1. Two-Dimensional Motion and Vectors
Chapter 2
Two-Dimensional Motion and Vectors

2. Vectors, Shmectors Objectives
1. Distinguish between a scalar and a vector.
2. Add and subtract vectors by using the graphical method.
3. Multiply and divide vectors by scalars.

3. Vectors, Schmectors
We can categorize quantities into scalars and vectors.
A scalar is a quantity that has magnitude, but no direction. Examples include speed, volume, and moles.
A vector is a physical quantity that has both direction and magnitude. Examples include velocity, acceleration, and displacement.
We will denote vectors in this class by placing an arrow above the variable.

4. Adding Vectors, Schmectors
When adding vectors, make sure they have the same units and describe similar qualities.
Do not add velocity and acceleration or velocity in km/h and velocity in m/s.
Chapter 1 discussed vector addition and subtraction in one dimension when we calculated the displacement of an object that moved in the positive and negative directions.
We can also add and subtract vectors graphically.

5. Adding Vectors, Schmectors
A resultant is a vector that represents the sum of two or more vectors.
When adding vectors graphically, first draw the two vectors to scale with arrows at the ends denoting direction of motion.
Next, draw a vector that connects the base of the first vector to the point of the second vector.
The magnitude of the resultant can be found by measuring it and multiplying it by your scale factor.

6. Adding Vectors, Schmectors
The direction of the resultant can be determined by measuring the angle between the resultant and the first vector or between the resultant and a chosen reference line.
This triangular method of vector addition is called the polygon method, the head-to-tail method, or the tip-to-tail method.

7. Properties of Vectors, Schmectors
Consider a situation in which two or more vectors act at the same point.
A resultant vector can be found that has the same total effect as the combination of the individual vectors.

8. Vectors, Schmectors Concept Check
Imagine looking down from the second level of an airport at a toy car moving at 0.80 m/s across a walkway that moves at 1.5 m/s. How can you determine the car's resultant velocity?

9. Okay, I get it now...
The car's resultant velocity will be the combination of the two independent motions.
We can draw a graphical representation of the two vectors with the car's velocity along the y-axis and the walkway's velocity along the x-axis.
Next, we draw a vector connecting the tail of the car's velocity vector to the head of the walkway's velocity vector and measure its length and angle.

10. Properties of Vectors, Schmectors
Vectors can be added in any order and the sum will always be the same.
When vectors are subtracted, you add the negative of the vector.
The negative direction will always be left (or west) or down.
The positive direction will always be right (or east) or up.
Vectors can be multiplied by scalars, with a vector being the result.

11. Vector, Schmector Operations
Objectives
1. Identify appropriate coordinate systems for solving problems with vectors.
2. Apply the Pythagorean Theorem and tangent function to calculate the magnitude and direction of a resultant vector.
3. Resolve vectors into components using sine and cosine functions.
4. Add vectors that are not perpendicular.

12. Coordinates
We use the standard x- and y-axes for diagramming the motion of an object in two dimensions.
There are no rules for applying coordinate systems to situations involving vectors. As long as you are consistent within a situation, the final answer will be correct.

13. Pythagorean Theorem and Vectors, Schmectors
We can use the Pythagorean Theorem to determine the magnitude of the resultant in a situation where the two vectors being added form a perpendicular.
c2 = a2 + b2
Or (length of hypotenuse)2 = (length of one leg)2 + (length of other leg)2

14. Off on a Tangent Again, Mrs. Beane?
We can determine the direction of the resultant of two vectors that interact at a right angle by using the tangent function.
tan Θ = opp
adj
tangent of angle = opposite leg
adjacent leg
Θ = tan-1(opp/adj)

15. Concept Check
A plane travels from Houston, Texas to Washington, DC, which is 1540 km east and 1160 km north of Houston. What is the total displacement of the plane?

