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, |\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): \vec{d} = \vec{r}{\text{final}} - \vec{r}{\text{initial}}.
Vector Addition and the Net Displacement
- Net displacement: \vec{c} = \vec{a} + \vec{b}.
- Addition is commutative: \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 \vec{d} = \vec{a} + \vec{b}.
Scalar Multiplication; Zero and Negative Vectors
- Scalar multiplication: \vec{b} = c \vec{a}. Magnitude scales by |c|; direction is the same as \vec{a} if c>0, opposite if c<0.
- Zero vector: \vec{0}; magnitude 0; direction undefined.
- Negative vector: -\vec{a} has the same magnitude as \vec{a} but opposite direction; \vec{a} + (-\vec{a}) = \vec{0}.
Vector Subtraction
- Subtraction as addition of a negative: \vec{a} - \vec{b} = \vec{a} + (-\vec{b}).
- Graphical subtraction: to subtract \vec{b} from \vec{a}, draw the tail of -\vec{b} at the tip of \vec{a}; the resultant is from the tail of \vec{a} to the tip of -\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): \vec{v}{P/T} = \vec{v}{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: |\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 \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 g.
- Vector decomposition allows quantitative calculations of the acceleration along the ramp and other 2D motion problems.
- 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.