Chapter 3Chapter 3

Vectors and Vectors and

Two-Dimensional MotionTwo-Dimensional Motion

Vector NotationVector Notation

When handwritten, use an arrow:When handwritten, use an arrow: When printed, will be in bold print: When printed, will be in bold print:

AA When dealing with just the When dealing with just the

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

A

Properties of VectorsProperties of Vectors

Equality of Two VectorsEquality of Two Vectors Two vectors are Two vectors are equalequal if they have if they have

the same magnitude and the same the same magnitude and the same directiondirection

Movement of vectors in a diagramMovement of vectors in a diagram Any vector can be moved parallel to Any vector can be moved parallel to

itself without being affecteditself without being affected

More Properties of VectorsMore Properties of Vectors

Negative VectorsNegative Vectors Two vectors are Two vectors are negativenegative if they if they

have the same magnitude but are have the same magnitude but are 180° apart (opposite directions)180° apart (opposite directions)

AA = - = -BB

Resultant VectorResultant Vector The The resultantresultant vector is the sum of a vector is the sum of a

given set of vectorsgiven set of vectors

Adding VectorsAdding Vectors

When adding vectors, their When adding vectors, their directions must be taken into directions must be taken into accountaccount

Units must be the same Units must be the same Graphical MethodsGraphical Methods

Use scale drawingsUse scale drawings Algebraic MethodsAlgebraic Methods

More convenientMore convenient

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

Choose a scale Choose a scale Draw the first vector with the appropriate Draw the first vector with the appropriate

length and in the direction specified, with length and in the direction specified, with respect to a coordinate systemrespect to a coordinate system

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

Graphically Adding Graphically Adding Vectors, cont.Vectors, cont.

Continue drawing Continue drawing the vectors “tip-to-the vectors “tip-to-tail”tail”

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

Measure the length Measure the length of of RR and its angle and its angle Use the scale factor to Use the scale factor to

convert length to convert length to actual magnitudeactual magnitude

Graphically Adding Graphically Adding Vectors, cont.Vectors, cont.

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

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

Alternative Graphical Alternative Graphical MethodMethod

When you have only When you have only two vectors, you may two vectors, you may use the use the Parallelogram Parallelogram MethodMethod

All vectors, including All vectors, including the resultant, are the resultant, are drawn from a common drawn from a common originorigin The remaining sides of The remaining sides of

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

Notes about Vector Notes about Vector AdditionAddition

Vectors obey the Vectors obey the Commutative Commutative Law of AdditionLaw of Addition The order in which The order in which

the vectors are the vectors are added doesn’t added doesn’t affect the resultaffect the result

Vector SubtractionVector Subtraction

Special case of Special case of vector additionvector addition

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

Continue with Continue with standard vector standard vector addition addition procedureprocedure

Multiplying or Dividing a Multiplying or Dividing a Vector by a ScalarVector by a Scalar

The result of the multiplication or division The result of the multiplication or division is a vectoris a vector

The magnitude of the vector is multiplied The magnitude of the vector is multiplied or divided by the scalaror divided by the scalar

If the scalar is positive, the direction of the If the scalar is positive, the direction of the result is the same as of the original vectorresult is the same as of the original vector

If the scalar is negative, the direction of If the scalar is negative, the direction of the result is opposite that of the original the result is opposite that of the original vectorvector

Components of a VectorComponents of a Vector

A A componentcomponent is is a parta part

It is useful to use It is useful to use rectangular rectangular componentscomponents These are the These are the

projections of the projections of the vector along the vector along the x- and y-axesx- and y-axes

Components of a Vector, Components of a Vector, cont.cont.

The x-component of a vector is the The x-component of a vector is the projection along the x-axisprojection along the x-axis

The y-component of a vector is the The y-component of a vector is the projection along the y-axisprojection along the y-axis

Then, Then,

cosAxA

sinAA y

yx AA A

More About Components More About Components of a Vectorof a Vector

The previous equations are valid The previous equations are valid only if only if θ is θ is measured with respect to the x-axismeasured with respect to the x-axis

The components can be positive or negative The components can be positive or negative and will have the same units as the original and will have the same units as the original vectorvector

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

May still have to find θ with respect to the May still have to find θ with respect to the positive x-axispositive x-axis

x

y12y

2x A

AtanandAAA

Adding Vectors Adding Vectors AlgebraicallyAlgebraically

Choose a coordinate system and Choose a coordinate system and sketch the vectorssketch the vectors

Find the x- and y-components of all Find the x- and y-components of all the vectorsthe vectors

Add all the x-componentsAdd all the x-components This gives RThis gives Rxx:: xx vR

Adding Vectors Adding Vectors Algebraically, cont.Algebraically, cont.

