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Motion in Two Dimensions

Chapter 4Chapter 4

©© 2005, 2014 A. Dzyubenko2005, 2014 A. Dzyubenko

©© 2004 Brooks/Cole2004 Brooks/Cole

Phys 221Phys 221

adzyubenko@csub.eduadzyubenko@csub.edu

http://www.csub.edu/~adzyubenko

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Displacement as a VectorDisplacement as a Vector

� The displacement is

defined as the change

in object’s position

� The position of an

object is described by

its position vector, r

if rrr −=∆

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Average VelocityAverage Velocity

� The average velocity during the time interval

∆t is defined as the displacement divided by

the time interval

t∆

∆≡

rv

� is a vector quantity directed along ∆r

� Independent of path taken

� Depends only on the initial and final

position vectors

v

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Instantaneous VelocityInstantaneous Velocity

� The instantaneous velocity is defined

as the limit of the average velocity as

Dt approaches zero:

� The direction at any point is along the

line tangent to the path at that point

dt

d

tt

rrv =

∆

∆≡

→∆ 0lim

� The magnitude of the instantaneous velocity vector

v = |v| is called speed, which is a scalar quantity

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AccelerationAcceleration

� The average acceleration is

defined as the change in the

instantaneous velocity vector

divided by the time interval

during which that change

occurs:

� The instantaneous acceleration is defined as the limiting

value of the ratio ∆v/∆t as ∆t approaches zero:

ttt if

if

∆

∆=

−

−≡

vvva

dt

d

tt

vva =

∆

∆≡

→∆ 0lim

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Ways an Object Might AccelerateWays an Object Might Accelerate

� The magnitude of the velocity (the speed)

can change

� The direction of the velocity can change

� Even though the magnitude is constant

� Both the magnitude and the direction of the

velocity vector can change simultaneously

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Quick QuizQuick Quiz

� Consider the following controls in an

automobile: gas pedal, brake, steering wheel.

The controls in this list that cause

acceleration of the car are

(a) all three controls

(b) the gas pedal and the brakes

(c) only the brake

(d) only the gas pedal

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TwoTwo--Dimensional Motion with Dimensional Motion with

Constant AccelerationConstant Acceleration

� The motion of a particle moving in the xy

plane is described by the position vector r

jir ˆˆ yx +=

� The velocity of a particle is

jijir

v ˆˆˆˆyx vv

dt

dy

dt

dx

dt

d+=+==

constantˆˆ =+= jia yx aa

+=

+=

tavv

tavv

yyiyf

xxixf

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Velocity as a Function of TimeVelocity as a Function of Time

tif avv +=

� The velocity of a particle equals the vector sum

of its initial velocity vi and additional velocity at

acquired at time t as a result of constant acceleration

( ) ( )

( ) ( )jiji

jiv

ˆˆˆˆ

ˆˆ

yxyixi

yyixxif

aavv

tavtav

+++=

+++=

t

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Position Vector Position Vector

as a Function of Timeas a Function of Time

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2

1

2

1tatvyytatvxx yyiifxxiif ++=++=

( ) ( ) ( ) 2ˆˆ2

1ˆˆˆˆ taatvvyx yxyixiii jijiji +++++=

jir ˆ2

1ˆ2

1 22

+++

++= tatvytatvx yyiixxiif

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Position Vector as a Function Position Vector as a Function

of Time, contof Time, cont

� The position vector is the vector

sum of (i) original position ri,

(ii) a displacement vit arising from the initial velocity

and

(iii) a displacement ½ at2 resulting from the constant

acceleration of the particle

2

2

1ttiif avrr ++=

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Component Form of Kinetic EquationsComponent Form of Kinetic Equations

2

2

1ttiif avrr ++=

++=

++=

2

2

2

1

2

1

tatvyy

tatvxx

yyiif

xxiif

+=

+=

tavv

tavv

yyiyf

xxixftif avv +=

� Two-dimensional motion at a constant acceleration is

equivalent to two, x- and y- , independent motions

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Projectile MotionProjectile Motion

