Transcript Vectors - Humble ISD

Vectors and Scalars
Physics 1 - L
Scalar Refresher:
A SCALAR is ANY quantity in
physics that has
MAGNITUDE, but NOT a
direction associated with
it.
Magnitude – A numerical
value with units.
Scalar
Example
Magnitude
Speed
20 m/s
Distance
10 m
Age
15 years
Heat
1000
calories
Vectors cont.
A VECTOR is ANY quantity in
physics that has BOTH
MAGNITUDE and
DIRECTION.
   
v , x, a, F
Vectors are typically
illustrated by drawing an
ARROW above the symbol.
The arrow is used to convey
direction and magnitude.
Vector
Velocity
Magnitude
& Direction
20 m/s, N
Acceleration 10 m/s/s, E
Force
5 N, West
Using Vectors
• It is a good habit to
label your vectors in a
diagram
• Always include an
arrowhead on your
diagram
F=10N
Adding Vectors
• Vectors can be added Using Scale
DiagramsGraphically OR
• Using Trigonometry (SOH-CAH-TOA)
•
• We will start with Graphical Vector
Addition
Parallel Vectors
VECTOR ADDITION – If 2 similar vectors point in the SAME direction, add
them.
• Example: A man walks 54.5 meters east, then another 30 meters
east. Calculate his displacement relative to where he started?
54.5 m, E
+
84.5 m, E
30 m, E
Notice that the SIZE of
the arrow conveys
MAGNITUDE and the
way it was drawn
conveys DIRECTION.
Parallel Vectors cont.
VECTOR ADDITION cont. - If 2 vectors are going in opposite
directions, you SUBTRACT.
• Example: A man walks 54.5 meters east, then 30 meters
west. Calculate his displacement relative to where he
started?
54.5 m, E
30 m, W
24.5 m, E
-
Graphical Vector Addition
Adding two vectors A and B GRAPHICALLY, also called,
TIP to TAIL METHOD


will give the sum of the vectors R ( Resultant Vector )
R is equal to the distance from the beginning to the
end point.
Arrows are used to represent the vectors



vectors will be drawn to scale (EX: 1 cm = 2 m)
the beginning of vector B is placed at the end of vector
A
The vector sum R can be drawn as the vector from the
beginning to the end point. ( indicated by the dotted
line )
This method can be used with multiple vectors using the tip to tail method to solve
for the R Vector.
You will use this method to complete the post vector scavenger hunt graphical
vector addition activity.
Combining Vectors
• The individual vectors
are called
components
 The “overall” vector is
called the resultant.
Graphical Vector addition
• Use your graph paper wisely! Think about
the vectors given and place them in the
coordinate system accordingly.
1. Use Tip to Tail method & graph vectors
2. Solve for the Resultant (displacement)
3. Solve for total distance traveled.
Graph the vectors using the tip to tail method
Use these vector Directions:
Start: Girls Bathroom 24
24 m, North
16 m, West
56 m, South
24 m, West
32 m, North
8 m, East
8 m, North
Finish: Mechanical 22
Lets learn how to read angles!
• Pass out Angle
worksheet
1st – Review Geometry Quadrants
• Where is Zero?
Did you correctly id zero?
Locate N, S, E, and W
Note: 12 u – is a scale for magnitude.
Vector A
12 u, 225°
OR
12 u, 45° W of S
OR
12 u, 45° S of W
90
ZERO
180
A
270
Review: Right Triangle Trigonometry
SOH CAH TOA
hypotenuse
C
opposite
B

A
adjacent
sin  = opposite
hypotenuse
cos  = adjacent
hypotenuse
tan  = opposite
adjacent
And don’t forget: Pythagorean Theorem
A2 + B2 = C2
Perpendicular Vectors
When 2 vectors are perpendicular, you must use the
Pythagorean Theorem.
A man walks 95 km, East then 55 km, north.
Calculate his RESULTANT DISPLACEMENT.
55 km, N
c2  a2  b2  c  a 2  b2
95 km,E
c  Resultant  952  552
c  12050  109.8 km
Calculator MODE Check
• Enter mode:
• Make sure your
calculator is in
• Degrees
• NOT RADIANS!
BUT…..what about the VALUE of the angle???
Just putting North of East on the answer is NOT specific enough for the
direction. We MUST find the VALUE of the angle.
Remember: SOH CAH TOA
109.8 km
55 km, N
 N of E
95 km,E
 - Greek symbol Theta
To find the value of the angle we
use a Trig function called TANGENT.
( TOA )
opposite side 55
Tan 

