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Copyright © by Holt, Rinehart and Winston. All rights reserved. ResourcesChapter menu Chapter 2,3,7 MASTER CLASS NOTES FOR MOTION Chapter 2,3 & 7 MASTER.

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Copyright © by Holt, Rinehart and Winston. All rights reserved. ResourcesChapter menu Chapter 1-7 MASTER CLASS NOTES FOR Chapter 1,2,3,5,6 & 7 MASTER NOTES.

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Copyright © by Holt, Rinehart and Winston. All rights reserved. ResourcesChapter menu Chapter 2 Table of Contents Section 1 Displacement and Velocity Section.

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Presentation on theme: "Copyright © by Holt, Rinehart and Winston. All rights reserved. ResourcesChapter menu Chapter 2,3,7 MASTER CLASS NOTES FOR MOTION Chapter 2,3 & 7 MASTER."— Presentation transcript:

1 Copyright © by Holt, Rinehart and Winston. All rights reserved. ResourcesChapter menu Chapter 2,3,7 MASTER CLASS NOTES FOR MOTION Chapter 2,3 & 7 MASTER NOTES

2 Copyright © by Holt, Rinehart and Winston. All rights reserved. ResourcesChapter menu Chapter 2 Table of Contents Section 1 Displacement and Velocity Section 2 Acceleration Section 3 Falling Objects Motion in One Dimension

3 Copyright © by Holt, Rinehart and Winston. All rights reserved. ResourcesChapter menu Section 1 Displacement and Velocity Chapter 2 Objectives Describe motion in terms of frame of reference, displacement, time, and velocity. Calculate the displacement of an object traveling at a known velocity for a specific time interval. Construct and interpret graphs of position versus time.

4 Copyright © by Holt, Rinehart and Winston. All rights reserved. ResourcesChapter menu Section 1 Displacement and Velocity Chapter 2 One Dimensional Motion To simplify the concept of motion, we will first consider motion that takes place in one direction. One example is the motion of a commuter train on a straight track. To measure motion, you must choose a frame of reference. A frame of reference is a system for specifying the precise location of objects in space and time.

5 Copyright © by Holt, Rinehart and Winston. All rights reserved. ResourcesChapter menu Section 1 Displacement and Velocity Chapter 2 Displacement x = x f – x i displacement = final position – initial position Displacement is a change in position. Displacement is not always equal to the distance traveled. The SI unit of displacement is the meter, m.

6 Copyright © by Holt, Rinehart and Winston. All rights reserved. ResourcesChapter menu Chapter 2 Positive and Negative Displacements Section 1 Displacement and Velocity

7 Copyright © by Holt, Rinehart and Winston. All rights reserved. ResourcesChapter menu Section 1 Displacement and Velocity Chapter 2 Average Velocity Average velocity is the total displacement divided by the time interval during which the displacement occurred. In SI, the unit of velocity is meters per second, abbreviated as m/s.

8 Copyright © by Holt, Rinehart and Winston. All rights reserved. ResourcesChapter menu Section 1 Displacement and Velocity Chapter 2 Velocity and Speed Velocity describes motion with both a direction and a numerical value (a magnitude). Speed has no direction, only magnitude. Average speed is equal to the total distance traveled divided by the time interval.

9 Copyright © by Holt, Rinehart and Winston. All rights reserved. ResourcesChapter menu Section 1 Displacement and Velocity Chapter 2 Interpreting Velocity Graphically –Object 1: positive slope = positive velocity –Object 2: zero slope= zero velocity –Object 3: negative slope = negative velocity For any position-time graph, we can determine the average velocity by drawing a straight line between any two points on the graph. If the velocity is constant, the graph of position versus time is a straight line. The slope indicates the velocity.

10 Copyright © by Holt, Rinehart and Winston. All rights reserved. ResourcesChapter menu Section 1 Displacement and Velocity Chapter 2 Interpreting Velocity Graphically, continued The instantaneous velocity at a given time can be determined by measuring the slope of the line that is tangent to that point on the position-versus-time graph. The instantaneous velocity is the velocity of an object at some instant or at a specific point in the objects path.

11 Copyright © by Holt, Rinehart and Winston. All rights reserved. ResourcesChapter menu Section 2 Acceleration Chapter 2 Objectives Describe motion in terms of changing velocity. Compare graphical representations of accelerated and nonaccelerated motions. Apply kinematic equations to calculate distance, time, or velocity under conditions of constant acceleration.

12 Copyright © by Holt, Rinehart and Winston. All rights reserved. ResourcesChapter menu Chapter 2 Changes in Velocity Acceleration is the rate at which velocity changes over time. Section 2 Acceleration An object accelerates if its speed, direction, or both change. Acceleration has direction and magnitude. Thus, acceleration is a vector quantity.

13 Copyright © by Holt, Rinehart and Winston. All rights reserved. ResourcesChapter menu Chapter 2 Changes in Velocity, continued Consider a train moving to the right, so that the displacement and the velocity are positive. The slope of the velocity-time graph is the average acceleration. Section 2 Acceleration –When the velocity in the positive direction is increasing, the acceleration is positive, as at A. –When the velocity is constant, there is no acceleration, as at B. –When the velocity in the positive direction is decreasing, the acceleration is negative, as at C.

14 Copyright © by Holt, Rinehart and Winston. All rights reserved. ResourcesChapter menu Chapter 2 Velocity and Acceleration Section 2 Acceleration

15 Copyright © by Holt, Rinehart and Winston. All rights reserved. ResourcesChapter menu Chapter 2 Motion with Constant Acceleration When velocity changes by the same amount during each time interval, acceleration is constant. The relationships between displacement, time, velocity, and constant acceleration are expressed by the equations shown on the next slide. These equations apply to any object moving with constant acceleration. These equations use the following symbols: x = displacement v i = initial velocity v f = final velocity t = time interval Section 2 Acceleration

16 Copyright © by Holt, Rinehart and Winston. All rights reserved. ResourcesChapter menu Chapter 2 Equations for Constantly Accelerated Straight-Line Motion Section 2 Acceleration

17 Copyright © by Holt, Rinehart and Winston. All rights reserved. ResourcesChapter menu Sample Problem Final Velocity After Any Displacement A person pushing a stroller starts from rest, uniformly accelerating at a rate of m/s 2. What is the velocity of the stroller after it has traveled 4.75 m? Section 2 Acceleration Chapter 2

18 Copyright © by Holt, Rinehart and Winston. All rights reserved. ResourcesChapter menu Sample Problem, continued Section 2 Acceleration 1. Define Given: v i = 0 m/s a = m/s 2 x = 4.75 m Unknown: v f = ? Diagram: Choose a coordinate system. The most convenient one has an origin at the initial location of the stroller, as shown above. The positive direction is to the right. Chapter 2

19 Copyright © by Holt, Rinehart and Winston. All rights reserved. ResourcesChapter menu Chapter 2 Sample Problem, continued Section 2 Acceleration 2. Plan Choose an equation or situation: Because the initial velocity, acceleration, and displacement are known, the final velocity can be found using the following equation: Rearrange the equation to isolate the unknown: Take the square root of both sides to isolate v f.

20 Copyright © by Holt, Rinehart and Winston. All rights reserved. ResourcesChapter menu Chapter 2 Sample Problem, continued Section 2 Acceleration Tip: Think about the physical situation to determine whether to keep the positive or negative answer from the square root. In this case, the stroller starts from rest and ends with a speed of 2.18 m/s. An object that is speeding up and has a positive acceleration must have a positive velocity. So, the final velocity must be positive. 3. Calculate Substitute the values into the equation and solve: 4. Evaluate The strollers velocity after accelerating for 4.75 m is 2.18 m/s to the right.

21 Copyright © by Holt, Rinehart and Winston. All rights reserved. ResourcesChapter menu Section 3 Falling Objects Chapter 2 Objectives Relate the motion of a freely falling body to motion with constant acceleration. Calculate displacement, velocity, and time at various points in the motion of a freely falling object. Compare the motions of different objects in free fall.

22 Copyright © by Holt, Rinehart and Winston. All rights reserved. ResourcesChapter menu Chapter 2 Free Fall Free fall is the motion of a body when only the force due to gravity is acting on the body. The acceleration on an object in free fall is called the acceleration due to gravity, or free-fall acceleration. Free-fall acceleration is denoted with the symbols a g (generally) or g (on Earths surface). Section 3 Falling Objects

23 Copyright © by Holt, Rinehart and Winston. All rights reserved. ResourcesChapter menu Chapter 2 Free-Fall Acceleration Free-fall acceleration is the same for all objects, regardless of mass. This book will use the value g = 9.81 m/s 2. Free-fall acceleration on Earths surface is –9.81 m/s 2 at all points in the objects motion. Consider a ball thrown up into the air. –Moving upward: velocity is decreasing, acceleration is – 9.81 m/s 2 –Top of path: velocity is zero, acceleration is –9.81 m/s 2 –Moving downward: velocity is increasing, acceleration is – 9.81 m/s 2 Section 3 Falling Objects

24 Copyright © by Holt, Rinehart and Winston. All rights reserved. ResourcesChapter menu Sample Problem Falling Object Jason hits a volleyball so that it moves with an initial velocity of 6.0 m/s straight upward. If the volleyball starts from 2.0 m above the floor, how long will it be in the air before it strikes the floor? Section 3 Falling Objects Chapter 2

25 Copyright © by Holt, Rinehart and Winston. All rights reserved. ResourcesChapter menu Sample Problem, continued Section 3 Falling Objects 1. Define Given:Unknown: v i = +6.0 m/s t = ? a = –g = –9.81 m/s 2 y = –2.0 m Diagram: Place the origin at the Starting point of the ball (y i = 0 at t i = 0). Chapter 2

26 Copyright © by Holt, Rinehart and Winston. All rights reserved. ResourcesChapter menu Chapter 2 Sample Problem, continued 2. Plan Choose an equation or situation: Both t and v f are unknown. Therefore, first solve for v f using the equation that does not require time. Then, the equation for v f that does involve time can be used to solve for t. Section 3 Falling Objects Rearrange the equation to isolate the unknown: Take the square root of the first equation to isolate v f. The second equation must be rearranged to solve for t.

27 Copyright © by Holt, Rinehart and Winston. All rights reserved. ResourcesChapter menu Chapter 2 Sample Problem, continued Tip: When you take the square root to find v f, select the negative answer because the ball will be moving toward the floor, in the negative direction. Section 3 Falling Objects 3. Calculate Substitute the values into the equation and solve: First find the velocity of the ball at the moment that it hits the floor.

28 Copyright © by Holt, Rinehart and Winston. All rights reserved. ResourcesChapter menu Chapter 2 Sample Problem, continued 4. Evaluate The solution, 1.50 s, is a reasonable amount of time for the ball to be in the air. Section 3 Falling Objects Next, use this value of v f in the second equation to solve for t.

29 Copyright © by Holt, Rinehart and Winston. All rights reserved. ResourcesChapter menu Multiple Choice Standardized Test Prep Chapter 2 Use the graphs to answer questions 1–3. 1. Which graph represents an object moving with a constant positive velocity? A. IC. III B. II D. IV

30 Copyright © by Holt, Rinehart and Winston. All rights reserved. ResourcesChapter menu Multiple Choice Standardized Test Prep Chapter 2 Use the graphs to answer questions 1–3. 1. Which graph represents an object moving with a constant positive velocity? A. IC. III B. II D. IV

31 Copyright © by Holt, Rinehart and Winston. All rights reserved. ResourcesChapter menu Multiple Choice, continued Standardized Test Prep Chapter 2 Use the graphs to answer questions 1–3. 2. Which graph represents an object at rest? F. IH. III G. II J. IV

32 Copyright © by Holt, Rinehart and Winston. All rights reserved. ResourcesChapter menu Multiple Choice, continued Standardized Test Prep Chapter 2 Use the graphs to answer questions 1–3. 2. Which graph represents an object at rest? F. IH. III G. II J. IV

33 Copyright © by Holt, Rinehart and Winston. All rights reserved. ResourcesChapter menu Multiple Choice, continued Standardized Test Prep Chapter 2 Use the graphs to answer questions 1–3. 3. Which graph represents an object moving with a constant positive acceleration? A. IC. III B. II D. IV

34 Copyright © by Holt, Rinehart and Winston. All rights reserved. ResourcesChapter menu Multiple Choice, continued Standardized Test Prep Chapter 2 Use the graphs to answer questions 1–3. 3. Which graph represents an object moving with a constant positive acceleration? A. IC. III B. II D. IV

35 Copyright © by Holt, Rinehart and Winston. All rights reserved. ResourcesChapter menu Multiple Choice, continued Standardized Test Prep Chapter 2 4.A bus travels from El Paso, Texas, to Chihuahua, Mexico, in 5.2 h with an average velocity of 73 km/h to the south.What is the buss displacement? F. 73 km to the south G. 370 km to the south H. 380 km to the south J. 14 km/h to the south

36 Copyright © by Holt, Rinehart and Winston. All rights reserved. ResourcesChapter menu Multiple Choice, continued Standardized Test Prep Chapter 2 4.A bus travels from El Paso, Texas, to Chihuahua, Mexico, in 5.2 h with an average velocity of 73 km/h to the south.What is the buss displacement? F. 73 km to the south G. 370 km to the south H. 380 km to the south J. 14 km/h to the south

37 Copyright © by Holt, Rinehart and Winston. All rights reserved. ResourcesChapter menu Multiple Choice, continued Standardized Test Prep Chapter 2 5.What is the squirrels displacement at time t = 3.0 s? A. –6.0 m B. –2.0 m C m D m Use the position-time graph of a squirrel running along a clothesline to answer questions 5–6.

38 Copyright © by Holt, Rinehart and Winston. All rights reserved. ResourcesChapter menu Multiple Choice, continued Standardized Test Prep Chapter 2 5.What is the squirrels displacement at time t = 3.0 s? A. –6.0 m B. –2.0 m C m D m Use the position-time graph of a squirrel running along a clothesline to answer questions 5–6.

39 Copyright © by Holt, Rinehart and Winston. All rights reserved. ResourcesChapter menu Multiple Choice, continued Standardized Test Prep Chapter 2 Use the position-time graph of a squirrel running along a clothesline to answer questions 5–6. 6.What is the squirrels average velocity during the time interval between 0.0 s and 3.0 s? F. –2.0 m/s G. –0.67 m/s H. 0.0 m/s J m/s

40 Copyright © by Holt, Rinehart and Winston. All rights reserved. ResourcesChapter menu Multiple Choice, continued Standardized Test Prep Chapter 2 6.What is the squirrels average velocity during the time interval between 0.0 s and 3.0 s? F. –2.0 m/s G. –0.67 m/s H. 0.0 m/s J m/s Use the position-time graph of a squirrel running along a clothesline to answer questions 5–6.

41 Copyright © by Holt, Rinehart and Winston. All rights reserved. ResourcesChapter menu Multiple Choice, continued Standardized Test Prep Chapter 2 7. Which of the following statements is true of acceleration? A. Acceleration always has the same sign as displacement. B. Acceleration always has the same sign as velocity. C. The sign of acceleration depends on both the direction of motion and how the velocity is changing. D. Acceleration always has a positive sign.

42 Copyright © by Holt, Rinehart and Winston. All rights reserved. ResourcesChapter menu Multiple Choice, continued Standardized Test Prep Chapter 2 7. Which of the following statements is true of acceleration? A. Acceleration always has the same sign as displacement. B. Acceleration always has the same sign as velocity. C. The sign of acceleration depends on both the direction of motion and how the velocity is changing. D. Acceleration always has a positive sign.

43 Copyright © by Holt, Rinehart and Winston. All rights reserved. ResourcesChapter menu Multiple Choice, continued Standardized Test Prep Chapter 2 8. A ball initially at rest rolls down a hill and has an acceleration of 3.3 m/s 2. If it accelerates for 7.5 s, how far will it move during this time? F. 12 m G. 93 m H. 120 m J. 190 m

44 Copyright © by Holt, Rinehart and Winston. All rights reserved. ResourcesChapter menu Multiple Choice, continued Standardized Test Prep Chapter 2 8. A ball initially at rest rolls down a hill and has an acceleration of 3.3 m/s 2. If it accelerates for 7.5 s, how far will it move during this time? F. 12 m G. 93 m H. 120 m J. 190 m

45 Copyright © by Holt, Rinehart and Winston. All rights reserved. ResourcesChapter menu Multiple Choice, continued Standardized Test Prep Chapter 2 9. Which of the following statements is true for a ball thrown vertically upward? A. The ball has a negative acceleration on the way up and a positive acceleration on the way down. B. The ball has a positive acceleration on the way up and a negative acceleration on the way down. C. The ball has zero acceleration on the way up and a positive acceleration on the way down. D. The ball has a constant acceleration throughout its flight.

46 Copyright © by Holt, Rinehart and Winston. All rights reserved. ResourcesChapter menu Multiple Choice, continued Standardized Test Prep Chapter 2 9. Which of the following statements is true for a ball thrown vertically upward? A. The ball has a negative acceleration on the way up and a positive acceleration on the way down. B. The ball has a positive acceleration on the way up and a negative acceleration on the way down. C. The ball has zero acceleration on the way up and a positive acceleration on the way down. D. The ball has a constant acceleration throughout its flight.

47 Copyright © by Holt, Rinehart and Winston. All rights reserved. ResourcesChapter menu Short Response Standardized Test Prep Chapter In one or two sentences, explain the difference between displacement and distance traveled.

48 Copyright © by Holt, Rinehart and Winston. All rights reserved. ResourcesChapter menu Short Response Standardized Test Prep Chapter In one or two sentences, explain the difference between displacement and distance traveled. Answer: Displacement measures only the net change in position from starting point to end point. The distance traveled is the total length of the path followed from starting point to end point and may be greater than or equal to the displacement.

