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Projectile Motion

Projectile Motion. Major Principles for All Circumstances Horizontal motion is constant velocity Vertical motion is constant downward acceleration a = g = -9.8 m/s 2. Projectile Motion. The Big Four + One More. In x x = v x t. In y y = v oy t + 1/2 gt 2 v y = v oy + gt

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Projectile Motion

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  1. Projectile Motion • Major Principles for All Circumstances • Horizontal motion is constant velocity • Vertical motion is constant downward acceleration • a = g = -9.8 m/s2

  2. Projectile Motion The Big Four + One More • In x • x = vxt • In y • y = voyt + 1/2 gt2 • vy = voy + gt • vy2 = voy2 + 2gy • y = 1/2(voy + vy)t • Ties them together Time & Launch Angle

  3. Projectile Motion Types of Projectile Problems • Type A - Half of a Parabola • Type B - Full Parabola - Symmetric • Type C - Partial or Asymmetric Parabola

  4. Projectile Motion Type A - Half of a Parabola • In the vertical direction • The object acts like a dropped object • Initial vertical velocity is zero; voy = 0

  5. Projectile Motion • t = (2y/g) Type A - Half of a Parabola WARNING: Be careful using shortcut formulas!!!! • To solve for time, often you will use... • y = voyt + 1/2 gt2 Therefore... since voy = 0 • y = 1/2 gt2

  6. Projectile Motion Type A - Half of a Parabola If the problem is reversed... Romeo throws a rock up to Juliet; hits window horizontally Because of symmetry, just solve the problem backwards, make voy = 0

  7. Projectile Motion Type B - Full Parabola Notice the ball lands back in the truck... only if the truck moves with constant velocity

  8. Projectile Motion Type B - Full Parabola If you solve for the full parabola... The vertical displacement is zero; y = 0 The time is the total hang time

  9. Projectile Motion Type B - Full Parabola If you solve for half the parabola... The vertical velocity at the peak is; vy = 0 The time is equal to half the hang time

  10. Projectile Motion Type B - Full Parabola The Range Formula WARNING: Use the triangle for velocities only!!!! vo voy = vosin vx = vocos

  11. Projectile Motion Type B - Full Parabola • x = vxt The Range Formula • x = (vocos)t voy = vosin vx = vocos

  12. Projectile Motion Type B - Full Parabola • x = vxt The Range Formula • x = (vocos)t vy = voy + gt -voy = voy + gt vy = -voy -2voy = gt voy = vosin -(2voy)/g = t -(2vosin)/g = t

  13. Projectile Motion Type B - Full Parabola • x = vxt The Range Formula • x = (vocos)t • x = (vocos)(-2vosin/g) -(2vosin)/g = t

  14. Projectile Motion Type B - Full Parabola • x = vxt The Range Formula • x = (vocos)t • x = (vocos)(-2vosin/g)

  15. Projectile Motion Type B - Full Parabola • x = vxt The Range Formula • x = (vocos)t • x = (vocos)(-2vosin/g) • x = -vo2(2sincos)/g Trig Identity: 2sincos = sin2 • x = -vo2sin2/g

  16. Projectile Motion Type B - Full Parabola • x = vxt The Range Formula • x = (vocos)t • x = (vocos)(-2vosin/g) • x = -vo2(2sincos)/g • x = -vo2sin2/g

  17. Projectile Motion Type B - Full Parabola • x = vxt The Range Formula • x = (vocos)t • x = (vocos)(-2vosin/g) WARNING: Be careful using shortcut formulas!!!! • x = -vo2(2sincos)/g • x = -vo2sin2/g

  18. Projectile Motion Type B - Full Parabola • Optimum Angle of 45 • Maximum range • Supplementary Angles • Equal ranges

  19. Projectile Motion Type C - Partial or Asymmetric Parabola • Some problems can be treated as two Type A problems • Each problem is unique, so take your time and... stick to your major principles from the beginning

  20. Projectile Motion Type C - Partial or Asymmetric Parabola • y = voyt + 1/2gt2 Very Unique Equation We don’t know time, but we must find out the height (y) of an object.

  21. Projectile Motion vo voy = vosin vx = vocos Type C - Partial or Asymmetric Parabola • y = voyt + 1/2gt2 • y = (vosin)t + 1/2gt2 Very Unique Equation

  22. Projectile Motion Type C - Partial or Asymmetric Parabola • y = voyt + 1/2gt2 • y = (vosin)t + 1/2gt2 Very Unique Equation x = vxt vo x = (vocos)t voy = vosin x/(vocos) = t vx = vocos

  23. Projectile Motion Type C - Partial or Asymmetric Parabola • y = voyt + 1/2gt2 • y = (vosin)t + 1/2gt2 Very Unique Equation • y = vosin(x/(vocos)) + 1/2g(x/(vocos))2 x/(vocos) = t

  24. Projectile Motion Type C - Partial or Asymmetric Parabola • y = voyt + 1/2gt2 • y = (vosin)t + 1/2gt2 Very Unique Equation • y = vosin(x/(vocos)) + 1/2g(x/(vocos))2 • y = x(sin/cos) + 1/2g(x2/(vo2cos2)) • y = xtan + gx2/(2vo2cos2)

  25. Projectile Motion Type C - Partial or Asymmetric Parabola • y = voyt + 1/2gt2 • y = (vosin)t + 1/2gt2 Works for all types of problems, !! Most useful with Type C!! Very Unique Equation • y = vosin(x/(vocos)) + 1/2g(x/(vocos))2 • y = x(sin/cos) + 1/2g(x2/(vo2cos2)) • y = xtan + gx2/(2vo2cos2)

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