Vectors and Motion: Physical Principles and Trigonometric Analysis

Fundamentals of Scalars and Vectors

• Scalar Quantities:

  • Physical quantities completely described by a single numerical value accompanied by an appropriate unit.

  • Can be positive, negative, or zero.

  • Examples of scalar quantities:

  • Mass (e.g., $6\,\text{kg}$)

  • Temperature (e.g., $30^\circ\text{C}$)

  • Time

• Vector Quantities:

  • Physical quantities possessing both a magnitude (size or length, answering "How far?" or "How fast?") and a direction (answering "Which way?").

  • Examples of vector quantities:

  • Velocity of a race car (e.g., $120\,\text{mph}$ to the west).

  • Force exerted when pushing a friend, which requires knowing both the push strength (magnitude) and direction.

• Magnitude of a Vector:

  • The size or length of a vector quantity.

  • Vector magnitude can be positive or zero ($\ge 0$), but can never be negative.

• Graphical Representation:

  • Represented graphically as an arrow pointing in the direction of the vector quantity.

  • The length of the drawn arrow is directly proportional to the magnitude of the vector quantity.

Symbols and Notation for Vectors

• Symbolic Distinctions:

  • Vector quantities are denoted symbolically by placing an arrow pointing to the right directly over the letter representing the quantity (e.g., $\vec{r}$ and $\vec{A}$).

  • Scalar quantities or pure magnitudes are represented by letters without arrows (e.g., $r$ and $A$).

  • In handwritten work, arrows must always be drawn over vector symbols to differentiate them from scalars, as both $\vec{r}$ and $r$ (or $\vec{A}$ and $A$) may appear in the same problem with distinct mathematical meanings.

• Direction of Vector Notation Arrow:

  • The arrow drawn above a vector symbol always points to the right (e.g., $\vec{r}$), regardless of the actual directional orientation of the vector quantity itself (never drawn as $\overleftarrow{r}$).

Displacement Vectors and Path Independence

• Definition of Displacement Vector:

  • A displacement vector specifies both how far an object moves and the direction of motion.

  • Drawn as a straight arrow from an object's initial position directly to its final position.

  • Denoted symbolically as $\vec{r}$.

  • Written in component form specifying magnitude and direction, such as $\vec{r} = (100\,\text{ft}, \text{east})$.

• Path Independence:

  • An object's displacement vector is drawn directly from its initial position to its final position, regardless of the actual physical path taken between those points.

  • Example (Sam's Trip): Sam moves from an initial location to a final location, resulting in a displacement vector $\vec{r}$.

  • Example (Jane's Trip): Jane starts on 12th Street and ends on Vine. Even if she walks east along 12th Street to an intersection and then north on Vine, her displacement vector is still the straight line drawn directly from her starting point on 12th Street to her endpoint on Vine.

Principles of Vector Addition

• Net Displacement Vector:

  • When an object undergoes multiple successive displacements, its overall change in position is represented by the net displacement vector $\vec{d}_{\text{net}}$.

  • Mathematically expressed as:     $\vec{d}_{\text{net}} = \vec{d}_1 + \vec{d}_2$

• Rules of Vector Addition:

  • Vector addition does not follow standard scalar addition rules.

  • Both the magnitudes and directions of all component vectors must be taken into account simultaneously.

  • Example (Sam's Two-Leg Trip): Sam starts at an intersection, walks $50\,\text{ft}$ east ($\vec{d}_1$), and then walks $100\,\text{ft}$ northeast through a vacant lot ($\vec{d}_2$). His total movement is represented by the net displacement vector $\vec{d}_{\text{net}} = \vec{d}_1 + \vec{d}_2$, pointing straight from his initial starting location to his final position.

• Tip-to-Tail Method (Tactics Box 1.4):

  • To add two vectors $\vec{A}$ and $\vec{B}$:

  1. Draw vector $\vec{A}$.

  2. Place the tail of vector $\vec{B}$ at the tip (head) of vector $\vec{A}$.

  3. Draw the vector sum $\vec{S}$ from the tail of $\vec{A}$ to the tip of $\vec{B}$.

Trigonometric Foundations for Vector Analysis

• Adding non-collinear vectors in two dimensions involves computing lengths and angles of right triangles using trigonometry.