� The path of a projectile (its trajectory) is always a

parabola

� Projectile motion: an object may move in both

the x- and y- directions simultaneously

g

v

� Assumptions of projectile motion

� The free-fall acceleration g is

constant and is directed

downward

� The effect of air resistance is

negligible

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Rules of Projectile MotionRules of Projectile Motion

� The x- and y-directions of motion can be treated

independently

� The x-direction is uniform motion

�ax = 0

� The y-direction is free fall

�ay = - g

� The initial velocity can be broken down into its

x- and y-components

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Projectile MotionProjectile Motion

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Some Details about Rules of Some Details about Rules of

Projectile MotionProjectile Motion

� x-direction

� ax = 0

� xf ( )tvtv iixi θcos==

� This is the only operative equation in

the x-direction since the velocity is

uniform in that direction

constcos == iiv θ� vxi

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More Details About the RulesMore Details About the Rules

� y-direction

� ay = - g

� free fall problem

� take the positive direction as upward

� uniformly accelerated motion, so the kinetic

equations all hold

iiv θsin=� vyi

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Projectile Trajectory is a ParabolaProjectile Trajectory is a Parabola

� Solve the following equations simultaneously

( )

( )

−=+=

=

)2(2

1sin

2

1

)1(cos

22gttvtatvy

tvx

iiyyif

iif

θ

θ

ii

f

v

xt

θcos=� From (1): substitute in (2)

( ) 2

22cos2

tan xv

gxy

ii

i

−=

θθ

2bxaxy −=

the equation of a parabola

that passes through the origin

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Position Vector of Projectile as a Position Vector of Projectile as a

Function of TimeFunction of Time

� If there were no acceleration, g = 0, the projectile would

continue to move along a straight path in the direction of vi

2

2

1ttiif gvrr ++=

0=ir

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Quick QuizQuick Quiz

� As a projectile thrown upwards moves in its

parabolic path, at what point along its path

are the velocity and acceleration vectors for

the projectile perpendicular to each other?

(a) nowhere

(b) the highest point

(a) the launch point

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Maximum Height of a ProjectileMaximum Height of a Projectile

tif avv +=

Note that at maximum height h vy=0

g

vt

gtv

iiA

Aii

θ

θ

sin

sin0

=

−=

Substitute in the equation

for a motion in the y-direction

( )2

sin

2

1sinsin

−=

g

v

g

vvh iiii

ii

θθθ g

vh ii

2

sin 22θ

=

22

Horizontal Range of a ProjectileHorizontal Range of a Projectile

( )

( )

.cossin2

sin2cos

2cos

2

g

v

g

vv

tvtvR

iii

iiii

AiiBxi

θθ

θθ

θ

=

=

==

� R is the horizontal range

� Total time of flight: tB= 2tA

g

vR ii θ2sin

2

=

tA

tB

θθθ cossin22sin =

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Max Horizontal Range?Max Horizontal Range?

g

vR ii θ2sin

2

=

12sinwhen i

2

max

=

==

θ

g

vRR i

o45=iθ

??

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Angle Dependence of Horizontal Angle Dependence of Horizontal

RangeRange

� Complementary values of Θi (Θ 1+ Θ 2 = 90º) result

in the same value of a horizontal range R of the projectile

complementary

angles

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Quick Quiz Quick Quiz

� Rank the launch angles for the five paths with respect

to the time of flight , from the shortest time of flight

to the longest

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Some Variations of Projectile Some Variations of Projectile

MotionMotion

� Object may be fired

horizontally

� The initial velocity is all

in the x-direction

� vi = vx and vy = 0

� All the general rules of

projectile motion apply

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Some Variations cont. Some Variations cont.

� Follow the general rules

for projectile motion

� Notice the origin chosen:

the numerical value of yf

has a negative sign!

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Uniform Circular MotionUniform Circular Motion

� An object traveling in a

circle, even though it moves

with a constant speed, will

have an acceleration

� The centripetal acceleration

is due to the change in the

direction of the velocity

ttt if

if

∆

∆=

−

−=

vvva

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Centripetal AccelerationCentripetal Acceleration

� The acceleration in uniform circular

motion is always perpendicular to the

path and points toward the center of

the circle

tr

v

t ∆

∆=

∆

∆=

||||||

rva

r

vac

2

=

� Centripetal refers to “center-seeking”

rv

|||| rv ∆=

∆

� Period T is the time of

one complete revolution v

rT

π2≡

Angular Speed

v

rT

π2≡

� Period T is the time of

one complete revolution

� The angle for one complete

revolution is radians

� Angular speed:

π2

T

πω

2=

rad/sr

v

r

v==

ππω

22

rv ω=v

2

rac

= = rω2

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Quick QuizQuick Quiz

� Which of the following correctly describes the

centripetal acceleration vector for a particle moving

uniformly in a circular path?

