vector unit
TRANSCRIPT
-
8/3/2019 Vector Unit
1/39
01/05/2012 1
VECTOR UNIT
GSS Science
-
8/3/2019 Vector Unit
2/39
Introduction
In this unit we will study all aspects of vectors,including vector addition equations.
We will then use the knowledge gained tosolve two dimensional vector problems suchas plane flights and force problems.
01/05/2012 2
-
8/3/2019 Vector Unit
3/39
Vector Unit:
Objectives
Define a vector
Identify resultant and component vectors on a diagram
Write a vector equation Use graphical methods to resolve a vector into two
perpendicular vectors
Use trig to add/subtract vectors
Resolve vectors into component vectors to add/subtract them Use cosine and sine law to add/subtract vectors
01/05/2012 3
-
8/3/2019 Vector Unit
4/39
Vectors
Two types of quantities exist in the world: Scalers a quantity with a value but without
direction i.e. speed = 12 m/s
Vectors a quantity with magnitude (value)and a direction (heading) i.e. velocity = 12 m/sdue north
Scalersinclude: Speed, energy, time, work, distance, mass
Vectors include: Velocity, acceleration, force, momentum,
displacement
01/05/2012 4
-
8/3/2019 Vector Unit
5/39
Vector Direction
Lets learn about directions for vectors.
We use a compass for basic direction/headings
01/05/2012 5
N
E
S
W
-
8/3/2019 Vector Unit
6/39
Vector Direction
If a vector is heading along one of the main heading (such asN, E, S, or W) just give it that direction
i.e. A train has a velocity of24 m/s (S)
01/05/2012 6
N
24 m/s
-
8/3/2019 Vector Unit
7/39
Vector Direction
If the vector is exactly mid-way between two main headingsthen we can use:
45o N of W or
NW, SW, SE, SW as the direction
01/05/2012
7
NNWNE
SW SE
45o
45o
W E
S
-
8/3/2019 Vector Unit
8/39
Vector Direction
But what about other directions? In this case we use a direction from one of the main headings (N, E, S,
or W)
This vector would become: 24 m/s at 32o N of E
01/05/2012 8
N
E
S
W
24 m/s
32o
The vector direction 32o
above the main EAST
direction. Hence we say32o N of E.
-
8/3/2019 Vector Unit
9/39
Vector Direction
There is another way of indicating this direction In this case we use a direction measured away
from the north direction
This vector would become: 24 m/s at 58o E of N
01/05/2012 9
N
E
S
W
24 m/s
58o
The vector direction 58o from
the main North direction.Hence we say 58o E of N.
-
8/3/2019 Vector Unit
10/39
Vector Direction
Every vector has two ways of indicating directions
You only need to state one of them
For example: 65 m at 22o N of W or 65 mat 68o W of N (same direction)
01/05/2012 10
N
E
S
W
65 m
22o
68o
Remember, you only need
one direction!
Note: The two directions must
always add up to 90o
. Why?
-
8/3/2019 Vector Unit
11/39
Vector Direction
Now try these: State the magnitude of each vector and give bothpossible directions
01/05/2012 11
35 N N
24o
N
45 m
13o
Vector = Vector =
35 N at 240 N of W
35 N at 66o W of N
45 m at 13o S of E
45 m at 77o E of S
-
8/3/2019 Vector Unit
12/39
Vector Direction
Bearing another direction system used byairplanes and ships for navigation
01/05/2012 12
N
0
o
90o
180o
270o
-
8/3/2019 Vector Unit
13/39
Vector Direction
This is easy: 24 Km bearing 125o (90o + 35o)
01/05/2012 13
N0o
90o
180o
270o 35o
24 Km bearing 125o
-
8/3/2019 Vector Unit
14/39
Vector Direction
Now try these:
01/05/2012 14
N
0o
180o
250 km
45o
60o
210 km
225 km
250 km bearing 45o
210 km bearing 150o
225 km bearing 270o
-
8/3/2019 Vector Unit
15/39
Adding & Subtracting Vectors
Graphing method:
Vectors in the same or opposite direction
Notes from Over-head
01/05/2012 15
-
8/3/2019 Vector Unit
16/39
Adding & Subtracting Vectors
When vectors are not in same or oppositedirection:
A little more work is involved:
You need a ruler & a protractor Over Head Notes
Do Graphing Vectors Worksheet
01/05/2012 16
-
8/3/2019 Vector Unit
17/39
Adding & Subtracting Vectors
Vector equations:
What does a + b =R really mean?
What does a - b=R really mean?
01/05/2012 17
Take these two vectors:
ab
a + b = R
R
(adding Vectors)
-
8/3/2019 Vector Unit
18/39
Adding & Subtracting Vectors
01/05/2012 18
Take the same two vectors:
ab
a - b = R
What does this mean?
Remember your math: a - b = a + (- b)In vectors (b) and (-b) vectors are related: They have the same
magnitude but have opposite directions!
a
- ba - b = a + (-b) = R
R
(Subtracting Vectors)
ddi b i
-
8/3/2019 Vector Unit
19/39
Adding & Subtracting Vectors
01/05/2012 19
Take the another two vectors: c
d
c + d = R c - d = R
c
R
c
d
- d
R
ddi & b i
-
8/3/2019 Vector Unit
20/39
Adding & Subtracting Vectors
01/05/2012 20
Take the another two vectors: e
e + f = R- e + f = R
R
R
e
f
e
f
f
ddi i
-
8/3/2019 Vector Unit
21/39
Vector Addition Component Method
Navigational vector problems are usually
solved by graphing, if you are a little offcourse no problem the airport has to bearound here somewhere!
