Modeling MotionModeling MotionModeling MotionModeling Motion

The nature of motion appears to be the The nature of motion appears to be the question with which we begin.question with which we begin.

-- Socrates-- Socrates

The nature of motion appears to be the The nature of motion appears to be the question with which we begin.question with which we begin.

-- Socrates-- Socrates

Our universe is in a constant state of motion.Our universe is in a constant state of motion.Our universe is in a constant state of motion.Our universe is in a constant state of motion.

Motion is the most common “physical” event Motion is the most common “physical” event around us.around us.Motion is the most common “physical” event Motion is the most common “physical” event around us.around us.

Motion has been studied by mankind for millennia.Motion has been studied by mankind for millennia.Motion has been studied by mankind for millennia.Motion has been studied by mankind for millennia.

Our modern understanding of motion did Our modern understanding of motion did not begin until not begin until GalileoGalileo (1564-1642)(1564-1642) first first formulated the concepts of motion in formulated the concepts of motion in mathematical terms mathematical terms ((via experimentsvia experiments))..

Our modern understanding of motion did Our modern understanding of motion did not begin until not begin until GalileoGalileo (1564-1642)(1564-1642) first first formulated the concepts of motion in formulated the concepts of motion in mathematical terms mathematical terms ((via experimentsvia experiments))..

Our understanding climaxed for several Our understanding climaxed for several centuries after centuries after IsaacIsaac NewtonNewton (1642-(1642-1727)1727) & his newly developed & his newly developed Calculus put the concepts of motion Calculus put the concepts of motion on a firm and rigorous footing.on a firm and rigorous footing.

Our understanding climaxed for several Our understanding climaxed for several centuries after centuries after IsaacIsaac NewtonNewton (1642-(1642-1727)1727) & his newly developed & his newly developed Calculus put the concepts of motion Calculus put the concepts of motion on a firm and rigorous footing.on a firm and rigorous footing.

As we begin, we must clearly define what we mean by the As we begin, we must clearly define what we mean by the term term motionmotion..

MotionMotion – the change in a objects position with respect to – the change in a objects position with respect to timetime

NOTENOTE: Position must always be stated with respect to some : Position must always be stated with respect to some stationary object or reference point in order to be stationary object or reference point in order to be meaningful!meaningful!

((Else, measurements of the same event would yield different resultsElse, measurements of the same event would yield different results))

What would be some examples of a stationary reference What would be some examples of a stationary reference point for a car traveling down the highway?point for a car traveling down the highway?

As we begin, we must clearly define what we mean by the As we begin, we must clearly define what we mean by the term term motionmotion..

MotionMotion – the change in a objects position with respect to – the change in a objects position with respect to timetime

NOTENOTE: Position must always be stated with respect to some : Position must always be stated with respect to some stationary object or reference point in order to be stationary object or reference point in order to be meaningful!meaningful!

((Else, measurements of the same event would yield different resultsElse, measurements of the same event would yield different results))

What would be some examples of a stationary reference What would be some examples of a stationary reference point for a car traveling down the highway?point for a car traveling down the highway?

Consistent and accurate measurements require a Consistent and accurate measurements require a ““coordinate systemcoordinate system”.”.

Coordinate SystemCoordinate System – a artificial reference grid imposed – a artificial reference grid imposed on a system in order to make on a system in order to make

measurementsmeasurements– Our ChoiceOur Choice– No Right or Wrong WayNo Right or Wrong Way

ExEx. . An object changing position from An object changing position from AA to to BB

Consistent and accurate measurements require a Consistent and accurate measurements require a ““coordinate systemcoordinate system”.”.

Coordinate SystemCoordinate System – a artificial reference grid imposed – a artificial reference grid imposed on a system in order to make on a system in order to make

measurementsmeasurements– Our ChoiceOur Choice– No Right or Wrong WayNo Right or Wrong Way

ExEx. . An object changing position from An object changing position from AA to to BB

Choose the simplestChoose the simplestsystem possible.system possible.

Every measurement of motion requires a reference point Every measurement of motion requires a reference point ANDAND a suitable coordinate system. a suitable coordinate system.

Reference FrameReference Frame – a coordinate system that is – a coordinate system that is considered stationary with respect to the object that considered stationary with respect to the object that is in motion.is in motion.

Motion can be classified into 3 main categories.Motion can be classified into 3 main categories.

