1. introd, math riview, motion in 1-D - Direktori File...

IntroductionIntroduction

General Physics (PHY 2130)

• Syllabus and teaching strategy

• Physics• Physics• Introduction• Mathematical review

• trigonometry• vectors

• Motion in one dimension

http://www.physics.wayne.edu/~apetrov/PHY2130/

Chapter 1

Lecturer: Prof. Alexey A. Petrov, Room 260 Physics Building,

Phone: 313-577-2739, or 313-577-2720 (for messages)

e-mail: [email protected], Web: http://www.physics.wayne.edu/~apetrov/

Office Hours: Monday 5:00-6:00 PM, at Oakland center

Tuesday 2:00-3:00 PM, on main campus, Physics Building, Room 260, or by appointment.

Grading: Reading Quizzes 15%

Quiz section performance/Homework 15%Best Hour Exam 20%

Syllabus and teaching strategy

Best Hour Exam 20%Second Best Hour Exam 20%Final 30% PLUS: 5% online homework

Reading Quizzes: It is important for you to come to class prepared!

BONUS POINTS: Reading Summaries

Homework: The quiz sessions meet once a week; quizzes will count towards your grade.

BONUS POINTS: online homework http://webassign.net

Exams: There will be THREE (3) Hour Exams (only two will count) and one Final Exam.

Additional BONUS POINTS will be given out for class activity.

I. Physics: IntroductionI. Physics: Introduction

►►Fundamental ScienceFundamental Science�� foundation of other physical sciencesfoundation of other physical sciences

►►Divided into five major areasDivided into five major areas�� MechanicsMechanics�� MechanicsMechanics

�� ThermodynamicsThermodynamics

�� ElectromagnetismElectromagnetism

�� RelativityRelativity

�� Quantum MechanicsQuantum Mechanics

1. Measurements1. Measurements

►►Basis of Basis of testingtesting theories in sciencetheories in science

►►Need to have consistent Need to have consistent systems of unitssystems of units for for the measurementsthe measurements

►►UncertaintiesUncertainties are inherentare inherent►►UncertaintiesUncertainties are inherentare inherent

►►Need Need rules for dealing with the uncertaintiesrules for dealing with the uncertainties

Systems of MeasurementSystems of Measurement

►►Standardized systemsStandardized systems�� agreed upon by some authority, usually a agreed upon by some authority, usually a

governmental bodygovernmental body

►►SI SI ---- SystSystééme Internationalme International►►SI SI ---- SystSystééme Internationalme International�� agreed to in 1960 by an international committeeagreed to in 1960 by an international committee

�� main system used in this coursemain system used in this course

�� also called also called mksmks for the first letters in the units for the first letters in the units of the fundamental quantitiesof the fundamental quantities

Systems of MeasurementsSystems of Measurements

►►cgscgs ---- Gaussian systemGaussian system

�� named for the first letters of the units it uses for named for the first letters of the units it uses for fundamental quantitiesfundamental quantities

►►US CustomaryUS Customary►►US CustomaryUS Customary

�� everyday units (ft, etc.)everyday units (ft, etc.)

�� often uses weight, in pounds, instead of mass often uses weight, in pounds, instead of mass as a fundamental quantityas a fundamental quantity

Basic Quantities and Their DimensionBasic Quantities and Their Dimension

►►Length [L]Length [L]

►►Mass [M]Mass [M]

►►Time [T]Time [T]

Why do we need standards?

LengthLength

►►UnitsUnits

�� SI SI ---- meter, mmeter, m

�� cgs cgs ---- centimeter, cmcentimeter, cm

�� US Customary US Customary ---- foot, ftfoot, ft�� US Customary US Customary ---- foot, ftfoot, ft

►►Defined in terms of a meter Defined in terms of a meter ---- the distance the distance traveled by light in a vacuum during a given traveled by light in a vacuum during a given time (1/299 792 458 s)time (1/299 792 458 s)

MassMass

►►UnitsUnits

�� SI SI ---- kilogram, kgkilogram, kg

�� cgs cgs ---- gram, ggram, g

�� USC USC ---- slug, slugslug, slug�� USC USC ---- slug, slugslug, slug

►►Defined in terms of kilogram, based on a Defined in terms of kilogram, based on a specific Ptspecific Pt--Ir cylinder kept at the Ir cylinder kept at the International Bureau of StandardsInternational Bureau of Standards

Standard KilogramStandard Kilogram

Why is it hidden under two glass domes?

TimeTime

►►UnitsUnits

�� seconds, sseconds, s in all three systemsin all three systems

►►Defined in terms of the oscillation of Defined in terms of the oscillation of radiation from a cesium atom radiation from a cesium atom radiation from a cesium atom radiation from a cesium atom

(9 192 631 700 times frequency of light emitted)(9 192 631 700 times frequency of light emitted)

Time MeasurementsTime Measurements

US “Official” Atomic ClockUS “Official” Atomic Clock

2. Dimensional Analysis2. Dimensional Analysis

►►DimensionDimension denotes the denotes the physical naturephysical nature of a of a quantity quantity

►►Technique to Technique to check the correctnesscheck the correctness of an of an equationequationequationequation

►►Dimensions (length, mass, time, combinations) Dimensions (length, mass, time, combinations) can be treated as algebraic quantitiescan be treated as algebraic quantities�� add, subtract, multiply, divideadd, subtract, multiply, divide

�� quantities added/subtracted only if have same unitsquantities added/subtracted only if have same units

►►Both sides of equation must have the same Both sides of equation must have the same dimensionsdimensions

Dimensional AnalysisDimensional Analysis

►► Dimensions for commonly used quantitiesDimensions for commonly used quantities

Length L m (SI)Area L2 m2 (SI)Volume L3 m3 (SI) Volume L3 m3 (SI) Velocity (speed) L/T m/s (SI)Acceleration L/T2 m/s2 (SI)

�� Example of dimensional analysis Example of dimensional analysis

distance = velocity · timeL = (L/T) · T

3. Conversions3. Conversions

►►When When units are not consistentunits are not consistent, you may , you may need to need to convertconvert to appropriate onesto appropriate ones

►►Units can be treated like algebraic Units can be treated like algebraic ►►Units can be treated like algebraic Units can be treated like algebraic quantities that can quantities that can cancel each other outcancel each other out

1 mile = 1609 m = 1.609 km 1 ft = 0.3048 m = 30.48 cm1m = 39.37 in = 3.281 ft 1 in = 0.0254 m = 2.54 cm

Example 1Example 1. Scotch tape:. Scotch tape:

Example 2Example 2. Trip to Canada:. Trip to Canada:Legal freeway speed limit in Canada is 100 km/h.

What is it in miles/h?

h

miles

km

mile

h

km

h

km62

609.1

1100100 ≈⋅=

PrefixesPrefixes

►►Prefixes correspond to powers of 10Prefixes correspond to powers of 10

►►Each prefix has a specific name/abbreviationEach prefix has a specific name/abbreviation

Power Prefix Abbrev.Power Prefix Abbrev.

1015 peta P109 giga G106 mega M103 kilo k10-2 centi P10-3 milli m10-6 micro µ10-9 nano n

Distance from Earth to nearest star 40 PmMean radius of Earth 6 MmLength of a housefly 5 mmSize of living cells 10 µmSize of an atom 0.1 nm

Example: An aspirin tablet contains 325 mg of acetylsalicylic acid. Express this mass in grams.

Solution:Given:

m = 325 mg Recall that prefix “milli” implies 10-3, som = 325 mg

Find:

m (grams)=?

Recall that prefix “milli” implies 10 , so

4. Uncertainty in Measurements4. Uncertainty in Measurements

►►There is uncertainty in There is uncertainty in every measurementevery measurement, , this uncertainty carries over through the this uncertainty carries over through the calculationscalculations

�� need a technique to account for this uncertaintyneed a technique to account for this uncertainty�� need a technique to account for this uncertaintyneed a technique to account for this uncertainty

►►We will use rules for We will use rules for significant figuressignificant figures to to approximate the uncertainty in results of approximate the uncertainty in results of calculationscalculations

Significant FiguresSignificant Figures

►► A A significant figuresignificant figure is one that is is one that is reliably knownreliably known

►► All nonAll non--zero digits are significantzero digits are significant

►► Zeros are significant whenZeros are significant when

�� between other nonbetween other non--zero digitszero digits�� between other nonbetween other non--zero digitszero digits

�� after the decimal point and another significant figureafter the decimal point and another significant figure

�� can be clarified by using scientific notationcan be clarified by using scientific notation

4

4

4

1074000.10.17400

107400.1.17400

1074.117400

×=

×=×= 3 significant figures

5 significant figures


Operations with Significant FiguresOperations with Significant Figures

►►AccuracyAccuracy ---- number of significant figuresnumber of significant figures

►►When multiplying or dividing, round the result When multiplying or dividing, round the result to the same accuracy as the to the same accuracy as the leastleast accurate accurate

meter stick: cm1.0±Example:

to the same accuracy as the to the same accuracy as the leastleast accurate accurate measurementmeasurement

►►When adding or subtracting, round the result to When adding or subtracting, round the result to the the smallest numbersmallest number of decimal places of any of decimal places of any term in the sumterm in the sum

Example:135 m + 6.213 m = 141 m

rectangular plate: 4.5 cmby 7.3 cmarea: 32.85 cm2 33 cm2


Example:

Order of MagnitudeOrder of Magnitude►►Approximation based on a number of assumptionsApproximation based on a number of assumptions

�� may need to modify assumptions if more precise results may need to modify assumptions if more precise results are neededare needed

Question:McDonald’s sells about 250 million packages of fries every year. Placed back-to-back, how far would the fries reach?

►►Order of magnitude is the power of 10 that appliesOrder of magnitude is the power of 10 that applies

Example: John has 3 apples, Jane has 5 apples.Their numbers of apples are “of the same order of magnitude”

every year. Placed back-to-back, how far would the fries reach?

Solution:There are approximately 30 fries/package, thus:

(30 fries/package)(250 . 106 packages)(3 in./fry) ~ 2 . 1010 in ~ 5 . 108 m,which is greater then Earth-Moon distance (4 . 108 m)!

II. Math Review: II. Math Review: Coordinate SystemsCoordinate Systems

►►Used to describe the position of a point in Used to describe the position of a point in spacespace

►►Coordinate system (frame)Coordinate system (frame) consists ofconsists of

�� a fixed reference point called the a fixed reference point called the originorigin�� a fixed reference point called the a fixed reference point called the originorigin

�� specific specific axes with scales and labelsaxes with scales and labels

�� instructions on how to label a pointinstructions on how to label a point relative to relative to the origin and the axesthe origin and the axes

Types of Coordinate SystemsTypes of Coordinate Systems

►►Cartesian Cartesian

►►Plane polarPlane polar

Cartesian coordinate systemCartesian coordinate system

►► also called rectangular also called rectangular coordinate systemcoordinate system

►► xx-- and yand y-- axesaxes

►► points are labeled (x,y)points are labeled (x,y)►► points are labeled (x,y)points are labeled (x,y)

Plane polar coordinate systemPlane polar coordinate system

�� origin and reference origin and reference line are notedline are noted

�� point is distance r from point is distance r from the origin in the the origin in the direction of angle direction of angle θθ, , direction of angle direction of angle θθ, , ccw from reference lineccw from reference line

�� points are labeled (r,points are labeled (r,θθ))

II. Math Review: II. Math Review: TrigonometryTrigonometry

sinhypotenuse

sideadjacent

hypotenuse

sideopposite

=

=

θ

θ

cos

sin

sin

sideadjacent

sideopposite

hypotenuse

=

=

θ

θ

tan

cos

�� Pythagorean TheoremPythagorean Theorem

Example: how high is the building?Example: how high is the building?

Known: angle and one side

Slide 13

Fig. 1.7, p.14

Known: angle and one sideFind: another side

Key: tangent is defined via two sides!

mmdistheight

dist

buildingofheight

3.37)0.46)(0.39(tantan.