49 Copyright © by Holt, Rinehart and Winston. All rights reserved. ResourcesChapter menu Short Response, continued Standardized Test Prep Chapter The graph shows the position of a runner at different times during a run. Use the graph to determine the runners displacement and average velocity: a. for the time interval from t = 0.0 min to t = 10.0 min b. for the time interval from t = 10.0 min to t = 20.0 min c. for the time interval from t = 20.0 min to t = 30.0 min d. for the entire run

50 Copyright © by Holt, Rinehart and Winston. All rights reserved. ResourcesChapter menu Short Response, continued Standardized Test Prep Chapter The graph shows the position of a runner at different times during a run. Use the graph to determine the runners displacement and average velocity. Answers will vary but should be approximately as follows: a. for t = 0.0 min to t = 10.0 min Answer: m, +4.0 m/s b. for t = 10.0 min to t = 20.0 min Answer: m, +2.5 m/s c. for t = 20.0 min to t = 30.0 min Answer: +900 m, +2 m/s d. for the entire run Answer: m, +2.7 m/s

51 Copyright © by Holt, Rinehart and Winston. All rights reserved. ResourcesChapter menu Short Response, continued Standardized Test Prep Chapter For an object moving with constant negative acceleration, draw the following: a. a graph of position vs. time b. a graph of velocity vs. time For both graphs, assume the object starts with a positive velocity and a positive displacement from the origin.

52 Copyright © by Holt, Rinehart and Winston. All rights reserved. ResourcesChapter menu Short Response, continued Standardized Test Prep Chapter For an object moving with constant negative acceleration, draw the following: a. a graph of position vs. time b. a graph of velocity vs. time For both graphs, assume the object starts with a positive velocity and a positive displacement from the origin. Answers:

53 Copyright © by Holt, Rinehart and Winston. All rights reserved. ResourcesChapter menu Short Response, continued Standardized Test Prep Chapter A snowmobile travels in a straight line. The snowmobiles initial velocity is +3.0 m/s. a. If the snowmobile accelerates at a rate of m/s 2 for 7.0 s, what is its final velocity? b. If the snowmobile accelerates at the rate of –0.60 m/s 2 from its initial velocity of +3.0 m/s, how long will it take to reach a complete stop?

54 Copyright © by Holt, Rinehart and Winston. All rights reserved. ResourcesChapter menu Short Response, continued Standardized Test Prep Chapter A snowmobile travels in a straight line. The snowmobiles initial velocity is +3.0 m/s. a. If the snowmobile accelerates at a rate of m/s 2 for 7.0 s, what is its final velocity? b. If the snowmobile accelerates at the rate of –0.60 m/s 2 from its initial velocity of +3.0 m/s, how long will it take to reach a complete stop? Answers: a m/s b. 5.0 s

55 Copyright © by Holt, Rinehart and Winston. All rights reserved. ResourcesChapter menu Extended Response Standardized Test Prep Chapter A car moving eastward along a straight road increases its speed uniformly from 16 m/s to 32 m/s in 10.0 s. a. What is the cars average acceleration? b. What is the cars average velocity? c. How far did the car move while accelerating? Show all of your work for these calculations.

56 Copyright © by Holt, Rinehart and Winston. All rights reserved. ResourcesChapter menu Extended Response Standardized Test Prep Chapter A car moving eastward along a straight road increases its speed uniformly from 16 m/s to 32 m/s in 10.0 s. a. What is the cars average acceleration? b. What is the cars average velocity? c. How far did the car move while accelerating? Answers: a. 1.6 m/s 2 eastward b. 24 m/s c. 240 m

57 Copyright © by Holt, Rinehart and Winston. All rights reserved. ResourcesChapter menu Extended Response, continued Standardized Test Prep Chapter A ball is thrown vertically upward with a speed of 25.0 m/s from a height of 2.0 m. a. How long does it take the ball to reach its highest point? b. How long is the ball in the air? Show all of your work for these calculations.

58 Copyright © by Holt, Rinehart and Winston. All rights reserved. ResourcesChapter menu Extended Response, continued Standardized Test Prep Chapter A ball is thrown vertically upward with a speed of 25.0 m/s from a height of 2.0 m. a. How long does it take the ball to reach its highest point? b. How long is the ball in the air? Show all of your work for these calculations. Answers: a s b s

59 Copyright © by Holt, Rinehart and Winston. All rights reserved. ResourcesChapter menu Chapter 3 Two-Dimensional Motion and Vectors

60 Copyright © by Holt, Rinehart and Winston. All rights reserved. ResourcesChapter menu Two-Dimensional Motion and Vectors Chapter 3 Table of Contents Section 1 Introduction to Vectors Section 2 Vector Operations Section 3 Projectile Motion Section 4 Relative Motion

61 Copyright © by Holt, Rinehart and Winston. All rights reserved. ResourcesChapter menu Section 1 Introduction to Vectors Chapter 3 Objectives Distinguish between a scalar and a vector. Add and subtract vectors by using the graphical method. Multiply and divide vectors by scalars.

62 Copyright © by Holt, Rinehart and Winston. All rights reserved. ResourcesChapter menu Chapter 3 Scalars and Vectors A scalar is a physical quantity that has magnitude but no direction. –Examples: speed, volume, the number of pages in your textbook A vector is a physical quantity that has both magnitude and direction. –Examples: displacement, velocity, acceleration In this book, scalar quantities are in italics. Vectors are represented by boldface symbols. Section 1 Introduction to Vectors

63 Copyright © by Holt, Rinehart and Winston. All rights reserved. ResourcesChapter menu Chapter 3 Graphical Addition of Vectors A resultant vector represents the sum of two or more vectors. Vectors can be added graphically. Section 1 Introduction to Vectors A student walks from his house to his friends house (a), then from his friends house to the school (b). The students resultant displacement (c) can be found by using a ruler and a protractor.

64 Copyright © by Holt, Rinehart and Winston. All rights reserved. ResourcesChapter menu Chapter 3 Triangle Method of Addition Vectors can be moved parallel to themselves in a diagram. Thus, you can draw one vector with its tail starting at the tip of the other as long as the size and direction of each vector do not change. The resultant vector can then be drawn from the tail of the first vector to the tip of the last vector. Section 1 Introduction to Vectors

65 Copyright © by Holt, Rinehart and Winston. All rights reserved. ResourcesChapter menu Chapter 3 Properties of Vectors Vectors can be added in any order. To subtract a vector, add its opposite. Multiplying or dividing vectors by scalars results in vectors. Section 1 Introduction to Vectors

66 Copyright © by Holt, Rinehart and Winston. All rights reserved. ResourcesChapter menu Section 2 Vector Operations Chapter 3 Objectives Identify appropriate coordinate systems for solving problems with vectors. Apply the Pythagorean theorem and tangent function to calculate the magnitude and direction of a resultant vector. Resolve vectors into components using the sine and cosine functions. Add vectors that are not perpendicular.

67 Copyright © by Holt, Rinehart and Winston. All rights reserved. ResourcesChapter menu Chapter 3 Coordinate Systems in Two Dimensions One method for diagraming the motion of an object employs vectors and the use of the x- and y-axes. Axes are often designated using fixed directions. In the figure shown here, the positive y-axis points north and the positive x-axis points east. Section 2 Vector Operations

68 Copyright © by Holt, Rinehart and Winston. All rights reserved. ResourcesChapter menu Chapter 3 Determining Resultant Magnitude and Direction In Section 1, the magnitude and direction of a resultant were found graphically. With this approach, the accuracy of the answer depends on how carefully the diagram is drawn and measured. A simpler method uses the Pythagorean theorem and the tangent function. Section 2 Vector Operations

69 Copyright © by Holt, Rinehart and Winston. All rights reserved. ResourcesChapter menu Chapter 3 Determining Resultant Magnitude and Direction, continued The Pythagorean Theorem Use the Pythagorean theorem to find the magnitude of the resultant vector. The Pythagorean theorem states that for any right triangle, the square of the hypotenusethe side opposite the right angleequals the sum of the squares of the other two sides, or legs. Section 2 Vector Operations

70 Copyright © by Holt, Rinehart and Winston. All rights reserved. ResourcesChapter menu Chapter 3 Determining Resultant Magnitude and Direction, continued The Tangent Function Use the tangent function to find the direction of the resultant vector. For any right triangle, the tangent of an angle is defined as the ratio of the opposite and adjacent legs with respect to a specified acute angle of a right triangle. Section 2 Vector Operations

71 Copyright © by Holt, Rinehart and Winston. All rights reserved. ResourcesChapter menu Chapter 3 Sample Problem Finding Resultant Magnitude and Direction An archaeologist climbs the Great Pyramid in Giza, Egypt. The pyramids height is 136 m and its width is m. What is the magnitude and the direction of the displacement of the archaeologist after she has climbed from the bottom of the pyramid to the top? Section 2 Vector Operations

72 Copyright © by Holt, Rinehart and Winston. All rights reserved. ResourcesChapter menu Chapter 3 Sample Problem, continued Section 2 Vector Operations 1. Define Given: y = 136 m x = 1/2(width) = 115 m Unknown: d = ? = ? Diagram: Choose the archaeologists starting position as the origin of the coordinate system, as shown above.