1.1. Translational MotionTranslational Motion2.2. Rotational MotionRotational Motion3.3. Periodic MotionPeriodic Motion

Every measurement of motion requires a reference point Every measurement of motion requires a reference point ANDAND a suitable coordinate system. a suitable coordinate system.

Reference FrameReference Frame – a coordinate system that is – a coordinate system that is considered stationary with respect to the object that considered stationary with respect to the object that is in motion.is in motion.

Motion can be classified into 3 main categories.Motion can be classified into 3 main categories.

1.1. Translational MotionTranslational Motion2.2. Rotational MotionRotational Motion3.3. Periodic MotionPeriodic Motion

Translational MotionTranslational MotionTranslational MotionTranslational Motion

• LinearLinear ( (straight-line or 1 dimensional motionstraight-line or 1 dimensional motion))

• ProjectileProjectile ( (arc’ed motionarc’ed motion))

NOTE:NOTE: When discussing translational motion, scientists often use the When discussing translational motion, scientists often use the term term TRAJECTORYTRAJECTORY to represent to represent the path an object moves alongthe path an object moves along..

• LinearLinear ( (straight-line or 1 dimensional motionstraight-line or 1 dimensional motion))

• ProjectileProjectile ( (arc’ed motionarc’ed motion))

NOTE:NOTE: When discussing translational motion, scientists often use the When discussing translational motion, scientists often use the term term TRAJECTORYTRAJECTORY to represent to represent the path an object moves alongthe path an object moves along..

Rotational MotionRotational MotionRotational MotionRotational Motion

• CircularCircular

• Non-CircularNon-Circular

• CircularCircular

• Non-CircularNon-Circular

Periodic MotionPeriodic MotionPeriodic MotionPeriodic Motion

• OscillationsOscillations

• VibrationsVibrations

• OscillationsOscillations

• VibrationsVibrations

Complex motion is a combination of 2 or more of Complex motion is a combination of 2 or more of the basic motion types.the basic motion types.Complex motion is a combination of 2 or more of Complex motion is a combination of 2 or more of the basic motion types.the basic motion types.

Ex.Ex.

Motion DiagramsMotion DiagramsMotion DiagramsMotion Diagrams

Film Strip ViewFilm Strip View

Composite View Composite View (Motion Diagram)(Motion Diagram)

Motion diagrams provide a way to Motion diagrams provide a way to visually interpret the change in an visually interpret the change in an objects motion with respect to time.objects motion with respect to time.

Motion Diagram ExamplesMotion Diagram ExamplesMotion Diagram ExamplesMotion Diagram Examples

No Motion No Motion (At Rest)(At Rest)

Rate of Motion is DecreasingRate of Motion is Decreasing

Rate of Motion is IncreasingRate of Motion is Increasing Constant Rate of MotionConstant Rate of Motion

Knowledge InventoryKnowledge InventoryKnowledge InventoryKnowledge Inventory

Cars Cars AA and and BB are traveling at different constant rates of motion. are traveling at different constant rates of motion.

Which car is going “faster” (Which car is going “faster” (has a higher rate of motionhas a higher rate of motion), ), AA or or BB??((Justify your reasoning – Justify your reasoning – Assume the time interval is the same for both carsAssume the time interval is the same for both cars))

Cars Cars AA and and BB are traveling at different constant rates of motion. are traveling at different constant rates of motion.

Which car is going “faster” (Which car is going “faster” (has a higher rate of motionhas a higher rate of motion), ), AA or or BB??((Justify your reasoning – Justify your reasoning – Assume the time interval is the same for both carsAssume the time interval is the same for both cars))

ANSWER: BANSWER: B

Scalars and VectorsScalars and VectorsScalar – a quantity with magnitude only

(just a number with units)

Vector – a quantity with magnitude and direction

Scalar – a quantity with magnitude only(just a number with units)

Vector – a quantity with magnitude and direction

What can we use to indicate direction?What can we use to indicate direction?NSEW, left/right/, up/down, +/- …NSEW, left/right/, up/down, +/- …

Ex.We are driving at 55 mph.

We are driving West at 55 mph.

Vectors are important because they can make good visual tools to simplify information when used in graphs.

Vectors are important because they can make good visual tools to simplify information when used in graphs.

