Projectile Motion

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Projectile Motion
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Objects in projectile motion follow a parabolic
path called a trajectory.
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If released at the same instant, a bullet shot
from a gun will hit the ground at the same time
as a bullet dropped from the same height.
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The horizontal and vertical motions of a
projectile are independent, meaning they do not
affect each other.
Horizontal motion
Vertical motion
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The 2-D motion of a projectile can be separated
into two 1-D motions horizontal and vertical.
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HPM Horizontal Projectile Motion
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The horizontal motion of a projectile is always
constant, if we neglect air resistance.
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For projectiles shot at 0, all of the initial
velocity is in the x direction. Thus, Vyi 0 m/s.
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For projectiles shot at 0, vertical displacement
and velocity will always be negative.
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To hit the target, when should you release the
package?
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Rules for Projectile Motion

• Treat horizontal and vertical as two separate
  sides of the problems
• TIME is the key, and the only variable that can
  be used for both horizontal and vertical
• Horizontal Motion is always constant
• vx is constant
• ax 0 m/s2
• Objects follow a parabolic shape

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Horizontal Projectile Motion

• All of the initial velocity is in the x
  direction, Vyi 0 m/s
• Vertical displacement and velocity will always be
  negative

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Example problem
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APM Angled Projectile Motion
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For projectiles shot at an angle, initial
velocity is both vertical and horizontal.
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Horizontal velocity and initial vertical velocity
can be found using trig functions.
Vi
Vx Vi Cos ?
Vyi
?
Vyi Vi Sin ?
Vx
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At the end of the problem, you can recombine
horizontal and vertical velocity to get the total
2-D velocity.
Vx
Vf
Vyf
Vx
Vf2 Vx2 Vyf2
?
Vyf
Vf
? tan-1(Vyf / Vx)
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For APM, vy at any height is the same while going
up and coming down except for direction.
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Velocities of projectile motionNote Vy 0 at
the highest point.
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With air resistance, the actual path is shorter.
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Angled Projectile Motion

• Initial velocity is both vertical and horizontal
• Use trig functions to find vyi and vx
• Vx Vi Cos ?
• Vyi Vi Sin ?
• Remember clues
• vy at the top is 0 m/s
• vy at any height is the same while going up and
  coming down except for direction

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Example Problem

• Happy Gilmore hits his shot at 55.0 m/s with an
  angle of 50.0 to the ground. How far did the
  ball travel before it lands?

• vi 55.0 m/s
• ? 50.0
• ay -9.81 m/s2
• ?x ?

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vi 55.0 m/s ay -9.81 m/s2 ?x ?? 50.0

• Find vx and vyi

• vx vi Cos ?
• 55.0 m/s Cos (50.0)
• 35.4 m/s

• vyi vi Sin ?
• 55.0 m/s Sin (50.0)
• 42.1 m/s

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vi 55.0 m/s ay -9.81 m/s2 ?x ??
50.0 vyi 42.1 m/s vx 35.4 m/s

• What do we need to find ?x?
• Time! Find time from the vertical side
• ?y vyi ?t ½ ay ?t2
• 0 m (42.1 m/s) ?t ½ (-9.81 m/s2) ?t2
• - (42.1 m/s) ?t ½ (-9.81 m/s2) ?t2
• (42.1 m/s) ½ (9.81 m/s2) ?t
• ?t 8.58 s

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vi 55.0 m/s ay -9.81 m/s2 ?x ??
50.0 vyi 42.1 m/s vx 35.4 m/s?t
8.58 s

• Now we can find ?x
• ?x vx ?t
• (35.4 m/s)(8.58 s)
• 304 m

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Where Should We Aim the Cannon?
At or Above the monkey?
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Above the Monkey
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At the Monkey
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Explanation of monkey experiment
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Ranges of projectiles versus angle.
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Another interesting application
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If speed is great enough
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Thats how the space shuttle and satellites orbit
the earth.