The document reviews vectors and vector operations. It discusses scalars and vectors, adding vectors graphically and by the tail-to-tip method, multiplying and dividing vectors by scalars, resolving vectors into x and y components, and projectile motion under gravity having independent horizontal and vertical motion. It also covers frames of reference and how velocity depends on the reference frame.

1. Lesson 3-1
Review of Vectors

2. Scalars and Vectors
 Recall a scalar does not have a direction
 A vector has BOTH magnitude and direction
 Vectors can be added graphically
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3. Similar Quantities
 When adding vectors, the units must match
 It would be meaningless to add a force vector to a
velocity vector
 They are essentially apples and oranges
 When vectors do have the same units, we
may add or subtract the vectors

4. Example
A student is walking to school. First, the
student walks 350m to a friend’s house. The
two then both walk 740 m to school.
The method to add the vectors is called the tail
to tip method. The vector we find is called
the resultant vector.

5. Moving Vectors
 Vectors can be moved parallel to each other
 Does not matter where the vectors are, as long as
they are addable, tail to tip
 Example
 Push a toy car across a moving sidewalk
 Say the sidewalk is moving at 1.5 m/s
 The car is pushed .8 m/s

6. Vector Addition and Subtraction
 Vector addition is commutative
 The order the vectors are added does not matter
 To subtract a vector, simply add the opposite

7. Multiplying and Dividing Vectors
 Multiplying or Dividing vectors by scalars
results in vectors
 Lets say we have the velocity of a race car
 We want to examine the properties of the car when it is
traveling twice as fast
 If vi is v, what is twice vi?
 What is half of vi
 What would be the new v if the car drove twice as fast in
the opposite direction?

8. Lesson 3-2
Vector Operations

9. Coordinate Systems
 Up to this point, we have only needed one
dimension to study our situations
 What if we wanted to study a ball being thrown at
45o above the ground?
 That path of motion does not fit any of our current
axis
 We will have to use a combination of the two axis
 Note: Orientation of the axis is up to you.

10. Determining the Resultant
 Trigonometry is very useful to find the
resultant vector.
 The Pythagorean Theorem
 Think of a tourist in Egypt walking up the side of a
pyramid
 Are they walking vertical?
 Are they walking horizontal?
 It is a combination of the two motions that produces one
final motion, somewhere between horizontal and vertical

11. Resultant
 The resultant of two vectors is also a vector
 That means the resultant must have:
 Magnitude
 Direction
 It is not enough to say the magnitude of the
resultant, it must have direction also.
 We will use the trig functions of sine, cosine
or tangent to find the direction

12. Resolving Vectors
 Any vector may be broken into x and y
components
 That is to say any vector may be RESOLVED into
its component vectors
 A horizontal vector has a 0 y component
 A vertical vector has a 0 x component
 A vector at 45o has equal x and y vectors

13. Examples
 Page 92, Film Crew
 Pg 93

14. Non-perpendicular Vectors
 Until now, all of our vectors have been
perpendicular to each other
 Things in real life are much, much less rigid
 Lets say a plane travels 50 km at an angle of
35o, then levels out and climbs at 10o for 220
km
 These vectors are not perpendicular, we cannot
use the Pythagorean Theorem, yet
 Resolve the vectors

15. Lesson 3-3
Projectile Motion

16. Two Dimensional Motion
 In the last section, we resolved vectors into x
and y components.
 We will apply the same ideas to something
thrown or flying in the air
 Think of a long jumper
 When she approaches her jump, she has only an
x component
 When she jumps, she has both x and y
components

17. Analyze Projectile Motion
 We can break the motion into the two
component vectors and apply the kinematic
equations one dimension at a time
 Any object thrown or launched into the air
that is subject to gravity is said to have
projectile motion
 Examples?

18. Projectile Path
 Projectiles follow a path called a parabola
 A common mistake is to assume projectiles fall
straight down
 Since there is vxi, there must be continuous
horizontal motion
Vx
V
Vy

19. Projectile Path
Vx
V
Vy
 Neglecting air resistance, is there anything to stop the
projectile in the horizontal direction?
 Velocity in the horizontal direction is constant
 In real life, horizontal velocity is not constant, but for our
purposes we will assume uniform, constant velocity

20. Projectile Path
 Projectile motion is simply free fall with horizontal
velocity
 If two similar objects fall to the ground from the
same height, one straight down while the other is
thrown out to the side, which hits first?
 It is very important to realize motion in the x
direction is completely independent of motion in the
y direction

21. Summary
 A projectile has horizontal velocity until the
object stops (hits the ground)
 A projectile will have a vertical velocity that is
ever changing due to gravity, until the
projectile stops (hits the ground)
 What is the only factor that is consistent in
the x AND y directions?
 Time

22. Projectile Path
Vx
V
Vy
Sample Pg 101, Practice Pg 102

23. Objects Launched at an Angle
 When an object is launched at an angle, the
object has both horizontal and vertical
velocity components
 This is similar to the motion of an object
thrown straight up with an initial vertical
velocity
 Example pg 103, Practice pg 104

24. Lesson 3-4
Relative Motion

25. Frames of Reference
 Velocities are different in different frames of
reference
 You are in a train traveling at 40 km/h
 Relative to the train, how fast are you moving?
 Someone outside sees the train pass, how fast do they
see you moving?
 The velocities are different because the
reference frames were different
 You – Train, Outside observer - Earth

26. Examples
 You are driving on the interstate at 80 km/h
and a car passes you at 90 km/h
 How fast does it seem the passing car is moving
to you? To someone on the side of the road?
 A semi-truck driving west at 85 km/h passes
a car on the other side of the road, driving
east at 75 km/h. To the trucker, how fast is
the car moving?

27. Examples
 A person standing on top of a train traveling at 20 km/h. They
throw a baseball. How fast does it look like the ball is moving to
a person standing on the ground when:
 The ball is thrown 10 km/h forward
 The ball is thrown 40 km/h backward
 The ball is thrown 20 km/h backward
 The ball is thrown straight up
Example pg 108, Practice pg 109