Vectors & 2D Motion - Lecture Notes

Math Review

Trigonometry & Coordinate Systems

• Trigonometric Functions:
  • Defined as ratios of sides of right triangles
  • Soh Cah Toa
    • Sine: $\sin \theta \equiv \frac{o}{h}$
    • Cosine: $\cos \theta \equiv \frac{a}{h}$
    • Tangent: $\tan \theta \equiv \frac{o}{a}$
  • Where:
    • $h$ = hypotenuse
    • $\theta$ = theta (angle)
    • $o$ = opposite
    • $a$ = adjacent
• Pythagorean Theorem:
  • $h^2 = o^2 + a^2$
  • Also known as $a^2 + b^2 = c^2$
• Cartesian Coordinates:
  • 1D requires one number to specify location
  • 2D requires two numbers, e.g., $(x, y)$
  • 3D requires three numbers, e.g., $(x, y, z)$
  • Use a right-handed coordinate system
• 2D Polar Coordinates: $(r, \theta)$
  • Two coordinates:
    • Radial distance: $r$
    • Angle counterclockwise from x-axis: $\theta$
  • Cartesian to polar conversions:
    • $r = \sqrt{x^2 + y^2}$
    • $\theta = \tan^{-1} \frac{y}{x}$
  • Polar to cartesian conversions:
    • $x = r \cos \theta$
    • $y = r \sin \theta$

Vectors

Describing Vectors

• In 2D or 3D, direction requires more than a sign; it's indicated by arrows.
• In math, these arrows are vectors.
• Examples of physics vectors:
  • Motion variables: Position, velocity, and acceleration
  • Forces
  • Momentum
• Scalars vs. Vectors:
  • Scalar: A single number (e.g., mass, GPA, height)
  • Vector: An arrow in space (e.g., position, velocity, acceleration)

Vectors Part 1

• Equivalent Descriptions of Vectors:
  • Magnitude & angles (like polar coordinates)
  • Components (like Cartesian coordinates)
    • Of tip with origin at base
  • Conversion formulas:
    • $r = \sqrt{x^2 + y^2}$
    • $\theta = \tan^{-1} \frac{y}{x}$
    • $x = r \cos(\theta)$
    • $y = r \sin(\theta)$
• Component Triangles:
  • Vector & components make a right triangle
  • Sides:
    • Magnitude = hypotenuse
    • Components = opposite & adjacent
  • Different triangles for different axes
    • Components align with axes
• Finding Vector Components:
  • Method #1: Standard formula with standard angle (counter-clockwise off +x axis)
    • May need to find standard angle
    • $x = r \cos(\theta)$
    • $y = r \sin(\theta)$
  • Method #2: Component triangles
    • Find $x$ and $y$ using provided angle
    • Select correct sign for $x$ and $y$
    • Standard $x$: + rightward, - leftward
    • Standard $y$: + upward, - downward
• Finding Vector Components: Easy Cases
  • Components easiest when vector is aligned with axes
  • Component options:
    • $\pm$ magnitude
    • Axis the vector points along
    • Zero
    • Other axes
• Vector Notation:
  • To indicate a quantity is a vector:
    • Arrow above symbol $\vec{A}$
    • Bold font A
    • Both $\vec{A}$
  • $\vec{A} + \vec{B} = \vec{C}$ means “A” + “B” = “C”, NOT the sum of lengths!
• Vector Notation: Components & magnitude
  • Components: Symbol with subscript $x$ or $y$
  • Vector in equation: Parentheses around components
  • Magnitude: Absolute value $\mid \vec{A} \mid$, or omitted arrow
  • $\vec{A} = (A<em>x, A</em>y)$
  • $\mid \vec{A} \mid = A = \sqrt{A<em>x^2 + A</em>y^2}$
• 3D Vectors:
  • Basic idea: A vector with $x$, $y$, and $z$ components
  • Notation: $\vec{A} = (A<em>x, A</em>y, A_z)$
  • Magnitude: $\mid \vec{A} \mid = \sqrt{A<em>x^2 + A</em>y^2 + A_z^2}$

Vector Operations

Addition & Subtraction

• Vector addition definition:
  • Move arrows tip to tail
  • Sum runs from first tail to last tip
  • Works with multiple vectors
• Vector subtraction definition:
  • Move arrows tail to tail
  • Difference runs from second tip to first tip
• Vector Addition in Equations:
  • $\vec{A} + \vec{B} = \vec{C}$
  • Add components separately!
  • $A<em>x + B</em>x = C_x$
  • $A<em>y + B</em>y = C_y$
• Vector Subtraction in Equations
  • $\vec{A} - \vec{B} = \vec{C}$
  • Subtract components separately!
  • $A<em>x - B</em>x = C_x$
  • $A<em>y - B</em>y = C_y$
• Vector Equations & Component Equations
  • Each vector equation is equivalent to multiple component equations
    • Replace vector with component
    • Equation for each dimension
    • 2D vectors = 2 equations
    • 3D vectors = 3 equations
  • Warning: Special rules for other operations (dot product and cross product)