,.

tan

==×=

=

oα

αα

II. Math Review: II. Math Review: Scalar and Vector Scalar and Vector

QuantitiesQuantities►► ScalarScalar quantities are completely described by quantities are completely described by

magnitude only (magnitude only (temperature, lengthtemperature, length,…),…)

►►VectorVector quantities need both magnitude (size) and quantities need both magnitude (size) and direction to completely describe themdirection to completely describe themdirection to completely describe themdirection to completely describe them

((force, displacement, velocityforce, displacement, velocity,…),…)

�� Represented by an arrow, the Represented by an arrow, the lengthlength of the arrow of the arrow is is proportional to the magnitudeproportional to the magnitude of the vectorof the vector

�� Head of the arrow represents the directionHead of the arrow represents the direction

Vector NotationVector Notation

►►When When handwrittenhandwritten, use an arrow:, use an arrow:

►►When When printedprinted, will be in bold print: , will be in bold print: AA

►►When dealing with just the magnitude of a When dealing with just the magnitude of a vector in print, an italic letter will be used: vector in print, an italic letter will be used: AA

Ar

vector in print, an italic letter will be used: vector in print, an italic letter will be used: AA

Properties of VectorsProperties of Vectors

►►Equality of Two VectorsEquality of Two Vectors

�� Two vectors are Two vectors are equalequal if they have the if they have the same same magnitudemagnitude and the and the same directionsame direction

►►Movement of vectors in a diagramMovement of vectors in a diagram►►Movement of vectors in a diagramMovement of vectors in a diagram

�� Any vector can be moved Any vector can be moved parallel to itselfparallel to itselfwithout being affectedwithout being affected

More Properties of VectorsMore Properties of Vectors

►►Negative VectorsNegative Vectors

�� Two vectors are Two vectors are negativenegative if they have the if they have the same magnitude but are 180same magnitude but are 180°° apart (opposite apart (opposite directions)directions)directions)directions)►► AA = = --BB

►►Resultant VectorResultant Vector

�� The The resultantresultant vector is the sum of a given set vector is the sum of a given set of vectorsof vectors

Adding VectorsAdding Vectors

►►When adding vectors, When adding vectors, their directions must their directions must be taken into accountbe taken into account

►►Units must be the sameUnits must be the same

►►Graphical MethodsGraphical Methods►►Graphical MethodsGraphical Methods

�� Use scale drawingsUse scale drawings

►►Algebraic MethodsAlgebraic Methods

�� More convenientMore convenient

Adding Vectors Graphically Adding Vectors Graphically (Triangle or Polygon Method)(Triangle or Polygon Method)

►►Choose a scale Choose a scale

►►Draw the first vector with the appropriate length Draw the first vector with the appropriate length and in the direction specified, with respect to a and in the direction specified, with respect to a coordinate systemcoordinate systemcoordinate systemcoordinate system

►►Draw the next vector with the appropriate length Draw the next vector with the appropriate length and in the direction specified, with respect to a and in the direction specified, with respect to a coordinate system whose origin is the end of coordinate system whose origin is the end of vector vector AA and parallel to the coordinate system and parallel to the coordinate system used for used for AA

Graphically Adding VectorsGraphically Adding Vectors

►► Continue drawing the Continue drawing the vectors “tipvectors “tip--toto--tail”tail”

►► The resultant is drawn The resultant is drawn from the origin of from the origin of AA to the to the end of the last vectorend of the last vector

Measure the length of Measure the length of RR►► Measure the length of Measure the length of RRand its angleand its angle�� Use the scale factor to Use the scale factor to

convert length to actual convert length to actual magnitudemagnitude

Graphically Adding VectorsGraphically Adding Vectors

►►When you have many When you have many vectors, just keep vectors, just keep repeating the process repeating the process until all are includeduntil all are included

►►The resultant is still The resultant is still ►►The resultant is still The resultant is still drawn from the origin drawn from the origin of the first vector to of the first vector to the end of the last the end of the last vectorvector

Alternative Graphical MethodAlternative Graphical Method

►► When you have only two When you have only two vectors, you may use the vectors, you may use the Parallelogram MethodParallelogram Method