Vector PropertiesVector Properties::

• Represented by bold faced letters or letters with arrows (carrots) over them

• Have magnitude and direction relative to a fixed origin

• Are additive (can be combined into 1 single vector)

• Can only add magnitudes that are in the same or opposite direction

Vector PropertiesVector Properties::

• Represented by bold faced letters or letters with arrows (carrots) over them

• Have magnitude and direction relative to a fixed origin

• Are additive (can be combined into 1 single vector)

• Can only add magnitudes that are in the same or opposite direction

resultant vector – sum of all the individual vectors

Ex. You start out driving E for 10 miles, then turn W for 7 miles. Where is your location relative to your original starting point?

(3 miles E) Ex.

You start out 10 miles W of OBU. You then walk 1 mile due N. How far W of OBU are you now?

(10 miles) NOTE: Only N/S can be added and Only E/W can be added. The resultant vectors for the previous examples can be expressed 2 ways: Method I (Algebraically)

Ex. 1: (3 miles) E

Ex. 2: (10 miles) W + (1 mile) N

Vectors can also be represented and added graphically without using ANYANY numbers at all!!!Vectors can also be represented and added graphically without using ANYANY numbers at all!!!

A vector can be represented graphically by:• Drawing an arrow to indicate its direction

• Altering the arrows length to indicate its magnitude

A vector can be represented graphically by:• Drawing an arrow to indicate its direction

• Altering the arrows length to indicate its magnitude

Vector Addition - Tail to Tip Method:

1.) Start by placing any vector with its tail at the origin (starting point) 2.) Place the tail of an unused vector to the tip of the previous vector 3.) After placing all the vectors in this fashion, the resultant vector is

found by drawing a straight line from the tail of the first vector to the tip of the last vector.

Ex.

Given the following vectors, find the resultant vector.

** NOTE: Order of placement (addition) does NOT matter!

Ex. Repeat the previous example but add the vectors in a different order to show that the resultant vector is the same.

.

.

Method II (Graphically)

Ex. 1: (3 miles) E

+ Order of Addition Doesn’t Matter + Ex. 2: (10 miles) W + (1 mile) N +

Order of Addition Doesn’t Matter +

ModelsModelsModelsModels

What is a Model?What is a Model?a simplified version of a real life physical system or event a simplified version of a real life physical system or event that would otherwise be too complicated to analyze in full that would otherwise be too complicated to analyze in full detaildetail

((approximation of realityapproximation of reality))

Why use models?Why use models? Reality is extremely complicated and complex.Reality is extremely complicated and complex.

Example: Model TrainsExample: Model Trains

Models cont.Models cont.

• What are the attributes of a What are the attributes of a GOODGOOD Model? Model?• Logically self-consistentLogically self-consistent

• Accurately predicts phenomena over a broad range of casesAccurately predicts phenomena over a broad range of cases

• Simple and elegant in designSimple and elegant in design

• What can be included in a model?What can be included in a model?

• Equations/symbolsEquations/symbols• WordsWords• DefinitionsDefinitions• AnalogiesAnalogies• DiagramsDiagrams• UnitsUnits• NumbersNumbers

Anything that accurately describes the Anything that accurately describes the phenomena & correctly represents or phenomena & correctly represents or explains the underlying physicsexplains the underlying physicsprinciple(s) involved.principle(s) involved.

Models cont.Models cont.

• How do you use/apply a model?How do you use/apply a model?a model can be applied to ANY system that meets the basic a model can be applied to ANY system that meets the basic requirements of the modelrequirements of the model

ExEx.. A model describing motion A model describing motion

GoodGood : Describing the motion of a car : Describing the motion of a car

BadBad: Describing how a burning candle releases heat: Describing how a burning candle releases heat ((you might could use it to model heat transferyou might could use it to model heat transfer))

• All models have limitations!All models have limitations!

What if there are several models proposed to describe the same event?What if there are several models proposed to describe the same event?

Occam’s Razor (1300’s)All things being equal, the simplest explanation tends to All things being equal, the simplest explanation tends to be the correct one.be the correct one.

This principle implies that one should not make more This principle implies that one should not make more assumptions than necessary.assumptions than necessary.