Add all the y-componentsAdd all the y-components This gives RThis gives Ryy: :

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

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

yy vR

2y

2x RRR

x

y1

R

Rtan

Motion in Two DimensionsMotion in Two Dimensions

Using + or – signs is not always Using + or – signs is not always sufficient to fully describe motion sufficient to fully describe motion in more than one dimensionin more than one dimension Vectors can be used to more fully Vectors can be used to more fully

describe motiondescribe motion Still interested in displacement, Still interested in displacement,

velocity, and accelerationvelocity, and acceleration

DisplacementDisplacement The position of an The position of an

object is described object is described by its position by its position vector, vector, rr

The The displacementdisplacement of the object is of the object is defined as the defined as the change in its change in its positionposition ΔΔrr = = rrff - - rrii

VelocityVelocity

The average velocity is the ratio of the The average velocity is the ratio of the displacement to the time interval for the displacement to the time interval for the displacementdisplacement

The instantaneous velocity is the limit of The instantaneous velocity is the limit of the average velocity as the average velocity as Δt approaches Δt approaches zerozero The direction of the instantaneous velocity is The direction of the instantaneous velocity is

along a line that is tangent to the path of the along a line that is tangent to the path of the particle and in the direction of motionparticle and in the direction of motion

t

rv

AccelerationAcceleration

The average acceleration is The average acceleration is defined as the rate at which the defined as the rate at which the velocity changesvelocity changes

The instantaneous acceleration is The instantaneous acceleration is the limit of the average the limit of the average acceleration as acceleration as Δt approaches zeroΔt approaches zero

t

va

Ways an Object Might Ways an Object Might AccelerateAccelerate

The magnitude of the velocity (the The magnitude of the velocity (the speed) can changespeed) can change

The direction of the velocity can The direction of the velocity can changechange Even though the magnitude is Even though the magnitude is

constantconstant Both the magnitude and the Both the magnitude and the

direction can changedirection can change

Projectile MotionProjectile Motion

An object may move in both the x An object may move in both the x and y directions simultaneouslyand y directions simultaneously It moves in two dimensionsIt moves in two dimensions

The form of two dimensional The form of two dimensional motion we will deal with is called motion we will deal with is called projectile motionprojectile motion

Assumptions of Projectile Assumptions of Projectile MotionMotion

We may ignore air frictionWe may ignore air friction We may ignore the rotation of the We may ignore the rotation of the

earthearth With these assumptions, an object With these assumptions, an object

in projectile motion will follow a in projectile motion will follow a parabolic pathparabolic path

Rules of Projectile MotionRules of Projectile Motion

The x- and y-directions of motion The x- and y-directions of motion can be treated independentlycan be treated independently

The x-direction is uniform motionThe x-direction is uniform motion aaxx = 0 = 0

The y-direction is free fallThe y-direction is free fall aayy = -g = -g

The initial velocity can be broken The initial velocity can be broken down into its x- and y-componentsdown into its x- and y-components

Projectile MotionProjectile Motion

Some Details About the Some Details About the RulesRules

x-direction x-direction aax x = 0= 0 x = vx = vxoxott

This is the only operative equation in the This is the only operative equation in the x-direction since there is uniform velocity x-direction since there is uniform velocity in that directionin that direction

constantvcosvv xooxo

More Details About the More Details About the RulesRules

y-directiony-direction free fall problemfree fall problem

a = -ga = -g take the positive direction as upwardtake the positive direction as upward uniformly accelerated motion, so the uniformly accelerated motion, so the

motion equations all holdmotion equations all hold

ooyo sinvv

Velocity of the ProjectileVelocity of the Projectile

The velocity of the projectile at any The velocity of the projectile at any point of its motion is the vector point of its motion is the vector sum of its x and y components at sum of its x and y components at that pointthat point

x

y12y

2x v

vtanandvvv

Some Variations of Some Variations of Projectile MotionProjectile Motion

An object may be An object may be fired horizontallyfired horizontally

The initial velocity The initial velocity is all in the x-is all in the x-directiondirection vvoo = v = vxx and v and vyy = 0 = 0

All the general All the general rules of projectile rules of projectile motion applymotion apply

Non-Symmetrical Non-Symmetrical Projectile MotionProjectile Motion

Follow the Follow the general rules for general rules for projectile motionprojectile motion

Break the y-Break the y-direction into direction into partsparts up and downup and down symmetrical back symmetrical back

to initial height to initial height and then the rest and then the rest of the heightof the height

Relative VelocityRelative Velocity

It may be useful to use a moving It may be useful to use a moving frame of reference instead of a frame of reference instead of a stationary onestationary one

It is important to specify the frame of It is important to specify the frame of reference, since the motion may be reference, since the motion may be different in different frames of different in different frames of referencereference

There are no specific equations to There are no specific equations to learn to solve relative velocity learn to solve relative velocity problemsproblems

Solving Relative Velocity Solving Relative Velocity ProblemsProblems

The pattern of subscripts can be The pattern of subscripts can be useful in solving relative velocity useful in solving relative velocity problemsproblems

Write an equation for the velocity Write an equation for the velocity of interest in terms of the of interest in terms of the velocities you know, matching the velocities you know, matching the pattern of subscriptspattern of subscripts

bcabac vvv