Vectors - Red Hook Central School District

```• What is wrong with the following statement?
I live exactly 15 miles from here. Meet me there
at 4 PM. I’ll cook dinner for you.
Vectors &amp; Scalars
Measurements can be vectors or scalars.
vector = magnitude (size) &amp; direction.
Scalar = magnitude (size) only.
Both have units.
List some measurements that
might have a direction.
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•
•
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Velocity
Time
Acceleration
Force
mass
Some Vector &amp; Scalars:
Vectors (direction)
Displacement
Velocity
Acceleration
Force
Momentum
Scalars (number unit)
no direction
distance
speed
temperature
time
mass
Displacement, d (s) and distance, d.
Displacement, d = change in position, Dx
Delta means subtract
Dx = xf – xi.
• Start at 2 cm end at 10 cm.
 Dx = xf – xi.
• d = 10 cm – 2 cm = 8 cm
• Start at 10 cm end at 2 cm.
 Dx = xf – xi.
• d = 2 cm – 10 cm = - 8 cm.
• The + or – sign says which direction.
• Same distance dif displacement.
Displacement (d)
• Straight line distance with direction from
starting point.
• Distance is the path length and no
direction.
Take a walk.
Representation of vectors:
Vectors represented diagrammatically
(graphically) by sketching scaled arrows.
1. Sketch &amp; label a vector arrow to
represent 9 m/s left.
Use a scale of 1 cm = 1m/s
sketch 9 cm arrow pointing left
Label
9 m/s left
-9 m/s
Scaled vector arrows can be sketched onto graph
axes to show direction:
scale labeled
in vector direction
d=8m
Magnitude shown
by length of arrow
Direction can be stated as compass directions or angles
in degrees (azimuth). Zero degrees is to the right or
east. Sketch into notes. Sketch the axes.
State this direction 3 different ways.
N.
30o.
W
S.
• 120o (azimuth)
• 30o W of N
o N of W.
•
60
E
Positive &amp; Negative Displacements
direction from start point
Positive
Negative
•
•
•
•
•
•
•
•
Right
East
North
Up
Left
West
South
Down
•
•
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•
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+30 m could be E, N, up, Right
- 25 could be
W,
S,
down,
Left
• A negative sign negates the direction:
• -40 km W
• +40 km, E.
Ex: State the vector – 25 m, W as a positive.
• +25 m, E
• Or simply 25 m, E
Hwk Worksheet Intro Vectors.
Sometimes, more than one vector
quantity are combined. In that case,
we must combine (add or subtract) the
individual vector components to find a
resultant vector.
Examples of combining vector
quantiti:
• Walk west and turn and walk north.
• Drive 20 km/h south then 50 km/h SE.
The resultant velocity depends on 2 pushes: the
engine push &amp; the wind push.
Combining Parallel Vectors at 0o or 180o.
•
•
•
•
20 m right and 5m right.
+20 m + 5m = + 25 m or 25 m right.
Object is 25 m right of start point.
• 20 m left and 5m right.
• - 20 m + 5m = - 15 m or 15 m left.
• Object is 2 m left of start point.
We can also use scaled arrows
sketched head to tail to represent
components and find resultant
vectors.
3. When vectors are at 0o or 180o (straight line). I walked
10 km, E, then 5 km, E. Find displacement. Sketch the
component and the resultant displacement vector arrows
with a scale 1 cm = 1 km .
0o Same direction
10 km, E +
5 km, E
= 15 km, E = Resultant
R = length from tail to tip of new arrow.
180o Opposite directions.
I walked 10 km, E, then 5 km, W.
10 km, E -
5 km, W
= 5 km, E = R
5 km, W
10 km, E
= 5 km, E = R
Mathematical operations can be done
on non-linear vectors – but not in the
usual way. Their direction has to be
taken into account.
We cannot simply add 40 km North +
20 km North East. The resulting
displacement is not 60 km.
Graphical analysis: a scaled diagram is used
for any number of, &amp; combination of
vectors.
Methods:
1) Parallelogram (Two vectors only)
2) Tail to tip (head to tail). For any number of vectors to
Parallelogram Method
• Football Vectors 3 min
• http://science360.gov/obj/tknvideo/0ca015f8-0d4c-4d0b-a31e257ba1445c32/science-nfl-football-vectors
Graphical Method to find Resultant
• Sketch scaled arrows one after the other.
1. Two people kick a ball at the same time or
concurrently. One gives it a velocity of 6.5 m/s
east, the other gives it a velocity of 4.5 m/s 30o N
of E. What is the final resultant velocity? Both
methods of solving.
4.5 m/s
30o
6.5 m/s
Tail to Tip:
Sketch a diagram with a scale of 1cm = 1 m/s.
Sketch each vector separately 1 at a time.
Place the tail of one of the vectors to the tip of the
other.
4.5 cm
30o
6.5 cm
30o
Now connect a straight line, the
resultant, from the tail of the unmoved
(1st)vector to the tip of the 2nd vector
(moved).
4.5 cm
R
30o
6.5 cm
Measure the resultant with your
ruler to get the magnitude.
