CHAPTERS 3 & 4CHAPTERS 3 & 4

3.1 Picturing Motion3.1 Picturing Motion

Motion DiagramsMotion Diagrams

A series of consecutive frames (frame by A series of consecutive frames (frame by frame) of the motion of an object. Similar frame) of the motion of an object. Similar to movie film (30 frames per second).to movie film (30 frames per second).

The Particle ModelThe Particle Model

Motion diagram of wheel with two different Motion diagram of wheel with two different dots, center of wheel and edge of wheel.dots, center of wheel and edge of wheel.

3.2 Where & When?3.2 Where & When?

Coordinate SystemsCoordinate Systems

2 dimensional2 dimensional

AKA x-y coordinate systemAKA x-y coordinate system

Vectors and ScalarsVectors and Scalars

Scalar Quantity=Scalar Quantity=

A quantity that tells you only the A quantity that tells you only the magnitude (size/amount) of something. magnitude (size/amount) of something.

A number with units.A number with units.

Examples:Examples:

42kg, 10042kg, 100ooC, 40s, 10hr, $89, 100m/s, C, 40s, 10hr, $89, 100m/s,

Vector QuantityVector Quantity

Vector Quantity=Vector Quantity=

Magnitude with direction.Magnitude with direction.

Example:Example:

46km/hr North, 15m/s SW, 58km46km/hr North, 15m/s SW, 58km→→

Vectors are drawn to scale, the larger Vectors are drawn to scale, the larger the vector, the larger the magnitude.the vector, the larger the magnitude.

Examples on board.Examples on board.

VectorsVectors

Time Intervals and DisplacementTime Intervals and Displacement

Displacement (Displacement (ΔΔd) d) ==The distance and direction between two The distance and direction between two positions.positions.

ΔΔd = dd = dff – d – dii

ddff = final position = final position

ddii = initial position = initial position

ΔΔd can be positive/negatived can be positive/negativeExamples on number line.Examples on number line.

Time Interval (Time Interval (ΔΔt) t) ==

The time required for an object to The time required for an object to complete some displacement. complete some displacement.

ΔΔt = tt = tff – ti – ti

ttff = final time = final time

ttii = initial time (usually t = initial time (usually tii = 0) = 0)

ΔΔt is always positivet is always positive

3.3 Velocity and Acceleration3.3 Velocity and Acceleration

VelocityVelocity

Speed vs velocity, is there a difference or Speed vs velocity, is there a difference or are the terms interchangeable?are the terms interchangeable?

Speed examples:Speed examples:

57m/s, 37km/hr, 17cm/yr, 68mph57m/s, 37km/hr, 17cm/yr, 68mph

speed is a scalar quantity.speed is a scalar quantity.

Velocity =Velocity =

Speed with direction, a vector quantity.Speed with direction, a vector quantity.

Examples:Examples:

57ms East, 37km/hr SE, 57ms East, 37km/hr SE,

17 cm/yr17 cm/yr→, 68mph West→, 68mph West

67m/s @ 30067m/s @ 300oo

75m/s @ 3875m/s @ 38oo N of W N of W

Average velocityAverage velocity

_ _ ΔΔdd ddff - d - dii

Avg vel (v)= Avg vel (v)= ΔΔt = tt = tf f – t– tii

________________

avg vel = vector quantity (speed&direction)avg vel = vector quantity (speed&direction)

Examples on number line.Examples on number line.

The frog jumps from 0m to The frog jumps from 0m to 9m in 3 seconds, what is 9m in 3 seconds, what is the frog’s avg vel?the frog’s avg vel?

ΔΔdd df – didf – di

avg vel = avg vel = ΔΔt = tf – ti =t = tf – ti =

9m – 0m9m – 0m 9m9m

3s – 0s = 3s = 3m/s3s – 0s = 3s = 3m/s

Other ExamplesOther Examples

1. A runner begins at the starting line and 1. A runner begins at the starting line and crosses the 80m finish line in 4 seconds. crosses the 80m finish line in 4 seconds. What is the runner’s average velocity?What is the runner’s average velocity?

A = 20m/sA = 20m/s

2. A car travels from the school, 200km 2. A car travels from the school, 200km West in 5hr. What is the car’s average West in 5hr. What is the car’s average velocity?velocity?

A = -40km/hrA = -40km/hr

Instantaneous Velocity = ?Instantaneous Velocity = ?

The speed and direction of an object The speed and direction of an object at a particular “instant” in time, how at a particular “instant” in time, how fast it is moving fast it is moving right nowright now..

Examples of instantaneous velocity =Examples of instantaneous velocity =

Speedometer, radar gun, tachometerSpeedometer, radar gun, tachometer

Motion Diagram of Golf BallMotion Diagram of Golf Ball

Examples on Board:Examples on Board:

Golf putting right/left (+v, -v, )Golf putting right/left (+v, -v, )

EQUATIONSEQUATIONS

V = vV = vii + at + at v = vel (inst/final)v = vel (inst/final)

vvii = initial = initial

velocityvelocity

VV22 = v = vii22 + 2a + 2aΔΔdd ΔΔd = dd = dff - d - dii

ddff = d = dii + v + vΔΔtt

Acceleration (a) =Acceleration (a) =

The rate of change in velocity (The rate of change in velocity (ΔΔv).v).

ΔΔvv vvff - v - vii

a = a = ΔΔt = tt = tff - t - ti i

Example: A car pulls out from a stop Example: A car pulls out from a stop sign, 20s later it is traveling West at sign, 20s later it is traveling West at 40m/s. What is the car’s acceleration.40m/s. What is the car’s acceleration.

ΔΔvv vvff - v - vii - -40m/s - 0m/s40m/s - 0m/s

a = a = ΔΔt = tt = tff - t - tii = 20s - 0s = 20s - 0s

-40m/s-40m/s

a = 20s = -2m/sa = 20s = -2m/s22

Examples of acceleration Examples of acceleration Motion DiagramsMotion Diagrams

Car speeding up, then constant velocity, Car speeding up, then constant velocity, then slowing down.then slowing down.

AllAll problems problems allall year for year for every every chapter chapter MUSTMUST have the following or have the following or points will be deducted. points will be deducted. HOMEWORK HOMEWORK INCLUDED!INCLUDED!

1. A sketch/drawing or FBD.1. A sketch/drawing or FBD.

2. Table of Known/unknown values.2. Table of Known/unknown values.

3. Write the equation(s) to be used.3. Write the equation(s) to be used.

4. Plug in numbers4. Plug in numbers

5. Show 5. Show ALLALL steps/work. steps/work.

More EquationsMore Equations

vvff22 = v = vii

22 + 2a + 2aΔΔdd

vvff22 = v = vii

22 + 2a(d + 2a(dff - d - dii))

4.1 Properties of Vectors4.1 Properties of Vectors

Graphical representation =Graphical representation =

An arrow is drawn to scale and at the An arrow is drawn to scale and at the proper direction.proper direction.

The length of the arrow represents The length of the arrow represents the magnitude of the vector.the magnitude of the vector.

VECTOR EXAMPLESVECTOR EXAMPLES

Resultant Vector (R) =Resultant Vector (R) =

The sum of 2 or more vectors.The sum of 2 or more vectors.

R = vector 1 + vector 2 + vector 3 +…R = vector 1 + vector 2 + vector 3 +…

Resultant Vector ExampleResultant Vector Example

Examples on boardExamples on board

1. 2 equal vectors1. 2 equal vectors

2. 2 equal/opposite vectors2. 2 equal/opposite vectors

3. 2 vectors 903. 2 vectors 90oo apart apart

Resultant Vector ExampleResultant Vector Example

Resultant Vector ExampleResultant Vector Example

From SVHS to home various From SVHS to home various examples.examples.