Scalar Multiplication

• Multiplying a vector by a scalar as repeated addition
  • Multiplying by 2 would double length
  • Multiplying by x would make magnitude x times longer
• Dividing a vector by a scalar is the same as multiplication
  • Dividing by 2 would halve length
    • Dividing by x would make magnitude x times shorter
• Multiplying a vector by a negative number
  • Subtraction = addition of a negative
  • Flipping direction flips component signs!
• Can swap subtraction for addition of flipped vector
  • Useful for chaining subtraction
• Scalar Multiplication Summary
  • Multiplies magnitude (graphical definition)
  • Multiplies each component (algebraic definition)
  • Negative signs flip direction (graphical definition)
  • $2 \vec{A} = 2(2,1) = (4,2)$
  • $-1 \vec{A} = -1(2,1) = (-2,-1)$

2D & 3D Motion

Position, Velocity and Acceleration Vectors

• 2D and 3D Reference Frames
  • 2 or 3 perpendicular axes creating a grid, like fictitious rulers
  • Different choices possible
    • Orientation
    • Location of origin
    • Motion relative to an object
  • Advice
    • Choose frame to align with vectors
• Position Vector, $\vec{r}$
  • Describe a location
  • Point from origin to location
  • SI units: Meters (m)
  • $\vec{r} = (x, y)$
• Displacement Vector, $\Delta \vec{r}$
  • $\Delta \vec{r} \equiv \vec{r}<em>f - \vec{r}</em>i$
  • Describes the change in position
  • Components
    • $\Delta \vec{r} = (\Delta x, \Delta y, \Delta z)$
• (Instantaneous) Velocity, $\vec{v} = (v<em>x, v</em>y)$
  • Points in direction of motion
  • Magnitude is speed
  • Mathematical definition:
    • $v_x \equiv \frac{dx}{dt}$
    • $v_y \equiv \frac{dy}{dt}$
  • Notes:
    • Often just called velocity
    • SI Unit: meters per second (m/s)
• Average Velocity, $\vec{v}_{avg} = (\frac{\Delta x}{\Delta t}, \frac{\Delta y}{\Delta t})$
  • On average, how quickly & in what direction is it moving over a specific interval of time
  • Mathematical definition
    • $\vec{v}_{avg}$ in same direction as $\Delta \vec{r}$
    • If $\vec{v}$ is constant, $\vec{v}_{avg} = \vec{v}$
  • SI units: $m/s$
  • $\vec{v}_{avg} = \frac{\Delta \vec{r}}{\Delta t}$
• Velocity & Trajectories
  • Trajectory: Path an object travels along
  • Velocity vectors are always tangent to the path of an object
    • Velocity vectors point in direction of $\Delta \vec{r}$ over short times
    • $\vec{v} \approx \vec{v}_{avg} = \frac{\Delta \vec{r}}{\Delta t}$
• (Instantaneous) Acceleration, $\vec{a} = (a<em>x, a</em>y)$
  • Speeding up, slowing down, & turning
  • Mathematical definition
    • $a<em>x \equiv \frac{dv</em>x}{dt}$
    • $a<em>y \equiv \frac{dv</em>y}{dt}$
  • Notes
    • Often just called acceleration
    • SI Unit: $\frac{m}{s^2}$
• Average Acceleration, $\vec{a}<em>{avg} = (\frac{\Delta v</em>x}{\Delta t}, \frac{\Delta v_y}{\Delta t})$
  • On average, how quickly & in what direction is it speeding up, slowing down, or turning over a specific interval of time
  • Mathematical definition
    • $\vec{a}_{avg}$ points in same direction as $\Delta \vec{v}$
    • If $\vec{a}$ is constant, $\vec{a}_{avg} = \vec{a}$
  • SI units: $\frac{m}{s^2}$
  • $\vec{a}_{avg} = \frac{\Delta \vec{v}}{\Delta t}$
• Interpreting Acceleration
  • $\vec{a}$ points in direction of $\Delta \vec{v}$ over very short times
  • Speeding up: $\vec{a}$ in same direction as $\vec{v}$
  • Slowing down: $\vec{a}$ in opposite direction as $\vec{v}$
  • Turning: $\vec{a}$ perpendicular to $\vec{v}$
• Interpreting Acceleration
  • Speeding up and turning (< 90° angle)
    • $\vec{a}$ somewhat perpendicular & toward $\vec{v}$
  • Slowing down and turning (> 90° angle)
    • $\vec{a}$ somewhat perpendicular & opposite $\vec{v}$
• Vectors with Units
  • Most vectors have units
    • Position Vector: m
    • Velocity Vector: m/s
    • Acceleration Vector: m/s$\vec{v}$
  • Drawn lengths via mental key
    • Rarely explicitly shown
  • Only compare lengths of same type