS VELOCITY A. Velocity of an object at time t is found by taking the DERIVATIVE of the position function. B. VELOCITY is a vector, which means it has direction. So, velocity can be positive, negative, or zero. C. SPEED is the absolute value of velocity, and cannot be negative. s t+ t s t D. Instantaneous Velocity Formula: lim t 0 t known as ds. dt s t v t. It is also 7/30/2018 1:34 AM 2.2A: Rates of Change 14
15 AVERAGE VELOCITY VS. INSTANTANEOUS VELOCITY Average Velocity Solve using Algebra Slope - (Not much Calculus needed) Instantaneous Velocity s t Take the Derivative 7/30/2018 1:34 AM 2.2A: Rates of Change 15
16 EXAMPLE 5 At time t 0, a diver jumps from a platform diving board that is 32 feet above the water. The position of the diver is given by s t 16t t + 32 where s is measured in feet and t is measured in seconds. (a) Graph the function on the graphing calculator and sketch the result below. (b) Find the average velocity (average rate of change) at which the diver is moving for the time interval 1, (c) Use your answers to (b) to estimate the instantaneous rate of change (instantaneous velocity) at which the diver is moving at time t 1 second. (d) When does the diver hit the water? (e) What is the diver s velocity at impact? What is their speed? (f) When does the diver stop moving upward and start their descent? 7/30/2018 1:34 AM 2.2A: Rates of Change 16
17 EXAMPLE 5A At time t 0, a diver jumps from a platform diving board that is 32 feet above the water. The position of the diver is given by s t 16t t + 32 where s is measured in feet and t is measured in seconds. (a) Graph the function on the graphing calculator and sketch the result below. 7/30/2018 1:34 AM 2.2A: Rates of Change 17
18 EXAMPLE 5B At time t 0, a diver jumps from a platform diving board that is 32 feet above the water. The position of the diver is given by s t 16t t + 32 where s is measured in feet and t is measured in seconds. (b) Find the average velocity (average rate of change) at which the diver is moving for the time interval 1, f ( a) f b s b a 1.5 s ft./ sec 7/30/2018 1:34 AM 2.2A: Rates of Change 18
19 EXAMPLE 5C At time t 0, a diver jumps from a platform diving board that is 32 feet above the water. The position of the diver is given by s t 16t t + 32 where s is measured in feet and t is measured in seconds. (c) Use your answers to (b) to estimate the instantaneous rate of change (instantaneous velocity) at which the diver is moving at time t 1 second. 2 s t 16t + 16t + 32 ds 2 v t 16t 16t 32 dt + + ds 2 ds ds v t 16t + 16t + 32 dt dt dt v t t /30/2018 1:34 AM 2.2A: Rates of Change 19
20 EXAMPLE 5C At time t 0, a diver jumps from a platform diving board that is 32 feet above the water. The position of the diver is given by s t 16t t + 32 where s is measured in feet and t is measured in seconds. (c) Use your answers to (b) to estimate the instantaneous rate of change (instantaneous velocity) at which the diver is moving at time t 1 second. v t 32t + 16 v t 32t + 16 v( t ) 32( 1) + 16 v t 16 ft. / sec. 7/30/2018 1:34 AM 2.2A: Rates of Change 20
21 EXAMPLE 5D At time t 0, a diver jumps from a platform diving board that is 32 feet above the water. The position of the diver is given by s t 16t t + 32 where s is measured in feet and t is measured in seconds. (d) When does the diver hit the water? s t 16t + 16t t + 16t+ 32 ( 2 t t ) t 2 t+ 1 t 2sec. 7/30/2018 1:34 AM 2.2A: Rates of Change 21
22 EXAMPLE 5E At time t 0, a diver jumps from a platform diving board that is 32 feet above the water. The position of the diver is given by s t 16t t + 32 where s is measured in feet and t is measured in seconds. (e) What is the diver s velocity at impact? What is their speed? v t 32t + 16 v ( 2) 32( 2) + 16 speed 48 ft. / sec 7/30/2018 1:34 AM 2.2A: Rates of Change 22
23 EXAMPLE 5F At time t 0, a diver jumps from a platform diving board that is 32 feet above the water. The position of the diver is given by s t 16t t + 32 where s is measured in feet and t is measured in seconds. (f) When does the diver stop moving upward and start their descent? v t 32t t t 16 t 1 sec. 2 7/30/2018 1:34 AM 2.2A: Rates of Change 23
24 EXAMPLE 6 The temperature T, in degrees Fahrenheit, of a cold potato placed in a hot oven is given by T f t, where t is the time in minutes since the potato was put in the oven. What is the meaning of the statement f 20 2? f ( t) T f '( 20) 2 At time t 20 minutes, the temperature of the potato is increasing at a rate of 2 F per minute. 7/30/2018 1:34 AM 2.2A: Rates of Change 24
25 AP FREE RESPONSE QUESTION 1 (NON-CALCULATOR) Let f be a function that is twice differentiable for all real numbers. The table above gives values of f for selected points in the closed interval, 2, 13. Estimate f 4. Show the work that leads to the answer. f f ' x '4 f b f f a b a 5 f x f(x) f f '4 '4 f '( 4) ( 5) ( 3) 6 2 7/30/2018 1:34 AM 2.2A: Rates of Change 25
26 AP FREE RESPONSE QUESTION 2 (NON-CALCULATOR) W gallons At 8 hours, there are 221 gallons of water W At 12 hours, there are 257 gallons of water 7/30/2018 1:34 AM 2.2A: Rates of Change 26
27 AP FREE RESPONSE QUESTION 2 (NON-CALCULATOR) W ' 15 f 16 f W ' W ' gallons / hour At t 15 hours, the rate of change of water flowing is 9.25 gallons per hour 7/30/2018 1:34 AM 2.2A: Rates of Change 27
28 ASSIGNMENT Page all, 105 7/30/2018 1:34 AM 2.2A: Rates of Change 28
A position graph will give the location of an object at a certain time.
Calculus 3.4 Notes A position graph will give the location of an object at a certain time. At t = 4, the car is 20 miles away from where it started. A position function is usually written as or. If the
More informationAP Physics B Summer Homework (Show work)
#1 NAME: AP Physics B Summer Homework (Show work) #2 Fill in the radian conversion of each angle and the trigonometric value at each angle on the chart. Degree 0 o 30 o 45 o 60 o 90 o 180 o 270 o 360 o
More information3. Answer the following questions with your group. How high do you think he was at the top of the stairs? How did you estimate that elevation?
J Hart Interactive Algebra 1 Classwork Exploratory Challenge 1. Watch the first 1:08 minutes of the video below and describe in words the motion of the man. Elevation vs. Time #2 [http://www.mrmeyer.com/graphingstories1/graphingstories2.mov.
More information3. Answer the following questions with your group. How high do you think he was at the top of the stairs? How did you estimate that elevation?
Classwork Exploratory Challenge 1. Watch the first 1:08 minutes of the video below and describe in words the motion of the man. Elevation vs. Time #2 [http://www.mrmeyer.com/graphingstories1/graphingstories2.mov.
More informationThe speed of an inline skater is usually described in meters per second. The speed of a car is usually described in kilometers per hour.
The speed of an inline skater is usually described in meters per second. The speed of a car is usually described in kilometers per hour. Speed How are instantaneous speed and average speed different? Average
More informationMHF 4U Unit 2 Rational Functions Outline
MHF 4U Unit Rational Functions Outline Day 1 Lesson Title Specific Epectations Rational Functions and Their Essential Characteristics C.1,.,.3 (Lesson Included) Rational Functions and Their Essential Characteristics
More informationCh. 2 & 3 Velocity & Acceleration
Ch. 2 & 3 Velocity & Acceleration Objective: Student will be able to Compare Velocity to Speed Identify what is acceleration Calculate velocity and acceleration from an equation and from slope of a graph.
More informationPage 2. Indicate with ticks ( ) in the table below which of the quantities are vectors and which are scalars. Velocity Speed Distance Displacement
Q1.(a) Indicate with ticks ( ) in the table below which of the quantities are vectors and which are scalars. Velocity Speed Distance Displacement vector scalar (b) A tennis ball is thrown vertically downwards
More informationx 2 = (60 m) 2 + (60 m) 2 x 2 = 3600 m m 2 x = m
3.1 Track Question a) Distance Traveled is 1600 m. This is length of the path that the person took. The displacement is 0 m. The person begins and ends their journey at the same position. They did not
More informationThe purpose of this experiment is to find this acceleration for a puck moving on an inclined air table.
Experiment : Motion in an Inclined Plane PURPOSE The purpose of this experiment is to find this acceleration for a puck moving on an inclined air table. GENERAL In Experiment-1 you were concerned with
More informationStudent Exploration: Distance-Time and Velocity-Time Graphs
Name: Date: Student Exploration: Distance-Time and Velocity-Time Graphs [NOTE TO TEACHERS AND STUDENTS: This lesson was designed as a follow-up to the Distance-Time Graphs Gizmo. We recommend you complete
More informationKinematics-Projectiles
1. A volleyball hit into the air has an initial speed of 10 meters per second. Which vector best represents the angle above the horizontal that the ball should be hit to remain in the air for the greatest
More informationPhysical Science You will need a calculator today!!
Physical Science 11.3 You will need a calculator today!! Physical Science 11.3 Speed and Velocity Speed and Velocity Speed The ratio of the distance an object moves to the amount of time the object moves
More informationMotion. 1 Describing Motion CHAPTER 2
CHAPTER 2 Motion What You ll Learn the difference between displacement and distance how to calculate an object s speed how to graph motion 1 Describing Motion 2(D), 4(A), 4(B) Before You Read Have you
More informationa. Determine the sprinter's constant acceleration during the first 2 seconds. b. Determine the sprinters velocity after 2 seconds have elapsed.
AP Physics 1 FR Practice Kinematics 1d 1 The first meters of a 100-meter dash are covered in 2 seconds by a sprinter who starts from rest and accelerates with a constant acceleration. The remaining 90
More informationPhysics 2204 Worksheet 6.5: Graphical Analysis of Non- Uniform Motion D-T GRAPH OF NON-UNIFORM MOTION (ACCELERATING) :
Physics 2204 Worksheet 6.5: Graphical Analysis of Non- Uniform Motion D-T GRAPH OF NON-UNIFORM MOTION (ACCELERATING) : The d-t graph for uniformly Accelerated motion is definitely not the same as a d-t
More informationREVIEW : KINEMATICS
1 REVIEW 5-4-16: KINEMATICS Kinematics-Defining Motion 1 A student on her way to school walks four blocks east, three blocks north, and another four blocks east, as shown in the diagram. Compared to the
More informationREAL LIFE GRAPHS M.K. HOME TUITION. Mathematics Revision Guides Level: GCSE Higher Tier
Mathematics Revision Guides Real Life Graphs Page 1 of 19 M.K. HOME TUITION Mathematics Revision Guides Level: GCSE Higher Tier REAL LIFE GRAPHS Version: 2.1 Date: 20-10-2015 Mathematics Revision Guides
More informationBall Toss. Vernier Motion Detector
Experiment 6 When a juggler tosses a ball straight upward, the ball slows down until it reaches the top of its path. The ball then speeds up on its way back down. A graph of its velocity vs. time would
More informationPerformance Task # 1
Performance Task # 1 Goal: Arrange integers in order. Role: You are a analyzing a Julie Brown Anderson s dive. Audience: Reader of article. Situation: You are interviewing for a job at a sports magazine.
More informationMotion, Vectors, and Projectiles Review. Honors Physics
Motion, Vectors, and Projectiles Review Honors Physics The graph below represents the relationship between velocity and time of travel for a toy car moving in a straight line. The shaded area under the
More informationFor example, the velocity at t = 10 is given by the gradient of the curve at t = 10, 10 t
R15 INTERPRET THE GRADIENT AT A POINT ON A CURVE AS THE INSTANTANEOUS RATE OF CHANGE; APPLY THE CONCEPTS OF AVERAGE AND INSTANTANEOUS RATE OF CHANGE (GRADIENTS OF CHORDS AND TANGENTS) IN NUMERICAL, ALGEBRAIC
More informationMotion in 1 Dimension
A.P. Physics 1 LCHS A. Rice Unit 1 Displacement, Velocity, & Acceleration: Motion in 1 Dimension In-Class Example Problems and Lecture Notes 1. Freddy the cat started at the 3 meter position. He then walked
More informationEg.#1 A diver steps off a 10. m. high diving board with an initial vertical velocity of zero and experiences an average acceleration 2
1.3.1 Acceleration due to Gravity Defined as: For many years, it was thought that higher mass objects fall towards the Earth more quickly than lower mass objects. This idea was introduced in approximately
More informationUnit 2 Review: Projectile Motion
Name: Unit 2 Review: Projectile Motion Date: 1. A projectile is fired from a gun near the surface of Earth. The initial velocity of the projectile has a vertical component of 98 meters per second and a
More informationHomework Helpers Sampler
Homework Helpers Sampler This sampler includes s for Algebra I, Lessons 1-3. To order a full-year set of s visit >>> http://eurmath.link/homework-helpers Published by the non-profit Great Minds. Copyright
More informationPractice Test: Vectors and Projectile Motion
ame: Practice Test: Vectors and Projectile Motion Part A: Multiple Choice [15 points] 1. A projectile is launched at an angle of 30 0 above the horizontal. eglecting air resistance, what are the projectile
More informationMovement and Position
Movement and Position Syllabus points: 1.2 plot and interpret distance-time graphs 1.3 know and use the relationship between average speed, distance moved and 1.4 describe experiments to investigate the
More information4-3 Rate of Change and Slope. Warm Up. 1. Find the x- and y-intercepts of 2x 5y = 20. Describe the correlation shown by the scatter plot. 2.
Warm Up 1. Find the x- and y-intercepts of 2x 5y = 20. Describe the correlation shown by the scatter plot. 2. Objectives Find rates of change and slopes. Relate a constant rate of change to the slope of
More information4-3 Rate of Change and Slope. Warm Up Lesson Presentation. Lesson Quiz
4-3 Rate of Change and Slope Warm Up Lesson Presentation Lesson Quiz Holt Algebra McDougal 1 Algebra 1 Warm Up 1. Find the x- and y-intercepts of 2x 5y = 20. x-int.: 10; y-int.: 4 Describe the correlation
More informationPhysics: 3. Velocity & Acceleration. Student Notes
Physics: 3. Velocity & Acceleration Please remember to photocopy 4 pages onto one sheet by going A3 A4 and using back to back on the photocopier Syllabus OP1 Perform simple calculations based on speed,
More information3. Approximately how far will an object near Earth's surface fall in 3.0 seconds? m m m m
Page 1 of 5 Sub work 10-10-02 Name 12-OCT-03 1. A car travels a distance of 98 meters in 10. seconds. What is the average speed of the car during this 10.-second interval? 1. 4.9 m/s 3. 49 m/s/ 2. 9.8
More information2015 AQA A Level Physics. Motion Introduction
2015 AQA A Level Physics Motion Introduction 9/22/2018 Distance and Displacement Distance is the actual path length that is taken Displacement is the change in position x = xf x 0 Where x is the displacement,
More informationWhere are you right now? How fast are you moving? To answer these questions precisely, you
4.1 Position, Speed, and Velocity Where are you right now? How fast are you moving? To answer these questions precisely, you need to use the concepts of position, speed, and velocity. These ideas apply
More information2. A car, starting from rest, accelerates in a straight-line path at a constant rate of 2.0 m/s 2. How far will the car travel in 12 seconds?
Name: Date: 1. Carl Lewis set a world record for the 100.0-m run with a time of 9.86 s. If, after reaching the finish line, Mr. Lewis walked directly back to his starting point in 90.9 s, what is the magnitude
More informationJeddah Knowledge International School. Science Revision Pack Answer Key Quarter 3 Grade 10
Jeddah Knowledge International School Science Revision Pack Answer Key 2016-2017 Quarter 3 Grade 10 Name: Section: ANSWER KEY- SCIENCE GRADE 10, QUARTER 3 1 1. What are the units for mass? A Kilograms
More informationPHYSICS 12 NAME: Kinematics and Projectiles Review
NAME: Kinematics and Projectiles Review (1-3) A ball is thrown into the air, following the path shown in the diagram. At 1, the ball has just left the thrower s hand. At 5, the ball is at its original
More informationHigher Projectile Motion Questions
Higher Projectile Motion Questions 1. a) Name the two components of motion in projectiles. b) What is the acceleration on Earth for each of these two components. 2. A pencil case is dropped vertically
More informationAdd this important safety precaution to your normal laboratory procedures:
Student Activity Worksheet Speed and Velocity Are You Speeding? Driving Question What is speed and how is it related to velocity? Materials and Equipment For each student or group: Data collection system
More informationPLEASE MARK YOUR ANSWERS WITH AN X, not a circle! 1. (a) (b) (c) (d) (e) 2. (a) (b) (c) (d) (e) (a) (b) (c) (d) (e) 4. (a) (b) (c) (d) (e)...
Math 10170, Exam 2 April 25, 2014 The Honor Code is in effect for this examination. All work is to be your own. You may use your Calculator. The exam lasts for 50 minutes. Be sure that your name is on
More informationCHAPTER 1. Knowledge. (a) 8 m/s (b) 10 m/s (c) 12 m/s (d) 14 m/s
CHAPTER 1 Review K/U Knowledge/Understanding T/I Thinking/Investigation C Communication A Application Knowledge For each question, select the best answer from the four alternatives. 1. Which is true for
More information½ 3. 2/3 V (1/3 (1/2V)+1/3(V)+1/3(1/2V))
TEST 2 Q 1 some HONORS review questions to try Define: displacement, velocity, average velocity, average speed, acceleration. Displacement: change in distance from start (with direction) Velocity: change
More informationA tennis player hits a ball at a height of 2.4 m. The ball has an initial horizontal velocity.
1991 Q31 A tennis player hits a ball at a height of 2.4 m. The ball has an initial horizontal velocity. The ball just passes over the net which is 0.6 m high and 6 m away from her. (Neglect air friction.)
More information1. A tiger leaps horizontally from a 7.5 meter high rock with a speed of 4.5 m/s. How far from the base of the rock will she land?
HONORS PHYSICS PROJECTILE MOTION 1. A tiger leaps horizontally from a 7.5 meter high rock with a speed of 4.5 m/s. How far from the base of the rock will she land? 2. A diver running 1.6 m/s dives out
More information2.5. All games and sports have specific rules and regulations. There are rules about. Play Ball! Absolute Value Equations and Inequalities
Play Ball! Absolute Value Equations and Inequalities.5 LEARNING GOALS In this lesson, you will: Understand and solve absolute values. Solve linear absolute value equations. Solve and graph linear absolute
More information2011 Excellence in Mathematics Contest Team Project Level I (Precalculus and above) School Name: Group Members:
011 Excellence in Mathematics Contest Team Project Level I (Precalculus and above) School Name: Group Members: Reference Sheet Formulas and Facts You may need to use some of the following formulas and
More informationtime v (vertical) time
NT4E-QRT20: PROJECTILE MOTION FOR TWO ROCKS VELOCITY AND ACCELERATION GRAPHS II Two identical rocks are thrown horizontally from a cliff with Rock A having a greater velocity at the instant it is released
More informationHONORS PHYSICS One Dimensional Kinematics
HONORS PHYSICS One Dimensional Kinematics LESSON OBJECTIVES Be able to... 1. use appropriate metric units and significant figures for given measurements 2. identify aspects of motion such as position,
More informationChapter 4: 2-D Kinematics
PHY 5 Ch 4. Solution Dr. Hael Shehadeh. Chapter 4: -D Kinematics Answers to Conceptual Questions. The component of velocit is first positive and then negative in a smmetric fashion. As a result, the average
More informationName Date. 5. In each pair, which rational number is greater? Explain how you know.
Master 3.18 Extra Practice 1 Lesson 3.1: What Is a Rational Number? 1. Which of the following numbers are equal to? 2. Write the rational number represented by each letter as a decimal. 3. Write the rational
More information2008 Excellence in Mathematics Contest Team Project B. School Name: Group Members:
2008 Excellence in Mathematics Contest Team Project B School Name: Group Members: Lassie - TEAM PROJECT B 2008 Excellence in Mathematics Contest The Team Project is a group activity in which the students
More informationPiecewise Functions. Updated: 05/15/10
Connecting Algebra 1 to Advanced Placement* Mathematics A Resource and Strategy Guide Updated: 05/15/ Objectives: Students will review linear functions and their properties and be introduced to piecewise
More informationCC Investigation 1: Graphing Proportions
CC Investigation 1: Graphing Proportions DOMAIN: Ratios and Proportional Relationships Problem 1.1 During the first basketball game of the season, Karl made 3 of his 5 free-throw attempts. Karl then made
More information6.6 Gradually Varied Flow
6.6 Gradually Varied Flow Non-uniform flow is a flow for which the depth of flow is varied. This varied flow can be either Gradually varied flow (GVF) or Rapidly varied flow (RVF). uch situations occur
More informationRegents Exam Practice: Measurement, Kinematics, Free Fall, PJM, and UCM
Regents Exam Practice: Measurement, Kinematics, Free Fall, PJM, and UCM 1. Which quantity and unit are correctly paired? 2. Which is a derived unit? meter second kilogram Newton 3. The fundamental unit
More informationVocabulary. Page 1. Distance. Displacement. Position. Average Speed. Average Velocity. Instantaneous Speed. Acceleration
Vocabulary Term Definition Distance Displacement Position Average Speed Average Velocity Instantaneous Speed Acceleration Page 1 Homer walked as follows: Starting at the 0,0 coordinate, he walked 12 meters
More informationLINEAR MOTION. General Review
LINEAR MOTION General Review 1. Velocity is to speed as displacement is to A. acceleration B. momentum C. time D. distance 1. Velocity is to speed as displacement is to A. acceleration B. momentum C. time
More informationThe diagram below represents the path of a stunt car that is driven off a cliff, neglecting friction.
1. A baseball is thrown at an angle of 40.0 above the horizontal. The horizontal component of the baseball s initial velocity is 12.0 meters per second. What is the magnitude of the ball s initial velocity?
More information1 An object moves at a constant speed of 6 m/s. This means that the object:
Slide 1 / 57 1 n object moves at a constant speed of 6 m/s. This means that the object: Increases its speed by 6 m/s every second ecreases its speed by 6 m/s every second oesn t move Has a positive acceleration
More information1. A rabbit can cover a distance of 80 m in 5 s. What is the speed of the rabbit?
Chapter Problems Motion at Constant Speed Class Work. A rabbit can cover a distance of 80 m in 5 s. What is the speed of the rabbit?. During the first 50 s a truck traveled at constant speed of 5 m/s.
More informationEXSC 408L Fall '03 Problem Set #2 Linear Motion. Linear Motion
Problems: 1. Once you have recorded the calibration frame for a data collection, why is it important to make sure the camera does not shut off? hat happens if the camera automatically shuts off after being
More informationmass of container full of air = g mass of container with extra air = g volume of air released = cm 3
1992 Q32 The air pressure inside the passenger cabin of an airliner is 9 x 10 4 Pa when the airliner is at its cruising height. The pressure of the outside atmosphere at this height is 4 x 10 4 Pa. Calculate
More information2011 Excellence in Mathematics Contest Team Project Level I (Precalculus and above)
011 Excellence in Mathematics Contest Team Project Level I (Precalculus and above) School Name: Solutions Group Members: Reference Sheet Formulas and Facts You may need to use some of the following formulas
More informationCalculus 12: Evaluation 3 Outline and Review
Calculus 12: Evaluation 3 Outline and Review You should be able to: 1. Differentiate various types of functions including trigonometric, exponential and logarithmic functions, 2. Solve various related
More informationSection 1. Objectives:
Chapter 2 Motion Objectives: Section 1 Use a frame of reference to describe motion Differentiate between Speed and Velocity Calculate the speed of an object Use graphs to describe speed Observing Motion
More information1. Scuba diver a scuba diver dives straight down into the water quantities: the depth of the dive the water pressure on the diver
HW - Functions and Rates name in #1-10, several situations are given below. for each, answer the following questions: a) does one quantity depend on another? if so, how? b) if the answer to (a) is yes,
More information1. Which one of the following is a vector quantity? A. time B. speed C. energy D. displacement
1. Which one of the following is a vector quantity? A. time B. speed C. energy D. displacement 2. A car is travelling at a constant speed of 26.0 m/s down a slope which is 12.0 to the horizontal. What
More informationChapter 11 Motion. Section 1
Chapter 11 Motion Objectives: Section 1 Use a frame of reference to describe motion Differentiate between Speed and Velocity Calculate the speed of an object Use graphs to describe speed 1 Observing Motion
More informationMoLE Gas Laws Activities
MoLE Gas Laws Activities To begin this assignment you must be able to log on to the Internet using Internet Explorer (Microsoft) 4.5 or higher. If you do not have the current version of the browser, go
More informationLEARNING OBJECTIVES. Overview of Lesson. guided practice Teacher: anticipates, monitors, selects, sequences, and connects student work
D Rate, Lesson 1, Conversions (r. 2018) RATE Conversions Common Core Standard N.Q.A.1 Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units
More informationRiverboat Simulator Activity
Riverboat Simulator Activity Purpose: The purpose of this activity is to analyze the relationship between the two vector components of motion for a river boat as it travels across a river in the presence
More informationPhysics 11 Unit III Practice Test Projectile Motion. Instructions: Pick the best answer available in Part A and Show all your work for Part B
Physics 11 Unit III Practice Test Projectile Motion Instructions: Pick the best answer available in Part A and Show all your work for Part B 1. Which of the following is constant for all projectiles? A.