73 Copyright © by Holt, Rinehart and Winston. All rights reserved. ResourcesChapter menu Chapter 3 Sample Problem, continued Section 2 Vector Operations Rearrange the equations to isolate the unknowns: 2. Plan Choose an equation or situation: The Pythagorean theorem can be used to find the magnitude of the archaeologists displacement. The direction of the displacement can be found by using the inverse tangent function.

74 Copyright © by Holt, Rinehart and Winston. All rights reserved. ResourcesChapter menu Chapter 3 Sample Problem, continued Section 2 Vector Operations 3. Calculate 4.Evaluate Because d is the hypotenuse, the archaeologists displacement should be less than the sum of the height and half of the width. The angle is expected to be more than 45 because the height is greater than half of the width.

75 Copyright © by Holt, Rinehart and Winston. All rights reserved. ResourcesChapter menu Chapter 3 Resolving Vectors into Components You can often describe an objects motion more conveniently by breaking a single vector into two components, or resolving the vector. The components of a vector are the projections of the vector along the axes of a coordinate system. Resolving a vector allows you to analyze the motion in each direction. Section 2 Vector Operations

76 Copyright © by Holt, Rinehart and Winston. All rights reserved. ResourcesChapter menu Chapter 3 Resolving Vectors into Components, continued Consider an airplane flying at 95 km/h. The hypotenuse (v plane ) is the resultant vector that describes the airplanes total velocity. The adjacent leg represents the x component (v x ), which describes the airplanes horizontal speed. The opposite leg represents the y component (v y ), which describes the airplanes vertical speed. Section 2 Vector Operations

77 Copyright © by Holt, Rinehart and Winston. All rights reserved. ResourcesChapter menu Chapter 3 Resolving Vectors into Components, continued The sine and cosine functions can be used to find the components of a vector. The sine and cosine functions are defined in terms of the lengths of the sides of right triangles. Section 2 Vector Operations

78 Copyright © by Holt, Rinehart and Winston. All rights reserved. ResourcesChapter menu Chapter 3 Adding Vectors That Are Not Perpendicular Section 2 Vector Operations Suppose that a plane travels first 5 km at an angle of 35°, then climbs at 10° for 22 km, as shown below. How can you find the total displacement? Because the original displacement vectors do not form a right triangle, you can not directly apply the tangent function or the Pythagorean theorem. d2 d1

79 Copyright © by Holt, Rinehart and Winston. All rights reserved. ResourcesChapter menu Chapter 3 Adding Vectors That Are Not Perpendicular, continued Section 2 Vector Operations You can find the magnitude and the direction of the resultant by resolving each of the planes displacement vectors into its x and y components. Then the components along each axis can be added together. As shown in the figure, these sums will be the two perpendicular components of the resultant, d. The resultants magnitude can then be found by using the Pythagorean theorem, and its direction can be found by using the inverse tangent function.

80 Copyright © by Holt, Rinehart and Winston. All rights reserved. ResourcesChapter menu Chapter 3 Sample Problem Adding Vectors Algebraically A hiker walks 27.0 km from her base camp at 35° south of east. The next day, she walks 41.0 km in a direction 65° north of east and discovers a forest rangers tower. Find the magnitude and direction of her resultant displacement Section 2 Vector Operations

81 Copyright © by Holt, Rinehart and Winston. All rights reserved. ResourcesChapter menu Chapter 3 1. Select a coordinate system. Then sketch and label each vector. Section 2 Vector Operations Given: d1 = 27.0 km 1 = –35° d2 = 41.0 km 2 = 65° Tip: 1 is negative, because clockwise movement from the positive x-axis is negative by convention. Unknown: d = ? = ? Sample Problem, continued

82 Copyright © by Holt, Rinehart and Winston. All rights reserved. ResourcesChapter menu Chapter 3 2. Find the x and y components of all vectors. Section 2 Vector Operations Make a separate sketch of the displacements for each day. Use the cosine and sine functions to find the components. Sample Problem, continued

83 Copyright © by Holt, Rinehart and Winston. All rights reserved. ResourcesChapter menu Chapter 3 3. Find the x and y components of the total displacement. Section 2 Vector Operations 4. Use the Pythagorean theorem to find the magnitude of the resultant vector. Sample Problem, continued

84 Copyright © by Holt, Rinehart and Winston. All rights reserved. ResourcesChapter menu Chapter 3 5. Use a suitable trigonometric function to find the angle. Section 2 Vector Operations Sample Problem, continued

85 Copyright © by Holt, Rinehart and Winston. All rights reserved. ResourcesChapter menu Section 3 Projectile Motion Chapter 3 Objectives Recognize examples of projectile motion. Describe the path of a projectile as a parabola. Resolve vectors into their components and apply the kinematic equations to solve problems involving projectile motion.

86 Copyright © by Holt, Rinehart and Winston. All rights reserved. ResourcesChapter menu Chapter 3 Projectiles Objects that are thrown or launched into the air and are subject to gravity are called projectiles. Projectile motion is the curved path that an object follows when thrown, launched,or otherwise projected near the surface of Earth. If air resistance is disregarded, projectiles follow parabolic trajectories. Section 3 Projectile Motion

87 Copyright © by Holt, Rinehart and Winston. All rights reserved. ResourcesChapter menu Chapter 3 Projectiles, continued Projectile motion is free fall with an initial horizontal velocity. The yellow ball is given an initial horizontal velocity and the red ball is dropped. Both balls fall at the same rate. –In this book, the horizontal velocity of a projectile will be considered constant. –This would not be the case if we accounted for air resistance. Section 3 Projectile Motion

88 Copyright © by Holt, Rinehart and Winston. All rights reserved. ResourcesChapter menu Chapter 3 Kinematic Equations for Projectiles How can you know the displacement, velocity, and acceleration of a projectile at any point in time during its flight? One method is to resolve vectors into components, then apply the simpler one-dimensional forms of the equations for each component. Finally, you can recombine the components to determine the resultant. Section 3 Projectile Motion

89 Copyright © by Holt, Rinehart and Winston. All rights reserved. ResourcesChapter menu Chapter 3 Kinematic Equations for Projectiles, continued To solve projectile problems, apply the kinematic equations in the horizontal and vertical directions. In the vertical direction, the acceleration a y will equal –g (–9.81 m/s 2 ) because the only vertical component of acceleration is free-fall acceleration. In the horizontal direction, the acceleration is zero, so the velocity is constant. Section 3 Projectile Motion

90 Copyright © by Holt, Rinehart and Winston. All rights reserved. ResourcesChapter menu Chapter 3 Kinematic Equations for Projectiles, continued Projectiles Launched Horizontally –The initial vertical velocity is 0. –The initial horizontal velocity is the initial velocity. Projectiles Launched At An Angle Section 3 Projectile Motion –Resolve the initial velocity into x and y components. –The initial vertical velocity is the y component. –The initial horizontal velocity is the x component.

91 Copyright © by Holt, Rinehart and Winston. All rights reserved. ResourcesChapter menu Chapter 3 Sample Problem Projectiles Launched At An Angle A zookeeper finds an escaped monkey hanging from a light pole. Aiming her tranquilizer gun at the monkey, she kneels 10.0 m from the light pole,which is 5.00 m high. The tip of her gun is 1.00 m above the ground. At the same moment that the monkey drops a banana, the zookeeper shoots. If the dart travels at 50.0 m/s,will the dart hit the monkey, the banana, or neither one? Section 3 Projectile Motion

92 Copyright © by Holt, Rinehart and Winston. All rights reserved. ResourcesChapter menu Chapter 3 1. Select a coordinate system. The positive y-axis points up, and the positive x- axis points along the ground toward the pole. Because the dart leaves the gun at a height of 1.00 m, the vertical distance is 4.00 m. Section 3 Projectile Motion Sample Problem, continued

93 Copyright © by Holt, Rinehart and Winston. All rights reserved. ResourcesChapter menu Chapter 3 2. Use the inverse tangent function to find the angle that the initial velocity makes with the x- axis. Section 3 Projectile Motion Sample Problem, continued

94 Copyright © by Holt, Rinehart and Winston. All rights reserved. ResourcesChapter menu Chapter 3 3. Choose a kinematic equation to solve for time. Rearrange the equation for motion along the x- axis to isolate the unknown t, which is the time the dart takes to travel the horizontal distance. Section 3 Projectile Motion Sample Problem, continued

95 Copyright © by Holt, Rinehart and Winston. All rights reserved. ResourcesChapter menu Chapter 3 4. Find out how far each object will fall during this time. Use the free-fall kinematic equation in both cases. For the banana, vi = 0. Thus: yb = ½ay( t) 2 = ½(–9.81 m/s 2 )(0.215 s) 2 = –0.227 m The dart has an initial vertical component of velocity equal to vi sin, so: yd = (vi sin )( t) + ½ay( t) 2 yd = (50.0 m/s)(sin )(0.215 s) +½(–9.81 m/s 2 )(0.215 s) 2 yd = 3.99 m – m = 3.76 m Section 3 Projectile Motion Sample Problem, continued

96 Copyright © by Holt, Rinehart and Winston. All rights reserved. ResourcesChapter menu Chapter 3 5. Analyze the results. Find the final height of both the banana and the dart. ybanana, f = yb,i+ yb = 5.00 m + (–0.227 m) ybanana, f = 4.77 m above the ground The dart hits the banana. The slight difference is due to rounding. Section 3 Projectile Motion ydart, f = yd,i+ yd = 1.00 m m ydart, f = 4.76 m above the ground Sample Problem, continued

97 Copyright © by Holt, Rinehart and Winston. All rights reserved. ResourcesChapter menu Section 4 Relative Motion Chapter 3 Objectives Describe situations in terms of frame of reference. Solve problems involving relative velocity.

