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Chapter 2
Vectors in One dimension
Language of motion
Origin: reference point
Position, d : The straight-line distance between
the origin and the object’s location including
magnitude and direction.
Ex 1: Position: 4.5 m [east of the origin] or 4.5 m [E].
Ex 2: Position: 2.0 m [south of the origin] or 2.0 m [S]
Scalar vs. Vector quantities
A scalar quantity has only magnitude.
Ex: length, area, volume, time, speed, mass,
temperature, etc.
A vector quantity has both magnitude and
direction.
Ex: displacement, direction, velocity,
acceleration, force, etc.
1.
Time, t = 30 s
2.
Displacement of 25 m North
3.
Velocity,
4.
Force,
v = 20 m/s, Right
F =6.50 x 107 N, 60o west of south
Distance vs. Displacement
Distance: the length of the path taken to move
from one position to another-scalar
Ex: distance of 70 m.
Displacement: a straight line between initial
and finial position; includes magnitude and
direction-vector
Ex: displacement of 30 m east
Ex: Find distance and displacement
Vectors
vectors are drawn using arrows which show
magnitude in the length as well as direction
the resultant is the sum of two or more vectors
to add vectors, arrange them head to tail then
draw the resultant from the tail of the first
vector to the head of the last vector
+
+
=
or
Note that the resultant is the same regardless of
the order of the vectors
Sign conventions
Ex 1: Contestants in a snowshoe race must move forward
10.0 m, untie a series of knots, move forward 5.0 m, solve a
puzzle, and finally move forward 25.0 m to the finish line.
Determine the resultant vector by adding the vectors
graphically.
1. Choose an appropriate scale and reference direction.
2. Draw the first vector and label its magnitude and direction
3. Place the tail of the second vector at the tip of the first
vector. The same for the rest of vectors.
4. Connect the tail of the first vector to the tip of the last
vector. The new vector pointing toward the tip of the last
vector is the resultant vector.
5. Find the magnitude of the resultant vector by measuring
with a ruler, then convert the measured value using the
scale.
Ex 2: Find the distance and displacement
travelled by the person.
Ex 3: Find the distance and displacement of a
dog that moved:[Use vector diagram and show on
it the arrow that presents the displacement]
- 10 m, west
- 6.0 m south
- 2.0 m east
- 3.0 m, north
- 8.0 m, east
Ex 4: A car travels 300 km east and then 200 km
west. Find the distance travelled by the car and
its displacement.[show your sign convention,
draw vector diagram, draw displacement
arrow]
Average speed vs. Average velocity
 Average speed = total distance/ time
 Average velocity = displacement/time
Ex: A person travels 25 m east then 10 m west
in 30 s. Find the average speed and average
velocity. [Provide vector diagrams]
Solve practice page 9#1-3
Solve Check and Reflect pg 10# 4-7
Vector Resultant (one dimension or linear)
1. Vectors in the Same Direction
If two vectors have the same direction, their
resultant has a magnitude equal to the sum of
their magnitudes and will also have the same
direction
RESULTANT = SUM
Ex 1:
A person runs 25 m south and then another 15 m
south.
a) What is the distance travelled?
b) What is the displacement?
Ex 2:
A plane has a velocity of 150 m/s [E].
The wind blows east at 75 m/s. Find the resultant
velocity.
2. Vectors in Opposite Directions
To add vector quantities that are in opposite
directions.
RESULTANT =DIFFERENCE
Ex 1:
A plane flies 200 km north and then turns around
and comes back 150 km.
a) What is the distance travelled?
b) What is the displacement?
Ex 2:
The skier moves from A to B to C to D. Use the
diagram to determine the distance traveled by the
skier and the resulting displacement during these
three minutes.
Solve practice problems pg 73#1 & pg 74#1-3 & pg
75#6-9
Solve Page 1&2 workbook
3. Motion in two dimensions
When you walk east,
west, north, south,
right, left, up, or
down, you are walking
in ONE DIMENSION.
When walking in an angled movement, you are
walking in TWO DIMENSIONS and you are making
2D vector.
Components of a vector- is the path you take from the
start (tail) to the end (head) using only +x, +y, -x, or –y.
The shortest path from the start to the end is the
resultant (sum).
End
‘
150 m
Start
120 m
‘
+x direction
y direction
90 m
Ex: Write the x and y components of the following:
y
x
Vector direction in 2-dimentions
How to indicate the direction of a vector in 2-D.
 Cartesian Method-angles are measured from
+x axis by moving counterclockwise.
+x
+x
Ex: Draw vectors in two dimensions
a) 25 km [120˚]
b) 10 m/s [180˚]
c) 225 m [75˚]
 Navigation Method- uses E, W, N, S to
indicate the direction of the vector and an angle
θ.
Example 12.5 m [30˚ W of N]
30˚
To draw the vector arrow start from the origin
then move towards the second direction, N,
then move towards the first, W, to get to the
head of the vector.
Ex: Draw vectors in two dimensions:
a) A boat sailing at 60 km [30˚ N of E]
b) A force applied 450 N [S]
c) The dog moved 10.2 m [25˚ W of S]
Vectors that lie along the same straight line in
one dimension are called collinear.
Vectors not along a straight line, 2-D, are noncollinear.
35˚
Collinear vectors
Non-Collinear vectors
Resultant of 2-D vectors at 90˚
(finding resultant):
1. Graphically (diagramming)
 Draw vector 1
vR
v2
 Draw vector 2 so that its tail
θ
at the tip of vector 1
v1
 Draw the resultant vector by
v2
connecting the tail of vector 1
vR
to the tip of vector 2 (head-to- v1
θ
tail)
 State the resultant vector (both magnitude
and direction)
NOTE: The resultant angle,θ, is at the tail of the
first vector and the tail of the resultant.
Ex: A camper walks 2.0 km [S], then 4.0 km [E],
and finally 1.0 km [N]. Find graphically her
resultant displacement.
2. Mathematically
Adding Vectors at 90˚
General equation for vector addition:
vR
Dv = v 1 +v 2
v2
v1
to add vectors that are at right angles to each
other, you must calculate:
a) the magnitude – use Pythagorean Theorem
c 2 = a2 + b2
b) the direction – compass direction for example
[35˚ N of E]…
Ex 1:
A plane flies 200 km north and then heads 150
km east. Calculate the displacement. (Draw a
diagram!!!)
Ex 2:
A plane flies 350 km west, 200 km south and 100
km east. Calculate the displacement.
Ex 3:
A disoriented dog runs 30 m north then 20 m
east then 10 m west. Find his displacement.
3. Determining Components
 first you need to be able to take any vector
and break it down into its x and y components
 start by forming a right angle triangle from
the vector…this shows each component
eg)
 then use trigonometry to calculate the
magnitude of the components ie) length of each
side of the right angle triangle, given the vector
Ex 1:
A plane flies at an angle of 40.0 N of E at a velocity
of 600 km/h. Find the east component and the
north component of the velocity.
Ex 2:
A cyclist’s velocity is 10 m/s [245o]. Determine the x
and y components of her velocity.
 now that you can break a vector into its
components, you can add two or more vectors using
the individual x and y components
1. Calculate the x and y components for each vector
2. Add all components in the x direction. Add all
components in the y direction
3. Find the magnitude of the resultant vector
(Pythagorean theorem)
4. Find the angle of the resultant vector
(trigonometric ratios)
Ex 1:
Add the following vectors: 12.0 m [30o] and 9.0 m
[155o].
Ex 2:
Use components to determine the displacement of
a cross-country skier who travelled 15.0 m [220o]
and then 25.0 m [335o].
Ex 3:
Give the magnitude and direction for the addition of
the following vectors :
200 N, 20 N of E and 300 N, 60 S of W
Ex 4:
Add the following vectors: 200 N, 45 N of W and
100 N, 30 N of E.
Ex 5:
Add the following vectors: 20 N, 40 N of E; 30 N,
due north; 50 N, 60 W of N
Relative motion
Objects sometimes move within a medium that
is moving:
 Airplanes, kites, sailboats moving within
moving air.
http://www.physicsclassroom.com/mmedia/vectors/plane.html
 Boats, swimmers moving under the effect of
water current.
http://www.physicsclassroom.com/mmedia/vectors/rb.html
The velocity of a moving object depends on the
location of the observer:
 The observer is on the moving object
(passenger riding on the bus)
 The observer is watching from a stationary
position.
Ex: The bus
http://sasklearning.gov.sk.ca/branches/elearning/tsl/resources/subject_area/science/physics_30_resources/lesson_one/relative_motion.shtml
Case 1: Collinear Relative motion (one
dimension)
Ex:
A west jet airplane travels with an air velocity
(velocity in still air) of 650 km/h [S] from Fort
McMurray to Edmonton. It encounters a wind
velocity of 43.2 km/h. What is the velocity of the
airplane relative to an observer on the ground if,
1) the wind is south wind blowing to the north.
2)
the wind is blowing towards the south.
Case 2: Non-collinear relative motion (two
dimensions)
Relative motion in air
Ex1:
A plane travels north at an air speed of
300 km/h. The wind blows west at 50.0 km/h.
a) At what angle should the plane be piloted
such that it will go straight north?
b) What is the velocity of the plane with respect
to the ground?
Relative motion in water
Ex2:
A boat is moving at 20 m/s in a river that has a
velocity of 5.0 m/s south. Calculate the resultant
velocity when:
a) the boat is moving downstream (same direction)
b) the boat is moving upstream (opposite direction)
c) the boat is moving east
Ex3:
A boat moving 4 m/s east across a 80 m wide
river encounters a current travelling 3.0 m/s
north. Calculate:
a) the resultant velocity of the boat
b) how long it takes to reach the other side
c) how far downstream the boat lands when it
reaches the other side
Projectile Motion
 a projectile is an object that travels
in air. Force of gravity is the only
force acting on it.
 it follows a curved path, called a
trajectory, which is due to its
horizontal and vertical velocity
 the horizontal distance the object
travels is called the range
 characteristics of projectile motion:
1. the horizontal velocity is constant
2. the vertical velocity changes with
the distance (height) the object falls
3. the horizontal velocity is
independent of the vertical velocity
 it doesn’t matter what the
horizontal speed is, gravity will affect
it in the same way…that is, it will take
the same time to fall
 at the point where the object starts
to fall (y direction) the vertical
velocity is 0 m/s
 to solve the problems, look at the x
and y directions separately
Ex 1:
A stone is thrown horizontally at 15 m/s from
the top of a cliff 44 m high.
a) How long does it take to reach the bottom of
the cliff?
b) How far from the base of the cliff does the
stone strike the ground?
c) What is the vertical speed as the stone hits
the ground?
Ex 2:
An object is thrown horizontally at a velocity of
10.0 m/s from the top of a 90.0 m high building.
a) How long does it take to hit the ground?
b) How far from the base of the building does it
strike the ground?
Ex 3:
An object is thrown horizontally at 20.0 m/s
from the top of a cliff. The object hits the
ground 48.0 m from the base of the cliff:
a) How high is the cliff?
b) What is the vertical speed as the object hits
the ground?