Comprehensive Study Guide: Vectors and 2D Motion

Fundamentals of Scalars and Vectors

• Scalar Quantity Definitions:

  • A scalar is a physical quantity specified entirely by a numerical magnitude accompanied by appropriate physical units.

  • Examples include time, temperature, mass, distance, and speed.

• Vector Quantity Definitions:

  • A vector is a physical quantity requiring both a magnitude (size or length) and a directional vector orientation in space for complete specification.

  • Examples include displacement, position, velocity, and acceleration.

  • Practical Example (Finding the Library): Directional navigation requires specifying both distance and angle/orientation. Knowing that a library is located $0.5\,\text{mile}$ away is insufficient without knowing the directional vector pointing toward the building.

Kinematic Vector Representations

• