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• Vavg = (ř2 – ř1)/(t2-t1) = Δř / Δt

• Δt is scalar so, V vector parallel to ř vector

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)

Average Acceleration

• âavg = (v’2 – v’1)/(t2-t1) = Δv’ / Δt

• â vector parallel to Δv’ vector

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

Parallel and PerpendicularComponents of Acceleration

Acceleration on Curve

• Different for a) constant speed, b) increasing speed, c) decreasing speed

Projectile Motion• Free Fall Problems in 2D or 3D are

“Projectile Motion” problems

• Projectile path is called a Trajectory

Acceleration during Projectile Motion

• The a vector is constant (g, gravity) and downward all along the projectile path

2D path, Acceleration Vector

Equations for PM

Uniform Circular Motion

What defines UCM?

• Constant SPEED (not velocity!)

• Constant Radius (R = c)

R

VV

x

y

(x,y)

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, y Position: 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

Velocity in Polar Form:

• Displacement is an Arc, S, of the Circle

• Displacement s = vt (like x = vt + xo)

but s = R = Rt, so:v = ωR

Average Acceleration in UCM:

• Average Acceleration, aavg = ΔV/Δt

• The Δ V vector points toward origin

Instantanous Acceleration in UCM

• This is called This is called Centripetal Acceleration.

• Like triangles, ΔR and ΔV:

But R = vt for small t

So:vt

vR

2 v

vv tR

vv

RR

Thus:

a = V2/R

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

Inertial Frames of Reference:• A Reference FrameReference 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.

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: