Physics 2048 Spring 2008

Lecture #4 Chapter 4 motion in 2D and 3D

Chapter 4 – 2D and 3D MotionChapter 4 – 2D and 3D Motion

I.I. DefinitionsDefinitions

II.II. Projectile motionProjectile motion

III.III. Uniform circular motionUniform circular motion

IV.IV. Relative motionRelative motion

Position vector:Position vector: extends from the origin of a coordinate system to the particle. extends from the origin of a coordinate system to the particle.

)1.4(ˆˆˆ kzjyixr

)2.4(ˆ)(ˆ)(ˆ)( 12121212 kzzjyyixxrrr

I.I. DefinitionsDefinitions

Average velocity:

)3.4(ˆˆˆ kt

zj

t

yit

x

t

rvavg

Displacement vector: Displacement vector: represents a particle’s position change during a certainrepresents a particle’s position change during a certain time interval.time interval.

Instantaneous velocity:Instantaneous velocity:

)4.4(ˆˆˆˆˆˆ kdt

dzj

dt

dyi

dt

dx

dt

rdkvjvivv zyx

-The direction of the instantaneous velocity of a The direction of the instantaneous velocity of a particle is always tangent to the particle’s path at particle is always tangent to the particle’s path at the particle’s positionthe particle’s position

Instantaneous accelerationInstantaneous acceleration:

Average acceleration:Average acceleration:)5.4(12

t

v

t

vvaavg

)6.4(ˆˆˆˆˆˆ kdt

dvj

dt

dvi

dt

dv

dt

vdkajaiaa zyxzyx

II. Projectile motionII. Projectile motion

Motion of a particle launched with initial velocity, vMotion of a particle launched with initial velocity, v00 and free fall acceleration and free fall acceleration

g.g.

- - Horizontal motion:Horizontal motion: aaxx=0 =0 v vxx=v=v0x0x= cte= cte

- - Vertical motion:Vertical motion: aayy= -g = -g

Range (R):Range (R): horizontal distance traveled by a horizontal distance traveled by a projectile before returning to launch height.projectile before returning to launch height.

)7.4()cos( 0000 tvtvxx x

)8.4(2

1)sin(

2

1 200

200 gttvgttvyy y

)9.4(sin 00 gtvvy )10.4()(2)sin( 02

002 yygvvy

The horizontal and vertical motions are independent from each other.The horizontal and vertical motions are independent from each other.

- Trajectory: - Trajectory: projectile’s path.projectile’s path.

)11.4()cos(2

)(tan

cos2

1

cossin

cos)8.4()7.4(

200

2

0

2

000000

00

v

gxxy

v

xg

v

xvy

v

xt

000 yx

- Horizontal range: - Horizontal range: R = x-x R = x-x00; y-y; y-y00=0.=0.

)12.4(2sincossin2

cos2

1tan

cos2

1

cos)sin(

2

1)sin(0

cos)cos(

0

202

000

022

0

2

0

2

000000

200

0000

g

vv

gR

v

RgR

v

Rg

v

Rvgttv

v

RttvR

(Maximum for a launch angle of 45º )

Overall assumption:Overall assumption: the air through which the projectile moves has no effect the air through which the projectile moves has no effect on its motion on its motion friction neglected. friction neglected.

## ## In Galileo’s Two New Sciences, the author states that “for elevations (angles of projection) which exceed In Galileo’s Two New Sciences, the author states that “for elevations (angles of projection) which exceed or fall short of 45or fall short of 45º by equal amounts, the ranges are equal…” Prove this statement.º by equal amounts, the ranges are equal…” Prove this statement.

45

45

45

2

1

290sin452sin'

290sin452sin

20

20

20

20

g

v

g

vR

g

v

g

vR

x

v0

x=R=R’?

y

θ=45º

02sin: max0

20 hatdg

vRRange

bababa

bababa

sincoscossin)sin(

sincoscossin)sin(

)2cos()2sin(90cos)2cos(90sin'

)2cos()2sin(90cos)2cos(90sin

20

20

20

20

g

v

g

vR

g

v

g

vR

III. Uniform circular motionIII. Uniform circular motion

Motion around a circle at Motion around a circle at constant speedconstant speed..

)13.4(2

r

va

)14.4(2

v

rT

tancos

sintan

sincos

ˆsinˆcosˆˆˆˆ

ˆˆˆ)cos(ˆ)sin(ˆˆ

222

222

22

x

y

yx

xypp

ppyx

a

aradiusalongdirecteda

r

v

r

vaaa

jr

vi

r

vjv

r

viv

r

vj

dt

dx

r

vi

dt

dy

r

v

dt

vda

jr

xvi

r

yvjvivjvivv

- Period of revolution- Period of revolution:

- Acceleration: - Acceleration: centripetalcentripetal

v0y

v0x

-Velocity:-Velocity: tangent to circle in the direction of motion. tangent to circle in the direction of motion.

Magnitude of velocity and acceleration constant. Magnitude of velocity and acceleration constant. Direction varies continuouslyDirection varies continuously.

1- 1- A cat rides a merry-go-round while turning with uniform circular motion. At time tA cat rides a merry-go-round while turning with uniform circular motion. At time t11= 2s, the cat’s velocity = 2s, the cat’s velocity

is: vis: v11= (3m/s)i+(4m/s)j, measured on an horizontal xy coordinate system. At time t= (3m/s)i+(4m/s)j, measured on an horizontal xy coordinate system. At time t22=5s its velocity is:=5s its velocity is:

vv22= (-3m/s)i+(-4m/s)j. What are (a) the magnitude of the cat’s centripetal acceleration and (b) the cat’s = (-3m/s)i+(-4m/s)j. What are (a) the magnitude of the cat’s centripetal acceleration and (b) the cat’s

average acceleration during the time interval taverage acceleration during the time interval t22-t-t11??

v1

v2

x

y

In 3s the velocity is reversed In 3s the velocity is reversed the cat reaches the opposite the cat reaches the opposite side of the circleside of the circle

mrsm

rs

v

rT

smv

77.4/5

32

/543 22

2222

/23.577.4

/25sm

m

sm

r

vac

2

2212

/33.3

ˆ)/67.2(ˆ)/2(3

ˆ)/8(ˆ)/6(

sma

jsmisms

jsmism

t

vva

avg

avg

IV. Relative motionIV. Relative motion

1D

Particle’s velocity depends on reference frameParticle’s velocity depends on reference frame

)15.4(BAPBPA vvv

Frame moves at constant velocityFrame moves at constant velocity

)16.4()()()( PBPABAPBPA aavdt

dv

dt

dv

dt

d

0

Observers on different frames of reference measure the same accelerationObservers on different frames of reference measure the same accelerationfor a moving particle if their relative velocity is constantfor a moving particle if their relative velocity is constant.