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Kinematics of 3- or 2-dimensional motion

Kinematics of 3- or 2-dimensional motion. z. Position vector:. Average velocity:. Instantaneous velocity:. y. x. Average acceleration:. Instantaneous acceleration:. a || → magnitude of velocity a ┴ → direction of velocity. Equations of 3-D Kinematics for Constant Acceleration.

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Kinematics of 3- or 2-dimensional motion

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  1. Kinematics of 3- or 2-dimensional motion z Position vector: Average velocity: Instantaneous velocity: y x Average acceleration: Instantaneous acceleration: a||→ magnitude of velocity a┴ → direction of velocity

  2. Equations of 3-D Kinematics for Constant Acceleration Result:3-D motion with constant acceleration is a superposition of three independent motions along x, y, and z axes.

  3. Projectile Motion ax=0 → vx=v0x=const ay= -g → vy= voy- gt x = x0 + vox t y = yo + voy t – gt2/2 v0x= v0 cos α0 v0y= v0 sin α0 tan α = vy / vx Exam Example 6: Baseball Projectile Data: v0=22m/s, α0=40o (examples 3.7-3.8, problems 3.12) Find: (a) Maximum height h; (b) Time of flight T; (c) Horizontal range R; (d) Velocity when ball hits the ground Solution: v0x=22m/s·cos40o=+17m/s; v0y=22m/s·sin40o=+14m/s • vy=0 → h = (vy2-v0y2) / (2ay)= - (14m/s)2 / (- 2 · 9.8m/s2) = +10 m • y = (v0y+vy)t / 2 → t = 2y / v0y= 2 · 10m / 14m/s = 1.45 s; T = 2t =2.9 s • R = x = v0x T = 17 m/s · 2.9 s = + 49 m • vx = v0x , vy = - v0y

  4. Motion in a Circle • Uniform circular motion: • v = const Centripetal acceleration: Magnitude: ac = v2 / r Direction to center: (b) Non-uniform circular motion: v ≠ const

  5. Exam Example 7: Ferris Wheel (problems 3.29) Data: R=14 m, v0 =3 m/s,a|| =0.5 m/s2 • Find: • Centripetal acceleration • Total acceleration vector • Time of one revolution T Solution: (a) Magnitude: ac =a┴ = v2 / r Direction to center: θ (b) (c)

  6. Relative Velocity c Flying in a crosswind Correcting for a crosswind

  7. Exam Example 8: Relative motion of a projectile and a target (problem 3.56) y Data: h=8.75 m, α=60o, vp0 =15 m/s, vtx =-0.45 m/s Find: (a) distance D to the target at the moment of shot, (b) time of flight t, (c) relative velocity at contact. Solution: relative velocity (c) Final relative velocity: (b) Time of flight (a) Initial distance x 0

  8. Principles of Special Theory of Relativity (Einstein 1905): • Laws of Nature are invariant for all inertial frames of reference. • (Mikelson-Morly’s experiment (1887): There is no “ether wind” ! ) • 2. Velocity of light c is the same for all inertial frames and sources. Relativistic laws for coordinates transformation and addition of velocities are not Galileo’s ones: y y’ Lorentz transformation x’ V x Proved by Fizeau experiment (1851) of light dragging by water Contraction of length: Slowing down of time: Twin paradox Slowing and stopping light in gases(predicted at Texas A&M)

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