Projectile Motion

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Chapter 3

• Projectile Motion

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3.1 Vector and Scalar Quantities

• Vector a quantity that has both magnitude and
  direction (velocity)
• Have special rules when adding/subtracting
  because of the direction component
• Scalar a quantity that is completely described
  by magnitude only (speed)
• Do not have special rules when adding/subtracting
• 1 kg cement added to 3 kg sand 4 kg of material
• 3 gallons of water poured from a bucket
  containing 5 gallons 2 gallons of water left in
  the bucket

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3.2 Velocity Vectors

• When drawing vectors
• Use an arrow to represent the magnitude and
  direction of a vector
• Use a ruler to draw vector to scale (be sure to
  give a key)
• Example 1cm 30m/s, then 3cm 90m/s

30m/s east
90m/s east
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3.2 Velocity Vectors

• An airplane is traveling 110km/h flying with the
  wind with a speed of 20km/h. How fast is the
  airplane actually traveling?
• An airplane is traveling 110km/h flying against
  the wind with a speed of 20km/h. How fast is the
  airplane actually traveling?
• An airplane is traveling north at 120km/h with a
  crosswind from the east with a speed of 50.0km/h.
• Will the wind affect the speed?
• If so, what is the actual speed of the plane?
• http//www.physicsclassroom.com/mmedia/vectors/pl
  .gif

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3.2 Velocity Vectors

• Resultant the vector sum of two or more
  component vectors the RESULT of adding two or
  more vectors
• Draw vectors head (arrow) to toe (no arrow)
• If vectors are at a right angle, use the
  Pythagorean theorem OR trigonometry

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3.3 Components of Vectors

• Component one of the vectors, often mutually
  perpendicular, whose sum is a resultant vector
• Any resultant vector may be regarded as the
  combination of two or more components
• Most often a vector has horizontal (x) and
  vertical (y) components
• Resolution the process of determining the
  component of a vector

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3.3 Components of Vectors

• Use the parallelogram rule to construct the
  resultants of the pairs of vectors.

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3.3 Components of Vectors

• Construct the vertical and horizontal components
  of the vectors.

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3.3 Components of Vectors

• Distance is different than displacement!
• You are a participant in the Indianapolis 500.
• What is your distance traveled?
• What is your displacement?

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3.3 Components of Vectors

• While following directions on a treasure map, a
  pirate walks 45.0m north and then turns and walks
  7.5m east. What single straight line
  displacement could the pirate have taken to reach
  the treasure?
• Find the components of the velocity of a
  helicopter traveling 95km/h at an angle of 35? to
  the ground.

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3.3 Components of Vectors

• An airplane flying parallel to the ground
  undergoes two consecutive displacements. The
  first is 75km 30.0? west of north, and the second
  is 155km 60.0? east of north. What is the total
  displacement of the airplane?
• A boat is crossing a river. If the boat is headed
  due north with a speed of 2.5m/s relative to the
  water and the rivers velocity is 3.0m/s to the
  east, what will the boats velocity relative to
  the Earth be?

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2 Dimensional Motion

• Break problems down into horizontal and vertical
  motion by using trig and/or Pythagorean theorem,
  then use kinematics equations.
• Starting from rest, a car accelerates at 2.0 m/s2
  up a hill that is inclined 5.5 above the
  horizontal.
• How far horizontally has the car traveled in 12
  s?
• How far vertically has the car traveled in 12 s?
• A bird is flying so that it is initially moving
  with a vertical speed of 3.5 m/s and accelerating
  horizontally at 8.0 m/s2.
• Assuming the birds acceleration is constant,
  find the horizontal and vertical distance the
  bird travels in 0.25 s.
• How long would it take the bird to move 2.0 m
  horizontally from its initial position?

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3.4 Projectile Motion

• Projectile any object that moves through space,
  acted on only by gravity (and air resistance, if
  any)
• Softballs, footballs, arrows, etc
• Projectiles near the surface of the earth follow
  a curved path
• To study the motion of a projectile, we will look
  at its horizontal and vertical motion
• The motion of a projectile is an arc
  (parabola)it is not linear
• The horizontal and vertical components of
  projectiles are independent of each other
• If two objects are released from the same
  heightone is dropped and the other is thrown
  horizontallythey will land on the ground at the
  same time!!!
• Projectile motion is free fall with an initial
  horizontal velocity
• Treat these problems as 2 dimensional motion
  problemswith constant horizontal velocity and a
  vertical acceleration of g (-9.8 m/s2)

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3.4 Projectile Motion

• Characteristics of Projectiles
• maintain a constant horizontal velocity
  (neglecting air resistance)
• experience a constant vertical acceleration (g
  10m/s2 downward)
• horizontal and vertical motion are completely
  independent of each othervelocity should be
  separated into horizontal and vertical components
• for a projectile starting and ending at the same
  height, the time it takes to rise to its highest
  point is equal to the time it takes to fall from
  its highest point back to its original position
• objects dropped from a moving vehicle have the
  same initial velocity as the moving vehicle

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3.4 Projectile Motion

• Consider a cannonball projected horizontally from
  a cannon from upon a very high cliff
• In the absence of gravity, the cannonball would
  continue its horizontal motion at a constant
  velocity.
• If it dropped from rest in the presence of
  gravity, the cannonball would accelerate
  downward, gaining speed by a rate of 10 m/s every
  second.
• This is consistent with our conception of
  free-falling objects accelerating at a rate we
  call the "acceleration of gravity."