No matter which route you take the No matter which route you take the displacement (displacement (ΔΔd) will be the same.d) will be the same.

Graphical Addition of VectorsGraphical Addition of Vectors

Vectors are drawn from tip to tailVectors are drawn from tip to tail

Resultant Vector (R) =Resultant Vector (R) =

The sum of two or more vectors.The sum of two or more vectors.

The order of adding the vectors does The order of adding the vectors does not matter, just like adding any other not matter, just like adding any other values.values.

Since the vectors are drawn to scale, Since the vectors are drawn to scale, the magnitude of the resultant ( R ) the magnitude of the resultant ( R ) can be measured with a ruler. can be measured with a ruler.

Resultant Vector ExampleResultant Vector Example

Resultant Vector ExampleResultant Vector Example

Resultant Vector ExampleResultant Vector Example

Example on board:Example on board:

2 different paths – use meter stick2 different paths – use meter stick

The resultant vectors (R), are equal, The resultant vectors (R), are equal, the path does not matter, when all the the path does not matter, when all the individual vectors are added together individual vectors are added together the resultants will be equal. the resultants will be equal.

What is the magnitude of R for the vectors What is the magnitude of R for the vectors below?below?

What is the magnitude of the R vector below?What is the magnitude of the R vector below?

What is the magnitude of the vectors below? The What is the magnitude of the vectors below? The red vector has a magnitude of 40, the purple red vector has a magnitude of 40, the purple vector 65, use the indicated angles.vector 65, use the indicated angles.

Can you use the Pythagorean Can you use the Pythagorean Theorem?Theorem?

NO!NO!

Why not?Why not?

Because the triangle is NOT a right Because the triangle is NOT a right triangle.triangle.

How can you solve for the resultant?How can you solve for the resultant?

You must use the LAW of Cosines.You must use the LAW of Cosines.

Law of Cosines EquationLaw of Cosines Equation

RR22 = = AA22 + B + B22 - 2ABcos - 2ABcosRR22 = 40 = 4022 + 65 + 6522 – 2(40)(65)(cos119 – 2(40)(65)(cos119oo))RR22 = 1600 + 4225 – 5200(-0.48409) = 1600 + 4225 – 5200(-0.48409)RR22 = 5825 + 2517.27 = 5825 + 2517.27RR22 = 8342.27 = 8342.27R = R = √¯8342.27√¯8342.27R = 91.37R = 91.37

Relative Velocities: Some Relative Velocities: Some ApplicationsApplications

You are in a bus traveling at a You are in a bus traveling at a velocity of 8m/s East. You walk velocity of 8m/s East. You walk towards the front of the bus at 3m/s, towards the front of the bus at 3m/s, what is your velocity relative to the what is your velocity relative to the street?street?

11m/s11m/s

You are in a bus traveling at a You are in a bus traveling at a velocity of 8m/s East. You walk velocity of 8m/s East. You walk towards the back of the bus at 3m/s, towards the back of the bus at 3m/s, what is your velocity relative to the what is your velocity relative to the street?street?

5m/s5m/s

A plane is traveling North at 800km/h, A plane is traveling North at 800km/h, the wind is blowing East at 150km/h. the wind is blowing East at 150km/h. What is the speed of the plane What is the speed of the plane relative to the ground?relative to the ground?

