Chapter 3 Intro to Vectors Notes
Section 1-3 Notes
Scalars & Vectors

Scalar – a quantity that has magnitude but no direction



Units must be expressed
Ex: mass, volume, speed, time, distance, counting numbers
Vector – a quantity that has magnitude and direction


Units & direction must be expressed
Ex: velocity, acceleration, force, displacement
Vector Addition – One Dimension
If an object’s motion involves more than one vector, these can be
added to find a net description of the motion
A person walks 8 km
East and then 6 km East.
Displacement = 14 km
East
A person walks 8 km East
and then 6 km West.
Displacement = 2 km
Module 5 - 2
Adding Vectors Graphically



When vectors are added the answer is called
the resultant
To add vectors graphically, they must be
drawn tail to tip
Ex: A bus travels north 10.0km and west
10.0km. What is the resultant displacement?
10.0km
R

10.0km
Displacement = 14.1km to the northwest

Ex: A bus travels north 10.0km and west
10.0km. What is the resultant displacement?

D= 14.1km, 135° from the horizontal
Graphical Method of Vector Addition
Tail to Tip Method

V1

V2

V1

V3

VR
Module 5 - 5

V2

V3
Adding Vectors Graphically – a little more complicated
with graph paper

A student walks 150m NE to his friend’s
house. The student then walks another
800m NW to get to the bus stop. What is the
displacement?
800m
R
150m
Properties of Vectors


Vectors can be moved parallel to themselves in a diagram to
get them tail to tip
Ex: A dog swims across a flowing river.
End
River velocity
End
River velocity
Dog velocity
Start


Vectors can be added in any order.
Ex: A bus travels from city A to city B.
OR


To subtract a vector, add its opposite.
Mathematically this means change from
positive to negative or vice versa.
Graphically it means switch the direction.
Ex: An airplane traveling at 50m/s is slowed
by a wind of 5m/s.
50m/s
45m/s
-5m/s
Ffr = 10N
Fp = 50N
Fnet = 40N to the right


Vectors that are multiplied by scalars result in
vectors
Ex: A customer tells a cab driver to drive
twice as fast
2 x 12m/s = 24m/s
Scalar
Vector
Practice Problems – use graph paper across from
your Cornell notes
1)
2)
A boat moves at 0.80m/s across a river that is flowing at
1.5m/s. What is the resultant velocity of the boat? (solve
graphically and algebraically)
A telephone pole support cable is in the way of some
construction workers. In order for the work to proceed, the
cable must be moved 2 meters closer to the pole. If the pole
is 10 meters tall and the cable is currently fastened to the
ground 8 meters from the pole, how much will the workers
need to cut off from the cable when they move it?
#2 Solution: diagram the problem
Solution
The original cable length is
or 12.8m.
By moving the cable 2 meters closer to the pole, we shorten the
overall length of the cable to
Or 11.7m. Therefore, 1.1m must be cut off the cable.
Ch 3 Homework #1
Questions: 1,5,8,9 on pg 67
Problems: 1,5,8,9 on pg. 68
Due October 26
Ch 3 Vector Quiz on Oct 26
Chapter 3-Resolving Vectors
Section 5,6,7 Notes
Pythagorean Theorem to Find a Resultant (vectors are
perpendicular)

Remember: c2 = a2 + b2
c is the resultant vector

Use the tangent function to find the direction


Tanθ = opp/adj
(c = hypotenuse)
or
θ = tan-1(opp/adj)

Example: A runner runs 8.5km to the east and
then runs 2.5km to the north. What is the
displacement?
R
θ
8.5km
2.5km
Resolving Vectors into Components

Components – the horizontal (x) or vertical (y)
parts of a vector
Vector
Vertical component
Horizontal component

The x and y components of a vector make a
right triangle with the vector


We can now describe motion in terms of the
horizontal motion and vertical motion
Ex: An airplane takes off at an angle of 25°
and a speed of 185mi/h. It’s ground speed is
168mi/h. It climbs up at 78mi/h.
185mi/h
78mi/h
25°
168mi/h
Signs of Components
y
Rx  
Rx  
Ry  
Ry  
x
Rx  
Ry  
Rx  
Ry  
Module 5 - 13
Properties of Right Triangles




SOH CAH TOA
sinθ = opp/hyp
cosθ = adj/hyp
tanθ = opp/adj
opp
hyp
θ
adj



SOH CAH TOA
Use the cosine function to get the x component
Use the sine function to get the y component
25°
185mi/h
Hyp = 185mi/h
25°
168mi/h
Adj = 168mi/h
Note: coordinate system has been
changed so all is positive
78mi/h
Opp = 78mi/h
Formulas for Resolving Components
 Vx
= V(cosθ)
Vy = V(sinθ)
Vx = 185(cos25°)
Vy = 185(sin25°)
Vx = 168
Vy = 78
185mi/h
25°
Steps for Adding Vectors that are not Perpendicular
θ
θ
θ
θ

R
We cannot use the pythagorean theorem, so
we must 1) After sketching, resolve each
vector into its x and y components
V2
V1y
V1
θ1
V1x
θ2
V2x
V2y
θ
R
θ


2)Add the x components and add the y
components
3)Use the Pythagorean theorem to find the
magnitude of the resultant
VR2 = Vx2 + Vy2

4)Find the direction of the resultant
θ = tan-1(Vy/Vx)

5)Does my answer make sense?
Practice Problems
7th edition (3-10,2,3)
3-9. A man in a rowboat is trying to cross a river that flows due west
with a strong current. The man starts on the south bank and is
trying to reach the north bank directly north from his starting point.
The boat must a)head due north, b)head due west, c)head in a
northwesterly direction, d)head in a northeasterly direction. Use a
sketch to justify your answer.
7th edition (3-2,3)
3-1. A rural mail carrier leaves the post office and drives 22.0km in a
northerly direction to the next town. She then drives in a direction
60.0° south of east for 47.0km to another town. What is her
displacement from the post office?
Ch 3 Homework #2
Questions
Problems
Ch 3 Test on
Physics Chapter 3 Projectile Motion
Section 3 Notes
Projectile Motion
The two dimensional motion of an object that is thrown or
launched into the air.
A projectile is an object
moving horizontally
under the influence of
Earth's gravity; its path
is a parabola.
Module 7 - 1
waterslide (1min):https://www.youtube.com/watch?v=W46UMzFEU24
Mythbusters (5min):https://www.youtube.com/watch?v=iHu6LVg-0Hs
Concept Facts about Projectile Motion
•Projectile motion is free fall with horizontal
velocity (Neglect air resistance)
•Consider motion only after release and before it hits
•Analyze the vertical and horizontal components
separately
•a=0 in the horizontal, so vx = constant
•ay = -g = -9.8m/s/s
•Object projected horizontally will reach the ground
at the same time as one dropped vertically
•If the ball returns to the y = 0 point, then v = v0.
•Range (x displacement) is determined by time it takes for projectile
to return to ground or other end point
Formulas for Projectile Motion
Horizontal Motion
vy vy0 gt
1 2
y
y
v
t
0
y
0t g
2
2
2
v

v
2
g
(y

y
)
y
y
0
0
Quadratic Formula:
ax2+bx+c = 0
x = -b± b2-4ac
2a
Vertical Motion
v0  vx0
x x0 vx0 t
Solving Projectile Motion Problems
1. Draw a picture and select a coordinate
system.
2. Write down all the given information.
3. Resolve any vectors (velocity) into x and y
components.
4. Choose the formula you
need.
5. Solve the problem.
Ex: James thought it was the perfect murder plot: push his wife
off a cliff and claim she fell. Ahh, but physics won, and James is
in jail. James pushed his wife straight out with a velocity of
0.559m/s. She fell 182m. How far from the cliff did she land?
Practice Problems
7th edition (3-6,7)
3-5. A child sits upright in a wagon which is moving to the right at
constant speed. The child extends her hand and throws an apple
straight upward while the wagon continues to move forward at
constant speed. If air resistance is neglected, will the apple land
a) behind the wagon, b)in the wagon, or c) in front of the wagon.
3-6. A boy on a small hill aims his water-balloon slingshot
horizontally, straight at a second boy hanging from a tree branch a
distance d away. The boy in the tree is the same height up in the
air as the water balloon sling shot on the hill. At the instant the
water balloon is released, the second boy lets go and falls from the
tree, hoping to avoid being hit. Use a diagram and 2-3 sentence
explanation to explain that he made the wrong move.
Practice Problems
7th edition (3-4,5)
3-3. A movie stunt driver on a motorcycle speeds horizontally off a
50.0m high cliff. How fast must the motorcycle leave the cliff-top if
it is to land on level ground below, 90.0m from the base of the cliff
where the cameras are.
3-4. A football is kicked at an angle θ=37.0° with a velocity of
20.0m/s. Calculate a) the maximum height, b) the time of travel
before the football hits the ground, c) how far away it hits the
ground, d)the velocity vector at its maximum height, and e) the
acceleration vector at maximum height. Assume the ball leaves
the foot at level ground and ignore air resistance and rotation of
the ball.
Practice Problems
7th edition (3-9)
3-8. Suppose the football in example 3-4 (θ=37.0° with a velocity of
20.0m/s) was a punt and left the punter’s foot at a height of 1.00m
above the ground. How far did the football travel before hitting the
ground? Set x0 = 0, y0 = 0.
Requires Quadratic
Old Practice Problems
1. People in movies often jump from buildings into
pools. If a person jumps from the 10th floor (30.0m) to
a pool that is 5.0m from the building, with what initial
speed must the person jump?
Practice #10
2. A golfer practices driving balls off a cliff into the water
below. The cliff is 15m from the water. If the ball is
launched at 51m/s at an angle of 15°, how far does
the ball travel horizontally before hitting then water?
Practice #10
3. A missile is shot at a 60.0° angle and a speed of
55.0m/s. It is trying to reach an aircraft that is flying at
450m. Will the missile be able to reach its target? If
not, what should be the initial velocity to reach its
target?