1. Physics: Problem Solving Chapter 4 Vectors

2. Physics: Problem Solving Chapter 4 Vectors

3. Chapter 4: Vectors Vector Review Trigonometry for Physics Vector Addition—Algebraic Vector Resolution

4. Chapter 4: Vectors Vector: A measurement with both magnitude and direction Magnitude: A numerical value Direction: +/– North, South, East, West Which is larger 2m/s or – 3 m/s?

5. Chapter 4: Vectors Vector: What does it mean when an object has…..? Velocity—negative and positive Acceleration—negative and positive

6. Chapter 4: Vectors Vectors can be added together both graphically and algebraically Graphic addition: using arrows of proper length and direction to add vectors together Algebraic addition: using trigonometry to add vectors together

7. Chapter 4: Vectors Graphic addition: “Butt-Head” Method 1.Draw first vector to scale (magnitude & direction) 2.Draw second vector to scale. Connect the “butt” of the second vector to the “head” of the first vector 3.Repeat Step #2 until you run out of vectors. 4.Draw Resultant (?). Connect the “butt” of the first vector with the “head” of the last vector—butt-butt, head-head

8. Chapter 4: Vectors Graphic addition: “Butt-Head” Method A dog walks 15 km east and then 10 km west find the sum of the vectors (resultant)

9. Chapter 4: Vectors Graphic addition: “Butt-Head” Method A dog walks 15 km east and then 10 km west find the sum of the vectors (resultant) 1.Draw first vector to scale (magnitude & direction)

10. Chapter 4: Vectors Graphic addition: “Butt-Head” Method A dog walks 15 km east and then 10 km west find the sum of the vectors (resultant) 2. Draw second vector to scale. Connect the “butt” of the second vector to the “head” of the first vector

11. Chapter 4: Vectors Graphic addition: “Butt-Head” Method A dog walks 15 km east and then 10 km west find the sum of the vectors (resultant) 4. Draw Resultant (?). Connect the “butt” of the first vector with the “head” of the last vector—butt-butt, head-head Measure: 5 km east

12. Chapter 4: Vectors Graphic addition: “Butt-Head” Method A dog walks 15 km east and then 10 km south find the sum of the vectors (resultant) Measure: 18 km 34  south of east

13. Chapter 4: Vectors Graphic addition: “Butt-Head” Method Examples A shopper walks from the door of the mall to her car 250 m down a row of cars, then turns 90  to the right and walks another 60 m. What is the magnitude and direction of her displacement from the door? Answer 257 meters 13.5  from door

14. Chapter 4: Vectors This presentation deals with adding vectors algebraically But first…..you need to learn some trigonometry!!!

15. Vector Addition-Graphic Let’s test our knowledge! (1-5)

16. All the Trig. you need to know for Physics (almost) This is a right triangle

17. All the Trig. you need to know for Physics (almost) This is an angle (  -theta) in a right triangle 

18. All the Trig. you need to know for Physics (almost) This is the hypotenuse (H) of a right triangle 

19. All the Trig. you need to know for Physics (almost) This is the side adjacent (A) to the angle (  -theta) in a right triangle 

20. All the Trig. you need to know for Physics (almost) This is the side opposite (O) the angle (  -theta) in a right triangle 

21. All the Trig. you need to know for Physics (almost) Summary Angle  Opposite (O) side Adjacent (A) side Hypotenuse (H)

22. All the Trig. you need to know for Physics (almost) SOH, CAH, TOA Trigonometry Used when working with right triangles only! Angle  Opposite (O) side Adjacent (A) side Hypotenuse (H)

23. All the Trig. you need to know for Physics (almost) SOH sin  = O/H Angle  Opposite (O) side Hypotenuse (H)

24. All the Trig. you need to know for Physics (almost) Example: Find the angle of a right triangle which has a hypotenuse of 12m and a side opposite the angle of 9m. Angle  = ? Opposite (O) side = 9m Hypotenuse (H) = 12m

25. All the Trig. you need to know for Physics (almost) Example: sin  = O/H sin  = 9/12 = 0.75  = sin –1 0.75 = 48.6° Angle  = ? Opposite (O) side = 9m Hypotenuse (H) = 12m

26. All the Trig. you need to know for Physics (almost) CAH cos  = A/H Angle  Adjacent (A) side Hypotenuse (H)

27. All the Trig. you need to know for Physics (almost) Example: Find the hypotenuse of a right triangle which has an angle of 35° and a side adjacent the angle of 7.5 m. Angle  = 35° Adjacent (A) side = 7.5m Hypotenuse (H) = ?

28. All the Trig. you need to know for Physics (almost) Example: cos  = A/H cos 35 = 7.5/H H = 7.5/cos 35° = 7.5/0.819 = 9.16m Angle  = 35° Adjacent (A) side = 7.5m Hypotenuse (H) = ?

29. All the Trig. you need to know for Physics (almost) TOA tan  = O/A Angle  Opposite (O) side Adjacent (A) side

30. All the Trig. you need to know for Physics (almost) Example: Find the side opposite the 35° angle of a right triangle which has a side adjacent the angle of 7.5 m. Angle  = 35° Opposite (O) side Adjacent (A) side = 7.5m

31. All the Trig. you need to know for Physics (almost) tan  = O/A tan 35 = O/7.5 O = (7.5)(tan 35) = (7.5)(.7)= 5.25m Angle  = 35° Opposite (O) side Adjacent (A) side = 7.5m

32. All the Trig. you need to know for Physics (almost) SOH, CAH, TOA Trigonometry Remember the Pythagorean Theorem Work on Trig. Worksheet—due tomorrow Angle  Opposite (O) side Adjacent (A) side Hypotenuse (H)

33. Vector Addition—Algebraic When adding vectors together algebraically using trigonometry it is important to use the Physics Problem Solving Technique

34. Vector Addition—Algebraic Example: A car moves east 45 km turns and travels west 30 km. What are the magnitude and direction of the car’s total displacement?

35. Vector Addition—Algebraic Example: A car moves east 45 km turns and travels west 30 km. What are the magnitude and direction of the car’s total displacement? 15 km east

36. Vector Addition—Algebraic Example: A car moves east 45 km turns and travels north 30 km. What are the magnitude and direction of the car’s total displacement? a 2 + b 2 = c 2 c = 54.1 km magnitude

37. Vector Addition—Algebraic Example: A car moves east 45 km turns and travels north 30 km. What are the magnitude and direction of the car’s total displacement? a 2 + b 2 = c 2 c = 54.1 km magnitude Tan  = o/a Tan  = 30/45  = Tan -1 0.667  = 33.7 

38. Vector Addition—Algebraic Example: A car moves east 45 km turns and travels north 30 km. What are the magnitude and direction of the car’s total displacement? 54.1 km 33.7  North of East (?)

39. Vector Addition—Algebraic Example: A car is driven east 125 km and then south 65 km. What are the magnitude and direction of the car’s total displacement? Answer: 141 km 27.5  south of east

40. Vector Addition—Algebraic Example: A boat is rowed directly across a river at a speed of 2.5 m/s. The river is flowing at a rate of 0.5 m/s. Find the magnitude and direction of the boat’s diagonal motion. Answer 2.55 m/s 11.3  measured from center of river (or 78.7  measured from shore)

41. Vector Resolution Vector Resolution—the process of breaking a single vector into its components Components—the two perpendicular vectors that when added together give a single vector Components are along the x-axis and y-axis

42. Vector Resolution Example A bus travels 23 km on a straight road that is 30  north of east. What are the east and north components of its displacement?

43. Vector Resolution Example A bus travels 23 km on a straight road that is 30  north of east. What are the east and north components of its displacement?

44. Vector Resolution Example A bus travels 23 km on a straight road that is 30  north of east. What are the east and north components of its displacement? cos  = a/h cos 30 = a/23 a = 23cos30 = 19.9 km east

45. Vector Resolution Example A bus travels 23 km on a straight road that is 30  north of east. What are the east and north components of its displacement? cos  = a/h cos 30 = a/23 a = 23cos30 = 19.9 km east sin  = o/h sin 30 = o/23 o = 23sin30 o = 11.5 km north

46. Vector Resolution Example A golf ball, hit from the tee, travels 325 m in a direction 25  south of east. What are the east and south components of its displacement? Answer East (cos 25 = a/325) 295 m South (sin 25 = o/325) 137 m

47. Vector Resolution Example An airplane flies at 65 m/s at 31  north of west. What are the north and west components of the plane’s velocity? Answer north (sin 31 = o/65) 33.5 m/s west (cos 31 = a/65) 55.7 m/s

48. Vector Addition and Resolution Let’s check our knowledge! (6-10)