Review
DisplacementAverage VelocityAverage 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 …
v v0 gt
y v0t 1
2gt 2
v 2 v02 2gy
v x
t
v v0 at
x v0t 1
2at 2
v 2 v02 2ax
a v
t
x x f x i
v v0 gt
y v0t 1
2gt 2
v 2 v02 2gy
Chapter 2
Motion in two dimensions
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, ...
P
Q
Initial Point
Terminal Point
magnitu
de is th
e length
direction is
this angle
How can we find the magnitude if we have the initial point and the terminal point? 22 , yx
11, yx
The distance formula
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!)
22 , yx
11, yx
Q
Terminal Point
direction is
this angle
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).
yx,
0,0If we subtract the initial point from the terminal point, we will have an equivalent vector with initial point at the origin.
P
Initial Point
A vector whose initial point is the origin is called a position vector
Equality of Two Vectors
Two vectors are equal if they have
the same magnitude & direction
Are the vectors here equal?
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.
Two vectors are equal if they have the same direction and magnitude (length).
Blue and orange vectors have same magnitude but different direction.
Blue and green vectors have same direction but different magnitude.
Blue and purple vectors have same magnitude and direction so they are equal.
Addition of vectors
Given two vectors , what is?
A &
B
A
B
A
B
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
A
B
R
A
B
R
Vector B50 m θ= 0O
Vector C30 m
Θ = 90O
Vector A30 m θ = 45O
A
B
C
Resultant = 9 x 10 = 90 meters
Angle is measured at 40o
To add the vectorsPlace them head to tail
A
B
C
D
A
BC
DR
A + + + =B C D R
ALL VECTORS MUST BE DRAWN TO
SCALE & POINTED INTHE PROPER DIRECTION
2 -the parallelogram method.
C = A + B2 2
Multiplying a Vector by a Scalar
• Given , what is ?
s
3s
s
s
s
s
s
s
Scalar multiplicationScalar multiplication: multiply vector by scalardirection stays samemagnitude stretched by given scalar(negative scalar reverses direction)
Vector Subtraction
A
B
C
DA + - - =B C D R
A+ + ( - ) + ( - ) =B C D R
-C
=
-D=
A
-D
R
B
-C
Example
Example
Example
Components of a Vector
Vector component:
A
A x
A y
where and are the components of the vector
Ax
Ay
A
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
Unit Vectors
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 .
Vector component
at
• 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 245 0 = - 25.4 m/S
• BY = 60 m/S x SIN 245 0 = - 54.4 m/S
VECTOR COMPONENTS
jiw 43
find the magnitude of the vector W
What is ?w
2 23 4 w 525
Example:
Example: The angle between where
and the positive x axis is :
1. 61°2. 29°3. 151°4. 209°5. 241°
A
A x
A y
Ax 25 & Ay 45
Vector component:
jiji 4352
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.
jiv 52
wv ji
Let's look at this geometrically:
i2
j5 v
i3
j4w
ij
When we want to know the magnitude of the vector (remember this is the length) we denote it
v 22 52
Can you see from this picture how to find the length of v?
29
jiw 43 Example:
ADDING & SUBTRACTING VECTORS USING COMPONENTS
Vector A30 m θ = 45O
Vector B 50 m θ = 0O
Vector C30 m Θ = 9 0O
ADD THE FOLLOWINGTHREE VECTORS USING
COMPONENTS
(1) RESOLVE EACH INTO X AND Y COMPONENTS
ADDING & SUBTRACTING VECTORS USING COMPONENTS
• AX = 30mx cos 450 = 21.2 m
• AY = 30 m x sin 450 = 21.2 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
(2) ADD THE X COMPONENTS OF EACH VECTOR ADD THE Y COMPONENTS OF EACH VECTOR
X = SUM OF THE Xs = 21.2 + 50 + 0 = +71.2 Y =SUM OF THE Ys = 21.2 + 0 + 30 = +51.2
(3) CONSTUCT A NEW RIGHT TRIANGLE USING THE X AS THE BASE AND Y AS THE OPPOSITE SIDE
X = +71.2
Y = +51.2
THE HYPOTENUSE IS THE RESULTANT VECTOR
(4) USE THE PYTHAGOREAN THEOREM TO THE LENGTH(MAGNITUDE) OF THE RESULTANT VECTOR
X = +71.2
Y = +51.2
(+71.2)2 + (+51.2)2 = 87.7
(5) FIND THE ANGLE (DIRECTION) USING INVERSETANGENT OF THE OPPOSITE SIDE OVER THE
ADJACENT SIDE
angle tan-1 (51.2/71.2)
Θ = 35.7 O
QUADRANT I
RESULTANT = 87.7 m θ = 35.7 O
SUBTRACTING VECTORS USING COMPONENTS
Vector A30 m θ = 45O
Vector C30 m θ = 90O
Vector B50 m θ = 0O
A - + =B C R
A + (- ) + =B C R
Vector A30 m θ = 45O
- Vector B
50 m θ = 180O
Vector C30 m θ = 90O
• 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 1800 = 0
• CX = 30 m x cos 900 = 0 m CY = 30 m x sin 900 = 30 m
X = SUM OF THE Xs = 21.2 + (-50) + 0 = -28.8 Y =SUM OF THE Ys = 21.2 + 0 + 30 = +51.2
X = -28.8
Y = +51.2
(2) ADD THE X COMPONENTS OF EACH VECTOR ADD THE Y COMPONENTS OF EACH VECTOR
X = SUM OF THE Xs = 21.2 + (-50) + 0 = -28.8 Y =SUM OF THE Ys = 21.2 + 0 + 30 = +51.2
(3) CONSTUCT A NEW RIGHT TRIANGLE USING THE X AS THE BASE AND Y AS THE OPPOSITE SIDE
X = -28.8
Y = +51.2
THE HYPOTENUSE IS THE RESULTANT VECTOR
X = -28.8
Y = +51.2angle
Θ=tan-1 (51.2/-28.8)θ = -60.6 0
(1800 –60.60 ) = 119.40
QUADRANT II
RESULTANT ( R) = 58.7 m θ = 119.4O
R = (-28.8)2 + (+51.2)2 = 58.7
jiji 4352
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.
jiv 52
wv ji
Let's look at this geometrically:
i2
j5 v
i3
j4w
ij
When we want to know the magnitude of the vector (remember this is the length) we denote it
v 22 52
Can you see from this picture how to find the length of v?
29
jiw 43 Example:
example
Example :
If we know the magnitude and direction of the vector, let's see if we can express the vector in a + b form.
5, 150 v
1505
As usual we can use the trig we know to find the length in the horizontal direction and in the vertical direction.
yxyx ˆ2
5ˆ
2
35ˆ150sinˆ150cos5 v
Example:
Example :F1 = 37N 54° N of E F2 = 50N 18° N of F3 = 67 N 4° W of S
F=F1+F2+F3
W
Ex : 2 – 10 A woman walks 10 Km north, turns toward the north west , and walks 5 Km further . What is her final position?
Example:
Example:
Example: