Download CHAPTERS 3 & 4

CHAPTERS 3 & 4
3.1 Picturing Motion
Motion Diagrams
A series of consecutive frames (frame by
frame) of the motion of an object. Similar
to movie film (30 frames per second).
The Particle Model
Motion diagram of wheel with two different
dots, center of wheel and edge of wheel.
3.2 Where & When?
Coordinate Systems
2 dimensional
AKA x-y coordinate system
Vectors and Scalars
Scalar Quantity=
A quantity that tells you only the
magnitude (size/amount) of something.
A number with units.
Examples:
42kg, 100oC, 40s, 10hr, $89, 100m/s,
Vector Quantity
Vector Quantity=
Magnitude with direction.
Example:
46km/hr North, 15m/s SW, 58km→
Vectors are drawn to scale, the larger
the vector, the larger the magnitude.
Examples on board.
Vectors
Time Intervals and Displacement
Displacement (Δd) =
The distance and direction between two
positions.
Δd = df – di
df = final position
di = initial position
Δd can be positive/negative
Examples on number line.
Time Interval (Δt) =
The time required for an object to
complete some displacement.
Δt = tf – ti
tf = final time
ti = initial time (usually ti = 0)
 Δt is always positive
3.3 Velocity and Acceleration
Velocity
Speed vs velocity, is there a difference or
are the terms interchangeable?
Speed examples:
57m/s, 37km/hr, 17cm/yr, 68mph
 speed is a scalar quantity.
Velocity =
Speed with direction, a vector quantity.
Examples:
57ms East, 37km/hr SE,
17 cm/yr→, 68mph West
67m/s @ 300o
75m/s @ 38o N of W
Average velocity
_ Δd
df - di
Avg vel (v)= Δt = tf – ti
________
avg vel = vector quantity (speed&direction)
Examples on number line.
The frog jumps from 0m to
9m in 3 seconds, what is
the frog’s avg vel?
Δd
df – di
avg vel = Δt = tf – ti =
9m – 0m
9m
3s – 0s = 3s = 3m/s
Other Examples
1. A runner begins at the starting line and
crosses the 80m finish line in 4 seconds.
What is the runner’s average velocity?
A = 20m/s
2. A car travels from the school, 200km
West in 5hr. What is the car’s average
velocity?
A = -40km/hr
Instantaneous Velocity = ?
The speed and direction of an object
at a particular “instant” in time, how
fast it is moving right now.
Examples of instantaneous velocity =
Speedometer, radar gun, tachometer
Motion Diagram of Golf Ball
Examples on Board:
Golf putting right/left
(+v, -v, )
EQUATIONS
V = vi + at
v = vel (inst/final)
vi = initial velocity
V2 = vi2 + 2aΔd
Δd = df - di
df = di + vΔt
Acceleration (a) =
The rate of change in velocity (Δv).
Δv vf - vi
a = Δt = tf - ti
Example: A car pulls out from a stop
sign, 20s later it is traveling West at
40m/s. What is the car’s acceleration.
Δv vf - vi
-40m/s - 0m/s
a = Δt = tf - ti =
20s - 0s
-40m/s
a = 20s
= -2m/s2
Examples of acceleration
Motion Diagrams
Car speeding up, then constant velocity,
then slowing down.
All problems all year for every
chapter MUST have the following or
points will be deducted. HOMEWORK
INCLUDED!
1. A sketch/drawing or FBD.
2. Table of Known/unknown values.
3. Write the equation(s) to be used.
4. Plug in numbers
5. Show ALL steps/work.
More Equations
vf2 = vi2 + 2aΔd
vf2 = vi2 + 2a(df - di)
4.1 Properties of Vectors
Graphical representation =
An arrow is drawn to scale and at the
proper direction.
The length of the arrow represents
the magnitude of the vector.
VECTOR EXAMPLES
Resultant Vector (R) =
The sum of 2 or more vectors.
R = vector 1 + vector 2 + vector 3 +…
Resultant Vector Example
Examples on board
1. 2 equal vectors
2. 2 equal/opposite vectors
3. 2 vectors 90o apart
Resultant Vector Example
Resultant Vector Example
From SVHS to home various
examples.
No matter which route you take the
displacement (Δd) will be the same.
Graphical Addition of Vectors
Vectors are drawn from tip to tail
Resultant Vector (R) =
The sum of two or more vectors.
The order of adding the vectors does
not matter, just like adding any other
values.
Since the vectors are drawn to scale,
the magnitude of the resultant ( R )
can be measured with a ruler.
Resultant Vector Example
Resultant Vector Example
Resultant Vector Example
Example on board:
2 different paths – use meter stick
The resultant vectors (R), are equal,
the path does not matter, when all the
individual vectors are added together
the resultants will be equal.
What is the magnitude of R for the vectors
below?
What is the magnitude of the R vector
below?
What is the magnitude of the vectors below? The
red vector has a magnitude of 40, the purple
vector 65, use the indicated angles.
Can you use the Pythagorean
Theorem?
NO!
Why not?
Because the triangle is NOT a right
triangle.
How can you solve for the resultant?
You must use the LAW of Cosines.
Law of Cosines Equation
R2 = A2 + B2 - 2ABcos
R2 = 402 + 652 – 2(40)(65)(cos119o)
R2 = 1600 + 4225 – 5200(-0.48409)
R2 = 5825 + 2517.27
R2 = 8342.27
R = √¯8342.27
R = 91.37
Relative Velocities: Some
Applications
You are in a bus traveling at a
velocity of 8m/s East. You walk
towards the front of the bus at 3m/s,
what is your velocity relative to the
street?
11m/s
You are in a bus traveling at a
velocity of 8m/s East. You walk
towards the back of the bus at 3m/s,
what is your velocity relative to the
street?
5m/s
A plane is traveling North at 800km/h,
the wind is blowing East at 150km/h.
What is the speed of the plane
relative to the ground?
1. Draw sketch of vectors
2. List known/unknown
3. Write equation
4. Plug in numbers
5. Solve showing all work/steps
Solve Example
Use Pythagorean Theorem A2 + B2 = C2
Vpg2 = Vp2 + Vw2
Vpg = √¯Vp2 + Vw2
Vpg = √¯(800km/h)2 + (150km/h)2
Vpg = √¯640,000km2/h2 + 22,500km2/h2
Vpg = √¯662,500km2/h2
Vpg = 813.9km/h
Boat/River Example
A river flows South at 8m/s, a boat travels
due East at 15m/s. Where will the boat
end up and what will the boat’s speed be
relative to the shore?
1. Draw sketch of vectors
2. List known/unknown
3. Write equation
4. Plug in numbers
5. Solve showing all work/steps
C2 = A2 + B2
R2 = VB2 + VR2
VBS2 = VB2 + VR2
VBS2 = (15m/s)2 + (8m/s)2
VBS = √¯(225m2/s2) + (64m2/s2)
VBS = √¯289m2/s2
VBS = 17m/s
4.2 Components of Vectors
Choosing a Coordinate System
Draw Coordinate system with a vector.
The angle () tells us the direction of the
vector.
The direction of the vector is defined as
the angle that the vector measures
counterclockwise from the positive x-axis.
Components
The vector “A” can be resolved into two
component vectors.
Ax = parallel to the x-axis
Ay = parallel to the y-axis
A = Ax + Ay
Vector Resolution = the process of
breaking down a vector into its x & y
components.
Components = the magnitude and
sign of the component vectors.
Algebraic calculations only involve the
components of vectors not the
vectors themselves.
ADJ Ax
Ax = Acos
cos = HYP = A
OPP AY
AY = Asin
sin = HYP = A
EXAMPLE PROBLEM-1
A car travels 72km on a straight road at
25o. What are the x and y component
vectors?
Ax = 65.25km
AY = 30.43km
Example Problem - 2
A bus travels 37km North, then 57km
East. What is the displacement and
direction of the resultant.
Example Problem - 3
A runner travels at 15m/s west, then
13m/s south. What is the magnitude and
direction (expressed all three ways) of the
runner’s velocity.
Algebraic Addition of Vectors
opp
sin = hyp
adj
cos= hyp
opp
tan= adj
Two or more vectors (A, B, C, …) may be
added by:
1. resolving each vector into its x & y
components
2. add the x-components together to form
Rx = Ax + Bx + Cx + …
3. add the y-components together to form
R Y = A Y + B Y + CY + …
R = R x + Ry
Add 3 vectors Example
Vector Addition Algebraically
What is the magnitude and direction
(expressed all three ways) of the following
vectors?
1) 14km east
4) 14km north
2) 10km east
5) 8km south
3) 7km west
6) 15km south