Projectile Motion

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Projectile Motion

• Projectile motion a combination of horizontal
  motion with constant horizontal velocity and
  vertical motion with a constant downward
  acceleration due to gravity.
• Projectile motion refers to the motion of an
  object that is thrown, or projected, into the air
  at an angle. We restrict ourselves to objects
  thrown near the Earths surface as the distance
  traveled and the maximum height above the Earth
  are small compared to the Earths radius so that
  gravity can be considered to be constant.

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Projectile Motion

• The motion of a projectile is determined only by
  the objects initial velocity and gravity.
• The vertical motion of a projected object is
  independent of its horizontal motion.
• The vertical motion of a projectile is nothing
  more than free fall.
• The one common variable between the horizontal
  and vertical motions is time.

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Path of a Projectile

• A projectile moves horizontally with constant
  velocity while being accelerated vertically. A
  right angle exists between the direction of the
  horizontal and vertical motion the resultant
  motion in these two dimensions is a curved path.
• The path of a projectile is called its
  trajectory.
• The trajectory of a projectile in free fall is a
  parabola.

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Path of a Projectile
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Path of a Projectile

• vo initial velocity or resultant velocity
• vx horizontal velocity
• vyi initial vertical velocity
• vyf final vertical velocity
• R maximum horizontal distance (range)
• x horizontal distance
• Dy change in vertical position
• yi initial vertical position
• yf final vertical position
• q angle of projection (launch angle)
• H maximum height
• ag gravity 9.8 m/s2

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Path of a Projectile
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Path of a Projectile

• The horizontal distance traveled by a projectile
  is determined by the horizontal velocity and the
  time the projectile remains in the air. The time
  the projectile remains in the air is dependent
  upon gravity.
• Immediately after release of the projectile, the
  force of gravity begins to accelerate the
  projectile vertically towards the Earths center
  of gravity.

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Path of a Projectile

• The velocity vector vo changes with time in both
  magnitude and direction. This change is the
  result of acceleration in the negative y
  direction (due to gravity). The horizontal
  component (x component) of the velocity vo
  remains constant over time because there is no
  acceleration along the horizontal direction
• The vertical component (vy) of the velocity vo is
  zero at the peak of the trajectory. However,
  there is a horizontal component of velocity, vx,
  at the peak of the trajectory.

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Path of a Projectile
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Path of a Projectile

• In the prior diagram, r is the position vector of
  the projectile. The position vector has x and y
  components and is the hypotenuse of the right
  triangle formed when the x and y components are
  plotted.
• The velocity vector vo?t would be the
  displacement of the projectile if gravity were
  not acting on the projectile.
• The vector 0.5?ag?t2 is the vertical displacement
  of the projectile due to the downward
  acceleration of gravity.
• Together, this determines the vertical position
  for the projectile
• ?y (vyt) (0.5agt2)

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Path of a Projectile
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Problem Solving Projectile Motion

• Analyze the horizontal motion and the vertical
  motion separately. If you are given the velocity
  of projection, vo, you may want to resolve it
  into its x and y components.
• Think for a minute before jumping into the
  equations. Remember that vx remains constant
  throughout the trajectory, and that vy 0 m/s at
  the highest point of any trajectory that returns
  downward.

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Horizontal velocity component

• vx is constant because there is no acceleration
  in the horizontal direction if air resistance is
  ignored.

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Vertical velocity component

• At the time of launch
• After the launch
• If vy positive, direction of vertical motion is
  up if vy negative, direction of vertical motion
  is down if vy 0, projectile is at highest
  point.

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Horizontal position component

• If you launch the projectile horizontally
• then vo vx
• vyi 0 m/s
• ? 0o

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Vertical position component
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Relationship Between Vertical and Horizontal
Position

• this equation is only valid for launch angles in
  the range 0? lt ? lt 90?

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Range (total horizontal displacement)
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Maximum Height
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When Do The Range Maximum Height Equations Work?

• Does not work when ?y ? 0.

• Works when ?y 0.

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Determining vo from vx and vy

• If the vertical and horizontal components of the
  velocity are known, then the magnitude and
  direction of the resultant velocity can be
  determined.
• Magnitude

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Determining vo from vx and vy

• Direction from the horizontal
• Direction from the vertical

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Range and Angle of Projection
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Range and Angle of Projection

• The range is a maximum at 45? because sin (245)
  1.
• For any angle ? other than 45?, a point having
  coordinates (x,0) can be reached by using either
  one of two complimentary angles for ?, such as 15
  ? and 75 ? or 30 ? and 50 ?.

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Range and Angle of Projection

• The maximum height and time of flight differ for
  the two trajectories having the same coordinates
  (x, 0).
• A launch angle of 90 (straight up) will result
  in the maximum height any projectile can reach.

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For Objects Shot Horizontally

• vx constant
• Dy negative
• Dy -height

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For Objects Shot Horizontally

• When hits at bottom
• Vyf should be negative
• vo resultant velocity

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For Objects Shot Horizontally

• q with horizontal
• q with vertical

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For Situations in Which Dy 0 m
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For Situations In Which Dy Positive
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For Situations In Which Dy Positive

• At any point in the flight

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For Situations In Which Dy Negative
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For Situations In Which Dy Negative

• At launch
• After launch
• When it hits ground