Section 1: Projectile Motion
A projectile’s horizontal motion is independent of its vertical
motion.
K
What I Know
W
What I Want to Find Out
L
What I Learned
Essential Questions
•
•
How are the vertical and horizontal motions of a projectile related?
What are the relationships between a projectile’s height, time in the air, initial
velocity, and horizontal distance traveled?
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Projectile Motion
Vocabulary
Review
New
•
•
•
motion diagram
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projectile
trajectory
Projectile Motion
Path of a Projectile
•
An object shot through the air is called a projectile.
•
After projectile has been given an initial thrust, (ignoring air resistance) it moves
through the air only under the force of gravity.
Its path through space is called its trajectory.
•
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Projectile Motion
Independence of Motion in Two Dimensions
Concepts in Motion
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Projectile Motion
vxi
Horizontally Launched Projectiles
ay
Δy
Δx
Use with Example Problem 1.
Problem
Miya’s pencil rolls off her desk and lands
0.50 m away. If the desk’s height is 1.0
m, how long did it take the pencil to
reach the floor? How fast was the pencil
going when it left the desk? Ignore the
pencil’s rotation.
SKETCH AND ANALYZE THE PROBLEM
• Sketch the situation.
• List the knowns and unknowns.
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y f  y i  v yit  21ay t 2
0  y i  0  21ay t 2
t 
Response
KNOWN
xi = 0 m
yi = 1.0 m
xf = 0.50 m yf = 0 m
ax = 0 m/s2 ay = −9.8 m/s2
vyi = 0 m/s
SOLVE FOR THE UNKNOWN
• Use the equation of motion in the y-direction
to find the time.
2y i

ay
21.0 m
9.8 m/s 
2
 0.45 s
UNKNOWN
t=?
vxi = ?
Projectile Motion
vxi
Horizontally Launched Projectiles
Δx
Use with Example Problem 1.
Problem
Miya’s pencil rolls off her desk and lands
0.50 m away. If the desk’s height is 1.0
m, how long did it take the pencil to
reach the floor? How fast was the pencil
going when it left the desk? Ignore the
pencil’s rotation.
SKETCH AND ANALYZE THE PROBLEM
• Sketch the situation.
• List the knowns and unknowns.
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SOLVE FOR THE UNKNOWN
• Use the equation of motion in the x-direction
to find the initial velocity.
xf  xi  v xit  21ax t 2
xf  0  v xit  0
v xi 
Response
KNOWN
xi = 0 m
yi = 1.0 m
xf = 0.50 m yf = 0 m
ax = 0 m/s2 ay = −9.8 m/s2
vyi = 0 m/s
ay
Δy
UNKNOWN
t = 0.45 s
vxi = ?
xf
0.50 m

 1.1 m/s
t
0.45 s
EVALUATE THE ANSWER
• The units are correct: seconds for time
and meters per second for velocity.
• The signs are correct: Time is positive and
the positive velocity agrees with our
diagram.
Projectile Motion
Angled Launches
•
•
When a projectile is launched at an
angle, the initial velocity has a
vertical component as well as a
horizontal component.
The adjoining figure shows the
separate vertical- and horizontalmotion diagrams for the trajectory of
the ball.
Animation
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Projectile Motion
Angled Launches
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There are three quantities
associated with a trajectory.
– Maximum height: the height of •
the projectile when the vertical
•
velocity is zero
– Range (R): the horizontal
distance that the projectile
travels when the initial and final
heights are the same
– Flight time: how much time the
projectile is in the air; also called
hang time
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Animation
Projectile Motion
Angled Launches
Response
Use with Example Problem 2.
SKETCH AND ANALYZE THE PROBLEM
• Sketch the situation.
• List the knowns and unknowns.
Problem
Courtney kicks a soccer ball that is at rest
on level ground and gives it an initial
velocity of 7.8 m/s at an angle of 32°
above the ground. Assume that forces due
to air drag on the ball are insignificant.
a. How long will the ball be in the air? 0.84 s
b. How high will the ball go?
c. What will be its range?
KNOWN
yi = yf = 0 m
θ = 32°
vyi = 4.1 m/s
ay = −9.8 m/s2
UNKNOWN
ymax = ?
R = xf = ?
xi = 0 m
vi = 7.8 m/s
vxi = 6.6 m/s
ax = 0 m/s2
tf = ?
SOLVE FOR THE UNKNOWN
• Part a: Use the relationship among
position, velocity, time, and acceleration.
 21 ayt f 2
 0  v yit f   21 ayt f 2
2v yi
24.1 m/s


y f  y i  v yit f 
0
tf
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ay

9.8 m/s2

 0.84 s
Projectile Motion
Angled Launches
Response
Use with Example Problem 2.
SKETCH AND ANALYZE THE PROBLEM
• Sketch the situation.
• List the knowns and unknowns.
Problem
Courtney kicks a soccer ball that is at rest
on level ground and gives it an initial
velocity of 7.8 m/s at an angle of 32°
above the ground. Assume that forces due
to air drag on the ball are insignificant.
a. How long will the ball be in the air? 0.84 s
b. How high will the ball go? 0.86 m
c. What will be its range?
xi = 0 m
vi = 7.8 m/s
vxi = 6.6 m/s
ax = 0 m/s2
tf = ?
KNOWN
yi = yf = 0 m
θ = 32°
vyi = 4.1 m/s
ay = −9.8 m/s2
UNKNOWN
ymax = ?
R = xf = ?
SOLVE FOR THE UNKNOWN
• Part b: Use the relationship among
position, velocity, time, and acceleration.
y max  y i  v yi  21t f  
 21 ay  21t f 
2
 4.1 m/s0.42 s   21 9.8 m/s 2 0.42 s
2
 0.86 m
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Projectile Motion
Angled Launches
Response
Use with Example Problem 2.
SKETCH AND ANALYZE THE PROBLEM
• Sketch the situation.
• List the knowns and unknowns.
Problem
Courtney kicks a soccer ball that is at rest
on level ground and gives it an initial
velocity of 7.8 m/s at an angle of 32°
above the ground. Assume that forces due
to air drag on the ball are insignificant.
a. How long will the ball be in the air? 0.84 s
b. How high will the ball go? 0.86 m
c. What will be its range? 5.5 m
xi = 0 m
vi = 7.8 m/s
vxi = 6.6 m/s
ax = 0 m/s2
tf = ?
KNOWN
yi = yf = 0 m
θ = 32°
vyi = 4.1 m/s
ay = −9.8 m/s2
UNKNOWN
ymax = ?
R = xf = ?
SOLVE FOR THE UNKNOWN
• Part c: Use the relationship among
position, velocity, and time.
R  xi  v xit f  6.6 m/s0.84 s  5.5 m
EVALUATE THE ANSWER
• The distance, height, and flight time are
reasonable for a kicked soccer ball.
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Projectile Motion
Forces from Air
•
Forces from the air can affect the trajectory of an object.
vi
No Effect from Air
vi
Force from Air in
Direction of vi
Fair
vi
Force from Air
Opposing vi
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Fair
Projectile Motion
Review
Essential Questions
•
•
How are the vertical and horizontal motions of a projectile related?
What are the relationships between a projectile’s height, time in the air, initial
velocity, and horizontal distance traveled?
Vocabulary
•
projectile
Copyright © McGraw-Hill Education
•
trajectory
Projectile Motion