98 Copyright © by Holt, Rinehart and Winston. All rights reserved. ResourcesChapter menu Chapter 3 Frames of Reference If you are moving at 80 km/h north and a car passes you going 90 km/h, to you the faster car seems to be moving north at 10 km/h. Someone standing on the side of the road would measure the velocity of the faster car as 90 km/h toward the north. This simple example demonstrates that velocity measurements depend on the frame of reference of the observer. Section 4 Relative Motion

99 Copyright © by Holt, Rinehart and Winston. All rights reserved. ResourcesChapter menu Chapter 3 Frames of Reference, continued Consider a stunt dummy dropped from a plane. (a) When viewed from the plane, the stunt dummy falls straight down. (b) When viewed from a stationary position on the ground, the stunt dummy follows a parabolic projectile path. Section 4 Relative Motion

100 Copyright © by Holt, Rinehart and Winston. All rights reserved. ResourcesChapter menu Chapter 3 Relative Velocity When solving relative velocity problems, write down the information in the form of velocities with subscripts. Using our earlier example, we have: v se = +80 km/h north (se = slower car with respect to Earth) v fe = +90 km/h north (fe = fast car with respect to Earth) unknown = v fs (fs = fast car with respect to slower car) Write an equation for v fs in terms of the other velocities. The subscripts start with f and end with s. The other subscripts start with the letter that ended the preceding velocity: v fs = v fe + v es Section 4 Relative Motion

101 Copyright © by Holt, Rinehart and Winston. All rights reserved. ResourcesChapter menu Chapter 3 Relative Velocity, continued An observer in the slow car perceives Earth as moving south at a velocity of 80 km/h while a stationary observer on the ground (Earth) views the car as moving north at a velocity of 80 km/h. In equation form: v es = –v se Thus, this problem can be solved as follows: v fs = v fe + v es = v fe – v se v fs = (+90 km/h n) – (+80 km/h n) = +10 km/h n A general form of the relative velocity equation is: v ac = v ab + v bc Section 4 Relative Motion

102 Copyright © by Holt, Rinehart and Winston. All rights reserved. ResourcesChapter menu Multiple Choice 1. Vector A has a magnitude of 30 units. Vector B is perpendicular to vector A and has a magnitude of 40 units. What would the magnitude of the resultant vector A + B be? A. 10 units B. 50 units C. 70 units D. zero Standardized Test Prep Chapter 3

103 Copyright © by Holt, Rinehart and Winston. All rights reserved. ResourcesChapter menu Multiple Choice 1. Vector A has a magnitude of 30 units. Vector B is perpendicular to vector A and has a magnitude of 40 units. What would the magnitude of the resultant vector A + B be? A. 10 units B. 50 units C. 70 units D. zero Standardized Test Prep Chapter 3

104 Copyright © by Holt, Rinehart and Winston. All rights reserved. ResourcesChapter menu Multiple Choice, continued 2. What term represents the magnitude of a velocity vector? F. acceleration G. momentum H. speed J. velocity Standardized Test Prep Chapter 3

105 Copyright © by Holt, Rinehart and Winston. All rights reserved. ResourcesChapter menu Multiple Choice, continued 2. What term represents the magnitude of a velocity vector? F. acceleration G. momentum H. speed J. velocity Standardized Test Prep Chapter 3

106 Copyright © by Holt, Rinehart and Winston. All rights reserved. ResourcesChapter menu Multiple Choice, continued Use the diagram to answer questions 3–4. 3. What is the direction of the resultant vector A + B? A. 15º above the x-axis B. 75º above the x-axis C. 15º below the x-axis D. 75º below the x-axis Standardized Test Prep Chapter 3

107 Copyright © by Holt, Rinehart and Winston. All rights reserved. ResourcesChapter menu Multiple Choice, continued Use the diagram to answer questions 3–4. 3. What is the direction of the resultant vector A + B? A. 15º above the x-axis B. 75º above the x-axis C. 15º below the x-axis D. 75º below the x-axis Standardized Test Prep Chapter 3

108 Copyright © by Holt, Rinehart and Winston. All rights reserved. ResourcesChapter menu Multiple Choice, continued Use the diagram to answer questions 3–4. 4. What is the direction of the resultant vector A – B? F. 15º above the x-axis G. 75º above the x-axis H. 15º below the x-axis J. 75º below the x-axis Standardized Test Prep Chapter 3

109 Copyright © by Holt, Rinehart and Winston. All rights reserved. ResourcesChapter menu Multiple Choice, continued Use the diagram to answer questions 3–4. 4. What is the direction of the resultant vector A – B? F. 15º above the x-axis G. 75º above the x-axis H. 15º below the x-axis J. 75º below the x-axis Standardized Test Prep Chapter 3

110 Copyright © by Holt, Rinehart and Winston. All rights reserved. ResourcesChapter menu Multiple Choice, continued Use the passage below to answer questions 5–6. A motorboat heads due east at 5.0 m/s across a river that flows toward the south at a speed of 5.0 m/s. 5. What is the resultant velocity relative to an observer on the shore ? A. 3.2 m/s to the southeast B. 5.0 m/s to the southeast C. 7.1 m/s to the southeast D m/s to the southeast Standardized Test Prep Chapter 3

111 Copyright © by Holt, Rinehart and Winston. All rights reserved. ResourcesChapter menu Multiple Choice, continued Use the passage below to answer questions 5–6. A motorboat heads due east at 5.0 m/s across a river that flows toward the south at a speed of 5.0 m/s. 5. What is the resultant velocity relative to an observer on the shore ? A. 3.2 m/s to the southeast B. 5.0 m/s to the southeast C. 7.1 m/s to the southeast D m/s to the southeast Standardized Test Prep Chapter 3

112 Copyright © by Holt, Rinehart and Winston. All rights reserved. ResourcesChapter menu Multiple Choice, continued Use the passage below to answer questions 5–6. A motorboat heads due east at 5.0 m/s across a river that flows toward the south at a speed of 5.0 m/s. 6. If the river is 125 m wide, how long does the boat take to cross the river? F. 39 s G. 25 s H. 17 s J. 12 s Standardized Test Prep Chapter 3

113 Copyright © by Holt, Rinehart and Winston. All rights reserved. ResourcesChapter menu Multiple Choice, continued Use the passage below to answer questions 5–6. A motorboat heads due east at 5.0 m/s across a river that flows toward the south at a speed of 5.0 m/s. 6. If the river is 125 m wide, how long does the boat take to cross the river? F. 39 s G. 25 s H. 17 s J. 12 s Standardized Test Prep Chapter 3

114 Copyright © by Holt, Rinehart and Winston. All rights reserved. ResourcesChapter menu Multiple Choice, continued 7. The pilot of a plane measures an air velocity of 165 km/h south relative to the plane. An observer on the ground sees the plane pass overhead at a velocity of 145 km/h toward the north.What is the velocity of the wind that is affecting the plane relative to the observer? A. 20 km/h to the north B. 20 km/h to the south C. 165 km/h to the north D. 310 km/h to the south Standardized Test Prep Chapter 3

115 Copyright © by Holt, Rinehart and Winston. All rights reserved. ResourcesChapter menu Multiple Choice, continued 7. The pilot of a plane measures an air velocity of 165 km/h south relative to the plane. An observer on the ground sees the plane pass overhead at a velocity of 145 km/h toward the north.What is the velocity of the wind that is affecting the plane relative to the observer? A. 20 km/h to the north B. 20 km/h to the south C. 165 km/h to the north D. 310 km/h to the south Standardized Test Prep Chapter 3

116 Copyright © by Holt, Rinehart and Winston. All rights reserved. ResourcesChapter menu Multiple Choice, continued 8. A golfer takes two putts to sink his ball in the hole once he is on the green. The first putt displaces the ball 6.00 m east, and the second putt displaces the ball 5.40 m south. What displacement would put the ball in the hole in one putt? F m southeast G m at 48.0º south of east H m at 42.0º south of east J m at 42.0º south of east Standardized Test Prep Chapter 3