More informationSHOT ON GOAL. Name: Football scoring a goal and trigonometry Ian Edwards Luther College Teachers Teaching with Technology
SHOT ON GOAL Name: Football scoring a goal and trigonometry 2006 Ian Edwards Luther College Teachers Teaching with Technology Shot on Goal Trigonometry page 2 THE TASKS You are an assistant coach with
More informationQUICK WARM UP: Thursday 3/9
Name: pd: Unit 6, QUICK WARM UP: Thursday 3/9 1) The slope of a distance vs. time graph shows an object s. 2) The slope of a position vs. time graph shows an object s. 3) Can an object have a constant
More informationThe High School Math Project Focus on Algebra. Bottles and Divers. (Rates of Change)
The High School Math Project Focus on Algebra Bottles and Divers (Rates of Change) Objective Students calculate the rate of change between two points on a curve by determining the slope of the line joining
More informationNote! In this lab when you measure, round all measurements to the nearest meter!
Distance and Displacement Lab Note! In this lab when you measure, round all measurements to the nearest meter! 1. Place a piece of tape where you will begin your walk outside. This tape marks the origin.
More information1 A Mangonel is a type of catapult used to launch projectiles such as rocks. A student made a working model of a Mangonel. crossbar. bucket.
1 A Mangonel is a type of catapult used to launch projectiles such as rocks. A student made a working model of a Mangonel. crossbar bucket arm rubber band string scale handle As the handle is turned, the
More informationMidterm Exam: Making a Study Guide
Name: Class: Physics Teacher: Mr. Szopiak Date: Midterm Exam: Making a Study Guide This worksheet will help you and your classmates put together a pretty comprehensive guide to your midterm studying. Your
More informationHOW FAST/FAR DOES FLY LINE FALL? N. Perkins of the University of Michigan, March 2003
HOW FAST/FAR DOES FLY LINE FALL? N. Perkins of the University of Michigan, March 003 This report summarizes a simple model for the free fall dynamics of a length of fly line. The line is assumed to remain
More informationMATH IN ACTION TABLE OF CONTENTS. Lesson 1.1 On Your Mark, Get Set, Go! Page: 10 Usain Bolt: The fastest man on the planet
MATH IN ACTION TABLE OF CONTENTS LESSON 1 WORLD RECORD SPEEDS LINEAR FUNCTIONS WITH PROPORTIONAL RELATIONSHIPS Focus on: SLOPE Lesson 1.1 On Your Mark, Get Set, Go! Page: 10 Usain Bolt: The fastest man
More informationUnit 1 Uniform Velocity & Position-Time Graphs
Name: Unit 1 Uniform Velocity & Position-Time Graphs Hr: Grading: Show all work, keeping it neat and organized. Show equations used and include units in all work. Vocabulary Distance: how far something
More information1. The graph below shows how the velocity of a toy train moving in a straight line varies over a period of time.
1. The graph below shows how the velocity of a toy train moving in a straight line varies over a period of time. v/m s 1 B C 0 A D E H t/s F G (a) Describe the motion of the train in the following regions
More informationObjectives. Materials TI-73 CBL 2
. Objectives Activity 18 To model the cooling rate of different sizes of animals To determine the effect of skin surface area on the cooling rate of animals Materials TI-73 Body Cooling Rate of Animals
More informationPhysics Acceleration and Projectile Review Guide
Physics Acceleration and Projectile Review Guide Name: Major Concepts 1-D motion on the horizontal 1-D motion on the vertical Relationship between velocity and acceleration https://www.khanacademy.org/science/physics/one-dimensional-motion/acceleration-tutorial/a/acceleratio
More information1.0 Converting. 1. What is the speed of a person walking at 3.1 mph in m/s? Show your work and box your answer (check your units)
1.0 Converting There are 1,609.34 meters in one mile. One meter equals 3.28 feet. I mph equals 0.44704 m/s 1. What is the speed of a person walking at 3.1 mph in m/s? Show your work and box your answer
More informationChapter 11 Motion. Displacement-. Always includes Shorter than distance
Chapter 11 Motion Section 1 - an object s change in position relative to a reference point. Observe objects in to other objects. international unit for. Frame of Reference Frame of reference- a system
More informationCHAPTER 10: LINEAR KINEMATICS OF HUMAN MOVEMENT
CHAPTER 10: LINEAR KINEMATICS OF HUMAN MOVEMENT 1. Vector mechanics apply to which of the following? A. displacement B. velocity C. speed D. both displacement and velocity 2. If velocity is constant, then
More informationFigure 1: A hockey puck travels to the right in three different cases.