►► All vectors, including the All vectors, including the resultant, are drawn from resultant, are drawn from resultant, are drawn from resultant, are drawn from a common origina common origin�� The remaining sides of the The remaining sides of the

parallelogram are sketched parallelogram are sketched to determine the diagonal, to determine the diagonal, RR

Notes about Vector AdditionNotes about Vector Addition

►►Vectors obey the Vectors obey the Commutative Law Commutative Law of Additionof Addition

�� The order in which the The order in which the vectors are added vectors are added vectors are added vectors are added doesn’t affect the resultdoesn’t affect the result

Vector SubtractionVector Subtraction

►► Special case of vector Special case of vector additionaddition

►► If If AA –– BB, then use , then use AA+(+(--BB))AA+(+(--BB))

►►Continue with standard Continue with standard vector addition vector addition procedureprocedure

Multiplying or Dividing a Vector Multiplying or Dividing a Vector by a Scalarby a Scalar

►► The The resultresult of the multiplication or division is a of the multiplication or division is a vectorvector

►► The The magnitudemagnitude of the vector is multiplied or divided by the of the vector is multiplied or divided by the scalarscalar

►► If the scalar is If the scalar is positivepositive, the , the directiondirection of the result is the of the result is the ►► If the scalar is If the scalar is positivepositive, the , the directiondirection of the result is the of the result is the samesame as of the original vectoras of the original vector

►► If the scalar is If the scalar is negativenegative, the , the directiondirection of the result is of the result is oppositeopposite that of the original vectorthat of the original vector

Components of a VectorComponents of a Vector

►►A A componentcomponent is a is a partpart

►► It is useful to use It is useful to use rectangular rectangular rectangular rectangular componentscomponents

�� These are the These are the projections of the projections of the vector along the xvector along the x-- and and yy--axesaxes

Components of a VectorComponents of a Vector

►►The xThe x--component of a vector is the component of a vector is the projection along the xprojection along the x--axisaxis

►►The yThe y--component of a vector is the component of a vector is the cosxA A θ=

►►The yThe y--component of a vector is the component of a vector is the projection along the yprojection along the y--axisaxis

►►Then, Then, sinyA A θ=

x yA A= +Aur ur ur

More About Components of a More About Components of a VectorVector

►►The previous equations are valid The previous equations are valid only if only if θ is θ is measured with respect to the xmeasured with respect to the x--axisaxis

►►The components can be positive or negative and The components can be positive or negative and will have the same units as the original vectorwill have the same units as the original vector

►►The components are the legs of the right triangle The components are the legs of the right triangle ►►The components are the legs of the right triangle The components are the legs of the right triangle whose hypotenuse is whose hypotenuse is AA

�� May still have to find θ with respect to the positive xMay still have to find θ with respect to the positive x--axisaxis

x

y12y

2x A

AtanandAAA −=θ+=

Adding Vectors AlgebraicallyAdding Vectors Algebraically

►►Choose a coordinate system and sketch the Choose a coordinate system and sketch the vectorsvectors

►►Find the xFind the x-- and yand y--components of all the components of all the vectorsvectorsvectorsvectors

►►Add all the xAdd all the x--componentscomponents

�� This gives RThis gives Rxx::

∑= xx vR

Adding Vectors AlgebraicallyAdding Vectors Algebraically

►►Add all the yAdd all the y--componentscomponents

�� This gives RThis gives Ryy: :

►►Use the Pythagorean Theorem to find the Use the Pythagorean Theorem to find the magnitude of the Resultant:magnitude of the Resultant:

∑= yy vR

22 RRR +=magnitude of the Resultant:magnitude of the Resultant:

►►Use the inverse tangent function to find the Use the inverse tangent function to find the direction of R:direction of R:

2y

2x RRR +=

x

y1

R

Rtan−=θ

III. Problem Solving StrategyIII. Problem Solving Strategy

Slide 13

Fig. 1.7, p.14

Known: angle and one sideFind: another sideKey: tangent is defined via two sides!

mmdistheight

dist

buildingofheight

3.37)0.46)(0.39(tantan.