117 Copyright © by Holt, Rinehart and Winston. All rights reserved. ResourcesChapter menu Multiple Choice, continued 8. A golfer takes two putts to sink his ball in the hole once he is on the green. The first putt displaces the ball 6.00 m east, and the second putt displaces the ball 5.40 m south. What displacement would put the ball in the hole in one putt? F m southeast G m at 48.0º south of east H m at 42.0º south of east J m at 42.0º south of east Standardized Test Prep Chapter 3

118 Copyright © by Holt, Rinehart and Winston. All rights reserved. ResourcesChapter menu Multiple Choice, continued Use the passage to answer questions 9–12. A girl riding a bicycle at 2.0 m/s throws a tennis ball horizontally forward at a speed of 1.0 m/s from a height of 1.5 m. At the same moment, a boy standing on the sidewalk drops a tennis ball straight down from a height of 1.5 m. 9. What is the initial speed of the girls ball relative to the boy? A. 1.0 m/s C. 2.0 m/s B. 1.5 m/s D. 3.0 m/s Standardized Test Prep Chapter 3

119 Copyright © by Holt, Rinehart and Winston. All rights reserved. ResourcesChapter menu Multiple Choice, continued Use the passage to answer questions 9–12. A girl riding a bicycle at 2.0 m/s throws a tennis ball horizontally forward at a speed of 1.0 m/s from a height of 1.5 m. At the same moment, a boy standing on the sidewalk drops a tennis ball straight down from a height of 1.5 m. 9. What is the initial speed of the girls ball relative to the boy? A. 1.0 m/s C. 2.0 m/s B. 1.5 m/s D. 3.0 m/s Standardized Test Prep Chapter 3

120 Copyright © by Holt, Rinehart and Winston. All rights reserved. ResourcesChapter menu Multiple Choice, continued Use the passage to answer questions 9–12. A girl riding a bicycle at 2.0 m/s throws a tennis ball horizontally forward at a speed of 1.0 m/s from a height of 1.5 m. At the same moment, a boy standing on the sidewalk drops a tennis ball straight down from a height of 1.5 m. 10. If air resistance is disregarded, which ball will hit the ground first? F. the boys ball H. neither G. the girls ball J. cannot be determined Standardized Test Prep Chapter 3

121 Copyright © by Holt, Rinehart and Winston. All rights reserved. ResourcesChapter menu Multiple Choice, continued Use the passage to answer questions 9–12. A girl riding a bicycle at 2.0 m/s throws a tennis ball horizontally forward at a speed of 1.0 m/s from a height of 1.5 m. At the same moment, a boy standing on the sidewalk drops a tennis ball straight down from a height of 1.5 m. 10. If air resistance is disregarded, which ball will hit the ground first? F. the boys ball H. neither G. the girls ball J. cannot be determined Standardized Test Prep Chapter 3

122 Copyright © by Holt, Rinehart and Winston. All rights reserved. ResourcesChapter menu Multiple Choice, continued Use the passage to answer questions 9–12. A girl riding a bicycle at 2.0 m/s throws a tennis ball horizontally forward at a speed of 1.0 m/s from a height of 1.5 m. At the same moment, a boy standing on the sidewalk drops a tennis ball straight down from a height of 1.5 m. 11. If air resistance is disregarded, which ball will have a greater speed (relative to the ground) when it hits the ground? A. the boys ball C. neither B. the girls ball D. cannot be determined Standardized Test Prep Chapter 3

123 Copyright © by Holt, Rinehart and Winston. All rights reserved. ResourcesChapter menu Multiple Choice, continued Use the passage to answer questions 9–12. A girl riding a bicycle at 2.0 m/s throws a tennis ball horizontally forward at a speed of 1.0 m/s from a height of 1.5 m. At the same moment, a boy standing on the sidewalk drops a tennis ball straight down from a height of 1.5 m. 11. If air resistance is disregarded, which ball will have a greater speed (relative to the ground) when it hits the ground? A. the boys ball C. neither B. the girls ball D. cannot be determined Standardized Test Prep Chapter 3

124 Copyright © by Holt, Rinehart and Winston. All rights reserved. ResourcesChapter menu Multiple Choice, continued Use the passage to answer questions 9–12. A girl riding a bicycle at 2.0 m/s throws a tennis ball horizontally forward at a speed of 1.0 m/s from a height of 1.5 m. At the same moment, a boy standing on the sidewalk drops a tennis ball straight down from a height of 1.5 m. 12. What is the speed of the girls ball when it hits the ground? F. 1.0 m/s H. 6.2 m/s G. 3.0 m/sJ. 8.4 m/s Standardized Test Prep Chapter 3

125 Copyright © by Holt, Rinehart and Winston. All rights reserved. ResourcesChapter menu Multiple Choice, continued Use the passage to answer questions 9–12. A girl riding a bicycle at 2.0 m/s throws a tennis ball horizontally forward at a speed of 1.0 m/s from a height of 1.5 m. At the same moment, a boy standing on the sidewalk drops a tennis ball straight down from a height of 1.5 m. 12. What is the speed of the girls ball when it hits the ground? F. 1.0 m/s H. 6.2 m/s G. 3.0 m/sJ. 8.4 m/s Standardized Test Prep Chapter 3

126 Copyright © by Holt, Rinehart and Winston. All rights reserved. ResourcesChapter menu Short Response Standardized Test Prep Chapter If one of the components of one vector along the direction of another vector is zero, what can you conclude about these two vectors?

127 Copyright © by Holt, Rinehart and Winston. All rights reserved. ResourcesChapter menu Short Response Standardized Test Prep Chapter If one of the components of one vector along the direction of another vector is zero, what can you conclude about these two vectors? Answer: They are perpendicular.

128 Copyright © by Holt, Rinehart and Winston. All rights reserved. ResourcesChapter menu Short Response, continued Standardized Test Prep Chapter A roller coaster travels 41.1 m at an angle of 40.0° above the horizontal. How far does it move horizontally and vertically?

129 Copyright © by Holt, Rinehart and Winston. All rights reserved. ResourcesChapter menu Short Response, continued Standardized Test Prep Chapter A roller coaster travels 41.1 m at an angle of 40.0° above the horizontal. How far does it move horizontally and vertically? Answer: 31.5 m horizontally, 26.4 m vertically

130 Copyright © by Holt, Rinehart and Winston. All rights reserved. ResourcesChapter menu Short Response, continued Standardized Test Prep Chapter A ball is thrown straight upward and returns to the throwers hand after 3.00 s in the air. A second ball is thrown at an angle of 30.0° with the horizontal. At what speed must the second ball be thrown to reach the same height as the one thrown vertically?

131 Copyright © by Holt, Rinehart and Winston. All rights reserved. ResourcesChapter menu Short Response, continued Standardized Test Prep Chapter A ball is thrown straight upward and returns to the throwers hand after 3.00 s in the air. A second ball is thrown at an angle of 30.0° with the horizontal. At what speed must the second ball be thrown to reach the same height as the one thrown vertically? Answer: 29.4 m/s

132 Copyright © by Holt, Rinehart and Winston. All rights reserved. ResourcesChapter menu Extended Response Standardized Test Prep Chapter A human cannonball is shot out of a cannon at 45.0° to the horizontal with an initial speed of 25.0 m/s. A net is positioned at a horizontal distance of 50.0 m from the cannon. At what height above the cannon should the net be placed in order to catch the human cannonball? Show your work.

133 Copyright © by Holt, Rinehart and Winston. All rights reserved. ResourcesChapter menu Extended Response Standardized Test Prep Chapter A human cannonball is shot out of a cannon at 45.0° to the horizontal with an initial speed of 25.0 m/s. A net is positioned at a horizontal distance of 50.0 m from the cannon. At what height above the cannon should the net be placed in order to catch the human cannonball? Show your work. Answer: 10.8 m

134 Copyright © by Holt, Rinehart and Winston. All rights reserved. ResourcesChapter menu Extended Response, continued Standardized Test Prep Chapter 3 Read the following passage to answer question 17. Three airline executives are discussing ideas for developing flights that are more energy efficient. Executive A: Because the Earth rotates from west to east, we could operate static flightsa helicopter or airship could begin by rising straight up from New York City and then descend straight down four hours later when San Francisco arrives below. Executive B: This approach could work for one-way flights, but the return trip would take 20 hours. continued on the next slide

135 Copyright © by Holt, Rinehart and Winston. All rights reserved. ResourcesChapter menu Extended Response, continued Standardized Test Prep Chapter 3 Executive C: That approach will never work. Think about it.When you throw a ball straight up in the air, it comes straight back down to the same point. Executive A: The ball returns to the same point because Earths motion is not significant during such a short time. 17. State which of the executives is correct, and explain why.

136 Copyright © by Holt, Rinehart and Winston. All rights reserved. ResourcesChapter menu Extended Response, continued Standardized Test Prep Chapter State which of the executives is correct, and explain why. Answer: Executive C is correct. Explanations should include the concept of relative velocitywhen a helicopter lifts off straight up from the ground, it is already moving horizontally with Earths horizontal velocity. (We assume that Earths motion is constant for the purposes of this scenario and does not depend on time.)

