physics 2048 spring 2008
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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.
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