1. Chapter 3: Two-Dimensional Motion and Vectors
Physics
Chapter 3:
Two-Dimensional Motion and Vectors

2. Two-Dimensional Motion and Vectors
Scalar and Vector Quantities
Scalar Quantity
Magnitude (number and units)
ex. 15km, 250C
Vector Quantity
Magnitude and Direction
ex. 15km north

3. Two-Dimensional Motion and Vectors
Represented by Arrows
Direction of the Arrow Represents the Direction of the Vector
The Length of the Arrow is Proportional to the Magnitude of the Vector

4. Two-Dimensional Motion and Vectors
Components
Two Perpendicular Components Add to Form Resultant (Displacement)
A Vector May be Broken Down Into Its “X” and “Y” Components
Resultant
Vector Sum of Two or More Vectors

5. Two-Dimensional Motion and Vectors
Components
Vector “r”
West Component
North Component

6. Two-Dimensional Motion and Vectors
Components

7. Two-Dimensional Motion and Vectors
Components

8. Two-Dimensional Motion and Vectors
Ex. Height Determination
Distance to Cliff = 500’
Angle = 34o

9. Two-Dimensional Motion and Vectors
Ex. Height Determination
Distance to Cliff = 500’
Angle = 34o

10. Two-Dimensional Motion and Vectors
Components
Determination of Direction
Observe Direction From Tail of Resultant to Right Angle Formed by X and Y Components

11. Two-Dimensional Motion and Vectors
Determination of Direction

12. Two-Dimensional Motion and Vectors
Adding and Subtracting Vectors
Adding Vectors Graphically
To add vectors A and B, place the tail of B at the head of A. The sum, C = A+B, is the vector extending from the tail of A to the head of B.

13. Two-Dimensional Motion and Vectors
Adding and Subtracting Vectors
Adding Vectors Graphically

14. Two-Dimensional Motion and Vectors
Adding and Subtracting Vectors
Adding Vectors Graphically
C = A + B

15. Two-Dimensional Motion and Vectors
Adding and Subtracting Vectors
Adding Vectors Graphically
C = A + B
C = B + A

16. Two-Dimensional Motion and Vectors
Adding and Subtracting Vectors
Adding Vectors Using Components

17. Two-Dimensional Motion and Vectors
Adding and Subtracting Vectors
Adding Vectors Using Components
A=5.00m
B=4.00m

18. Two-Dimensional Motion and Vectors
Adding and Subtracting Vectors
Adding Vectors Using Components
A=5.00m
B=4.00m

19. Two-Dimensional Motion and Vectors
Adding and Subtracting Vectors
Adding Vectors Using Components
A=5.00m
B=4.00m
Ax=2.50m
Ay=4.33m
Bx=3.76m
By=1.37m

20. Two-Dimensional Motion and Vectors
Adding and Subtracting Vectors
Adding Vectors Using Components

21. Two-Dimensional Motion and Vectors
Adding and Subtracting Vectors
Subtracting Vectors
D = A - B
D = A + (-B)
Now the same rules for adding apply 

22. Two-Dimensional Motion and Vectors
Horizontal and Vertical Motions are Independent

23. Two-Dimensional Motion and Vectors
An Object with a Horizontal Velocity will Continue with the Same Velocity in the Horizontal Direction Even While an Independent Vertical Motion Acts on the Object
Each Motion Continues as if the Other Motion Were Not Present

24. Two-Dimensional Motion and Vectors
Motion in Two Dimensions
Constant Velocity
vi = 26 m/s
What is the turtle’s position after 5.0s?

25. Two-Dimensional Motion and Vectors
Motion in Two Dimensions
Constant Velocity
vi = 26 m/s
t = 5.0s

26. Two-Dimensional Motion and Vectors
Motion in Two Dimensions
Constant Velocity
vi = 26 m/s
t = 5.0s
What are the x and y components of the vector?

27. Two-Dimensional Motion and Vectors
Motion in Two Dimensions
Constant Velocity
vi = 26 m/s
t = 5.0s
q = 25o

28. Two-Dimensional Motion and Vectors
Motion in Two Dimensions
Constant Velocity
vi = 26 m/s
What is the turtle’s displacement after 5.0s?

29. Two-Dimensional Motion and Vectors
Motion in Two Dimensions
Constant Velocity
vi = 26 m/s
What is the turtle’s displacement after 5.0s?
Well, We Could Find the x and y Components First.

30. Two-Dimensional Motion and Vectors
Motion in Two Dimensions
Constant Velocity
vi = 26 m/s
t = 5.0s

31. Two-Dimensional Motion and Vectors
Motion in Two Dimensions
Constant Velocity
vi = 26 m/s
t = 5.0s
vix = 23.56m/s
viy = 10.99m/s

32. Two-Dimensional Motion and Vectors
Motion in Two Dimensions
Constant Acceleration
Remember this?
We Can use This on Each Component

33. Two-Dimensional Motion and Vectors
Motion in Two Dimensions
Constant Acceleration
And Remember This?
We Can use This on Each Component Too

34. Two-Dimensional Motion and Vectors
Motion in Two Dimensions
Constant Acceleration
And Remember This?
We Can use This on Each Component Too

35. Two-Dimensional Motion and Vectors
Problem
As you walk to class with a constant speed of 1.60 m/s, you are moving in a direction that is 15.0° north of east. How long does it take you to move 30.0 m north?

36. Two-Dimensional Motion and Vectors
Solution
v = 1.60m/s
q = 15.0o
y = 30.0m
t = ?

37. Two-Dimensional Motion and Vectors
Problem

38. Two-Dimensional Motion and Vectors
Problem
Two canoeists start paddling at the same time and head toward a small island in a lake. Canoeist 1 paddles with a speed of 1.35m/s at an angle of 45° north of east. Canoeist 2 starts on the opposite shore of the lake, a distance of 1.5 km due east of canoeist 1. In what direction relative to north must canoeist 2 paddle to reach the island?

39. Two-Dimensional Motion and Vectors
Solution
Canoeist 1’s 45° path determines an isosceles right triangle whose legs measure 1.0 km. So canoeist 2’s path determines a right triangle whose legs measure 1.0 km and 0.5 km. Then for canoeist 2…

40. Two-Dimensional Motion and Vectors
Problem
Two canoeists start paddling at the same time and head toward a small island in a lake. Canoeist 1 paddles with a speed of 1.35m/s at an angle of 45° north of east. Canoeist 2 starts on the opposite shore of the lake, a distance of 1.5 km due east of canoeist 1. What speed must canoeist 2 have if the two canoes are to arrive at the island at the same time?

41. Two-Dimensional Motion and Vectors
Solution
v1 = 1.35m/s
d1 =
d2 =
q1 = 45o
v2 = ?

42. Two-Dimensional Motion and Vectors
Homework
Pages
Problems
6 (a, o below x-axis b, o above x-axis c, o below x-axis d, o clockwise from positive x-axis)
9 58.4o S of E)
27 (2.81km E, 1.31km N)
o S of W)

43. Two-Dimensional Motion and Vectors
Projectile Motion
A Projectile is an Object that is Launched into Motion by Any Means and Then Allowed to Follow a Path Determined Solely by the Influence of Gravity

44. Two-Dimensional Motion and Vectors
Projectile Motion
Assumptions…
Ignore Air Resistance
g = -9.81m/s2
Ignore the Earth’s Rotation

45. Two-Dimensional Motion and Vectors
Projectile Motion
Free Fall /w Initial Horizontal Velocity
Parabolic Path

46. Two-Dimensional Motion and Vectors
Projectile Motion
Acceleration
Horizontal Acceleration is Always Zero (Constant Velocity)
Vertical Acceleration is Always –g

47. Two-Dimensional Motion and Vectors
Projectile Motion
Equations
ax = 0, ay = -g
Displacement

48. Two-Dimensional Motion and Vectors
Projectile Motion
Equations
ax = 0, ay = -g
Velocity

49. Two-Dimensional Motion and Vectors
Projectile Motion
Zero Launch Angle

50. Two-Dimensional Motion and Vectors
Projectile Motion
Zero Launch Angle
q = 0o

51. Two-Dimensional Motion and Vectors
Projectile Motion
Zero Launch Angle
Where Will an Object Land?
y = 0 = ground
h = release height

52. Two-Dimensional Motion and Vectors
Projectile Motion

53. Two-Dimensional Motion and Vectors
Projectile Motion
General Launch Angle

54. Two-Dimensional Motion and Vectors
Projectile Motion
General Launch Angle

55. Two-Dimensional Motion and Vectors
Projectile Motion
Time of Flight
Full
At Specific dx

56. Two-Dimensional Motion and Vectors
Projectile Motion
Maximum Range (R)
Projectile Range is Dependant on the Angle q that the Projectile is Fired from Horizontal

57. Two-Dimensional Motion and Vectors
Projectile Motion
Maximum Range (R)
With Air Resistance

58. Two-Dimensional Motion and Vectors
Projectile Motion
Maximum Height (ymax)

59. Two-Dimensional Motion and Vectors
Projectile Motion
Symmetry of Projectile Motion
Parabolic Flight
Peak of Flight = ½ Time of Flight
Vertical Impact Velocity = -Vertical Launch Velocity
Impact Angle = Launch Angle

60. Two-Dimensional Motion and Vectors
Problem
What is the acceleration of a projectile when it reaches its highest point?

61. Two-Dimensional Motion and Vectors
Solution
The only acceleration on a projectile is gravity. At any point, the acceleration on a projectile is equal to g.

62. Two-Dimensional Motion and Vectors
Problem
A particle passes through the origin with a velocity of 6.2m/s in the y axis. If the particle’s acceleration is -4.4m/s2 in the x axis what are its x and y positions after 5.0 s?

63. Two-Dimensional Motion and Vectors
Solution
ax = -4.4m/s2
vy = 6.2m/s
t = 5.0s
x = ?
y = ?

64. Two-Dimensional Motion and Vectors
Problem
Playing shortstop, you pick up a ground ball and throw it to second base. The ball is thrown horizontally, with a speed of 22m/s, directly toward point A. When the ball reaches the second baseman 0.45 s later, it is caught at point B. How far were you from the second baseman and what is the distance of the vertical drop?

65. Two-Dimensional Motion and Vectors
Solution
vx = 22m/s
t = 0.45s

66. Two-Dimensional Motion and Vectors
Problem
A soccer ball is kicked with a speed of 9.50m/s at an angle of 25.0° above the horizontal. If the ball lands at the same level from which it was kicked, how long was it in the air?

67. Two-Dimensional Motion and Vectors
At the peak of the ball’s trajectory,
Solution so
The total time is twice this. 2t = 2( s) = s

68. Two-Dimensional Motion and Vectors
Problem
A golfer gives a ball a maximum initial speed of 30.0m/s. What is the highest tree the ball could clear on its way to the hole-in-one?

69. Two-Dimensional Motion and Vectors
Solution
vi = 30.0m/s
q = 45o
ymax = ?

70. Two-Dimensional Motion and Vectors
Problem

71. Two-Dimensional Motion and Vectors
Problem
Astronomers have discovered several volcanoes on Io, a moon of Jupiter. One of them, named Loki, ejects lava to a maximum height of 2.00x105m. What is the initial speed of the lava?
(The acceleration of gravity on Io is 1.80m/s2)

72. Two-Dimensional Motion and Vectors
Solution
ymax = 2.00x105m/s
g = 1.80m/s2
At the lava’s maximum height,

73. Two-Dimensional Motion and Vectors
Problem
A ball thrown straight upward returns to its original level in 2.50s. A second ball is thrown at an angle of 40.0° above the horizontal. What is the initial speed of the second ball if it also returns to its original level in 2.50s?

74. Two-Dimensional Motion and Vectors
Solution
t1 = 2.50s
t2 = 2.50s
q1 = 90o
q2 = 40.0o
vo = ?
at the ball’s maximum height.

75. Two-Dimensional Motion and Vectors
Relative Motion
Solved by the use of Vector Addition and Subtraction

76. Two-Dimensional Motion and Vectors
Relative Motion

77. Two-Dimensional Motion and Vectors
Relative Motion?

78. Two-Dimensional Motion and Vectors
Relative Motion
Climbing 0.20 m/s (vpt)
Train speed is 0.70 m/s (vtg)
What is the Relative Velocity of the Person on the Train? (vpg)

79. Two-Dimensional Motion and Vectors
Relative Motion
vpt = 0.20 m/s
vtg = 0.70 m/s
vpg = ?

80. Two-Dimensional Motion and Vectors
Problem
You are driving up a long inclined road. After 1.7 miles you notice that signs along the roadside indicate that your elevation has increased by 550 ft. What is the angle of the road above the horizontal?

81. Two-Dimensional Motion and Vectors
Solution

82. Two-Dimensional Motion and Vectors
Problem
You are driving up a long inclined road. After 1.7 miles you notice that signs along the roadside indicate that your elevation has increased by 550 ft. How far do you have to drive to gain an additional 150 ft of elevation?

83. Two-Dimensional Motion and Vectors
Solution

84. Two-Dimensional Motion and Vectors
Problem
A lighthouse that rises 49 ft above the surface of the water sits on a rocky cliff that extends 19 ft from its base. A sailor on the deck of a ship sights the top of the lighthouse at an angle of 30.0° above the horizontal. If the sailor’s eye level is 14 ft above the water, how far is the ship from the rocks?

85. Two-Dimensional Motion and Vectors
Solution
h = 49’
r = 19’
s = 14’
d = ?
y = h – s
x = d + r

86. Two-Dimensional Motion and Vectors
Problem
The x and y components of a vector r are rx=14m and ry= -9.5m respectively. If both rx and ry are doubled, how do the previous answers change?

87. Two-Dimensional Motion and Vectors
Solution
The direction does not change. The magnitude is doubled.

88. Two-Dimensional Motion and Vectors
Homework
Pages
Problems
35 (a, 2.77x105m b, 284s)
39 (10.8m)
41 (80m; 210m)
49 (a, 14.1o N of W b, 199km/hr)
51 (a, 23.2o upstream b, 8.72m/s across)