Measure the angle to get the
direction of the resultant.
R=10.6 m/s
b=13o
Negative vectors are in the
opposite direction of positive ones.
-10 m/s East = +10 m/s West
- 36 km 20o N of E = +36 km 20o S of W.
What does –10 m South mean?
+10 m North
Subtraction: Just reverse the direction of the
negative vector &amp; add graphically (make
• 12 km East – 6 km south
• 12 km East + 6 km north.
13 m/s north – 5 m/s 20o N of E =
13 m/s north +5 m/s 20o S of W
Equilibrant is a vector that
“neutralizes” the resultant.
It is equal and opposite the resultant.
Ex: R = 25 m/s South,
Equilibrant = 25 m/s North or
(-25m/s S)
Wksht Prb Vector Sketching
Parallelogram Method
The parallelogram method is similar, but
instead of moving a vector &amp; sketching a
triangle, you turn your two vector
components into a parallelogram.
You may sketch both vectors from the
origin. But you must turn the shape
into a parallelogram
4.5 km
40o
8.0 km
When the parallelogram is complete,
sketch the resultant between the two
original component vectors corner to
corner. Measure the resultant and the
new angle.
Try this:
A hiker walks 4.5 km at 40o. He
then turns and walks 8 km due
east. Find his resultant
displacement and direction.
12 m/s, 14 o.
Note: the tail to tip (head to tail) vector
diagram may be to resolve any components
even more than two.
The parallelogram method may be used to
resolve only two vector components.
The Pythagorean theorem may only be used
for vectors at right angles.
Graphical Method
• Make a scale – plan the line lengths
• Sketch graph axes
• Sketch 1st vector to scale in appropriate direction
from origin – place arrowhead at the end.
• Sketch the 2nd vector in the appropriate direction
to the tip (arrowhead) of the 1st vector.
• Continue until all vectors sketched.
• Connect the beginning (origin) to the tip of the last
vector sketched w/ a straight line.
• Measure the line (resultant) and convert the scale
back to appropriate units. Use protractor to find
direction.
Hwk read 3-1 do page 87 show
scaled vector sketches.
Youtube Vector Diagram Lesson 13 minutes
Situations where No Diagrams
Needed
• Vectors along a line-add or subtract.
• Vectors at right angles. Use Pythagorean
Theorem.
If vectors at 0o or 180o to each other –
What is the resultant displacement for the
following:
• 15 m North + 5 m North.
20 m, N
• 15 m North + 5 m South.
10 m, N
Vectors at right angles
a and b are the components,
and c is the resultant.
Ex 1: Find Resultant displacement &amp; direction of R using
Pythagorean:
10 km
5 km
q
52 + 10 2 = R2
R = 11 km
tan q = 5 km
10 km
tan-1 0.5 = q
q = 26.6o S of W
Ex 2: A dog walks 4 km to the east
and then 9 km to the north. What
is his resultant displacement?
• 9.8 km
• 66o N of E
Ex 3: A pirate in search of treasure
follows a map and walks 45-m north,
turns and walks 7.5-m east. Use
trigonometry to find the displacement.
• 45.6-m, 9.5o E of North
Ex 4: Spiderman sprints forward for 115 m
and then scales a vertical building straight
up for 136 m.
A. Make a rough sketch.
B. Find his resultant displacement.
do
pg 91 #2 - 4
and pg 114 #22 - 24
Resolution of Resultant to
Components
All 2-d vectors can be described as the sum
of perpendicular vectors.
Instead of combining vector components to
give resultant, we take resultant &amp;resolve it
(break it ) into perpendicular components.
A plane takes off at an angle. Make a rough
sketch of its velocity vector relative to the
ground.
It is sometimes useful to break or “resolve” the
vector into horizontal &amp; vertical components.
Vector a can be broken down, or resolved
into 2 perpendicular components: ay &amp; ax.
ay is opposite the angle q, so
ay = a sin q .
ax is adjacent to angle q, so
ax = a cos q .
Ex 1: How fast must a car be moving to stay beneath a
plane taking off at 105 km/h at 25o to the ground?
• Need horizontal component of plane v.
• vx = v cos q.
• vx = 105 km/h cos 25 = 95 km/h
What is the vertical plane velocity from the
previous example?
• vy = v sin q
• 105 km/h sin 25o
• 44 km/h
Ex 2: The landing speed of the space shuttle is
99.7 m/s. If the shuttle is landing at an angle of
15o to the horizontal:
a. How fast is it descending?
b. What is its horizontal velocity?
• a. 25.8 m/s
• b. 96.3 m/s
Hwk Text pg 94 # 3 -7
show all work
Finding Resultant Algebraically
In what situation can we simply add or
subtract vectors to find a resultant without
making a diagram or using Pythagorean?
To find resultant of 2 or more vectors, we
can resolve each vector into the X and Y
components.
Find the resultant of the 2 vectors below:
Each vector can be resolved to X &amp; Y
components.
The X &amp; Y components can be added:
To find the resultant.
Example Problem
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