Projectile Motion

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Projectile Motion

• Chapter 3.3

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Objectives

• Recognize examples of projectile motion
• Describe the path of a projectile as a parabola
• Resolve vectors into their components and apply
  the kinematic equations to solve problems
  involving projectile motion

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Projectile Motion

• How can you know the displacement, velocity and
  acceleration of a ball at any point in time
  during its flight?
• Use the kinematic equations of course! ?

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Vector Components p.98 (Running vs Jumping)
While running, the person is only moving in one
dimension Therefore, the velocity only has one
component. V
While jumping, the person is moving in two
dimensions Therefore, the velocity has two
components. Vy Vx
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Definition of Projectile Motion

• Objects that are thrown or launched into the air
  and are subject to gravity are called projectiles
• Examples?
• Thrown Football, Thrown Basketball, Long Jumper,
  etc

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Path of a projectile

• Neglecting air resistance, the path of a
  projectile is a parabola
• Projectile motion is free fall with an initial
  horizontal velocity
• At the top of the parabola, the velocity is not
  0!!!!!!

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Vertical and Horizontal Motion
Horizontal Motion Vertical Motion
Velocity Vx Displacement ?x Velocity Vy Displacement ?y
Because gravity does not act in the horizontal direction, Vx is always constant! Gravity acts vertically, therefore a -9.81 m/s2
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Equations for projectiles launched horizontally
Horizontal Motion Vertical Motion
?xVxt Vx is constant! a0 Vy,i0 (initial velocity in y direction is 0)
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Revised Kinematic Eqns for projectiles launched
horizontally
Horizontal Motion Vertical Motion
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Finding the total velocity

• Use the pythagorean theorem to find the resultant
  velocity using the components (Vx and Vy)
• Use SOH CAH TOA to find the direction

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Example p. 102 2

• A cat chases a mouse across a 1.0 m high table.
  The mouse steps out of the way and the cat slides
  off the table and strikes the floor 2.2 m from
  the edge of the table. What was the cats speed
  when it slid off the table? What is the cats
  velocity just before it hits the ground?

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What do we know and what are we looking for?

• ?x 2.2 m
• ?y -1.0m (bc the cat falls down)

What are we looking for??
1.0 m
2.2m
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How do we find Vx?

• Equation for horizontal motion
• We have xso we need t.
• How do we find how long it takes for the cat to
  hit the ground?
• Use the vertical motion kinematic equations.

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Vertical Motion

• ?y -1.0m
• a-9.81 m/s2
• What equation should we use?
• Rearrange the equation, to solve for t then plug
  in values.

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Horizontal equation

• Rearrange and solve for Vx
• Cats Speed is 4.89 m/s

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Cliff example

• A boulder rolls off of a cliff and lands 6.39
  seconds later 68 m from the base of the cliff.
• What is the height of the cliff?
• What is the initial velocity of the boulder?
• What is the velocity of the boulder just as it
  strikes the ground?

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How high is the cliff?

• ?y ? a-9.81 m/s2
• t 6.39 s Vx?
• Vy,i 0 ?x 68 m

The cliff is 200 m high
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What is the initial velocity of the boulder?

• The boulder rolls off the cliff horizontally
• Therefore, we are looking for Vx

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Important Concepts for Projectiles Launched
Horizontally
Horizontal Components Vertical Components
Horizontal Velocity is constant throughout the flight Horizontal acceleration is 0 Initial vertical velocity is 0 but increases throughout the flight Vertical acceleration is constant -9.81 m/s2
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Projectiles Launched at An Angle
Projectiles Launched Horizontally Projectiles Launched at an Angle
Vx is constant Initial Vy is 0 Vx is constant Initial Vy is not 0

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Components of Initial Velocity for Projectiles
Launched at an angle
Use soh cah toa to find the Vx,i and Vy,i
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Revise the kinematic equations again
Horizontal Motion Vertical Motion

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Example p. 104 3

• A baseball is thrown at an angle of 25 relative
  to the ground at a speed of 23.0 m/s. If the ball
  was caught 42.0 m from the thrower, how long was
  it in the air? How high was the tallest spot in
  the balls path?

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What do we know?
?x 42.0 m ? 25 Vi 23.0 m/s Vy at top 0 ?t? ?y?
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What can we use to solve the problem?

• Find t using the horizontal eqn
• ?xvx?t vicos(?)t
• How to find ?y?
• Vy,f 0 at top of the balls path
• What equation should we use?

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Cliff example

• A girl throws a tennis ball at an angle of
  60North of East from a height of 2.0 m. The
  balls range is 90 m and it is in flight for 6
  seconds.
• What is the initial horizontal velocity of the
  ball?
• What is the initial vertical velocity of the
  ball?
• What is the total initial velocity of the ball?
• How high above the initial position does the ball
  get?
• What is the vertical velocity of the ball 2
  seconds after it is thrown?

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What is the initial horizontal velocity of
the ball?

• ?x 90 m
• T60
• Total time 6 s
• Horizontal velocity is constant Vx

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What is the initial vertical velocity of the ball?
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What is the total initial velocity of the ball?
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How high above the ground does the ball get?

• At the top of the parabola, Vy is 0so use the
  revised kinematic equations
• Add 2m to get the height above the ground 36.65 m

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What is the vertical velocity of the ball 2
seconds after it is thrown?

• Vy,i26 m/s
• a -9.81 m/s2
• t 2 seconds

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Important Concepts for Projectiles Launched at an
Angle

• At the top of the parabola, neither the objects
  velocity nor its acceleration is 0!!!!!
• Only Vy is 0
• Vx is constant throughout the flight
• Horizontal acceleration is always 0
• Vertical acceleration is always -9.81 m/s2