1-Dimensional Kinematics

PhysicsUniform Acceleration Equations
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1-Dimensional Kinematics

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  1. 1-Dimensional Kinematics

    Slide 1 - 1-Dimensional Kinematics

    • Honors Physics
  2. Changes in Position

    Slide 2 - Changes in Position

    • Displacement
    • ∆ x : change in position
    • Velocity: rate of change in position
    • vav = ∆ x
    • ∆ t
    • “average” : over an interval
    • Slope on “x vs. t” graph: velocity
  3. Velocity

    Slide 3 - Velocity

    • “instantaneous” : at a given moment
    • v = ∆ x as ∆ t  zero
    • ∆ t
    • v = lim ∆ x a.k.a. dx
    • ∆ t dt
    • Slope of tangent to curve on “x vs. t” graph
    • ∆ t  zero
  4. Changes in Velocity

    Slide 4 - Changes in Velocity

    • ∆v : change in velocity
    • Acceleration: rate of change in velocity
    • aav = ∆ v
    • ∆ t
    • “average” : over an interval
    • Slope on “v vs. t” graph: acceleration
    • Area under “v vs. t” graph: displacement
  5. Acceleration

    Slide 5 - Acceleration

    • “instantaneous” : at a given moment
    • a = ∆v as ∆t  zero
    • ∆t
    • a = lim ∆ v a.k.a. dv
    • ∆ t dt
    • Slope of tangent to curve on “v vs. t” graph
    • ∆ t  zero
  6. Changes in Acceleration

    Slide 6 - Changes in Acceleration

    • Jerk: rate of change in acceleration
    • Slope on “a vs. t” graph: Jerk
    • Area under “a vs. t” graph: change in velocity
  7. Uniformly Accelerated Motion

    Slide 7 - Uniformly Accelerated Motion

    • Since vav = vi + vf
    • 2
    • and vav = ∆ x thus.. ∆ x = vav ∆ t
    • ∆ t
    • ∆ x = vi + vf ∆ t
    • 2
  8. Uniformly Accelerated Motion

    Slide 8 - Uniformly Accelerated Motion

    • aav = ∆v = vf – vi
    • ∆t tf – ti
    • a = vf – vi
    • t – 0
    • vf = vi + at
  9. Uniformly Accelerated Motion

    Slide 9 - Uniformly Accelerated Motion

    • Since vav = ½ (vi + vf)
    • and vf = vi + at
    • vav = ½ (vi + vf) = ½ [vi + (vi + at)] = vi + ½ at
    • Since ∆ x = vav ∆ t
    • ∆ x = (vi + ½ a ∆ t) ∆ t
    • ∆ x = vi ∆ t + ½ a (∆ t)2
  10. Uniformly Accelerated Motion

    Slide 10 - Uniformly Accelerated Motion

    • Since ∆ x = vi t + ½ at2
    • And a = vf – vi so: t = vf – vi
    • t a
  11. Uniformly Accelerated Motion

    Slide 11 - Uniformly Accelerated Motion

    • Since ∆ x = vi t + ½ at2
    • And a = vf – vi so: t = vf – vi
    • t a
    • ∆ x = vi vf – vi + ½ a vf – vi 2
    • a a
    • vf 2 = vi 2 + 2a ∆ x
    • (
    • )
    • )
    • (
  12. Uniformly Accelerated Motion

    Slide 12 - Uniformly Accelerated Motion

    • Full Equations
    • ∆ x = [(vi + vf)/2]∆ t
    • vf = vi + at
    • ∆ x = vi ∆ t + ½ a (∆ t)2
    • vf 2 = vi 2 + 2a ∆ x
    • Zero Initial Velocity
    • ∆ x = (vf/2)∆ t
    • vf = at
    • ∆ x = ½ a (∆ t)2
    • vf 2 = 2a ∆ x