,.

tan

==×=

=

oα

α

Problem Solving StrategyProblem Solving Strategy

►►Read the problemRead the problem

�� identify type of problem, principle involvedidentify type of problem, principle involved

►►Draw a diagramDraw a diagram

�� include appropriate values and coordinate include appropriate values and coordinate �� include appropriate values and coordinate include appropriate values and coordinate systemsystem

�� some types of problems require very specific some types of problems require very specific types of diagramstypes of diagrams

Problem Solving cont.Problem Solving cont.

►►Visualize the problemVisualize the problem

►►Identify informationIdentify information

�� identify the principle involvedidentify the principle involved

�� list the data (given information)list the data (given information)�� list the data (given information)list the data (given information)

�� indicate the unknown (what you are looking for)indicate the unknown (what you are looking for)

Problem Solving, cont.Problem Solving, cont.

►►Choose equation(s)Choose equation(s)

�� based on the principle, choose an equation or based on the principle, choose an equation or set of equations to apply to the problemset of equations to apply to the problem

�� solve for the unknownsolve for the unknown�� solve for the unknownsolve for the unknown

►►Solve the equation(s)Solve the equation(s)

�� substitute the data into the equationsubstitute the data into the equation

�� include unitsinclude units

Problem Solving, finalProblem Solving, final

►► Evaluate the answerEvaluate the answer�� find the numerical resultfind the numerical result

�� determine the units of the resultdetermine the units of the result

►►Check the answerCheck the answer►►Check the answerCheck the answer�� are the units correct for the quantity being found?are the units correct for the quantity being found?

�� does the answer seem reasonable? does the answer seem reasonable? ►►check order of magnitudecheck order of magnitude

�� are signs appropriate and meaningful?are signs appropriate and meaningful?

IV. Motion in One DimensionIV. Motion in One Dimension

DynamicsDynamics

►►The branch of physics involving the motion The branch of physics involving the motion of an object and the relationship between of an object and the relationship between that motion and other physics conceptsthat motion and other physics concepts

►►KinematicsKinematics is a part of dynamicsis a part of dynamics►►KinematicsKinematics is a part of dynamicsis a part of dynamics�� In kinematics, you are interested in the In kinematics, you are interested in the

descriptiondescription of motionof motion

�� NotNot concerned with the cause of the motionconcerned with the cause of the motion

Position and DisplacementPosition and Displacement

►► PositionPosition is defined in terms is defined in terms of a of a frame of referenceframe of reference

Frame A: Frame A: xxii>0 >0 and and xxff>0 >0

A

Frame B: Frame B: x’x’ii<0 <0 butbut x’x’ff>0 >0

►►One dimensional, so One dimensional, so generally the generally the xx-- or yor y--axisaxis

By’

x’O’xi’ xf’

Position and DisplacementPosition and Displacement

►► PositionPosition is defined in terms is defined in terms of a of a frame of referenceframe of reference

�� One dimensional, so One dimensional, so generally the generally the xx-- or yor y--axisaxis

►►DisplacementDisplacement measures the measures the ►►DisplacementDisplacement measures the measures the change in positionchange in position�� Represented as Represented as ∆∆xx (if (if

horizontal) or horizontal) or ∆∆yy (if vertical)(if vertical)

�� Vector quantityVector quantity►►+ or + or -- is generally sufficient to is generally sufficient to

indicate direction for indicate direction for oneone--dimensional motiondimensional motion

UnitsUnits

SISI Meters (m)Meters (m)

CGSCGS Centimeters (cm)Centimeters (cm)

US CustUS Cust Feet (ft)Feet (ft)

DisplacementDisplacement(example)(example) �� DisplacementDisplacement measures measures

the the change in positionchange in position

�� represented as represented as ∆∆xx or or ∆∆yy

mm

xxx if

10801

−=

−=∆

m

mm

70

1080

+=−=�

m

mm

xxx if

60

80202

−=−=

−=∆

�

Distance or Displacement?Distance or Displacement?

