1. Review Displacement Average Velocity Average Acceleration
Instantaneous velocity (acceleration) is the slope of the line tangent to the curve of the position (velocity) -time graph
For constant acceleration… For constant gravitational acceleration…

2. Motion in two dimensions
Chapter 2
Motion in two dimensions

3. 2.1: An introduction to vectors
Many quantities in physics, like displacement, have a magnitude and a direction. Such quantities are called VECTORS.
Other quantities which are vectors: velocity, acceleration, force, momentum, ...
Many quantities in physics, like distance, have a magnitude only. Such quantities are called SCALARS.
Other quantities which are scalars: speed, temperature, mass, volume, ...

4. How can we find the magnitude if we have the initial point and the terminal point?Q
The distance formula
Terminal Point
magnitude is the length
direction is this angle
Initial Point P
How can we find the direction? (Is this all looking familiar for each application? You can make a right triangle and use trig to get the angle!)

5. Although it is possible to do this for any initial and terminal points, since vectors are equal as long as the direction and magnitude are the same, it is easiest to find a vector with initial point at the origin and terminal point (x, y). Q
Terminal Point
A vector whose initial point is the origin is called a position vector
direction is this angle
Initial Point P
If we subtract the initial point from the terminal point, we will have an equivalent vector with initial point at the origin.

6. Equality of Two Vectors
Two vectors are equal if they have the same magnitude & direction
Are the vectors here equal?

7. A vector is a quantity that has both magnitude and direction
A vector is a quantity that has both magnitude and direction. It is represented by an arrow. The length of the vector represents the magnitude and the arrow indicates the direction of the vector.
Blue and orange vectors have same magnitude but different direction.
Blue and purple vectors have same magnitude and direction so they are equal.
Blue and green vectors have same direction but different magnitude.
Two vectors are equal if they have the same direction and magnitude (length).

9. Addition of vectors
Given two vectors , what is ?

10. Graphical Techniques of Vector Addition
Two vectors can be added using these method:
1- tip to tail method.
2- the parallelogram method.
1-“Tip-to-Tail Method”
Two vectors can be added by
placing the tail of the 2nd on
the tip of the 1st

12. Drawing Vectors to Scale
Vector A
30 m θ = 45O
Drawing Vectors to Scale
To add the vectors
Place them head to tail
Vector B
50 m θ= 0O C
Vector C
30 m
Θ = 90O A B
Angle is measured at 40o
Resultant = 9 x 10 = 90 meters

13. Adding Vectors ALL VECTORS MUST BE DRAWN TO SCALE & POINTED IN
THE PROPER
DIRECTION A D R B C B C A D = A B C D R

14. 2- the parallelogram method.

17. 2 2
C = A + B

20. Multiplying a Vector by a Scalar
Given , what is ?
Scalar multiplication
Scalar multiplication: multiply vector by scalar
direction stays same
magnitude stretched by given scalar
(negative scalar reverses direction)

21. Vector Subtraction

23. Subtracting Vectors -C -D -C -D A B B R A C = D = + - - = A B C D R= A B C D R
( - ) + ( - ) = A B C D R

25. Example

26. Example

27. Example

30. Components of a Vector

31. Vector component:
where and are the components of the vector

32. Unit Vectors
A unit vector is a vector that has a magnitude of 1, with no units.
Its only purpose is to point
We will use x , y for our Unit Vectors
x means x – direction, y is y – direction,
We also put little “hats” (^) on x , y to show that they are unit vectors

33. Notes about Components
The previous equations are valid only if Ѳ is measured with respect to the X-axis.
The components can be positive or negative and will have the same units as the original vector .

34. Vector component

35. at

36. VECTOR COMPONENTS
WHAT ARE THE X AND Y COMPONENTS OF A VECTOR 40 m θ=60O ?
AX = 40 m x COS 600 = 20 m
AY = 40 m x SIN 600 = 34.6 m
WHAT ARE THE X AND Y COMPONENTS OF A VECTOR 60 m/s θ = 2450 ?
BX = 60 m/S x COS = m/S
BY = 60 m/S x SIN = m/S

37. Example :
find the magnitude of the vector W

38. Example : The angle between where and the positive x axis is: 61° 29°
151°
209°
241°

39. Vector component:

40. If we want to add vectors that are in the form a i + b j, we can just add the i components and then the j components.
Example :
When we want to know the magnitude of the vector (remember this is the length) we denote it
Let's look at this geometrically:
Can you see from this picture how to find the length of v?

