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Projectiles in Two Dimensions

Projectiles in Two Dimensions. projectile – an object that moves only under the influence of gravity. trajectory – the path taken by a projectile The trajectory of all two dimensional projectiles is in the shape of a parabola.

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Projectiles in Two Dimensions

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  1. Projectiles in Two Dimensions projectile – an object that moves only under the influence of gravity. trajectory – the path taken by a projectile The trajectory of all two dimensional projectiles is in the shape of a parabola. All projectiles accelerate vertically by the same rate (-9.80 m/s2), regardless of the horizontal velocity. The horizontal acceleration is zero. The horizontal and vertical velocity components of projectiles are independent. The velocity is always tangent to the path of the projectile. Projectile Animation

  2. Horizontal Projectiles • A projectile with no initial vertical velocity.

  3. Another Horizontal Projectile Example Horizontal Projectile

  4. A Third Horizontal Projectile Example Horizontal Projectile

  5. Horizontal Projectiles vx • A horizontal projectile is a projectile given an initial horizontal velocity with no initial vertical velocity (viy = 0) vix θ v vx y vy θ x vy v vx θ ymax vy v vx θ v vy vx θ Range = R =xmax v vy

  6. Two Dimensional Projectile Equations ymax = maximum vertical displacement R = Range or xmax (maximum horizontal displacement) vix = initial horizontal velocity viy = initial vertical velocity vy = vertical velocity at any time after the initial vx = horizontal velocity at any time other than the initial v = resultant velocity magnitude (speed) θ = the angle that the resultant velocity makes with the horizontal y = vertical displacement x = horizontal displacement tT = total flight time vy =viy+gt vy2=viy2+2gy y=viyt+ ½ gt2 x=vixt=vxt R = xmax = vix(tT) = vx(tT) v = v, θ

  7. Angled Projectiles A projectile with both an initial horizontal and vertical velocity. vi y max viy vix Range = R = xmax

  8. Angled Projectiles vy=0 vy v vx y vy v θ vy v vx vx x v θ θ vy vx vy v θ vx v θ vx vy vy v ymax θ vx vi vx θ viy vy v θ vix vx θ Range = R =xmax v vy vix=vi cos θ viy=vi sin θ

  9. Another Angled Projectile Example • As long the object has both an initial horizontal and vertical velocity • it is considered an angled projectile. Why does the ball land in the truck? • The ball land in the truck because it has the same horizontal velocity as the truck.

  10. Maximum Height and Flight Time vy=0 Half of total flight time (t1/2) Maximum height ymax tT t = 0 Flight Time Maximum Height vy =viy+gt vy2=viy2+2gy vy=0 At half the flight time vy=0 At maximum height 0=viy+g(t1/2) 0=viy2+2g(ymax) Half of the total flight time Maximum height Total flight time

  11. Maximum Height and Flight Time vy=0 Half of total flight time (t1/2) Maximum height ymax tT t = 0 Flight Time Equations: Equations are valid only for symmetric flight paths

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