Ex. Analyzing the motion of a baseballEx. Analyzing the motion of a baseball

• RealityReality• Mass of the ballMass of the ball• Motion of the earthMotion of the earth• Rotation of the ballRotation of the ball• Atomic structure of the ballAtomic structure of the ball• Effects of gravityEffects of gravity• Biochemistry of the bodyBiochemistry of the body• Muscle behaviorMuscle behavior• Effects of the airEffects of the air• Initial speed of the ballInitial speed of the ball• Launch angleLaunch angle• Shape of the ballShape of the ball• …

ModelModel Mass of the ballMass of the ball Initial velocityInitial velocity GravityGravity Air resistance *Air resistance * Rotation *Rotation *

** Required for a more detailed look Required for a more detailed look at the motion of a baseballat the motion of a baseball

((i.e. curveball, slideri.e. curveball, slider…)…)

Modeling Translational MotionModeling Translational MotionModeling Translational MotionModeling Translational Motion

To describe the motion of a To describe the motion of a ‘solid or rigid’‘solid or rigid’ object, all that is object, all that is necessary is to track a single, fixed point on the object.necessary is to track a single, fixed point on the object.

Ex. Ex. Modeling the Motion of a Car that is Slowing DownModeling the Motion of a Car that is Slowing Down

To describe the motion of a To describe the motion of a ‘solid or rigid’‘solid or rigid’ object, all that is object, all that is necessary is to track a single, fixed point on the object.necessary is to track a single, fixed point on the object.

Ex. Ex. Modeling the Motion of a Car that is Slowing DownModeling the Motion of a Car that is Slowing Down

Match the motion diagram with its possible description.Match the motion diagram with its possible description.

A: A dust particle settling to the ground at a constant speedA: A dust particle settling to the ground at a constant speed

B:B: A ball dropped from the roof of a building A ball dropped from the roof of a building

C:C: A rocket descending slowly in order to make a soft landing A rocket descending slowly in order to make a soft landing

Match the motion diagram with its possible description.Match the motion diagram with its possible description.

A: A dust particle settling to the ground at a constant speedA: A dust particle settling to the ground at a constant speed

B:B: A ball dropped from the roof of a building A ball dropped from the roof of a building

C:C: A rocket descending slowly in order to make a soft landing A rocket descending slowly in order to make a soft landing

Knowledge InventoryKnowledge InventoryKnowledge InventoryKnowledge Inventory

BB AA CC

When modeling a rigid object as a single point, we also When modeling a rigid object as a single point, we also treat the object as if all of its mass were concentrated at treat the object as if all of its mass were concentrated at that point. This modeling trick is called the that point. This modeling trick is called the particle particle modelmodel..

Particle ModelParticle Model

A simplification in which the mass of an object is treated A simplification in which the mass of an object is treated as if all of it were concentrated at a single point.as if all of it were concentrated at a single point.

When modeling a rigid object as a single point, we also When modeling a rigid object as a single point, we also treat the object as if all of its mass were concentrated at treat the object as if all of its mass were concentrated at that point. This modeling trick is called the that point. This modeling trick is called the particle particle modelmodel..

Particle ModelParticle Model

A simplification in which the mass of an object is treated A simplification in which the mass of an object is treated as if all of it were concentrated at a single point.as if all of it were concentrated at a single point.

Limitations of the Particle ModelLimitations of the Particle Model Most effective when describing the translational motionMost effective when describing the translational motion of ‘rigid’ objects. of ‘rigid’ objects.

Possible ways to measure a Possible ways to measure a change in positionchange in position..

Distance vs. DisplacementDistance vs. Displacement

Possible ways to measure a Possible ways to measure a change in positionchange in position..

Distance vs. DisplacementDistance vs. Displacement

• DistanceDistance ( (scalarscalar))the total path length traveledthe total path length traveled

• DisplacementDisplacement ( (vectorvector))the net change in positionthe net change in position

• DistanceDistance ( (scalarscalar))the total path length traveledthe total path length traveled

• DisplacementDisplacement ( (vectorvector))the net change in positionthe net change in position

44 4444

44

distdist = 4 = 4 distdist = 16 = 16distdist = 8 = 8

4444 4444

dispdisp = 4 = 4 dispdisp = 0 = 0dispdisp = 4 = 422

Ex. Ex. There and Back AgainThere and Back Again

A person walks 70 m East and then walks back along the A person walks 70 m East and then walks back along the same same path 30 m West. What is the total path 30 m West. What is the total distancedistance traveled and the traveled and the net net displacementdisplacement relative to the starting point? relative to the starting point?

Ex. Ex. There and Back AgainThere and Back Again

A person walks 70 m East and then walks back along the A person walks 70 m East and then walks back along the same same path 30 m West. What is the total path 30 m West. What is the total distancedistance traveled and the traveled and the net net displacementdisplacement relative to the starting point? relative to the starting point?

