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1. Class 9 – Review of motion in 2D and 3DClass 9 – Review of motion in 2D and 3D
Chapter 4 - Monday September 13thChapter 4 - Monday September 13th
•Exam instructions
•Brief review
•Example problems
Reading: pages 58 thru 75 (chapter 4) in HRWReading: pages 58 thru 75 (chapter 4) in HRW
Read and understand the sample problemsRead and understand the sample problems
•Exam tonight: 8:20-10:20pmExam tonight: 8:20-10:20pm

2. ˆ ˆ ˆi j kr x y z= + +
r
2 1
ˆ ˆ ˆi j kr r r x y z∆ = − = ∆ + ∆ + ∆
r r r
ˆ ˆ ˆi j k
dr dx dy dz
v
dt dt dt dt
= = + +
r
r
ˆ ˆ ˆi j kyx z
dvdv dv dv
a
dt dt dt dt
= = + +
r
r
Two-dimensional kinematicsTwo-dimensional kinematics
Acceleration:Acceleration:
Position:Position:
Displacement:Displacement:
Velocity:Velocity:

3. ( )
( )
( ) ( )
0 0 0
0 0
21
0 0 0 2
0 0
22
0 0 0
cos 4 21
cos
sin 4 22
sin 4 23
sin 2 4 24
x
y
y
x x v t
v v
y y v t gt
v v gt
v v g y y
θ
θ
θ
θ
θ
− = −
=
− = − −
= − −
= − − −
Projectile motionProjectile motion

4. Uniform circular motionUniform circular motion
•AlthoughAlthough vv does not change, thedoes not change, the
direction of the motion does,direction of the motion does, i.e.i.e.
the velocity (a vector) changes.the velocity (a vector) changes.
•Thus, there is an accelerationThus, there is an acceleration
associated with the motion.associated with the motion.
•We call this a centripetalWe call this a centripetal
acceleration.acceleration.
2
2
Acceleration: Period:
1 1
Frequency : ; 2
2
v r
a T
r v
v v
f f
T r r
π
ω π
π
= =
= = = =
•SinceSince vv does not change, the acceleration must bedoes not change, the acceleration must be
perpendicular to the velocity.perpendicular to the velocity.

5. cos sin
ˆ ˆi j
p p
p p
x r y r
r x y
θ θ= =
= +
r
Analyzing the motionAnalyzing the motion
( )2 2
t
f t t
T
θ π π ω= = =
Uniform motion:
Note : 180 2 radianso
π = 
cos sinp px r t y r tω ω= =

6. sin cos
sin cos
ˆ ˆi j
x y
x y
x y
v r t v r t
v v t v v t
v v v
ω ω ω ω
ω ω
= − =
= − =
= +
r
Analyzing the motionAnalyzing the motion
cos sinp px r t y r tω ω= =
p p
x y
dx dy
v v
dt dt
= =
[remember, ]
v
v r
r
ω ω= =

7. AccelerationAcceleration
ˆ ˆi j
ˆ ˆ= i j
yx
x y
dvdvdv
a
dt dt dt
a a
  
= = + ÷  ÷
   
+
r
r
( ) ( )
2 2
2 22 2
cos sinx y
v v
a a a v
r r
θ θ ω= + = + = =
sin cos
cos sin
x y
x y
v v t v v t
a v t a v t
ω ω
ω ω ω ω
= − =
= − = −
recall : cos sinp px r t y r tω ω = = 
2 2
cos sinx y
v v
a t a t
r r
ω ω= − = −
v
r
ω = ⇒

8. Relative motionRelative motion
PA PB BAr r r= +
r r r
Position:Position:
PA PB BA
PA PB BA PB R
dr dr dr
v v v v v
dt dt dt
= = + = + = +
r r r
r r r r r
Velocity:Velocity:
PA PB R
PA PB
dv dv dv
a a
dt dt dt
= = + =
r r r
r r
Acceleration (if frame B is “inertial”):Acceleration (if frame B is “inertial”):

9. Made BY:
uMang

10. Made BY:
uMang