►►Distance may be, but is not necessarily, the Distance may be, but is not necessarily, the magnitude of the displacementmagnitude of the displacement

Distance(blue line)

Displacement(yellow line)

PositionPosition--time graphstime graphs

� Note:position-time graph is not necessarily a straight line, eventhough the motion is along x-direction

ConcepTest 1ConcepTest 1

An object (say, car) goes from one point in space to another. After it arrives to its destination, itsdisplacement is

1. either greater than or equal to 2. always greater than3. always equal to4. either smaller or equal to5. either smaller or larger

than the distance it traveled.

Please fill your answer as question 1ofGeneral Purpose Answer Sheet





Please fill your answer as question 2 ofGeneral Purpose Answer Sheet

ConcepTest 1 (answer)ConcepTest 1 (answer)




�

Note: displacement is a vector from the final to initial points, distance is total path traversed

Average VelocityAverage Velocity

►► It takes time for an object to undergo a It takes time for an object to undergo a displacementdisplacement

►►The The average velocityaverage velocity is is raterate at which the at which the displacement occursdisplacement occurs

xxx if −=∆=

rrrr

►► It is a It is a vectorvector, , directiondirection will be will be the same asthe same as the the direction of the direction of the displacementdisplacement ((∆∆tt is always positive)is always positive)�� + or + or -- is sufficient for oneis sufficient for one--dimensional motiondimensional motion

t

xx

t

xv if

average ∆−

=∆∆=r

More About Average VelocityMore About Average Velocity

►►Units of velocity:Units of velocity:

UnitsUnits

SISI Meters per second (m/s)Meters per second (m/s)

►►Note:Note: other units may be given in a problem, other units may be given in a problem, but generally will need to be converted to thesebut generally will need to be converted to these

SISI Meters per second (m/s)Meters per second (m/s)

CGSCGS Centimeters per second (cm/s)Centimeters per second (cm/s)

US CustomaryUS Customary Feet per second (ft/s)Feet per second (ft/s)

Example:Example:

s

m

t

xv average 10

7011

+=∆∆=r

r

Suppose that in both cases truck covers the distance in 10 seconds:

smst

710

+=∆

sms

m

t

xv average

610

6022

−=

−=∆

∆=r

r

SpeedSpeed

►►Speed is a Speed is a scalarscalar quantityquantity

�� same units as velocitysame units as velocity

�� speed = total distance / total timespeed = total distance / total time

►►May be, but is not necessarily, the May be, but is not necessarily, the ►►May be, but is not necessarily, the May be, but is not necessarily, the magnitude of the velocitymagnitude of the velocity

Graphical Interpretation of Average VelocityGraphical Interpretation of Average Velocity

►►Velocity can be determined from a positionVelocity can be determined from a position--time graphtime graph

s

m

t

xvaverage 0.3

40+=∆∆=r

r

►►Average velocityAverage velocity equals the equals the slopeslope of the line of the line joining the initial and final positionsjoining the initial and final positions

smst

130.3

+=∆

Instantaneous VelocityInstantaneous Velocity

►► Instantaneous velocityInstantaneous velocity is defined as the is defined as the limit limit of the average velocityof the average velocity as the time interval as the time interval becomes infinitesimally short, or as the time becomes infinitesimally short, or as the time interval approaches zerointerval approaches zerointerval approaches zerointerval approaches zero

►►The instantaneous velocity indicates what is The instantaneous velocity indicates what is happening at every point of timehappening at every point of time

t

xx

t

xv if

ttinst ∆

−=

∆∆=

→∆→∆

rrrr

00limlim

Uniform VelocityUniform Velocity

►►UniformUniform velocity is velocity is constantconstant velocityvelocity

►►The instantaneous velocities are always the The instantaneous velocities are always the same same

�� All the instantaneous velocities will also equal All the instantaneous velocities will also equal �� All the instantaneous velocities will also equal All the instantaneous velocities will also equal the average velocitythe average velocity

Graphical Interpretation of Instantaneous Graphical Interpretation of Instantaneous VelocityVelocity

►► Instantaneous velocityInstantaneous velocity is the is the slopeslope of the of the tangenttangent to the curve at the time of interestto the curve at the time of interest

►►The The instantaneous speedinstantaneous speed is the magnitude of is the magnitude of the instantaneous velocitythe instantaneous velocity