137 Copyright © by Holt, Rinehart and Winston. All rights reserved. ResourcesChapter menu Chapter 3 Graphical Addition of Vectors Section 1 Introduction to Vectors

138 Copyright © by Holt, Rinehart and Winston. All rights reserved. ResourcesChapter menu Chapter 3 Adding Vectors That Are Not Perpendicular Section 2 Vector Operations

139 Copyright © by Holt, Rinehart and Winston. All rights reserved. ResourcesChapter menu Chapter 3 Projectiles Section 3 Projectile Motion

140 Copyright © by Holt, Rinehart and Winston. All rights reserved. ResourcesChapter menu Chapter 3 Frames of Reference Section 4 Relative Motion

141 Copyright © by Holt, Rinehart and Winston. All rights reserved. ResourcesChapter menu Chapter 7 Circular Motion and Gravitation

142 Copyright © by Holt, Rinehart and Winston. All rights reserved. ResourcesChapter menu Circular Motion and Gravitation Chapter 7 Table of Contents Section 1 Circular Motion Section 2 Newtons Law of Universal Gravitation Section 3 Motion in Space Section 4 Torque and Simple Machines

143 Copyright © by Holt, Rinehart and Winston. All rights reserved. ResourcesChapter menu Chapter 7 Objectives Solve problems involving centripetal acceleration. Solve problems involving centripetal force. Explain how the apparent existence of an outward force in circular motion can be explained as inertia resisting the centripetal force. Section 1 Circular Motion

144 Copyright © by Holt, Rinehart and Winston. All rights reserved. ResourcesChapter menu Chapter 7 Tangential Speed The tangential speed (v t ) of an object in circular motion is the objects speed along an imaginary line drawn tangent to the circular path. Tangential speed depends on the distance from the object to the center of the circular path. When the tangential speed is constant, the motion is described as uniform circular motion. Section 1 Circular Motion

145 Copyright © by Holt, Rinehart and Winston. All rights reserved. ResourcesChapter menu Chapter 7 Centripetal Acceleration Section 1 Circular Motion

146 Copyright © by Holt, Rinehart and Winston. All rights reserved. ResourcesChapter menu Chapter 7 Centripetal Acceleration The acceleration of an object moving in a circular path and at constant speed is due to a change in direction. An acceleration of this nature is called a centripetal acceleration. Section 1 Circular Motion

147 Copyright © by Holt, Rinehart and Winston. All rights reserved. ResourcesChapter menu Chapter 7 Centripetal Acceleration, continued (a) As the particle moves from A to B, the direction of the particles velocity vector changes. (b) For short time intervals, v is directed toward the center of the circle. Centripetal acceleration is always directed toward the center of a circle. Section 1 Circular Motion

148 Copyright © by Holt, Rinehart and Winston. All rights reserved. ResourcesChapter menu Chapter 7 Centripetal Acceleration, continued You have seen that centripetal acceleration results from a change in direction. In circular motion, an acceleration due to a change in speed is called tangential acceleration. To understand the difference between centripetal and tangential acceleration, consider a car traveling in a circular track. –Because the car is moving in a circle, the car has a centripetal component of acceleration. –If the cars speed changes, the car also has a tangential component of acceleration. Section 1 Circular Motion

149 Copyright © by Holt, Rinehart and Winston. All rights reserved. ResourcesChapter menu Chapter 7 Centripetal Force Consider a ball of mass m that is being whirled in a horizontal circular path of radius r with constant speed. The force exerted by the string has horizontal and vertical components. The vertical component is equal and opposite to the gravitational force. Thus, the horizontal component is the net force. This net force, which is is directed toward the center of the circle, is a centripetal force. Section 1 Circular Motion

150 Copyright © by Holt, Rinehart and Winston. All rights reserved. ResourcesChapter menu Chapter 7 Centripetal Force, continued Newtons second law can be combined with the equation for centripetal acceleration to derive an equation for centripetal force: Section 1 Circular Motion

151 Copyright © by Holt, Rinehart and Winston. All rights reserved. ResourcesChapter menu Chapter 7 Centripetal Force, continued Centripetal force is simply the name given to the net force on an object in uniform circular motion. Any type of force or combination of forces can provide this net force. –For example, friction between a race cars tires and a circular track is a centripetal force that keeps the car in a circular path. –As another example, gravitational force is a centripetal force that keeps the moon in its orbit. Section 1 Circular Motion

152 Copyright © by Holt, Rinehart and Winston. All rights reserved. ResourcesChapter menu Chapter 7 Centripetal Force, continued If the centripetal force vanishes, the object stops moving in a circular path. A ball that is on the end of a string is whirled in a vertical circular path. –If the string breaks at the position shown in (a), the ball will move vertically upward in free fall. –If the string breaks at the top of the balls path, as in (b), the ball will move along a parabolic path. Section 1 Circular Motion

153 Copyright © by Holt, Rinehart and Winston. All rights reserved. ResourcesChapter menu Chapter 7 Describing a Rotating System To better understand the motion of a rotating system, consider a car traveling at high speed and approaching an exit ramp that curves to the left. As the driver makes the sharp left turn, the passenger slides to the right and hits the door. What causes the passenger to move toward the door? Section 1 Circular Motion

154 Copyright © by Holt, Rinehart and Winston. All rights reserved. ResourcesChapter menu Chapter 7 Describing a Rotating System, continued Section 1 Circular Motion As the car enters the ramp and travels along a curved path, the passenger, because of inertia, tends to move along the original straight path. If a sufficiently large centripetal force acts on the passenger, the person will move along the same curved path that the car does. The origin of the centripetal force is the force of friction between the passenger and the car seat. If this frictional force is not sufficient, the passenger slides across the seat as the car turns underneath.

155 Copyright © by Holt, Rinehart and Winston. All rights reserved. ResourcesChapter menu Section 2 Newtons Law of Universal Gravitation Chapter 7 Objectives Explain how Newtons law of universal gravitation accounts for various phenomena, including satellite and planetary orbits, falling objects, and the tides. Apply Newtons law of universal gravitation to solve problems.

156 Copyright © by Holt, Rinehart and Winston. All rights reserved. ResourcesChapter menu Chapter 7 Gravitational Force Orbiting objects are in free fall. To see how this idea is true, we can use a thought experiment that Newton developed. Consider a cannon sitting on a high mountaintop. Section 2 Newtons Law of Universal Gravitation Each successive cannonball has a greater initial speed, so the horizontal distance that the ball travels increases. If the initial speed is great enough, the curvature of Earth will cause the cannonball to continue falling without ever landing.

157 Copyright © by Holt, Rinehart and Winston. All rights reserved. ResourcesChapter menu Chapter 7 Gravitational Force, continued The centripetal force that holds the planets in orbit is the same force that pulls an apple toward the groundgravitational force. Gravitational force is the mutual force of attraction between particles of matter. Gravitational force depends on the masses and on the distance between them. Section 2 Newtons Law of Universal Gravitation

158 Copyright © by Holt, Rinehart and Winston. All rights reserved. ResourcesChapter menu Chapter 7 Gravitational Force, continued Newton developed the following equation to describe quantitatively the magnitude of the gravitational force if distance r separates masses m 1 and m 2 : Section 2 Newtons Law of Universal Gravitation The constant G, called the constant of universal gravitation, equals –11 Nm 2 /kg.

159 Copyright © by Holt, Rinehart and Winston. All rights reserved. ResourcesChapter menu Chapter 7 Newtons Law of Universal Gravitation Section 2 Newtons Law of Universal Gravitation

160 Copyright © by Holt, Rinehart and Winston. All rights reserved. ResourcesChapter menu Chapter 7 Gravitational Force, continued The gravitational forces that two masses exert on each other are always equal in magnitude and opposite in direction. This is an example of Newtons third law of motion. One example is the Earth-moon system, shown on the next slide. As a result of these forces, the moon and Earth each orbit the center of mass of the Earth-moon system. Because Earth has a much greater mass than the moon, this center of mass lies within Earth. Section 2 Newtons Law of Universal Gravitation

161 Copyright © by Holt, Rinehart and Winston. All rights reserved. ResourcesChapter menu Chapter 7 Newtons Law of Universal Gravitation Section 2 Newtons Law of Universal Gravitation

162 Copyright © by Holt, Rinehart and Winston. All rights reserved. ResourcesChapter menu Chapter 7 Applying the Law of Gravitation Newtons law of gravitation accounts for ocean tides. High and low tides are partly due to the gravitational force exerted on Earth by its moon. The tides result from the difference between the gravitational force at Earths surface and at Earths center. Section 2 Newtons Law of Universal Gravitation

163 Copyright © by Holt, Rinehart and Winston. All rights reserved. ResourcesChapter menu Chapter 7 Applying the Law of Gravitation, continued Cavendish applied Newtons law of universal gravitation to find the value of G and Earths mass. When two masses, the distance between them, and the gravitational force are known, Newtons law of universal gravitation can be used to find G. Once the value of G is known, the law can be used again to find Earths mass. Section 2 Newtons Law of Universal Gravitation

164 Copyright © by Holt, Rinehart and Winston. All rights reserved. ResourcesChapter menu Chapter 7 Gravity is a field force. Gravitational field strength, g, equals F g /m. The gravitational field, g, is a vector with magnitude g that points in the direction of F g. Gravitational field strength equals free-fall acceleration. Section 2 Newtons Law of Universal Gravitation The gravitational field vectors represent Earths gravitational field at each point. Applying the Law of Gravitation, continued

165 Copyright © by Holt, Rinehart and Winston. All rights reserved. ResourcesChapter menu Chapter 7 Applying the Law of Gravitation, continued weight = mass gravitational field strength Because it depends on gravitational field strength, weight changes with location: Section 2 Newtons Law of Universal Gravitation On the surface of any planet, the value of g, as well as your weight, will depend on the planets mass and radius.

