Using Parametric Equations to Model Motion

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Section 8-7

• Using Parametric Equations to Model Motion

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Vocabulary

• Projectiles objects that are launched
• Trajectory the path of the projectile
• Range horizontal distance that a projectile
  travels
• Motion described in terms of its position,
  velocity, and acceleration

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Parametric Equations

• X tvcos?
• Y tvsin? ½ gt2 h0

t time v initial velocity g gravity, -9.8
m/s2 or -32 ft/s2 h0 initial height
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Example 1

• A football player kicks a ball with an initial
  velocity of 40 ft/s at a 40 angle with the
  ground.
• How far has the ball traveled after 0.8 seconds?
• First write the equations
• X t(40)cos40 30.64t
• Y t(40)sin40 ½ (-32)t2 25.71t 16t2

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Continued

• Plug in 0.8 for t
• X 30.64(.8) 24.5 ft. (horiz.)
• Y 25.71(.8) 16 (.8)2 10.3 ft. (vert.)

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Example 2

• Hole 4 is 220 yds. to the pin. The initial
  velocity is 160 ft/s with a 28 7-wood.
• Can the ball reach the pin?
• Write the equations
• X t(160)cos28 141.3t
• Y t(160)sin28 ½ (-32)t2 75.1t 16t2

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Continued

• First find t when y 0.
• 0 75t 16t2
• 0 t (75 16t)
• t 0 or t 75/16 4.7 seconds
• Now find x after 4.7 seconds
• X 141.3t
• X 141.3(4.7) 662 feet
• 662 ft. 220.7 yards
• So Yes!?

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Example 3

• An archer shoots an arrow with an initial
  velocity of 60 m/s at an angle of 4.8 with the
  horizontal.
• The target is 70 meters away and the bow is held
  1.4 meters off the ground.
• How far off the ground will the arrow be when it
  hits the target?

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Continued

• First write the equations
• X t(60)cos4.8 59.8t
• Y t(60)sin4.8 ½ (-9.8)t2 1.4
• 5.0t 4.9t2 1.4
• Now, find out how long it takes to hit the
  target.
• 70 59.8t
• t 1.17 seconds

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Continued

• Plug in t 1.17 into the equation for y.
• Y 5.0t 4.9t2 1.4
• 5.0(1.17) 4.9(1.17)2 1.4
• .54
• Therefore, the arrow will be .54 meters off the
  ground when it hits the target.

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Homework

• Page 453-455
• 8-12, 16-17