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Physics 2011

Physics 2011. Chapter 3: Motion in 2D and 3D. Describing Position in 3-Space. A vector is used to establish the position of a particle of interest. The position vector, r , locates the particle at some point in time. Average Velocity in 3-D. V avg = (ř 2 – ř 1 )/(t 2 -t 1 ) = Δ ř / Δ t

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Physics 2011

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  1. Physics 2011 Chapter 3: Motion in 2D and 3D

  2. Describing Position in 3-Space • A vector is used to establish the position of a particle of interest. • The position vector, r, locates the particle at some point in time.

  3. Average Velocity in 3-D • Vavg = (ř2 – ř1)/(t2-t1) = Δř/ Δt • Δt is scalar so, V vector parallel to ř vector

  4. Instantaneous Velocity • V’ = lim (Δř/ Δt) as Δt  0 = dř/ dt • 3 Components : V’x = dx / dt, etc • Magnitude, |V’| = SQRT( Vx^2 + Vy ^2 + Vz^2)

  5. Average Acceleration • âavg = (v’2 – v’1)/(t2-t1) = Δv’/ Δt • â vector parallel to Δv’vector

  6. Instantaneous Acceleration • â = lim (Δv’/ Δt) as Δt  0 = dv’/ dt • Has the 3 components: âx = d vx/ dt, etc • These components could also be written with respect to position vector: âx = d2x / dt2, etc

  7. Parallel and PerpendicularComponents of Acceleration

  8. Acceleration on Curve • Different for a) constant speed, b) increasing speed, c) decreasing speed

  9. Projectile Motion • Free Fall Problems in 2D or 3D are “Projectile Motion” problems • Projectile path is called a Trajectory

  10. Acceleration during Projectile Motion • The a vector is constant (g, gravity) and downward all along the projectile path

  11. 2D path, Acceleration Vector

  12. Equations for PM

  13. Uniform Circular Motion

  14. What defines UCM? • Constant SPEED (not velocity!) • Constant Radius (R = c) y V (x,y) R x

  15. UCM using Polar Coordinates • The polar coordinate system (magnitude and angle) is a natural way of describing UCM, where R and speed are constant:

  16. Velocity in Polar Form: • Displacement is an Arc, S, of the Circle • Displacement s = vt (like x = vt + xo) but s = R = Rt, so: v = ωR

  17. Average Acceleration in UCM: • Average Acceleration, aavg = ΔV/Δt • The Δ V vector points toward origin

  18. Thus: Instantanous Acceleration in UCM • This is called Centripetal Acceleration. • Like triangles, ΔR and ΔV: But R = vt for smallt So: a = V2/R

  19. Relative Motion • First thing: A Frame of Reference • Since Einstein, a distinction has to be made between references that behave classically and those that allow Relativity • Classical frames of reference are called Intertial

  20. Inertial Frames of Reference: • A Reference Frame is the place you measure from. • It allows you to nail down your (x,y,z) axes • An Inertial Reference Frame (IRF) is one that is not accelerating. • We will consider only IRFs in this course. Stationary or constant velocity • Valid IRFs can have fixed velocities with respect to each other. • More about this later when we discuss forces. • For now, just remember that we can make measurements from different vantage points.

  21. Consider the Frames in Relative Motion: • A plane flies due south from Duluth to MPLS at 100 m/s in a 15 m/s crosswind:

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