IMPORTANT NOTICES
•Review sessions (both in NPB 1001, i.e. here):
Thursday, Sept. 9th (Hill), 6:15-8:10pm
Friday, Sept. 10th (Woodard), 6:15-8:10pm
•Subject: chapters 1-4 (similar coverage each night)
•Exam 1 – Mon. Sept. 13, 8:20-10:20pm
(Important instructions for exam in Fri./Mon. classes)
Chapter 1 and 2 WebAssign homework deadlines
are tonight at 11pm (chapter 3 on Friday, 11pm).
Class 7 - Motion in 2D and 3D
Chapter 4 - Wednesday September 8th
•2D/3D Position, displacement, velocity and acceleration
•Example problems
•Projectile motion, with demonstrations
•Example problems
Reading: pages 58 thru 70 (chapter 4) in HRW
Read and understand the sample problems
Assigned problems from Chapter 4:
6, 8, 14, 20, 30, 40, 46, 52, 56, 62 (due Sun. 12th)
Position and displacement
•The equations of motion that we introduced in chapter 2
apply equally to two- and three-dimensional motion.
•All we need to do is to break the motion up into
components, and treat each component independently, i.e.
we will have two sets of equations in 2D, and three sets in
3D.
Position:
r  xˆi  yˆj  zkˆ
•Coefficients x, y, and z give the particle's coordinates
relative to the origin.
Displacement:
r  r2  r1  xˆi  yˆj  zkˆ
  x2  x1  ˆi   y2  y1  ˆj   z2  z1  kˆ
Important note!
•In chapter 3, you were told that the sum of two
displacement vectors is given as follows:
r  a  b  r3  r1
a  r2  r1
b  r3  r2
a  b  r3  r1
B
b
a
r
A
r1
r2
C
r3
•Displacement vectors are added graphically by placing the
tail of one vector at the head of the other.
Important note!
•Position vectors represent coordinates.
•The displacement between two position vectors is given by
the difference between the ending position and the
starting position:
2
1
r  r  r
r1
r2
O
Average and instantaneous velocity
displacement r2  r1 r
average velocity 


time interval
t
t
•In component form:
vavg
xˆi  yˆj  zkˆ x ˆ y ˆ z ˆ


i
j
k
t
t
t
t
•Instantaneous velocity:

•Or:

dr d ˆ ˆ ˆ
dx ˆ dy ˆ dz ˆ
v

xi  yj  zk  i 
j k
dt dt
dt
dt
dt
v  vx ˆi  v y ˆj  vz kˆ
dx
dy
dz
vx  ; v y  ; vz 
dt
dt
dt
Average and instantaneous acceleration
aavg
change in velocity v2  v1 v



time interval
t
t
•In component form:
aavg 
vx ˆi  v y ˆj  vz kˆ
t
vx ˆ v y ˆ vz ˆ

i
j
k
t
t
t
•Instantaneous acceleration:


dv
dv d
dv
dvz ˆ
y ˆ
x ˆ
ˆ
ˆ
ˆ
a

vx i  v y j  vz k 
i
j
k
dt dt
dt
dt
dt
•Or:
dv
dv
dvz
y
x
ˆ
ˆ
ˆ
a  ax i  a y j  az k ax 
; ay 
; az 
dt
dt
dt
Projectile motion
•Motion in a vertical plane
where the only influence is
the constant acceleration due
to gravity.
•In projectile motion, the
horizontal motion and vertical
motion are independent of
each other, i.e. they do not
affect each other.
•This feature allows us to break the motion into two
separate one-dimensional problems: one for the
horizontal motion; the other for the vertical motion.
•We will assume that air resistance has no effect.
Examples of projectile
motion
Demonstration
Back to projectile motion
•This demonstration, which I
showed in class, illustrates the
fact that vertical motion is
unaffected by horizontal
motion, i.e. the two balls
accelerate at the same constant
rate, irrespective of their
horizontal component of motion.
•In all of the projectile motion
problems that we will consider,
we shall assume that the only
acceleration is due to gravity
(a=-g) which acts along the yaxis.
Analyzing the motion
v0 x  v0 cos 0
v0 y  v0 sin  0
Initial coordinates x0 and y0
Initial velocity  v0 xˆi  v0 y ˆj
Remember these formulae?
Horizontal motion
vx  v0 x  axt
ax  0

vx  v0 cos 0
x  x0  v0 xt  axt
1
2
2
x  x0   v0 cos 0  t
Vertical motion
(2  15)
y  y0  v0 yt  12 a yt 2
y  y0   v0 sin  0  t  gt
1
2
2
(4  22)
v y  v0 y  a yt
(2  11)
v y  v0 sin  0  gt
(4  23)
Projectile equations of motion
x  x0   v0 cos 0  t
4  21
vx  v0 cos 0
y  y0   v0 sin  0  t  gt
1
2
2
4  22
v y  v0 sin  0  gt
4  23
v   v0 sin  0   2 g  y  y0 
4  24
2
y
2
The effects of air
The physics
professor's home run
always goes further
than the
professional's