1. Chapter 3 – Two Dimensional Motion and Vectors

2. 3 – 1: Objectives Distinguish between a scalar and a vector
Add and subtract vectors using the graphical method
Multiply and Divide Vectors by Scalars

3. Every physical quantity is either a scalar or a vector quantity
Scalar: a physical quantity that can be completely specified by its magnitude (a number) with appropriate units.
Examples: mass, speed, distance and volume
Vector: a physical quantity that has both magnitude and direction
Examples: position, displacement, velocity, and acceleration

4. Notation used to represent vector quantities
Book uses boldface type to represent vector quantities v a x ∆x
Handwritten – place a “vector symbol” over the variable v a x ∆x

5. Vectors can be represented by diagrams
Arrows are used to show a vector quantity that points in the direction of the vector.
The length of the arrow represents the magnitude
50 m/s, East
25 m/s, East
Notice, the 50 m/s vector is twice as long as the 25 m/s vector

6. Draw 2 vectors that represent 10 m east and 15 m west
Notice: The arrow head is pointing in the required direction and the lengths are drawn to a chosen scale where each unit represents 5 m.

7. Vector Addition – Graphical Method
1. Vectors to be added are physically placed tip – to – tail (the tip of one vector touches the tail of the next vector) in any order
NOTE: Within a diagram, vectors can be moved (translated) for the purpose of vector addition, as long as the direction and the length remain the same.

8. Resultant Vector the sum of 2 or more vectors
the solution to a vector addition problem
also called vector sum

9. Finding a Resulant Vector
Found graphically by drawing another vector that begins at the tail of the first vector and ends at the tip of the last vector that is being added.
--NOT TIP-TO-TAIL! Beginning to end.

10. Graphical Vector Addition in One - Dimension
Tip – to - tip
Tail – to - tail
Tip – to - tail
NOTES: technically if all vectors are in one – dimension, they would be drawn on top of each other, these are separated slightly for clarity.
The magnitude of the resultant vector can be found by measuring the length and converting the number to the proper units using the given scale.
The direction is shown by the arrow tip.

11. The resultant vector is obviously 5 m west.
The diagram shown on the previous page shows 2 displacement vectors that were being added (10 m east and 15 m west)
The resultant vector is obviously 5 m west.
In one – dimension it is certainly easier to use the magnitude and a +/- sign for direction to add the vectors
Ex. (+10 m) + (-15 m) = -5 m
The resultant vector is 5 m to the west!

12. Graphically Add the following 3 displacement vectors (1-dimensional)
Choose an appropriate scale and draw the graphical solution to this vector addition problem
225 m north, 175 m south, and 125 m south

13. Graphical Vector Addition in 2 Dimensions
The graphical procedure is the same as in 1 – dimension
Vectors to be added are physically placed tip – to – tail (the tip of one vector touches the tail of the next vector) in any order
The resultant vector is found graphically by drawing another vector that begins at the tail of the first vector and ends at the tip of the last vector

14. Vector 1
Plus vector 2
Resultant vector

15. The direction is described differently.
The magnitude of the resultant vector can be found by measuring the length and converting the number to the proper units using the given scale. (exactly the same as in 1 – dimension)
The direction is described differently.
The direction of a 2 – dimensional vector is graphically determined with a protractor and is measured counter-clockwise (CCW) from the +x - axis

16. Vector 1
Plus vector 2
Resultant vector
2. Measure the direction CCW from the + x - axis
1. Place an x/y coordinate system at THE TAIL of your resultant vector θ

17. Important comment!
If given a vector diagram where the vectors are not drawn tip - to – tail, you can move a vector in a diagram so that you can set up a tip – to – tail situation! Proceed as before.