(a) constant and always perpendicular

to the velocity vector for the particle

(b) constant and always parallel to the

velocity vector for the particle

(c) of constant magnitude and always

perpendicular to the velocity vector for the particle

(d) of constant magnitude and always parallel to the

velocity vector for the particle

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Motion Along an Arbitrary Motion Along an Arbitrary

Curved PathCurved Path

� The total acceleration vector changes in direction

and in magnitude from point to point

� Velocity changes both in direction and in

magnitude

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Motion Along an Arbitrary Motion Along an Arbitrary

Curved Path, contCurved Path, cont

� The total acceleration

vector a can be resolved

into two components :

� a radial component ar

along the radius of the

model circle

� a tangential component

at perpendicular to this

radius

tr aaa +=

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Tangential AccelerationTangential Acceleration

� The tangential acceleration component causes

the change in the speed of the particle

dt

dat

|| v=

� The direction of at :

� the same as v if v is increasing

�opposite v if v is decreasing

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Radial AccelerationRadial Acceleration

� The radial acceleration component arises from

the change in direction of the velocity vector

r

vaa cr

2

−=−=

� r is the radius of curvature of the path at the point at

question

� the negative sign: the centripetal acceleration is

opposite to the radial unit vector r

� At a given speed, ar is large when the radius of

curvature is small

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Total Acceleration in Total Acceleration in

Terms of Unit VectorsTerms of Unit Vectors

� Define unit vectors:

�r is lying along the radius vector

and directed radially outward from

the center of the circle

� Θ is tangent to the circle (a vector!)

rθv

aaa ˆˆ|| 2

r

v

dt

drt −=+=

22

rt aaa +=

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Quick QuizQuick Quiz

� A particle moves along a path and its

speed increases with time. In which of the

following cases are its acceleration and

velocity vectors perpendicular everywhere

along the path?

(a) the path is circular

(b) the path is straight

(c) the path is a parabola

(d) never

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Relativity of MotionRelativity of Motion

� How observations of motion made by different

observers in different frames of reference are

related each other

The man is walking on the moving beltway

the relative velocity of the

two frames of reference

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Relative VelocityRelative Velocity

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

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

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� Two observers moving relative to each other generally

do not agree on the outcome of a measurement

Relative VelocityRelative Velocity

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Galilean Coordinate TransformationGalilean Coordinate Transformation

� Reference frame S´ is

moving relative to S with

constant velocity v0

� At t = 0

the origins of the frames S

and S´ coincide in space

t0vrr −=′

or

t0vrr +′=

fixed frame

moving frame� At time t

� r is the particle position

vector relative to S

� r´ is the particle position

vector relative to S´

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Galilean Velocity TransformationGalilean Velocity Transformation

� Differentiate with respect to time

t0vrr −=′

0vr

−=′

dt

dr

dt

d

� The velocity v of a particle measured in a fixed frame

of reference S can be related to the velocity v´ of the

same particle measured in a moving frame S´ by

0vvv −=′ or0vvv +′=

where v0 is the velocity of S relative to S´

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Relative AccelerationRelative Acceleration

� The acceleration of the particle measured by an observer in one frame of reference is the same as measured by any other observer moving with constant velocity relative to the first frame

0vvv −=′

� Differentiate with respect to time

dt

d

dt

d

dt

d 0vvv−=

′

� Because v0 is constant 00 =dt

dvaa =′

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Solving Relative Velocity ProblemsSolving Relative Velocity Problems

� The pattern of subscripts can

be useful in solving relative

velocity problems

� Write an equation for the

velocity of interest in terms

of the velocities you know,

matching the pattern of

subscripts

rEbrbE vvv +=