• Right Triangle Trigonometric Ratios:

  • Sine function:     $\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}$

  • Cosine function:     $\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}$

  • Tangent function:     $\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}$

• Pythagorean Theorem:

  • Relates the side lengths of a right triangle to its hypotenuse:     $\text{hypotenuse} = \sqrt{\text{opposite}^2 + \text{adjacent}^2}$

Worked Examples and Practice Problems

• Stop to Think 1.6:

  • Evaluates distance $x$ in a right triangle using given trigonometric relations and options:

  • $26\,\text{cm}$

  • $20\,\text{cm}$

  • $17\,\text{cm}$

  • $15\,\text{cm}$

• Example 1.6: How Far North and East? (Alex's Navigation):

  • Problem: Alex navigates using a compass, walking at an angle of $60^\circ$ north of east for a total distance of $100\,\text{m}$. Calculate how far north and how far east she is from her starting point.

  • Strategize & Prepare: Sketch a right triangle with the $100\,\text{m}$ displacement as the hypotenuse. The distance north is the opposite side ($O$), and the distance east is the adjacent side ($A$).

  • Solve:

  • Distance North:       $O = (100\,\text{m}) \sin(60^\circ) = (100\,\text{m})(0.866) = 87\,\text{m}$

  • Distance East:       $A = (100\,\text{m}) \cos(60^\circ) = (100\,\text{m})(0.500) = 50\,\text{m}$

  • Assess: Both calculated distances ($87\,\text{m}$ and $50\,\text{m}$) are strictly less than the total distance of $100\,\text{m}$. The distance east is less than the distance north, which aligns with a trajectory directed $60^\circ$ north of east. This process breaks displacement down into perpendicular vector components.

• Example 1.7: How Far Away is Anna?:

  • Problem: Anna walks $90\,\text{m}$ due east and then $50\,\text{m}$ due north. Determine her net displacement from her starting point.

  • Strategize & Prepare: Set up a coordinate system with Anna's initial location at the origin. Draw displacement vectors $\vec{d}_1 = (90\,\text{m}, \text{east})$ and $\vec{d}_2 = (50\,\text{m}, \text{north})$ tip-to-tail.

  • Solve:

  • Net displacement magnitude via the Pythagorean theorem:       $r = \sqrt{(50\,\text{m})^2 + (90\,\text{m})^2} = \sqrt{2500\,\text{m}^2 + 8100\,\text{m}^2} = \sqrt{10600\,\text{m}^2} \approx 100\,\text{m}$

  • Direction angle via the inverse tangent function:       $\theta = \tan^{-1}\left(\frac{50\,\text{m}}{90\,\text{m}}\right) = \tan^{-1}(0.5556) \approx 29^\circ$

  • Net Displacement Vector:       $\vec{r} = (100\,\text{m}\text{, } 29^\circ\text{ north of east})$

  • Assess: A hypotenuse length of $100\,\text{m}$ for right-triangle sides of $50\,\text{m}$ and $90\,\text{m}$ is consistent. An angle of $29^\circ$ is appropriately less than $45^\circ$ since the opposite side ($50\,\text{m}$) is shorter than the adjacent side ($90\,\text{m}$).

Velocity Vectors and Motion Diagrams

• Definition of Velocity Vector:

  • Velocity is a vector quantity specifying both speed (how fast) and direction of motion.

  • Represented symbolically by $\vec{v}$.

  • Points in the exact direction of the object's motion.

  • Length (magnitude) of $\vec{v}$ is proportional to speed.

• Representation on Motion Diagrams:

  • Vectors connecting consecutive dots on a motion diagram point in the direction of motion.

  • Greater displacement between points indicates higher speed, requiring longer velocity vectors.

  • Arrows connecting dots in motion diagrams are labeled as velocity vectors $\vec{v}$ rather than displacement vectors.

• Average vs. Instantaneous Velocity:

  • Vectors drawn between discrete positions on motion diagrams represent average velocity vectors over that time interval.

  • For an accelerating object, instantaneous velocity is slightly less than average velocity at the start of a time interval and slightly greater at the end.

Worked Example: Motion Diagram for Projectile Motion

• Example 1.8: Drawing a Ball's Motion Diagram (Jake and Saeed):

  • Problem: Jake hits a ball at a $60^\circ$ angle relative to the horizontal, and Saeed catches it. Draw a motion diagram showing velocity vectors.

  • Strategize & Prepare:

  • Define precise boundaries for the motion: begin the instant after the ball leaves Jake's bat (already in motion) and end the instant it contacts Saeed's hand (still in motion).

  • Model the ball as a particle along a curved arc.

  • Solve:

  • Ascending trajectory: Velocity vectors $\vec{v}$ point upward and decrease in length as the ball slows down.

  • Descending trajectory: Velocity vectors $\vec{v}$ point downward and increase in length as the ball speeds up.

  • Each vector $\vec{v}$ differs in magnitude and direction throughout the flight, demonstrating non-constant-velocity motion.

  • Assess: Velocity vectors transition from pointing upward initially to pointing downward at the end, matching observed physical motion when tossing a ball back and forth.