1. Draw sketch of vectors1. Draw sketch of vectors

2. List known/unknown2. List known/unknown

3. Write equation3. Write equation

4. Plug in numbers4. Plug in numbers

5. Solve showing 5. Solve showing allall work/steps work/steps

Solve ExampleSolve Example

Use Pythagorean Theorem AUse Pythagorean Theorem A22 + B + B2 2 = C= C22

VpgVpg22 = Vp = Vp22 + Vw + Vw22

Vpg = Vpg = √¯Vp√¯Vp22 + Vw + Vw22

Vpg = √¯(800km/h)Vpg = √¯(800km/h)22 + (150km/h) + (150km/h)22

Vpg = √¯640,000kmVpg = √¯640,000km22/h/h22 + 22,500km + 22,500km22/h/h22

Vpg = √¯662,500kmVpg = √¯662,500km22/h/h22

Vpg = 813.9km/hVpg = 813.9km/h

Boat/River ExampleBoat/River Example

A river flows South at 8m/s, a boat travels A river flows South at 8m/s, a boat travels due East at 15m/s. Where will the boat due East at 15m/s. Where will the boat end up and what will the boat’s speed be end up and what will the boat’s speed be relative to the shore?relative to the shore?1. Draw sketch of vectors1. Draw sketch of vectors2. List known/unknown2. List known/unknown3. Write equation3. Write equation4. Plug in numbers4. Plug in numbers5. Solve showing 5. Solve showing allall work/steps work/steps

CC22 = A = A22 + B + B22

RR22 = V = VBB22 + V + VRR

22

VVBSBS22 = V = VBB

2 2 + V+ VRR22

VVBSBS22 = (15m/s) = (15m/s)22 + (8m/s) + (8m/s)22

VVBSBS = = √¯(225m√¯(225m22/s/s22) + (64m) + (64m22/s/s22))

VVBSBS = √¯289m = √¯289m22/s/s22

VVBSBS = 17m/s = 17m/s

4.2 Components of Vectors4.2 Components of Vectors

Choosing a Coordinate SystemChoosing a Coordinate System

Draw Coordinate system with a vector.Draw Coordinate system with a vector.

The angle (The angle () tells us the direction of the ) tells us the direction of the vector.vector.

The direction of the vector is defined as The direction of the vector is defined as the angle that the vector measures the angle that the vector measures counterclockwise from the positive x-axis.counterclockwise from the positive x-axis.

ComponentsComponents

The vector “A” can be resolved into two The vector “A” can be resolved into two component vectors.component vectors.

AAxx = parallel to the x-axis = parallel to the x-axis

AAyy = parallel to the y-axis = parallel to the y-axis

A = AA = Axx + A + Ayy

Vector Resolution = the process of Vector Resolution = the process of breaking down a vector into its x & y breaking down a vector into its x & y components.components.

Components = the magnitude and Components = the magnitude and sign of the component vectors.sign of the component vectors.

AlgebraicAlgebraic calculations only involve calculations only involve the components of vectors the components of vectors notnot the the vectors themselves.vectors themselves.

ADJ ADJ AAxx

AAxx = Acos = Acos coscos = HYP = A = HYP = A

OPPOPP AAYY

AAYY = Asin = Asin sinsin = HYP = A = HYP = A

EXAMPLE PROBLEM-1EXAMPLE PROBLEM-1

A car travels 72km on a straight road at A car travels 72km on a straight road at 2525oo. What are the x and y component . What are the x and y component vectors?vectors?

AAxx = 65.25km A = 65.25km AYY = 30.43km = 30.43km

Example Problem - 2Example Problem - 2

A bus travels 37km North, then 57km A bus travels 37km North, then 57km East. What is the displacement and East. What is the displacement and direction of the resultant. 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 VectorsAlgebraic Addition of Vectors

oppoppsinsin = hyp = hyp

adjadjcoscos= hyp= hyp

oppopptantan= adj= adj

Two or more vectors (A, B, C, …) may be Two or more vectors (A, B, C, …) may be added by:added by:

1. resolving each vector into its x & y 1. resolving each vector into its x & y componentscomponents

2. add the x-components together to form 2. add the x-components together to form

RRxx = A = Axx + B + Bxx + C + Cxx + … + …

3. add the y-components together to form 3. add the y-components together to form

RRYY = A = AYY + B + BYY + C + CYY + … + …

R = RR = Rxx + R + Ryy

Add 3 vectors ExampleAdd 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