Average vs Instantaneous VelocityAverage vs Instantaneous Velocity

Average velocity Instantaneous velocity


The graph shows position as a function of timefor two trains running on parallel tracks. Which of the following is true:

1. at time tB both trains have the same velocity 1. at time tB both trains have the same velocity 2. both trains speed up all the time3. both trains have the same velocity at some time before tB4. train A is longer than train B5. all of the above statements are true

Please fill your answer as question 3ofGeneral Purpose Answer Sheet

A

B

time

position

tB




Please fill your answer as question 4 ofGeneral Purpose Answer Sheet

A

B

time

position

tB

ConcepTest 2 (answer)ConcepTest 2 (answer)



Note:the slopeof curve B is parallelto line A at some point t< tB

A

B

time

position

tB

Average AccelerationAverage Acceleration

►►Changing velocity (nonChanging velocity (non--uniform) means an uniform) means an acceleration is presentacceleration is present

►►Average accelerationAverage acceleration is the is the rate of change of rate of change of the velocitythe velocity

►►Average acceleration is a Average acceleration is a vectorvector quantityquantity

t

vv

t

va if

average ∆−

=∆∆=

rrrr

Average AccelerationAverage Acceleration

►►When the When the signsign of the of the velocityvelocity and the and the accelerationacceleration are the are the samesame (either positive or (either positive or negative), then negative), then the speed is increasingthe speed is increasing

►►When the When the signsign of the of the velocityvelocity and the and the ►►When the When the signsign of the of the velocityvelocity and the and the accelerationacceleration are are oppositeopposite, , the speed is the speed is decreasingdecreasing

UnitsUnits

SISI Meters per second squared (m/sMeters per second squared (m/s22))

CGSCGS Centimeters per second squared (cm/sCentimeters per second squared (cm/s22))

US CustomaryUS Customary Feet per second squared (ft/sFeet per second squared (ft/s22))

Instantaneous and Uniform Instantaneous and Uniform AccelerationAcceleration

►► Instantaneous accelerationInstantaneous acceleration is the is the limitlimit of the of the average acceleration as the time interval goes average acceleration as the time interval goes to zeroto zero

vv −∆rrr

►►When the instantaneous accelerations are When the instantaneous accelerations are always the same, the acceleration will be always the same, the acceleration will be uniformuniform�� The instantaneous accelerations will all be equal The instantaneous accelerations will all be equal

to the average accelerationto the average acceleration

t

vv

t

va if

ttinst ∆

−=

∆∆=

→∆→∆

rrrr

00limlim

Graphical Interpretation of Graphical Interpretation of AccelerationAcceleration

►► Average accelerationAverage acceleration is the is the slopeslope of the line connecting of the line connecting the the initial and final velocitiesinitial and final velocitieson a velocityon a velocity--time graphtime graphon a velocityon a velocity--time graphtime graph

►► Instantaneous accelerationInstantaneous accelerationis the is the slopeslope of the of the tangenttangentto the curve of the velocityto the curve of the velocity--time graphtime graph

Example 1: Motion DiagramsExample 1: Motion Diagrams

►►Uniform velocityUniform velocity (shown by red arrows (shown by red arrows maintaining the same size)maintaining the same size)

►►Acceleration equals zeroAcceleration equals zero

Example 2:Example 2:

►► Velocity and acceleration are in the Velocity and acceleration are in the same directionsame direction

►► Acceleration is uniform (blue arrows maintain the same Acceleration is uniform (blue arrows maintain the same length)length)

►► Velocity is increasing (red arrows are getting longer)Velocity is increasing (red arrows are getting longer)

Example 3:Example 3:

►► Acceleration and velocity are in Acceleration and velocity are in opposite directionsopposite directions

►► Acceleration is uniform (blue arrows maintain the same Acceleration is uniform (blue arrows maintain the same length)length)

►► Velocity is decreasing (red arrows are getting shorter)Velocity is decreasing (red arrows are getting shorter)

1. introd, math riview, motion in 1-D - Direktori File...

Documents

Transcript of 1. introd, math riview, motion in 1-D - Direktori File...