 0.5789
adjacent side 95
  Tan 1 (0.5789)  30
FYI: Tan -1 refers to the inverse – Use 2nd TAN
on the calculator.
What if you are missing a component?
Suppose a person walked 65 m, 25 degrees East of North. What were his
horizontal and vertical components?
The goal: ALWAYS MAKE A RIGHT TRIANGLE!
65 m
Y=?
To solve for components, we often use the trig
functions sine and cosine. ( SOH CAH )
25
adjacent side
opposite side
sine 
hypotenuse
hypotenuse
adj  hyp cos 
opp  hyp sin 
cosine 
adj  x  65cos 25  58.91m, N
opp  y  65sin 25  27.47m, E
Example
A bear, searching for food wanders 35 meters east then 20 meters north. Frustrated, he
wanders another 12 meters west then 6 meters south. Calculate the bear's displacement.
-
12 m, W
-
=
6 m, S
23 m, E
=
14 m, N
20 m, N
35 m, E
14 m, N
R

23 m, E
The Final Answer:
R  14 2  232  26.93m
14
Tan 
 .6087
23
  Tan 1 (0.6087)  31.3
26.93m @31.3 NofE
Vector Addition-Analytical
To add vectors mathematically follow the steps:
1.
2.
3.
4.
5.
6.
7.
8.
Write - SOH CAH TOA on the top of your paper.
Make sure your calculator is in DEGREE MODE.
Resolve the vectors to be added into their x- and y- components by
making a sketch.
Draw the X component – What quadrant is it in? I, II, III, IV
What sign will it have? + or Draw the Y component - What quadrant is it in? I, II, III, IV
What sign will it have? + or Ax – on the x-axis – solve using cosine ( CAH )
Ay – on the y axis – solve using sin ( SOH )
After solving for Ax and Ay – check yourself using the Pythagorean
Theorem.
Vector Add. – Analytical cont.
9. Add the x- components together to get the x-component of
the resultant (Rx ) by Adding Like Terms : Rx = Ax + Bx
10. Add the y- components together to get the y-component of
the resultant (Ry ) by Adding Like Terms: Ry = Ay + By
11. Use the Pythagorean Theorem and the x and y
components together to find the magnitude of resultant.
12. Use inverse tangent ( 2nd TAN )to find the angle and then
adjust to find the direction of the resultant from Zero East.
Vector Resolution Example
Given:
1. vector A at
angle  from
horizontal.
y
Hypo
A
Ay

Ax
Adj
x
Opp
2. Resolve A
Use trig functions:
SOH CAH
into its
components.
(Ax and Ay) cos =A /A so… A = A
x
x
cos
sin= Ay/A so… Ay = A sin
Vector Addition-Analytical
Example:
Given: Vector A is 90 u, 30O and vector B is 50 u, 125O.
Find the resultant R = A + B mathematically.
B
By
Ax= 90 cos 30O = 77.9 u
A
55O
Bx
Ay
30O
Ax
First, calculate the
x and y components
of each vector.
Ay= 90 sin 30O = 45u
Bx = 50 cos 55O =- 28.7u
By = 50 sin55O = 41u
Note: Bx will be negative
because it is acting along
the neg x axis.
Vector Addition-Analytical cont.
Find Rx and Ry:
Rx = Ax + Bx
Ry = Ay + By
Rx = 77.9 - 28.7 = 49.2 u
R
Ry

Rx
Ry = 41 + 45 = 86 u
Then find R:
R2 = Rx2 + Ry2
R2 = (49.2)2+(86)2 , so… R = 99.1 u
To find direction(angle) of R:
 = tan-1 ( Ry / Rx )
 = tan-1 ( 86 / 49.2 ) = 60.2O
Stating the final answer
1. All vectors must be stated with a
magnitude and direction.
2. Angles must be adjusted by adding
compass directions( i.e. N of E) or
adjusted to be measured from East
(or 0° or the positive x-axis).
3. When added graphically, the length of
resultant must be multiplied by scale
to find the magnitude.