- The document outlines topics to be covered in a class review on motion in 2D and 3D, including exam instructions, example problems, and key concepts like position, velocity, acceleration, projectile motion, uniform circular motion, and relative motion. - Key formulas are provided for acceleration, period, frequency, analyzing circular motion, and relative motion between reference frames. - An exam on these topics from chapters 4 and 5 of the textbook will take place that evening from 8:20-10:20pm. WebAssign registration is required to complete online homework problems.

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
WebAssign is NO LONGER FREE!! You must register

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”):