166 Copyright © by Holt, Rinehart and Winston. All rights reserved. ResourcesChapter menu Chapter 7 Sample Problem, continued 1. Define Given: r 1 = 361 km = m Unknown: T = ?v t = ? 2. Plan Choose an equation or situation: Use the equations for the period and speed of an object in a circular orbit. Section 3 Motion in Space

167 Copyright © by Holt, Rinehart and Winston. All rights reserved. ResourcesChapter menu Chapter 7 Sample Problem, continued Use Table 1 in the textbook to find the values for the radius (r2) and mass (m) of Venus. r2 = mm = kg Find r by adding the distance between the spacecraft and Venuss surface (r1) to Venuss radius (r2). r = r1 + r2 r = m m = m Section 3 Motion in Space

168 Copyright © by Holt, Rinehart and Winston. All rights reserved. ResourcesChapter menu Chapter 7 Sample Problem, continued 3. Calculate 4. Evaluate Magellan takes ( s)(1 min/60 s) 94 min to complete one orbit. Section 3 Motion in Space

169 Copyright © by Holt, Rinehart and Winston. All rights reserved. ResourcesChapter menu Chapter 7 Weight and Weightlessness To learn about apparent weightlessness, imagine that you are in an elevator: –When the elevator is at rest, the magnitude of the normal force acting on you equals your weight. –If the elevator were to accelerate downward at 9.81 m/s 2, you and the elevator would both be in free fall. You have the same weight, but there is no normal force acting on you. –This situation is called apparent weightlessness. –Astronauts in orbit experience apparent weightlessness. Section 3 Motion in Space

170 Copyright © by Holt, Rinehart and Winston. All rights reserved. ResourcesChapter menu Chapter 7 Weight and Weightlessness Section 3 Motion in Space

171 Copyright © by Holt, Rinehart and Winston. All rights reserved. ResourcesChapter menu Multiple Choice 1. An object moves in a circle at a constant speed. Which of the following is not true of the object? A. Its acceleration is constant. B. Its tangential speed is constant. C. Its velocity is constant. D. A centripetal force acts on the object. Standardized Test Prep Chapter 7

172 Copyright © by Holt, Rinehart and Winston. All rights reserved. ResourcesChapter menu Multiple Choice 1. An object moves in a circle at a constant speed. Which of the following is not true of the object? A. Its acceleration is constant. B. Its tangential speed is constant. C. Its velocity is constant. D. A centripetal force acts on the object. Standardized Test Prep Chapter 7

173 Copyright © by Holt, Rinehart and Winston. All rights reserved. ResourcesChapter menu Multiple Choice, continued Use the passage below to answer questions 2–3. A car traveling at 15 m/s on a flat surface turns in a circle with a radius of 25 m. 2. What is the centripetal acceleration of the car? F m/s 2 G m/s 2 H. 9.0 m/s 2 J. zero Standardized Test Prep Chapter 7

174 Copyright © by Holt, Rinehart and Winston. All rights reserved. ResourcesChapter menu Multiple Choice, continued Use the passage below to answer questions 2–3. A car traveling at 15 m/s on a flat surface turns in a circle with a radius of 25 m. 2. What is the centripetal acceleration of the car? F m/s 2 G m/s 2 H. 9.0 m/s 2 J. zero Standardized Test Prep Chapter 7

175 Copyright © by Holt, Rinehart and Winston. All rights reserved. ResourcesChapter menu Multiple Choice, continued Use the passage below to answer questions 2–3. A car traveling at 15 m/s on a flat surface turns in a circle with a radius of 25 m. 3. What is the most direct cause of the cars centripetal acceleration? A. the torque on the steering wheel B. the torque on the tires of the car C. the force of friction between the tires and the road D. the normal force between the tires and the road Standardized Test Prep Chapter 7

176 Copyright © by Holt, Rinehart and Winston. All rights reserved. ResourcesChapter menu Multiple Choice, continued Use the passage below to answer questions 2–3. A car traveling at 15 m/s on a flat surface turns in a circle with a radius of 25 m. 3. What is the most direct cause of the cars centripetal acceleration? A. the torque on the steering wheel B. the torque on the tires of the car C. the force of friction between the tires and the road D. the normal force between the tires and the road Standardized Test Prep Chapter 7

177 Copyright © by Holt, Rinehart and Winston. All rights reserved. ResourcesChapter menu Multiple Choice, continued 4. Earth (m = kg) orbits the sun (m = kg) at a mean distance of m. What is the gravitational force of the sun on Earth? (G = Nm 2 /kg 2 ) F N G N H –2 N J –8 N Standardized Test Prep Chapter 7

178 Copyright © by Holt, Rinehart and Winston. All rights reserved. ResourcesChapter menu Multiple Choice, continued 4. Earth (m = kg) orbits the sun (m = kg) at a mean distance of m. What is the gravitational force of the sun on Earth? (G = Nm 2 /kg 2 ) F N G N H –2 N J –8 N Standardized Test Prep Chapter 7

179 Copyright © by Holt, Rinehart and Winston. All rights reserved. ResourcesChapter menu Multiple Choice, continued 5. Which of the following is a correct interpretation of the expression ? A. Gravitational field strength changes with an objects distance from Earth. B. Free-fall acceleration changes with an objects distance from Earth. C. Free-fall acceleration is independent of the falling objects mass. D. All of the above are correct interpretations. Standardized Test Prep Chapter 7

180 Copyright © by Holt, Rinehart and Winston. All rights reserved. ResourcesChapter menu Multiple Choice, continued 5. Which of the following is a correct interpretation of the expression ? A. Gravitational field strength changes with an objects distance from Earth. B. Free-fall acceleration changes with an objects distance from Earth. C. Free-fall acceleration is independent of the falling objects mass. D. All of the above are correct interpretations. Standardized Test Prep Chapter 7

181 Copyright © by Holt, Rinehart and Winston. All rights reserved. ResourcesChapter menu Multiple Choice, continued 12. Which of the following statements is correct? F. Mass and weight both vary with location. G. Mass varies with location, but weight does not. H. Weight varies with location, but mass does not. J. Neither mass nor weight varies with location. Standardized Test Prep Chapter 7

182 Copyright © by Holt, Rinehart and Winston. All rights reserved. ResourcesChapter menu Multiple Choice, continued 12. Which of the following statements is correct? F. Mass and weight both vary with location. G. Mass varies with location, but weight does not. H. Weight varies with location, but mass does not. J. Neither mass nor weight varies with location. Standardized Test Prep Chapter 7

183 Copyright © by Holt, Rinehart and Winston. All rights reserved. ResourcesChapter menu Short Response 14. Explain how it is possible for all the water to remain in a pail that is whirled in a vertical path, as shown below. Standardized Test Prep Chapter 7

184 Copyright © by Holt, Rinehart and Winston. All rights reserved. ResourcesChapter menu Short Response 14. Explain how it is possible for all the water to remain in a pail that is whirled in a vertical path, as shown below. Standardized Test Prep Chapter 7 Answer: The water remains in the pail even when the pail is upside down because the water tends to move in a straight path due to inertia.

185 Copyright © by Holt, Rinehart and Winston. All rights reserved. ResourcesChapter menu Short Response, continued 15. Explain why approximately two high tides take place every day at a given location on Earth. Standardized Test Prep Chapter 7

186 Copyright © by Holt, Rinehart and Winston. All rights reserved. ResourcesChapter menu Short Response, continued 15. Explain why approximately two high tides take place every day at a given location on Earth. Answer: The moons tidal forces create two bulges on Earth. As Earth rotates on its axis once per day, any given point on Earth passes through both bulges. Standardized Test Prep Chapter 7

187 Copyright © by Holt, Rinehart and Winston. All rights reserved. ResourcesChapter menu Chapter 7 Centripetal Acceleration Section 1 Circular Motion

188 Copyright © by Holt, Rinehart and Winston. All rights reserved. ResourcesChapter menu Chapter 7 Centripetal Force Section 1 Circular Motion

189 Copyright © by Holt, Rinehart and Winston. All rights reserved. ResourcesChapter menu Chapter 7 Keplers Laws Section 3 Motion in Space


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