Question (12) Average velocity and position (update) for three hockey pucks You view the motion of a hockey puck in a video and mark its location every. The resulting image for three different hockey pucks
More informationLINEAR AND ANGULAR KINEMATICS Readings: McGinnis Chapters 2 and 6 DISTANCE, DISPLACEMENT, SPEED, VELOCITY, AND ACCELERATION:
LINEAR AND ANGULAR KINEMATICS Readings: McGinnis Chapters 2 and 6 1 DISTANCE, DISPLACEMENT, SPEED, VELOCITY, AND ACCELERATION: How far? Describing change in linear or angular position Distance (Scalar
More informationUNDERSTANDING A DIVE COMPUTER. by S. Angelini, Ph.D. Mares S.p.A.
UNDERSTANDING A DIVE COMPUTER by S. Angelini, Ph.D. Mares S.p.A. Dive Computer UNDERSTANDING A DIVE COMPUTER The decompression algorithm in a dive computer is an attempt to replicate the effects of a dive
More informationChapter : Linear Motion 2
Text: Chapter 2.5-2.9 Think and Explain: 4-8 Think and Solve: 2-4 Chapter 2.5-2.9: Linear Motion 2 NAME: Vocabulary: constant acceleration, acceleration due to gravity, free fall Equations: s = d t v =
More informationApplying Hooke s Law to Multiple Bungee Cords. Introduction
Applying Hooke s Law to Multiple Bungee Cords Introduction Hooke s Law declares that the force exerted on a spring is proportional to the amount of stretch or compression on the spring, is always directed
More informationNAME:... SCHOOL: LINEAR MOTION. Answer ALL questions in this paper in the spaces provided.
NAME:.... SCHOOL: DATE:... LINEAR MOTION INSTRUCTIONS TO CANDIDATES Answer ALL questions in this paper in the spaces provided. 1. Two forces that act on a moving cyclist are the driving force and the resistive
More informationRiverboat Simulator Activity Sheet
Riverboat Simulator Activity Sheet Purpose: The purpose of this activity is to analyze the relationship between the two vector components of motion for a river boat as it travels across a river in the
More informationHigher, Lower; Faster, Slower? Student Data Page Activity 4B Part 2
Activity Materials: (Per Group): Higher, Lower; Faster, Slower? Student Data Page Activity 4B Part 2 30 cm ramp made of cardboard Meter stick Table Ring stand 16 Washers or weight set Pom-Pom Granny Model
More informationMotion Graphing Packet
Name: Motion Graphing Packet This packet covers two types of motion graphs Distance vs. Time Graphs Velocity vs. Time Graphs Describing the motion of an object is occasionally hard to do with words. Sometimes
More informationQUESTION 1. Sketch graphs (on the axes below) to show: (1) the horizontal speed v x of the ball versus time, for the duration of its flight;
QUESTION 1 A ball is thrown horizontally from a cliff with a speed of 10 ms -1 shown in the diagram at right. Neglecting the effect of air resistance and taking gravitational acceleration to be g +9.8ms
More informationProblem 5: Platform Diving
Problem 5: Platform Diving In the 22 World Cup Trials, Kyle Prandi set up a diving record with a back 3 ½ somersault pike from the 1 m board. He pushed off from the board at an angle of θ = 46 with an
More information1ACE Exercise 4. Name Date Class
1ACE Exercise 4 Investigation 1 4. A farm wants to add a small rectangular petting zoo for the public. They have a fixed amount of fencing to use for the zoo. This graph shows the lengths and areas of
More informationPhysics: Principles and Applications, 6e Giancoli Chapter 3 Kinematics in Two Dimensions; Vectors. Conceptual Questions
Physics: Principles and Applications, 6e Giancoli Chapter 3 Kinematics in Two Dimensions; Vectors Conceptual Questions 1) Which one of the following is an example of a vector quantity? A) distance B) velocity
More information