PHYSICS 220
Lecture 06
Projectile Motion
Textbook Sections 4.2
Lecture 6
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1
Relative Velocity
•  We often assume that our reference frame is attached to the Earth. What
happen when the reference frame is moving at a constant velocity with
respect to the Earth?
•  The motion can be explained by including the relative velocity of the
reference frame in the description of the motion.
The ground velocity of an
Example airplane
airplane is the vector sum of
the air velocity and the wind
velocity. Using the air as the
intermediate reference frame,
ground speed is:
Lecture 6
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2
iClicker
You are on a train traveling 40 mph North. If you
walk 5 mph sideways across the car (W), what is
your speed relative to the ground?
A) < 40 mph
B) 40 mph
C) >40 mph
40 mph N + 5 mph W = 41 mph NW
!
40

Lecture 6
5
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3
Exercise
Three swimmers can swim equally fast relative to the water. They
have a race to see who can swim across a river in the least time.
Relative to the water, Beth (B) swims perpendicular to the flow, Ann
(A) swims upstream, and Carly (C) swims downstream. Which
swimmer wins the race?
A) Ann
B) Beth
C) Carly
correct
y
!
x
t = d / vy
Ann vy = v cos()
A B C
Beth vy = v
Carly vy = v cos()
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4
Exercise
What angle should Ann take to get directly to the other
side if she can swim 5 mph relative to the water, and the
river is flowing at 3 mph?
VAnn,ground = Vann,water+Vwater,ground
y
!
x
sin() = |Vwater,ground|/ |Vann,water|
sin() = 3/5
Lecture 6
A B C
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5
2-Dimensions
•  X and Y are
INDEPENDENT!
y
x
•  Break 2-D problem into two
1-D problems
Lecture 6
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6
Position, Velocity and Acceleration
•  Position, Velocity and Acceleration are Vectors!
x-direction
y-direction
•  x and y directions are INDEPENDENT!
Lecture 6
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7
Velocity in Two Dimensions
A ball is rolling on a horizontal surface at 5 m/s. It then rolls
up a ramp at a 25 degree angle. After 0.5 seconds, the ball
y
has slowed to 3 m/s.
What is the magnitude of the change in velocity?
x
x-direction
y-direction
vix = 5 m/s
vfx = 3 m/s cos(25)
viy = 0 m/s
vfy = 3 m/s sin(25)
vx = 3cos(25)–5 =-2.28m/s
vy = 3sin(25)=+1.27 m/s
3 m/s
5 m/s
Lecture 6
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8
Acceleration in Two Dimensions
A ball is rolling on a horizontal surface at 5 m/s. It then rolls
up a ramp at a 25 degree angle. After 0.5 seconds, the ball
y
has slowed to 3 m/s.
What is the average acceleration?
x-direction
x
y-direction
3 m/s
5 m/s
Lecture 6
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9
Kinematics in Two Dimensions
x = x0 + v0xt + 1/2 axt2
y = y0 + v0yt + 1/2 ayt2
vx = v0x + axt
vy = v0y + ayt
vx2 = v0x2 + 2ax x
vy2 = v0y2 + 2ay y
x and y motions are independent!
They share a common time t
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Projectile Motion
x-direction: ax = 0
x = x0 + v0x t
vx = v0x
y-direction: ay = -g
y = y0 + v0y t - ½ gt2
vy = v0y – g t
vy2 = v0y2 – 2 g y
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Velocity of a Projectile
Velocity components of a projectile
Lecture 6
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12
The Range of a Kickoff
A place-kicker kicks a football at an angle of =400
above the horizontal axis. The initial speed of the
ball is v0=22 m/s. Ignore air resistance and find the
range R that the ball attains.
v0=22m/s
H
=400
R
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The Range of a Kickoff
•  Use the equations
x = x0 + v0xt
vx = v0x
y = y0 + v0yt - 1/2 gt2
vy = v0y - gt
vy2 = v0y2 - 2g y
•  The range is a characteristic of the horizontal motion
•  You need v0x and v0y but you have been given v0
v0x = v0cos  =
(22m/s)cos 400 = 17 m/s
Lecture 6
v0
v0y
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!
v0x
14
The Range of a Kickoff
•  We could be done if we know the time of flight of the kickoff
•  The time of flight can be determined from y equations. For
example the time to get to height H is
v
v0y
0
!
v0x
v0y =v0sin =
(22m/s)sin 400=14 m/s
•  Therefore the time to determine the range is 2.9 s
•  The range depends on the angle  at which the football is kicked.
Maximum range is reached for =450
Lecture 6
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15
iClicker
A ball is thrown into the air and follows a parabolic trajectory.
At the highest point in the trajectory,
A)  The velocity is zero, but the acceleration is not zero.
B)  Both the velocity and acceleration are zero.
C)  The acceleration is zero, but the velocity is not zero.
D)  Neither the acceleration nor the velocity is zero.
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Range of a Projectile
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Shooting the Monkey
You are a vet trying to shoot a tranquilizer dart into a monkey hanging
from a branch in a distant tree. You know that the monkey is very
nervous, and will let go of the branch and start to fall as soon as your gun
goes off. On the other hand, you also know that the dart will not travel in
a straight line, but rather in a parabolic path like any other projectile. In
order to hit the monkey with the dart, where should you point the gun
before shooting?
A) Right at the monkey
B) Below the monkey
C) Above the monkey
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Shooting the Monkey
Dart hits the
monkey!
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