1. Two-Dimensional Motion and VectorsSection 1 Preview Section 1 Introduction to VectorsIntroduction to Vectors Section 2 Vector OperationsVector Operations Section 3 Extra QuestionsExtra Questions

2. Two-Dimensional Motion and VectorsSection 1 What do you think? How are measurements such as mass and volume different from measurements such as velocity and acceleration? How can you add two velocities that are in different directions?

3. Two-Dimensional Motion and VectorsSection 1 Introduction to Vectors Scalar - a quantity that has magnitude but no direction –Examples: volume, mass, temperature, speed Vector - a quantity that has both magnitude and direction –Examples: acceleration, velocity, displacement, force

4. Two-Dimensional Motion and VectorsSection 1 Vector Properties Vectors are generally drawn as arrows. –Length represents the magnitude –Arrow shows the direction Resultant - the sum of two or more vectors

5. Two-Dimensional Motion and VectorsSection 1 Finding the Resultant Graphically Method –Draw each vector in the proper direction. –Establish a scale (i.e. 1 cm = 2 m) and draw the vector the appropriate length. –Draw the resultant from the tip of the first vector to the tail of the last vector. –Measure the resultant. The resultant for the addition of a + b is shown to the left as c.

6. Two-Dimensional Motion and VectorsSection 1 Vector Addition Vectors can be moved parallel to themselves without changing the resultant. –the red arrow represents the resultant of the two vectors

7. Two-Dimensional Motion and VectorsSection 1 Vector Addition Vectors can be added in any order. –The resultant (d) is the same in each case Subtraction is simply the addition of the opposite vector.

8. Two-Dimensional Motion and VectorsSection 1 Click below to watch the Visual Concept. Visual Concept Properties of Vectors

9. Two-Dimensional Motion and VectorsSection 1 Sample Resultant Calculation A toy car moves with a velocity of.80 m/s across a moving walkway that travels at 1.5 m/s. Find the resultant speed of the car.

10. Two-Dimensional Motion and VectorsSection 1 Now what do you think? How are measurements such as mass and volume different from measurements such as velocity and acceleration? How can you add two velocities that are in different directions?

11. Two-Dimensional Motion and VectorsSection 2 What do you think? What is one disadvantage of adding vectors by the graphical method? Is there an easier way to add vectors?

12. Two-Dimensional Motion and VectorsSection 2 Vector Operations Use a traditional x-y coordinate system as shown below on the right. The Pythagorean theorem and tangent function can be used to add vectors. –More accurate and less time-consuming than the graphical method

13. Two-Dimensional Motion and VectorsSection 2 Pythagorean Theorem and Tangent Function

14. Two-Dimensional Motion and VectorsSection 2 Vector Addition - Sample Problems 12 km east + 9 km east = ? –Resultant: 21 km east 12 km east + 9 km west = ? –Resultant: 3 km east 12 km east + 9 km south = ? –Resultant: 15 km at 37° south of east 12 km east + 8 km north = ? –Resultant: 14 km at 34° north of east

15. Two-Dimensional Motion and VectorsSection 2 Resolving Vectors Into Components

16. Two-Dimensional Motion and VectorsSection 2 Resolving Vectors into Components Opposite of vector addition Vectors are resolved into x and y components For the vector shown at right, find the vector components v x (velocity in the x direction) and v y (velocity in the y direction). Assume that that the angle is 20.0˚. Answers: –v x = 89 km/h –v y = 32 km/h

17. Two-Dimensional Motion and VectorsSection 2 Adding Non-Perpendicular Vectors Four steps –Resolve each vector into x and y components –Add the x components (x total =  x 1 +  x 2 ) –Add the y components (y total =  y 1 +  y 2 ) –Combine the x and y totals as perpendicular vectors

18. Two-Dimensional Motion and VectorsSection 2 Click below to watch the Visual Concept. Visual Concept Adding Vectors Algebraically

19. Two-Dimensional Motion and VectorsSection 2 Classroom Practice A camper walks 4.5 km at 45° north of east and then walks 4.5 km due south. Find the camper’s total displacement. Answer –3.4 km at 22° S of E

20. Two-Dimensional Motion and VectorsSection 2 Now what do you think? Compare the two methods of adding vectors. What is one advantage of adding vectors with trigonometry? Are there some situations in which the graphical method is advantageous?

21. Two-Dimensional Motion and VectorsSection 4 Classroom Practice Problem A plane flies northeast at an airspeed of 563 km/h. (Airspeed is the speed of the aircraft relative to the air.) A 48.0 km/h wind is blowing to the southeast. What is the plane’s velocity relative to the ground? Answer: 565.0 km/h at 40.1° north of east How would this pilot need to adjust the direction in order to to maintain a heading of northeast?

22. Section 4Two-Dimensional Motion and Vectors Now what do you think? Suppose you are traveling at a constant 80 km/h when a car passes you. This car is traveling at a constant 90 km/h. –How fast is it going, relative to your frame of reference? –How fast is it moving, relative to Earth as a frame of reference? Does velocity always depend on the frame of reference? Does acceleration depend on the frame of reference?

23. Section 4Two-Dimensional Motion and Vectors Preview Multiple Choice Short Response Extended Response

24. Section 4Two-Dimensional Motion and Vectors 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

25. Section 4Two-Dimensional Motion and Vectors 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

26. Section 4Two-Dimensional Motion and Vectors 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

27. Section 4Two-Dimensional Motion and Vectors 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. 10.0 m/s to the southeast

28. Section 4Two-Dimensional Motion and Vectors 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

29. Section 4Two-Dimensional Motion and Vectors 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

30. Section 4Two-Dimensional Motion and Vectors 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. 11.40 m southeast G. 8.07 m at 48.0º south of east H. 3.32 m at 42.0º south of east J. 8.07 m at 42.0º south of east

31. Section 4Two-Dimensional Motion and Vectors Short Response 13. If one of the components of one vector along the direction of another vector is zero, what can you conclude about these two vectors?

32. Section 4Two-Dimensional Motion and Vectors Short Response, continued 14. A roller coaster travels 41.1 m at an angle of 40.0° above the horizontal. How far does it move horizontally and vertically?

33. Section 4Two-Dimensional Motion and Vectors Short Response, continued 14. 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