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 V avg = (ř 2 – ř 1 )/(t 2 -t 1 ) = Δ ř / Δt Δt is scalar so, V vector parallel to ř vector

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

5. Average Acceleration â a vg = (v’ 2 – v’ 1 )/(t 2 -t 1 ) = Δ v’ / Δt â vector parallel to Δ v’ vector

6. Instantaneous Acceleration â = lim (Δ v’ / Δt) as Δt  0 = d v’ / dt Has the 3 components: â x = d v x / dt, etc These components could also be written with respect to position vector: â x = d 2 x / dt 2, etc

7. Parallel and Perpendicular Components 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) R V x y (x,y)

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 : Cartesian:Polar: Position: x, yPosition: R, θ Velocity: Vx = dx/dt, Vy = dy/dt Velocity: dR/dt, dθ/dt (let ω=dθ/dt) Vx,Vy always changing dR/dt =0 dθ/dt=ω=constant

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

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

18. Instantanous Acceleration in UCM This is calledThis is called Centripetal Acceleration. Like triangles, ΔR and ΔV: But  R = v  t for small  t So: Thus: a = V 2 /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: Reference FrameA 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: