Chapter 3 Preview: and 3.1 Vectors in Motion (Two Dimensions)

Overview

  • Chapter 3 preview: vectors in motion in two dimensions; main tool is vectors. Dark green vectors = velocity; light green components = horizontal and vertical velocities.
  • Goals: learn to find vector components and use them to solve problems (projectile motion, circular motion); use vectors to analyze two-dimensional motion; relate to free fall.

Vectors and Components

  • A vector is a quantity with magnitude and direction; magnitude is represented without an arrow. For velocity, ∣v⃗∣=v|\vec{v}| = v.
  • Direction is given by the orientation of the arrow. The components show motion along coordinate axes, e.g., horizontal and vertical components.

Displacement and Magnitude

  • Displacement is the straight-line vector from the initial to final position. Magnitude equals the straight-line distance.
  • Two vectors are equal if they have the same magnitude and direction, independent of where they are drawn.
  • Displacement (example): d⃗=r⃗<em>final−r⃗</em>initial\vec{d} = \vec{r}<em>{\text{final}} - \vec{r}</em>{\text{initial}}.

Vector Addition and the Net Displacement

  • Net displacement: c⃗=a⃗+b⃗\vec{c} = \vec{a} + \vec{b}.
  • Addition is commutative: a⃗+b⃗=b⃗+a⃗\vec{a} + \vec{b} = \vec{b} + \vec{a}.

Graphical and Parallelogram Addition

  • Tip-to-tail (graphical) addition: place the tail of one vector at the tip of the other; the resultant is from the tail of the first to the tip of the second.
  • Parallelogram rule: if two vectors share the same initial point, their sum is the diagonal of the parallelogram formed by the two vectors; equivalently, the diagonal equals d⃗=a⃗+b⃗\vec{d} = \vec{a} + \vec{b}.

Scalar Multiplication; Zero and Negative Vectors

  • Scalar multiplication: b⃗=ca⃗\vec{b} = c \vec{a}. Magnitude scales by ∣c∣|c|; direction is the same as a⃗\vec{a} if c>0, opposite if c<0.
  • Zero vector: 0⃗\vec{0}; magnitude 0; direction undefined.
  • Negative vector: −a⃗-\vec{a} has the same magnitude as a⃗\vec{a} but opposite direction; a⃗+(−a⃗)=0⃗\vec{a} + (-\vec{a}) = \vec{0}.

Vector Subtraction

  • Subtraction as addition of a negative: a⃗−b⃗=a⃗+(−b⃗)\vec{a} - \vec{b} = \vec{a} + (-\vec{b}).
  • Graphical subtraction: to subtract b⃗\vec{b} from a⃗\vec{a}, draw the tail of −b⃗-\vec{b} at the tip of a⃗\vec{a}; the resultant is from the tail of a⃗\vec{a} to the tip of −b⃗-\vec{b}.

Relative Motion

  • Relative motion example: motion with respect to one frame plus the motion of that frame relative to another equals the motion with respect to the third frame.
  • Formal relation (velocity): v⃗<em>P/T=v⃗</em>P/A+v⃗A/T\vec{v}<em>{P/T} = \vec{v}</em>{P/A} + \vec{v}_{A/T}.
  • Concept: motion of a propeller with respect to air plus air motion with respect to the table gives motion with respect to the table.

Two-Dimensional Motion and Components

  • In 1D we did straight-line motion; now motion is curved and requires two components (horizontal and vertical).
  • A velocity vector has magnitude and direction, shown as an arrow. Magnitude: ∣v⃗∣=v|\vec{v}| = v.
  • To analyze, decompose motion into components along chosen axes; velocity and position are described using vectors.

Gravity, Free Fall, and Ramp Example (Concepts)

  • Free fall acceleration is g≈9.8 m s−2g \approx 9.8\ \mathrm{m\,s^{-2}} downward.
  • When an object moves on a ramp, gravity contributes a component along the ramp; the acceleration along the ramp is a component of the free-fall acceleration and is less than gg.
  • Vector decomposition allows quantitative calculations of the acceleration along the ramp and other 2D motion problems.

Summary of Toolkit

  • Key ideas: vectors, components, vector addition/subtraction, scalar multiplication, zero/negative vectors, and relative motion.
  • In 2D motion, resolve into components and apply vector algebra to analyze.