18. Vector 2
Vector 1
Vector 1
Resultant Vector
Vector 2

19. Hints about vector addition
When adding vectors:
1. The vectors must represent the same physical quantity (you can’t add velocity and displacement)
2. The vector quantities must have the same units (you can’t add m and km, you must convert first)

20. Resultant Vector
The resultant vector represents a SINGLE vector that produces the same RESULT as the other vectors (addends) acting together

21. Example (Displacement)
Walking 3 m east and then 4 m north puts you at the same final position as walking 5 m at an angle of 53º
4 m
5 m
53º
3 m

22. Sample problem
140 m
120 m
Find the resultant displacement.

23. A person rows due east across the Delaware River at 8. 0 m/s
A person rows due east across the Delaware River at 8.0 m/s. The current carries the boat downstream (south) at 2.5 m/s. What is the person’s resultant velocity?

24. Graphical Vector Addition Practice
Worksheet
Rulers
Protractors

25. Review Problems
Two ropes are tied to a tree to be cut down. The first rope pulls on the tree with a force of 350 N west. The second pulls at 425 N at 320 degrees. What’s the resultant force?
A person drives through town 6 blocks north, then 3 blocks east. They run into a one way street and have to travel 1 block south to go 2 more blocks east. Finally, the person parks and walks 2 blocks north to the destination. What is the person’s displacement?

26. Part II

27. Properties of Vectors
1. Vectors may be translated in a diagram (moved parallel to themselves)
2. Vectors may be added in any order (Vector addition is commutative)
3. To subtract a vector, add its opposite.
The opposite of a vector has the same magnitude and points in the opposite direction. (+/- 180º)
4.Multiplying or dividing vectors by scalars results in vectors

28. 2. Vectors may be added in any order (Vector addition is commutative)

29. 3. To subtract a vector, add its opposite.
A - B
A + (-B) A A -B B

30. 4.Muliplying or dividing vectors by scalars results in vectors
Notice: The magnitude is multiplied or divided but the direction remains the same.
A ball is thrown 25 m at an angle of 30º
Two times this displacement vector is 50 m at an angle to 30º

31. Sample problems Given the following vectors:
A = 50 m South B = 80 m East
C = ° D = °
Find:
A – C D + B -2A
3. ½ B – 4A

32. 3-2 Vector Operations 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

33. Geometry / Trigonometry Review
Pythagorean Theorem – The square of the hypotenuse of a right triangle is equal to the sum of the squares of its legs
c2 = a2 + b2 c a b

34. Trigonometric Ratios:
Hypotenuse (c)
Leg (b)
Leg (a) C A B
Adjacent side
Opposite side
Opposite side
Cos θ =
Tan θ =
Sin θ =
hypotenuse
Adjacent side
hypotenuse

35. Using Trig. Ratios
Given an acute angle of a right triangle, to find the ratio of 2 specific sides of the triangle, enter the appropriate function (sine, cosine, tangent) of the angle in your calculator.
Sin(20º)=b/c
Cos(20º) = a/c
Tan(20º) = b/a a b c
20º

36. To find an acute angle of a right triangle, enter the inverse of the appropriate function of the ratio of the 2 corresponding sides.
θ = sin-1(a/c)
θ = cos-1(b/c)
θ = tan-1(a/b) a b c θ

37. Trigonometry Review Practice Worksheet

38. Part III

39. Vector Addition – Analytical Method
Case #1 (easiest method):
Adding 2 Vectors that are perpendicular
Resultant, R A B θ
R = magnitude of the resultant vector
R = A2 + B2
The angle, θ, of the triangle can be found using the tan-1 function and THEN CONVERT it to the direction measured CCW from the +x - axis

40. Example for Case #1 Add the following 2 velocity vectors.
5 m/s west (180º) and 8 m/s north (90º)
R2 =
R = 9.4 m/s
θ = tan-1 (8/5)
θ = 58º
The direction (measured CCW from the +x – axis) is found by subtracting 180 – 58 = 122º R
8 m/s θ
5 m/s
R = 9.4 m/s <122º

41. Case #2: Adding more than 2 perpendicular vectors
First, find the vector sum of all of the horizontal vectors, call this Rx.
Second, find the vector sum of all of the vertical vectors, call this Ry.
Find the vector sum of Rx and Ry
By following the method from Case #1

42. Example of Case #2
A boyscout walks 8 m east, 2 m north, 6 m east, 10 m south, 3 m east, 5 m south and 3 m west.

43. Horizontal Vectors
Vertical Vectors
+ 8m
+ 6m
+ 3m
- 3m
+ 2m
10m 5m
-13 m
+14 m

44. Rx Ry R θ
R2 =
R = 19.1 m
θ = tan-1 (13/14)
θ = 43º
The direction (measured CCW from the +x – axis) is found by subtracting 360 – 43 = 317º
R = 19.1 m <317º

45. Vector Resolution (opposite process of adding 2 vectors)
Any vector acting at an angle can be replaced with 2 vectors that act perpendicular to each other, one horizontal and one vertical. (The 2 vectors working together are equivalent to the single vector acting at an angle.)

46. Step 1
Sketch the given vector with the tail located at the origin of an x-y coordinate system. (Ex. 25 m at an angle of 36º)
25 m
36º

47. Step 2
Draw a line segment from the tip of the vector perpendicular to the x-axis
25 m
36º
Notice, you now have a right triangle with a known hypotenuse and known angle measurements

48. Step 3
Replace the perpendicular sides of the right triangle with vectors drawn tip – to - tail
25 m

49. Step 4
Use sine and cosine functions to find the horizontal and vertical components of the given vector.
25 m Ry
36º
Cos(36) = Rx/25
Rx = 25cos(36)
Rx = 20.2 m
sin(36) = Ry/25
Ry = 25sin(36)
Ry = 14.7 m Rx

50. Important
Remember that the 2 components acting together gives the same result as the single vector acting at an angle.
****The 2 components can be used to REPLACE the single vector****

51. Example #2 Find the components of 16m at 200º 200º 20º
You have 2 choices at this point.
You can use the directional angle of 200 and not worry about the sign of the components (the calculator will do it for you).
OR, you can use 20 and YOU must remember to put – signs when the component points down or to the left

52. Example #2 Find the components of 16m at 200º 200º 20º
Rx = 16cos(200) = -15 m
Ry = 16sin(200) = -5.5 m
Rx = -16cos(20) = -15 m
Ry = -16sin(20) = -5.5 m

53. Case #3 – Adding Vectors at Angles (not perpendicular)
When vectors to be added are not perpendicular, they do not form sides of a right triangle.

54. Look at the geometry for the situation
Resultant Vector
Rx2
Ry2
Ry1 + Ry2
Rx1
Ry1
Rx1 + Rx2

55. Notice, the length of the horizontal component of the resultant vector is equal to the sum of the lengths of the horizontal components of the vectors that are being added together.
This is also true for the vertical component.

56. Steps for solving Case #3 Problems
1. Resolve each vector that is being added (addends) into components.
2. Add all the horizontal components together and all the vertical components together (Case #2)
3. Use the Pythagorean Theorem and trig ratios to find the resultant vector (Case #1)

57. Example for Case #3
Add these 2 vectors together: 10 m/s at 0º and 12 m/s at 25º (Find the resultant vector, R at θ) R
12 m/s θ
25º
10 m/s

58. Example for Case #3 Find components of each vector x y Vector 1
10cos(0)
10sin(0) R
Vector 2
12cos(25)
12sin(25)
12 m/s θ
25º
10 m/s

59. Example for Case #3 Add horizontal and vertical components x y
Vector 1
10cos(0)
10sin(0) R
Vector 2
12cos(25)
12sin(25)
12 m/s θ
25º 21
5.1
10 m/s

60. Example for Case #3
Find the magnitude of the resultant vector using the Pythagorean Theorem
R2 = R
5.1
R = 21.6 m/s
θ = tan-1 (5.1/21)
θ = 14º θ 21
R = 21. m/s at 14º