Transcript PPt

Projectile Motion Horizontally
Projectile Motion
 Once
a difficult problem for
cannoneers
 If
the vertical and horizontal
components are separated, the problem
becomes simple.
Projectile Motion
– simple as a ball
rolling across the table
 Use equations like x = v x t
 Horizontal
Projectile Motion
– simple as a free falling
gravity problem
 Use equations ∆y = (1/2)gt2
and vf = gt
 Vertical
Projectile Motion
NOTE:
Air and other resistance is
being ignored for the time
being.
Projectile Motion
Projectile Motion of a Cannon Ball
Projected Horizontally
Question 1
Consider these diagrams to answer the
following questions:
 a. Which shows the initial velocity?
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a
 b. Which shows the acceleration vector?
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b
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Question 2
What determines the time for the cannon ball to hit the ground?
Question 2 Solution
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The time for the cannon ball to hit the
ground is determined by the height from
which it drops.
Question 3
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Supposing a snowmobile is equipped with a flare
launcher which is capable of launching a sphere
vertically (relative to the snowmobile). If the
snowmobile is in motion and launches the flare
and maintains a constant horizontal velocity after
the launch, then where will the flare land (neglect
air resistance)?
a. in front of the snowmobile
b. behind the snowmobile
c. in the snowmobile
Solution for Question 3
C
 The flare will land inside the snowmobile.
The flare has the same horizontal velocity
as the snowmobile that it was initially in.
 The velocity does not change
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Simulation:
Constant Velocity in the Horizontal Direction
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Horizontal & Vertical Velocities
Horizontal & Vertical Displacements
Summary of Projectile Motion
Horizontally
Horizontal Motion Vertical Motion
Forces
(Present?- Yes or No) If
present, what direction
No
Yes
The force of gravity acts
downward
Acceleration
No
“g” is downward at -9.81 m/s2
(Present?- Yes or No)
If present, what direction?)
Velocity
(Constant or Changing)
Yes
Constant
Changing
(by -9.81 m/s each second)
Projectile Motion at Zero Angle
An ugly giant rolls a billiard ball with uniform velocity 30 m/s
across the top of his desk. The ball rolls off the edge of the
desk and lands on the floor 120 m from the edge of the desk.
How high is the table?
Height of Table
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Time of fall
vx = x/t
30 m/s = 120 m / t
t = 4.0 s
Height of table
∆y = vit + ½ gt2
∆y = ½ (-9.81 m/s2)(4.0s)2
∆y = -78 m………78m
An ugly giant rolls a billiard ball with uniform velocity 30 m/s
across the top of his desk. The ball rolls off the edge of the
desk and lands on the floor 120 m from the edge of the desk.
At what speed will it hit the ground?
Velocity When It Hits The Ground
Horizontally speed is constant.
 30 m/s = vx
 Vertical Speed
 vf = vi + gt
 vf = (-9.81 m/s2 )4.0 s
 vf = 39 m/s
 Resultant speed = 49 m/s
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Characteristics & Assumptions
Planar motion
 Air resistance: negligible
 Gravity: downward
 No horizontal acceleration
 Constant vertical acceleration
 Parabolic trajectory
 Independent horizontal and vertical motions
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Question 4
Anna Litical drops a ball from rest from the top of
80.-meter high cliff. How much time will it take for the
ball to reach the ground and at what height will the ball
drop after each second of motion?
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Solution
∆y = vit + ½ gt2
 -80.m = ½ (-9.81 m/s2)t2
 -160m = (-9.81 m/s2) t2
 t = 4.0 s
 Each sec the displacements are as follows:
∆y = vit + ½ gt2 = ½ (-9.81m/s2)(1sec)2
 -4.9m, -20m,-44m,-78m
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Question 5
A cannonball is launched horizontally from the top of an
80.-meter high cliff. How much time will it take for the
ball to reach the ground and at what height will the ball be
after each second of travel?
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Question 5 Solution
∆y = vit + ½ gt2
 -80.m = ½ (-9.81 m/s2)t2
 -160m = (-9.81 m/s2) t2
 t = 4.0 s
 Each sec the displacements are as follows:
∆y = vit + ½ gt2 = ½ (-9.81m/s2)(1sec)2
 -4.9m, -20m,-44m,-78m
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Question 6
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A pool ball leaves a 0.60-meter high table
with an initial horizontal velocity of
2.4 m/s. Predict the time required for the
pool ball to fall to the ground and the
horizontal distance between the table's edge
and the ball's landing location.
Solution for Question 6
Vertical Time
 ∆y = vit + ½ gt2
 -.60m = ½ (-9.81 m/s2)t2
 -1.20m = (-9.81 m/s2) t2
 t = .35 s
 Horizontal Distance
 x = v (t) = 2.4 m/s ( .3497s) = .84m
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Question 7
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A soccer ball is kicked horizontally off a
22.0-meter high hill and lands a distance of
35.0 meters from the edge of the hill.
Determine the initial horizontal velocity of
the soccer ball.
Solution Question 7
Vertical Time
 ∆y = vit + ½ gt2
 -22m = ½ (-9.81 m/s2)t2
 -44m = (-9.81 m/s2) t2
 t = 2.1 s
 Initial Horizontal Velocity
 vx = x/t = 35 m/2.1178s = 17 m/s
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Question 8
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You accidentally throw your car keys
horizontally at 8.0 m/s from a cliff 64 m
high. How far from the base of the cliff
should you look for the keys?
Question 9
A toy car runs of the edge of a table that is
1.225 m high. If the car lands 0.400 m from
the base of the table,
 a. how long did it take the car to fall?
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b. how fast was the car going on the table?
Question 10
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Divers in Acapulco dive from a cliff that is
61 m high. If the rocks below the cliff
extend outward for 23 m, what is the
minimum horizontal velocity a diver must
have to clear the rocks?
Question 11
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A dart player throws a dart horizontally at a
speed of 12.4 m/s. The dart hits the board
0.32 m below the height from which it was
thrown. How far away is the player from the
board?
Question 12
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A ball thrown horizontally from a 13 m high
building strikes the ground 5.0 m from the
building. With what velocity was the ball
thrown?
Question 13
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A person leaps horizontally from the top of
a tower and lands 17.0 m from the base of
the tower. If the speed at which the person
was projected was 9.50 m/s, how high is the
tower?
Monkey & Hunter Experiment
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PROJECTILE MOTION
Projectile Motion
Projectile Motion
Velocity
Projectile Motion
Projectile Motion
Projectile Motion
Trajectory
 Up to a 45 degree angle – greater
distance at greater angle
 After 45 degree – greater height, less
distance
 45 degree gives the greatest distance
 Any two angles that add to 90 degrees
will hit the same place
Projectile Motion
Air Resistance
Monkey and the Hunter
http://physics.bu.edu/~duffy/semester1/c04_
monkeyhunter.html
 http://cccmkc.edu.hk/~kei-yhk/vx_vy1.html
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Analysis of Projectile Motion
Motion
Horizontal
Vertical
Initial
Velocity
v cos 
v sin 
Acceleration
0
-g
(-ve for
downwards)
Time taken
t
t
Distance
moved
Final
Velocity
x = vt cos  y = vt sin  gt2/2
v cos 
v sin  - gt
Graphs for a Projectile Motion
Range R
v sin 2
R
g
2
Simulation
Terminal Velocity
Sky-diver in free fall
Terminal Velocity
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Abc
Terminal velocity is reached when f(v) = g
Terminal Velocity: v-t Graph