70 70 mm E E

30 30 mm W W

Distance = Distance =

Displacement = Displacement =

40 40 mm E E

100 m100 m

40 m E40 m E

ExEx.. Road TripRoad Trip ((Part IPart I))

A college students road trip through several states. What is A college students road trip through several states. What is the the students total distance and displacement relative to OBUstudents total distance and displacement relative to OBU??

ExEx.. Road TripRoad Trip ((Part IPart I))

A college students road trip through several states. What is A college students road trip through several states. What is the the students total distance and displacement relative to OBUstudents total distance and displacement relative to OBU??

OBU – LR = 60 miOBU – LR = 60 mi

LR – OKC = 350 miLR – OKC = 350 mi

OKC – Big D = 200 miOKC – Big D = 200 mi

Big D – OBU = 240 miBig D – OBU = 240 mi

Distance =Distance =

Displacement =Displacement =

850 mi850 mi

0 mi0 mi

ExEx.. Road TripRoad Trip ((Part IIPart II))

What is the students total distance and displacement What is the students total distance and displacement relative to relative to OBU if they stop in DallasOBU if they stop in Dallas??

ExEx.. Road TripRoad Trip ((Part IIPart II))

What is the students total distance and displacement What is the students total distance and displacement relative to relative to OBU if they stop in DallasOBU if they stop in Dallas??

OBU – LR = 60 miOBU – LR = 60 mi

LR – OKC = 350 miLR – OKC = 350 mi

OKC – Big D = 200 miOKC – Big D = 200 mi

Big D – OBU = 240 miBig D – OBU = 240 mi

Distance =Distance =

Displacement =Displacement =

610 mi610 mi

240 mi240 mi

ExEx. . The LakeThe Lake

How many How many minimumminimum distances are there around a lake distances are there around a lake between two points? between two points?

How many displacements are there between two points?How many displacements are there between two points?

ExEx. . The LakeThe Lake

How many How many minimumminimum distances are there around a lake distances are there around a lake between two points? between two points?

How many displacements are there between two points?How many displacements are there between two points?

There are an There are an infinite infinite number of trajectories (distances) between twonumber of trajectories (distances) between twopoints, each with a potentially unique distance. But, there can only points, each with a potentially unique distance. But, there can only ever be ever be ONE ONE displacement!displacement!

AA

BB

Crater Lake

2211

SummarySummarySummarySummary

• Observations and attempts to describe motion have Observations and attempts to describe motion have been around for millennia.been around for millennia.

• To measure motion quantitatively, suitable To measure motion quantitatively, suitable reference reference framesframes are necessary. are necessary.

• The simplest way to model the motion of objects is to The simplest way to model the motion of objects is to treat them as treat them as point particlespoint particles in conjunction with visual in conjunction with visual aids called aids called motion diagramsmotion diagrams..

• DistanceDistance and and DisplacementDisplacement are two distinct ways of are two distinct ways of measuring changes in position.measuring changes in position.

• ModelsModels are simplified versions of reality used to predict are simplified versions of reality used to predict or describe the behavior of an object or event.or describe the behavior of an object or event.

• Observations and attempts to describe motion have Observations and attempts to describe motion have been around for millennia.been around for millennia.

• To measure motion quantitatively, suitable To measure motion quantitatively, suitable reference reference framesframes are necessary. are necessary.

• The simplest way to model the motion of objects is to The simplest way to model the motion of objects is to treat them as treat them as point particlespoint particles in conjunction with visual in conjunction with visual aids called aids called motion diagramsmotion diagrams..

• DistanceDistance and and DisplacementDisplacement are two distinct ways of are two distinct ways of measuring changes in position.measuring changes in position.

• ModelsModels are simplified versions of reality used to predict are simplified versions of reality used to predict or describe the behavior of an object or event.or describe the behavior of an object or event.

My dear brothers, take note My dear brothers, take note of this: Everyone should be of this: Everyone should be quick to listen, slow to speak quick to listen, slow to speak and slow to become angry, and slow to become angry, for man’s anger does not for man’s anger does not bring about the righteous bring about the righteous life that God desires.life that God desires.

James 1:19-20James 1:19-20

My dear brothers, take note My dear brothers, take note of this: Everyone should be of this: Everyone should be quick to listen, slow to speak quick to listen, slow to speak and slow to become angry, and slow to become angry, for man’s anger does not for man’s anger does not bring about the righteous bring about the righteous life that God desires.life that God desires.

James 1:19-20James 1:19-20