But building bridges or airplanes need a moreprecision to adding vectors in solvingconstruction problems.
01/05/2012 21
t dditi C t th d
-
8/3/2019 Vector Unit
22/39
Vector Addition Component Method
Imagine this conversation betweentwo engineers:
Engineer #1 - Well the graphing
method of adding vectors gave me anapproximate load limit for the bridge.I guess it was a little low. You bettercall in emergency vehicles and
cranes Engineer #2 - I guess so I told you
we should have used the componentmethod!
01/05/2012 22
V t Additi
-
8/3/2019 Vector Unit
23/39
Vector Addition Component Method
Any vector can be re-made into two orthonganol (right-angled) vectors
These are called component vectors
This is some vector a:
01/05/2012 23
a
26o
V t Additi C t M th d
-
8/3/2019 Vector Unit
24/39
Vector Addition Component Method
This is the horizontal component vector (ax)
01/05/2012 24
a
26o
ax
V t Additi
-
8/3/2019 Vector Unit
25/39
Vector Addition Component Method
This is the vertical component vector (ay)
01/05/2012 25
a
26o
ay
V t Additi C t M th d
-
8/3/2019 Vector Unit
26/39
Vector Addition Component Method
These two vector can be shown as the addition of two right-angled vectors
One is the horizontal component (ax)
One is the vertical component (ay)
01/05/2012 26
Head to Tail
Method
a
26o
ax
ay
Vector Addition Component Method
-
8/3/2019 Vector Unit
27/39
Vector Addition Component Method
The two vectors are sometimes shown in tail totail addition method
01/05/2012 27
a
26o
ax
ayTail to Tail
Method
Vector Addition Component Method
-
8/3/2019 Vector Unit
28/39
Vector Addition Component Method
In summary there are two ways to show the addition of vectorsand they both (of course) end up with the same resultant vector
01/05/2012 28
a
26o
ax
ay
Tail to Tail Method
a
26o
ax
ay
Head to Tail Method
or
Vector Addition Component Method
-
8/3/2019 Vector Unit
29/39
Vector Addition Component Method
Combing the two methods results in parallelogram!! It doesnt matter which way it is shown, you end up at the same
end-point!
01/05/2012
29
a
26o
ay
ax
26o
There are two identical
triangles here.
ax
ay
Head to TailTail to Tail
End point!
Vector Addition Component Method
-
8/3/2019 Vector Unit
30/39
Vector Addition Component Method
You have to use a little trig. (sine, cosine and tangent) towork with these vector components
01/05/2012 30
a
26o
ay = a x sin 26o
ax = a x cos 26o
Vector Addition Component Method
-
8/3/2019 Vector Unit
31/39
Vector Addition Component Method
Finding resultant vector: Using some basic trigonometry
Sine, Cosine, and Tangent
Right angle triangles
Pythagorean Theory
Over Head Notes
Go on to the Vector Component Worksheet
01/05/2012 31
Vector Addition Cosine Law
-
8/3/2019 Vector Unit
32/39
Vector Addition Cosine Law
This method works when the componentvectors are not at right angles to each other.
You have to use two other trigonometryLaws:
Cosine law to find magnitude
Sine Law to find direction
01/05/2012 32
Vector Addition Cosine Law
-
8/3/2019 Vector Unit
33/39
Vector Addition Cosine Law
Examine the following two vectors:
01/05/2012 33
b
c
These two vectors are not at right angles to one another.
So how can we add these two vectors?
Vector Addition Cosine Law
-
8/3/2019 Vector Unit
34/39
Vector Addition Cosine Law
To find the resultant vector R we use thecosine law:
R = (b2 + c2 (2 x b x c x cos )
01/05/2012 34
R
b
c
Vector Addition Cosine Law
-
8/3/2019 Vector Unit
35/39
Vector Addition Cosine Law
To use the cosine law you need to know the angle opposite tothe resultant vector that you want to find:
01/05/2012 35
R
b
c
This will allow you to find the magnitude of the resultant
vector, R, by using the cosine law.
Opposite angle
to the resultant
Vector Addition Cosine Law
-
8/3/2019 Vector Unit
36/39
Vector Addition Cosine Law
To find the direction of the resultant vector usethe sine law:
01/05/2012 36
Sin B Sin C Sin= =b c
b
c
R
B
C
R
Vector Addition Cosine Law
-
8/3/2019 Vector Unit
37/39
Vector Addition Cosine Law
Over Head Notes
Do the Vector: Cosine Law Problem Worksheet
01/05/2012 37
-
8/3/2019 Vector Unit
38/39
Concluding Statements
You now should have a good grasp of dealingwith vectors.
We will be using these pesky vectorsthroughout the Physics 12 course.
Make sure you understand these topics andthat you are fairly comfortable with adding
vectors.
01/05/2012 38
-
8/3/2019 Vector Unit
39/39
The End
of this Topic.
On to Vector Kinematics!