Chapter 3
Two-Dimensional
Motion
Projectiles launched at
an angle
Some Variations of Projectile
Motion


An object may be fired
horizontally
The initial velocity is all
in the x-direction


vi = vx and vy = 0
All the general rules of
projectile motion apply
Projectile Motion at an angle
How are they different?

Projectiles Launched Horizontally
–
–

The initial vertical velocity is 0.
The initial horizontal velocity is the total initial velocity.
Projectiles Launched At An Angle
–
–
–
Resolve the initial velocity into x and y components.
The initial vertical velocity is the y component.
The initial horizontal velocity is the x component.
Some Details About the Rules

x-direction
•
•
•
ax = 0
vx = vix = vicosΘi = constant
x = vixt
•
•
This is the only equation in the x-direction since there is
constant velocity in that direction
Initial velocity still equals final velocity
More Details About the Rules

y-direction


viy = visinΘi
Free fall problem





a=g
Object slows as it goes up (-9.8m/s2)
Uniformly accelerated motion, so the motion
equations all hold
JUST LIKE STOMP ROCKETS
Symmetrical
Problem-Solving Strategy





Resolve the initial velocity into x- and ycomponents
Treat the horizontal and vertical motions
independently
Make a chart again, showing horizontal and
vertical motion
Choose to investigate up or down.
Follow rules of kinematics equations
Solving Launched Projectile Motion
vi = (r , Θ) = (vix , viy)
Horizontal
Vertical
a=
0
a=
+/- 9.8m/s2 = g
vi =
vix
vi =
0 or viy
vf =
vix
vf =
viy or 0
t=
#
t=
#
x=
# = range
x=
Max height = y
UP or DOWN INVESTIGATION… Where do the
resolved components go?
Projectile Motion at an angle
Example 1:
The punter for the Steelers punts the football with
a velocity of 27 m/s at an angle of 30. Find the
ball’s hang time, maximum height, and distance
traveled (range) when it hits the ground.
(Assume the ball is kicked from ground level.)
Looking for:
Total time (t)
Max height (y)
Range (x)
Given:
vi = (27m/s, 30o)
What do we do with the given info?
vi = (27m/s, 30o)
What are
the units?
vi = (23.4m/s, 13.5m/s)
“resolved” vector
27m/s
m/s
Viy = 27sin30
30o
Vix = 27cos30
Vix = 23.4
Vix = 23.4m/s
Viy = 13.5
Viy = 13.5m/s
So where does this info “fit” in the
chart?
Horizontal
Vertical
a=
0
a=
vi =
23.4m/s
vi = Viy if solving “up” = 13.5m/s
vf =
23.4m/s
vf = Viy if solving “down” = 13.5m/s
t=
t=
x=
x=
Pick a “side” to solve – symmetry
Up:
Horizontal
Vertical
a=
0
a = - 9.8m/s2
vi =
23.4m/s
vi = 13.5m/s
vf =
23.4m/s
vf = 0
t=
t=
x=
y=
vf2 = vi2 + 2gy
vf = vi + gt
y = 9.3m
t = 1.38s
Projectile Motion at an angle
Pick a “side” to solve – symmetry
Down:
Horizontal
Vertical
a=
0
a = 9.8m/s2
vi =
23.4m/s
vi = 0
vf =
23.4m/s
vf = 13.5m/s
t=
t=
x=
y=
vf2 = vi2 + 2gy
vf = vi + gt
y = 9.3m
t = 1.38s
Projectile Motion at an angle
On the horizontal side




Max height occurs midway through the flight.
We found t = 1.38s both directions (up and
down).
How long is the projectile in the air?
DOUBLE this time for total air time
t = 1.38 x 2 = 2.76s
What about range?
x = vt
x = (23.4)(2.76)
x = 64.6m = RANGE
This tells us…

Now we only need an initial velocity vector to
determine all of the information we need to
have a detailed description of where an
object is in its path.
vi = (r , Θ) = (vix , viy)
Maximum Range vs. Maximum Height

What angle of a launched projectile gets the
maximum height?
90o

What angle of a launched projectile gets the
maximum range?
45o
Projectile Motion at Various Initial
Angles

Complementary
values of the initial
angle result in the
same range
–

The heights will be
different
The maximum
range occurs at a
projection angle of
45o
Non-Symmetrical Projectile Motion


Follow the general rules for
projectile motion
Break the y-direction into
parts
– up and down
– symmetrical back to initial
height and then the rest of
the height
your homework …


We are going to see what kind of job
Hollywood writers and producers would do
on their NECAP assessments…
Watching a clip of the Bus Jump, use the
timer provided to time the flight of the bus
and then do the actual calculations for
homework.