1. Class 9 – Review of motion in 2D and 3D
Chapter 4 - Monday September 13th
•Exam instructions
•Brief review
•Example problems
Reading: pages 58 thru 75 (chapter 4) in HRW
Read and understand the sample problems
•Exam tonight: 8:20-10:20pm
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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 kinematics
Acceleration:
Position:
Displacement:
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
Projectile motion
4. Uniform circular motion
•Although v does not change, the
direction of the motion does, i.e.
the velocity (a vector) changes.
•Thus, there is an acceleration
associated with the motion.
•We call this a centripetal
acceleration.
2
2
Acceleration: Period:
1 1
Frequency : ; 2
2
v r
a T
r v
v v
f f
T r r
•Since v does not change, the acceleration must be
perpendicular to the velocity.
5. cos sin
ˆ ˆi j
p p
p p
x r y r
r x y
r
Analyzing the motion
2 2
t
f t t
T
Uniform motion:
Note: 180 2 radianso
cos sinp px r t y r t
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
r
Analyzing the motion
cos sinp px r t y r t
p p
x y
dx dy
v v
dt dt
[remember, ]
v
v r
r
7. Acceleration
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
sin cos
cos sin
x y
x y
v v t v v t
a v t a v t
recall : cos sinp px r t y r t
2 2
cos sinx y
v v
a t a t
r r
v
r
8. Relative motion
PA PB BAr r r
r r r
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:
PA PB R
PA PB
dv dv dv
a a
dt dt dt
r r r
r r
Acceleration (if frame B is “inertial”):
