"Vectors and Scalars " (9 класс)

Слайд #1

VECTORS
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Слайд #2

VECTORS vs. SCALARS
SCALARS
Quantities that can be described by magnitude only
Does not depend on direction
For example: TIME

You don’t say “I will arrive at Riverside in 5 minutes to the right!”

Слайд #3

VECTORS vs. SCALARS
OTHER EXAMPLES OF SCALARS
MASS
TEMPERATURE
ENERGY
VOLUME
DENSITY
LENGTH
DISTANCE
SPEED

Слайд #4

VECTORS vs. SCALARS
VECTORS
Quantities that can be described by both magnitude and direction
RECALL:
(+) upward, right, east and north

(-) downward, left, west and south
DIRECTION

Слайд #5

VECTORS vs. SCALARS
EXAMPLES OF VECTORS
FORCE
ACCELERATION
VELOCITY
DISPLACEMENT
MOMENTUM

Слайд #6

VECTORS vs. SCALARS
REPRESENTING VECTORS

Слайд #7

VECTOR ADDITION
Graphical Method

Analytical Method

Component Method

Слайд #8

VECTOR ADDITION - Graphical
Example 1: Tip to Tail


3 cm
2 m
5 cm
Resultant Vector

Слайд #9

2 cm
3 cm
-1 cm
Resultant Vector
VECTOR ADDITION - Graphical
Example 2: Tip to Tail

Слайд #10

Example 3: Tip to Tail


3 cm
4 cm
VECTOR ADDITION - Graphical
~5 cm
𝜃

Слайд #11

TRY THIS ON YOUR OWN (Tip to Tail)


2. You sail a boat due east at 12 m/s. The wind is blowing due north at 5 m/s. Determine the resultant vector. (Scale: 1 cm = 1 m/s)
VECTOR ADDITION - Graphical
1. Two forces with magnitudes of 4 N to the +x axis and 8 N at an angle of 40°. Find the magnitude of the resultant vector. (Scale: 1 cm = 1 N

Слайд #12

VECTOR ADDITION - Graphical

Слайд #13

Parallelogram Method
This time we’ll add red & blue by placing the tails together and drawing a parallelogram with dotted lines. The resultant’s tail is at the same point as the other tails. It’s tip is at the intersection of the dotted lines.
Note: Opposite sides of a parallelogram are congruent.

Слайд #14

VECTOR ADDITION - Graphical

Слайд #15

VECTOR ADDITION - Graphical

Слайд #16

VECTOR ADDITION - Graphical
Adding More than 2 Vectors


𝐴 =5 𝑐𝑚, +𝑥
𝐵 =3 𝑐𝑚, +𝑦
𝐶 =4 𝑐𝑚, 110 0
𝐴 + 𝐵 + 𝐶

Слайд #17

𝜃
𝑅

Слайд #18

𝐴
𝐵
𝑅
𝐴 + 𝐵
𝐵 + 𝐴
𝐵
𝐴
𝑅

Слайд #19

Vector Addition has commutative property.
𝐴 + 𝐵 = 𝐵 + 𝐴

Слайд #20

Opposite of a Vector
v
- v
If v is 17 m/s up and to the right, then -v is 17 m/s down and to the left. The directions are opposite; the magnitudes are the same.

Слайд #21

Vector Subtraction
red - blue
blue - red
Put vector tails together and complete the triangle, pointing to the vector that “comes first in the subtraction.”
Why it works: In the first diagram, blue and black are tip to tail, so
blue + black = red
 red – blue = black.

Note that red - blue is the opposite of blue - red.

Слайд #22

𝐴 − 𝐵
𝐴 +(− 𝐵 )
𝐴 =5 𝑐𝑚, 30 0
𝐵 =8 𝑐𝑚, +𝑥
− 𝐵 =8 𝑐𝑚, −𝑥
𝑅

Слайд #23

Comparison of Vectors
15 N
43 m
0.056 km
27 m/s
Which vector is bigger?
The question of size here doesn’t make sense. It’s like asking, “What’s bigger, an hour or a gallon?” You can only compare vectors if they are of the same quantity. Here, red’s magnitude is greater than blue’s, since 0.056 km = 56 m > 43 m, so red must be drawn longer than blue, but these are the only two we can compare.

Слайд #24

VECTOR ADDITION
Graphical Method

Analytical Method

Component Method

Слайд #25

Example 1: Along the same axis
You walk 10 meters to the right and then continues 6 m to the same direction.


10 m
6 m
+
+
16 m
Resultant Vector
VECTOR ADDITION - Analytical
+

Слайд #26

Example 2: Along the same axis
You walk 6 meters to the North and then heads back 10 m to the South.


6 m
10 m
+
-
-4 m
Resultant Vector
VECTOR ADDITION - Graphical

Слайд #27

VECTOR ADDITION
Example 3: Along different axes
You walk 3 meters to the North and then heads 4 m to the East.


3.00 m
4.00 m
Right Triangle
Pythagorean
Theorem
𝑅 2 = 𝐴 2 + 𝐵 2
𝑅 2 = 3 2 + 4 2
𝑅 2 =9+16
𝑅 2 =25


𝑅=5.00 𝑚

Слайд #28

VECTOR ADDITION
Direction of the resultant vector???


3.00 m
4.00 m
Right Triangle
tan 𝜃 = 3.00 4.00
𝜃
θ= 36.9 0
𝑅 =5.00 𝑚, 36.9 0

Слайд #29

VECTOR ADDITION
Example 4: Along different axes
You walk 6 meters to the South and then heads 8 m to the West.


6 m
8 m
Right Triangle
Pythagorean
Theorem
𝑅 2 = 𝐴 2 + 𝐵 2
𝑅 2 = 6 2 + 8 2
𝑅 2 =36+64
𝑅 2 =100


𝑅=10 𝑚

Слайд #30

Pythagorean Theorem
34 m/s
30.814 m/s
25
14.369 m/s
Since components always form a right triangle, the Pythagorean theorem holds: (14.369)2 + (30.814)2 = (34)2.
Note that a component can be as long, but no longer than, the vector itself. This is because the sides of a right triangle can’t be longer than the hypotenuse.

Слайд #31

VECTOR ADDITION
Graphical Method

Analytical Method

Component Method

Слайд #32

VECTOR ADDITION – Component

Слайд #33

VECTOR ADDITION – Component

Слайд #34

VECTOR ADDITION – Component

Слайд #35

VECTOR ADDITION – Component
Adding 2 or more vectors using component method

Слайд #36

VECTOR ADDITION – Component
Adding 2 or more vectors using component method

Слайд #37

Solve the following problem using the component method.
10 km at 30
6 km at 120

Слайд #38

Solving Vector Problems using the Component Method
Each vector is replaced by 2 perpendicular vectors called components.
Add the x-components and the y-components to find the x- and y-components of the resultant.
Use the Pythagorean theorem and the tangent function to find the magnitude and direction of the resultant.

Слайд #39

Advantages and Disadvantages
of the Graphical Method
Can add any number of vectors at once
Uses simple tools
No mathematical equations needed

Must be correctly draw to scale and at appropriate angles
Subject to human error
Time consuming

Слайд #40

Advantages and Disadvantages
of the Analytical Method
Does not require drawing to scale.
More precise answers are calculated.
Works for any type of triangle if appropriate laws are used.
Can only add 2 vectors at a time.
Must know many mathematical formulas.
Can be quite time consuming.

Слайд #41

Advantages of the
Component Method:
Can be used for any number of vectors.
All vectors are added at one time.
Only a limited number of mathematical equations must be used.
Least time consuming method for multiple vectors.