16. Concept Check X = 1540 km y = 1160 km d = √(1540km2 + 1160km2)
Θ = tan-1(1160 km/1540 km)
Θ = 37.0° north of east

17. Concept Check
Emily passes a soccer ball 6.0 m directly across the field to Kara. Kara then kicks the ball 14.5 m directly down the field to Luisa. What is the ball's total displacement as it travels between Emily and Luisa?

18. Concept Check Δx=6.0 m Δy=-14.5 m c2=a2+b2 d2=Δx2+Δy2 d=√(Δx2+Δy2)
d=√{(6.0 m)2+(-14.5 m)2}
d=16 m
Θ=tan-1(opp/adj)
Θ=tan-1(Δy/Δx)
Θ=tan-1(-14.5m/6.0m)=-67.5°

19. Resolving Vectors into Components
The components of a vector are the projections of a vector along the axis of a coordinate system.
The y component is parallel to the y-axis and the x component is parallel to the x-axis.
Components of vectors are called projections in math class.
You can break a vector down into its components to analyze its motion; this process is called resolving the vector.

20. Resolving Vectors
We use the sine and cosine functions to find the magnitude of the components of a vector.
sin Θ = opp/hyp
cos Θ = adj/hyp

21. Concept Check
An arrow is shot from a bow at an angle of 25° above the horizontal with an initial speed of 45 m/s. Find the horizontal and vertical components of the arrow's initial velocity.

22. Concept Check v=45 m/s Θ=25° vx=? Vy=? sin Θ =vy/v cos Θ =vx/v
vy=v sin Θ vx=v cos Θ
vy=(45 m/s) sin (25) vx=(45 m/s) cos (25)
vy=19 m/s vx=41 m/s

23. Concept Check
How fast must a truck travel to stay beneath an airplane that is moving 105 km/h at an angle of 25° to the ground?

24. Concept Check
95 km/h

25. Concept Check
What is the magnitude of the vertical component of the velocity of the plane in the previous problem?

26. Concept Check
44 km/h

27. Concept Check
A truck drives up a hill with a 15° incline. If the truck has a constant speed of 22 m/s, what are the horizontal and vertical components of the truck's velocity?

28. Concept Check
21 m/s; 5.7 m/s

29. Concept Check
What are the horizontal and vertical components of a cat's displacement when the cat has climbed 5m directly up a tree?

30. Concept Check
0 m; 5 m

31. Vectors that are not Perpendicular
When adding vectors that do not form a right triangle, you must resolve each of the vectors into its x and y components.
The components along each axis are then added together.
The resultants magnitude can then be found by using the Pythagorean Theorem and the direction can be found by using the tangent function.

32. Concept Check
A plane flies 118 km at 15.0° south of east and then flies 118 km at 35.0° west of north. Find the magnitude and direction of the total displacement of the plane.

33. Concept Check
d=81km
Θ=55°

34. Concept Check
A football player runs directly down the field for 35 m before turning to the right at an angle of 25° from his original direction and running an additional 15 m before getting tackled. What is the magnitude and direction of the runners total displacement?

35. 49 m at 7.3° to the right of downfield

36. Concept Check
A plane travels 2.5 km at an angle of 35° to the ground and changes direction and travels 5.2 km at an angle of 22° to the ground. What is the magnitude and direction of the plane's total displacement?

37. 7.5 km at 26° above the horizontal
Concept Check
7.5 km at 26° above the horizontal

38. Projectile Motion Objectives
1. Recognize examples of projectile motion.
2. Describe the path of a projectile as a parabola.
3. Resolve vectors into their components and apply the kinematic equations to solve problems involving projectile motion.

39. Vectors and Kinematic Equations
We can resolve vectors into their components, apply the kinematic equations to each component, and recombine the components to determine the result.
We will use components to simplify projectile motion
The curved path that an object follows when thrown,launched, or otherwise projected near the surface of Earth.

40. Paths of Projectiles
Projectiles move in a path that curves in a parabola.
If an object has an initial horizontal velocity, there will be horizontal motion for the entire motion of the projectile.
We will consider the horizontal velocity of an object to be constant. (neglect air resistance)
We divide the motion of a projectile into vertical and horizontal components to examine them.

41. Vertical Motion of a Projectile that Falls From Rest
To analyze the vertical motion of a projectile, we use the following:
vy,f=ayΔt
vy,f2=2ayΔy
Δy= ½ ay(Δt)2

42. Horizontal Motion of a Projectile
Remember that an object's horizontal velocity is considered to remain constant, so
vx=vx,i=constant
Δx=vxΔt

43. Projectile Motion
Realize that projectile motion is free fall with an initial horizontal velocity.
To find the velocity of a projectile at any point during its flight, find the vector that has the known components.
Specifically, use the Pythagorean theorem to find the magnitude, and use the tangent function to find the direction of the velocity.

44. Concept Check
A baseball rolls off a 0.70 m high desk and strikes the floor 0.25 m away from the base of the desk. How fast was the ball rolling?

45. Concept Check
0.66 m/s

46. Concept Check
A cat chases a mouse across a 1.0 m high table. The mouse steps out of the way, and the cat slides off the table and strikes the floor 2.2 m from the edge of the table. When the cat slid off the table, what was its speed?

47. Concept Check
4.9 m/s

48. Concept Check
A pelican flying along a horizontal path drops a fish from a height of 5.4 m. The fish travels 8.0 m horizontally before it hits the water. What is the pelican's speed?

49. Concept Check
7.6 m/s

50. Concept Check
If the pelican in item 3 was traveling at the same speed but was only 2.7 m above the water, how far would the fish travel horizontally before hitting the water?

51. Concept Check
5.6 m

52. Projectiles Launched at an Angle
When an object is launched an at angle to the horizontal, the projectile has an initial vertical and horizontal velocity.
Given that the initial velocity vector makes an angle Θ with the horizontal, we can resolve the vector into its components such that
vx=vx,i=vicosΘ=constant
Δx=(vicosΘ)Δt
vy,f=visinΘ + ayΔt
vy,f2=vi2(sinΘ)2 + 2ayΔy

53. Projectiles Launched at an Angle
Δy = (visinΘ)Δt + ½ ay(Δt)2

54. Concept Check
In a scene in an action movie, a stuntman jumps from the top of one building to the top of another building 4.0 m away. After a running start, he leaps at a velocity of 5.0 m/s at an angle of 15° with respect to the flat roof. Will he make it to the other roof, which is 2.5 m shorter than the building he jumps from?

55. Concept Check
Yes, Δy = -2.3 m

56. Concept Check
A golfer hits a golf ball at an angle of 25.0° to the ground. If the golf ball covers a horizontal distance of m, what is the ball's maximum height? (Hint: At the top of its flight, the ball's vertical velocity component will be zero.)

57. Concept Check
35.1 m

58. Concept Check
A baseball is thrown at an angle of 25° relative to the ground at a speed of 23.0 m/s. If the ball was caught m from the thrower, how long was it in the air? How high did the ball travel before being caught?

59. Concept Check
2.0 s; 4.8 m

60. Concept Check
Salmon often jump waterfalls to reach their breeding grounds. One salmon starts 2.00 m from a waterfall that is 0.55 m tall and jumps at an angle of 32.0°. What must be the salmon's minimum speed to reach the waterfall?

61. Concept Check
6.2 m/s

62. Relative Motion Objectives
1. Describe situations in terms of frame of reference.
2. Solve problems involving relative velocity.

63. Frames of Reference
When an object in motion is being observed by a person in motion, the observed velocity or displacement will be skewed.
Depending on the location from which you observe an object's motion, the motion can appear different.
When calculating the relative velocity of an object, use vac= vab+vbc
The subscripts denote “with respect to”

64. Relative Velocity
Consider a situation in which a car traveling at 90 km/h is passing a car traveling at 80 km/h. What would be the velocity of the faster car with respect to the slower car?

65. Relative Velocity
10 km/h

66. Concept Check
A plane flies northeast at an airspeed of km/h. (Airspeed is the speed of an 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?

67. Concept Check
565.0 km/h at 40.1° north of east.