1. Two-Dimensional Motion and Vectors

2. 3.1 Introduction to Vectors

3. Scalars & Vectors Scalar quantities have magnitude only
Speed, volume, # students
Vectors have magnitude & direction
Velocity, force, weight, displacement

4. Representing Vectors Boldface type: v is a vector; v is not
Symbol with arrow:
Arrow drawn to scale:
-----> 5m/s, 0° (east)
> 10 m/s, 0° (east)
< m/s, 180° (west)

5. Resultant Vectors Are the sum of two or more vectors Are “net” vectors
Can be determined by various methods
Graphical addition
Mathematically
Pythagorean theorem
Trigonometry

6. Graphical Addition of Vectors
Parallelogram method
Head-to-tail method
Draw vectors to scale, with direction, in head-to-tail fashion
Resultant drawn from tail of first vector to head of last vector
Measure length and angle to determine magnitude and direction

7. Head-to-Tail Addition of Vectors

8. Head-to-Tail Addition of Vectors

9. Other Properties of Vectors
Vectors may be moved parallel to themselves when constructing vector diagrams
Vectors may be added in any order
Vectors are subtracted by adding its opposite
Vectors multiplied by scalars are still vectors

10. Vectors may be added in any order

11. Direction of Vectors
Degrees are measured counterclockwise from x-axis

12. Direction of Vectors
Direction may be described with reference to N, S, E, or West axis
30° West of North
60° North of West
Or 120°

13. 3.2 Vector Operations
Adding parallel vectors
Simple arithmetic

14. Perpendicular Vectors & the Pythagorean Theorem
If two vectors are perpendicular, then you can use the Pythagorean theorem to determine magnitude only
Example: if
Δx = 40 km/h East
Δy = 100 km/h North
Then
d = ( )½
d ≈ 108 km/h
But what is the direction?

15. Basic Trig Functions
hyperphysics.phy-astr.gsu.edu

16. Tangent Function When two vectors are perpendicular:
Use Pythagorean theorem to find the magnitude of the resultant vector
Use the tangent function to find the direction of the resultant vector

17. Sample Problem
Indiana Jones climbs a square pyramid that is 136 m tall. The base is 2.30 x 102 m wide. What was the displacement of the archeologist?
What is the angle of the pyramid

18. Resolving Vectors Vectors can be “resolved” into x- and y- components
Resolve = Decompose = Break down
Trig functions are used to resolve vectors

19. Resolving Vectors into X & Y Components
For the vector A
Horizontal component = Ax = A·cos θ
Vertical component =
Ay = A·sin θ
hyperphysics.phy-astr.gsu.edu

20. Resolving a Vector
A helicopter travels 95 35º above horizontal. Find the x- and y- components of its velocity.

21. Adding Non-perpendicular Vectors A + B = R

22. Resolving Vectors into x and y Components
hyperphysics.phy-astr.gsu.edu

23. Adding vectors that are not perpendicular
Two or more vectors can be added by decomposing each vector
Add all x components to determine Rx
Add all y components to determine Ry
Determine magnitude of the resultant R using Pythagorean theorem
Determine direction angle θ of resultant using tan-1

24. Decomposition of Vectors
Vector Analysis
Vector Analysis Worksheet
Magnitude (units)
Direction (deg)
Decomposition of Vectors
Vector A 6 25
Ax =
5.44
Ay =
2.54
Vector B 3 60
Bx =
1.50
By =
2.60
Vector C
Cx =
0.00
Cy =
Vector D
Dx =
Dy =
Resultant
8.6
36.5
Rx =
6.94
Ry =
5.13

25. 3.3 Projectile Motion phet.colorado.edu
Objectives
Recognize examples of projectile motion
Describe the path of a projectile as a parabola
Resolve vectors into components and apply kinematic equations to solve projectile motion problems

26. Projectile Motion
Motion of objects moving in two dimensions under the influence of gravity
Baseball, arrow, rocket, jumping frog, etc.
Projectile trajectory is a parabola

27. Projectile motion Assumptions of our problems
Horizontal velocity is constant, i.e.
Air resistance is ignored
Projectile motion is free fall with a horizontal velocity

28. Motion of a projectile Equations relating to vertical motion:
∆y = vyi(∆t) + ½ag(∆t)2
vyf = vyi + ag∆t
vyf2 = vyi2 + 2ag∆y
Equations relating to horizontal motion:
∆x = vx∆t
vx = vxi = constant
(an assumption relating to Newton’s 1st law of motion)

29. Horizontal Projectile
vx = 5.0 m/s
t (s) vy vx Δy Δx
vy =
agt
vx = 1
Δy =
1/2agt2 2
Δx =
vxΔt 3 4 5

30. Horizontal Launch Comparison of vx & vy vectors

31. Driving off a Cliff
A stunt driver on a motorcycle speeds horizontally of a 50.0m high cliff. How fast must the motorcycle leave the cliff in order to land on level ground below, 90.0m from the base of the cliff? Ignore air resistance.
Sketch the problem
List knowns & unknowns
Apply relevant equations

32. Driving off a Cliff
Known: ∆x = 90.0m; ∆y = -50.0m; ax = 0; ay = -g = m/s2; vyi = 0
Unknown: vx; ∆t
Strategy: vx = ∆x/∆t
Since ∆tx must = ∆ty determine ∆t from the vertical drop
Now solve for vx using ∆t

33. Effect of Gravity on Ballistic Launch
[physicsclassroom.com]

34. Projectile Motion: Horizontal vs Angled
Horizontal Launch
Ballistic Launch

35. Sample Projectile MotionProblem
A ball is thrown with an initial velocity of 50.0 m/s at an angle of 60º.
How long will it be in the air?
How high will it go?
How far will it go?

36. Sample Problem
A ball is thrown with an initial velocity of 50.0 m/s at an angle of 60º.
How long will it be in the air?
Known: vi = 50.0 m/s, θ = 60º, a = -g = m/s2;
Find: Δt, total time in the air

37. Sample Problem
A ball is thrown with an initial velocity of 50 m/s at an angle of 60.
How high will it go?

38. Sample Problem
A ball is thrown with an initial velocity of 50 m/s at an angle of 60º.
How far will it go?

39. 3.4 Relative Motion Objectives
Describe motion in terms of frames of reference
Solve problems involving relative velocity

40. Frames of Reference Motion is relative to frame of reference
To an observer in the plane, the ball drops straight down (vx = 0)
To an observer on the ground, the ball follows a parabolic projectile path
(vx ≠ 0)
Frame of reference: a coordinate (defined by the observer) system for specifying the precise location of objects in space
A frame of reference is a “point of view” from which motion is described

41. Relative Velocity
Relative velocity of one object to another is determined from the velocities of each object relative to another frame of reference

42. Example See problem 1, page 109
Use subscripts to indicate relative velocities
vbe = vbt + vte
vbe = -15 m/s + 15 m/s
vbe = 0 m/s

43. Example
Car A travels 40 mi/h north; Car B travels 30 mi/h south. What is the velocity of Car A relative to Car B?
vae = 40; vbe = -30; veb = +30
Find vab
vab = vae + veb
vab =
vab = 70 mi/h North

44. Relative Velocity
One car travels 90 km/h north, another travels 80 km/h north. What is the speed of the fast car relative to the slow car?
vfe = 90 km/h north; vse = 80 km/h north
vfs = vfe + ves
vfs = 90 + (-)80
vfs = 10 km/h north