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3.4 Projectile Motion

• We project the cannonball horizontally in the
  presence of gravity, then the cannonball would
  maintain the same horizontal motion as before - a
  constant horizontal velocity.
• Gravity will act upon the cannonball to cause the
  same vertical motion as before - a downward
  acceleration. The cannonball falls the same
  amount of distance as it did when it was merely
  dropped from rest.
• The projectile travels with a constant horizontal
  velocity and a downward vertical acceleration.
• http//www.physicsclassroom.com/mmedia/vectors/bds
  .gif

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3.4 Projectile Motion

• Suppose there is a mark on the wall and you are
  using a sling shot to hit the mark with a ball.
  Where should you aim?
• Higher than the mark on the wallthe ball will
  fall (free fall) vertically as it travels through
  the air
• Suppose it takes 1s for the ball to get to the
  wall how high should you aim?
• 5m above the mark
• d ½ (10 m/s2)(1s)2 5m

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3.4 Projectile MotionLaunched Horizontally

• Vertical Motion of a Projectile
• vyf vyo ayt vyf2 vyo2 2aydy dy vyot
  ½ ayt2
• Horizontal Motion of a Projectile
• vx vox constant dx vxt
• http//www.physicsclassroom.com/mmedia/vectors/hlp
  .gif
• A baseball rolls off a 0.70m high desk and
  strikes the floor 0.25m away from the base of the
  desk. How fast was the ball rolling?
• 2. A 10g marble rolls down a ramp and off of a
  table with a horizontal velocity of 1.2m/s. The
  marble fell in a cup placed 0.51m away from the
  tables edge. How high is the table?

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3.4 Projectile Motion

• Bert is standing on a ladder picking apples in
  his grandfathers orchard. As he pulls each
  apple off the tree, he tosses it into a basket
  that sits on the ground 3.0 m below at a
  horizontal distance of 2.0 m from Bert. How fast
  must Bert throw the apples horizontally in order
  for them to land in the basket?

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3.4 Projectile Motion

• Suppose an airplane drops a flare while it is
  moving with a constant horizontal speed at an
  elevated height. Assuming that air resistance is
  negligible, where will the flare land relative to
  the plane?
• a. directly below the plane
• b. below the plane and ahead of it
• c. below the plane and behind it.
• The answer is A. The horizontal motion of the
  flare remains the same, therefore the flare will
  always be positioned directly below the plane.
  The force of gravity causes the flare to fall,
  but does not affect the flares horizontal motion.
• http//www.physicsclassroom.com/mmedia/vectors/pap
  .gif
• http//www.physicsclassroom.com/mmedia/vectors/tb.
  gif

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3.5 Upwardly Launched Projectilesat an angle

• Use trigonometry to find the horizontal (x) and
  vertical (y) components of the velocity
• Separate these componentsTIME is the same for
  both
• http//www.physicsclassroom.com/mmedia/vectors/nhl
  p.gif
• Vertical Equations (a 10 m/s2) Horizontal
  Equations
• vf vo at d vt
• vf2 vo2 2ad vf vo
• d vot ½ at2
• d ½ (vo vf)t
• A golfer hits a golf ball at an angle of 25.0? to
  the ground with a speed of 55.0 m/s. Find the
  time of flight and the horizontal range of the
  golf ball.

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3.5 Upwardly Launched Projectilesat an angle

• A circus performer was fired out of a cannon with
  a speed of 24.0m/s at and angle of 40.0? to the
  horizontal. If he landed in a net 56.6m away at
  the same height from which he was fired, how long
  was he in the air?
• A football is kicked with an initial velocity of
  25 m/s at an angle of 45-degrees with the
  horizontal. Determine the time of flight, the
  horizontal displacement, and the peak height of
  the football.

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3.5 Upwardly Launched Projectiles

• The diagram below shows the trajectory for a
  projectile launched non-horizontally from an
  elevated position on top of a cliff. The initial
  horizontal and vertical components of the
  velocity are 8 m/s and 20 m/s as shown in the
  diagram. Positions of the object at 1-second
  intervals are shown.
• Determine the horizontal and vertical velocities
  at each instant shown in the diagram.

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3.6 Fast Moving Projectiles--Satellites

• Satellite a projectile traveling fast enough to
  fall around the Earth rather than into it
• If a ball were thrown fast enough and it curved
  path matched the Earths curved path, then the
  ball would become a satellite
• Satellites experience gravity even though they
  are far away from the Earth
• Satellites are launched to high altitudes where
  air resistance is negligible