41. ADDING & SUBTRACTING VECTORS USING COMPONENTS
Vector A
30 m θ = 45O
ADD THE FOLLOWING
THREE VECTORS USING
COMPONENTS
RESOLVE EACH INTO
X AND Y COMPONENTS
Vector B
50 m θ = 0O
Vector C
30 m Θ = 9 0O

42. ADDING & SUBTRACTING VECTORS USING COMPONENTS
AX = 30mx cos 450 = 21.2 m
AY = 30 m x sin 450 = m
BX = 50 m x cos 00 = 50 m
BY = 50 m x sin 00 = 0 m
CX = 30 m x cos 900 = 0 m
CY = 30 m x sin 900 = 30 m

43. (2) ADD THE X COMPONENTS OF EACH VECTOR
ADD THE Y COMPONENTS OF EACH VECTOR
 X = SUM OF THE Xs = = +71.2
 Y =SUM OF THE Ys = = +51.2
(3) CONSTUCT A NEW RIGHT TRIANGLE USING THE
 X AS THE BASE AND  Y AS THE OPPOSITE SIDE
 Y = +51.2
 X = +71.2
THE HYPOTENUSE IS THE RESULTANT VECTOR

44. (4) USE THE PYTHAGOREAN THEOREM TO THE LENGTH
(MAGNITUDE) OF THE RESULTANT VECTOR
 X = +71.2
 Y = +51.2
angle
tan-1 (51.2/71.2)
Θ = 35.7 O
QUADRANT I
(+71.2)2 + (+51.2)2 = 87.7
(5) FIND THE ANGLE (DIRECTION) USING INVERSE
TANGENT OF THE OPPOSITE SIDE OVER THE
ADJACENT SIDE
RESULTANT = 87.7 m θ = 35.7 O

45. SUBTRACTING VECTORS USING COMPONENTS
Vector A
30 m θ = 45O A = B C R A
+ ( ) = B C R
Vector A
30 m θ = 45O
Vector B
50 m θ = 0O
- Vector B
50 m θ = 180O
Vector C
30 m θ = 90O
Vector C
30 m θ = 90O

46. AX = 30 m x cos 450 = 21.2 m AY = 30 m x sin450 = 21.2 m
RESOLVE EACH INTO X AND Y COMPONENTS
X-comp y-comp
AX = 30 m x cos 450 = 21.2 m AY = 30 m x sin450 = 21.2 m
BX = 50 m x cos1800 = - 50 m BY = 50 m x sin = 0
CX = 30 m x cos 900 = 0 m CY = 30 m x sin 900 = 30 m
 X = SUM OF THE Xs = (-50) + 0 =  Y =SUM OF THE Ys = = +51.2
 Y = +51.2
 X = -28.8

47. (2) ADD THE X COMPONENTS OF EACH VECTOR
ADD THE Y COMPONENTS OF EACH VECTOR
 X = SUM OF THE Xs = (-50) + 0 = -28.8
 Y =SUM OF THE Ys = = +51.2
(3) CONSTUCT A NEW RIGHT TRIANGLE USING THE
 X AS THE BASE AND  Y AS THE OPPOSITE SIDE
 Y = +51.2
 X = -28.8
THE HYPOTENUSE IS THE RESULTANT VECTOR

48. (-28.8)2 + (+51.2)2 = 58.7 angle Θ=tan-1 (51.2/-28.8) θ = -60.6 0
 X = -28.8
 Y = +51.2
angle
Θ=tan-1 (51.2/-28.8)
θ =
(1800 –60.60 ) =
QUADRANT II
R =
(-28.8)2 + (+51.2) = 58.7
RESULTANT ( R) = m θ = 119.4O

49. If we want to add vectors that are in the form a i + b j, we can just add the i components and then the j components.
Example :
When we want to know the magnitude of the vector (remember this is the length) we denote it
Let's look at this geometrically:
Can you see from this picture how to find the length of v?

50. example

51. Example :

53. If we know the magnitude and direction of the vector, let's see if we can express the vector in a + b form.
Example :
As usual we can use the trig we know to find the length in the horizontal direction and in the vertical direction.

54. F1 = 37N 54° N of E F2 = 50N 18° N of F3 = 67 N 4° W of S
Example :
F1 = 37N 54° N of E F2 = 50N 18° N of F3 = 67 N 4° W of S
F=F1+F2+F3 W

55. Ex : 2 – 10
A woman walks 10 Km north, turns toward the north west , and walks 5 Km further . What is her final position?

56. Example :

57. Example :

58. Example :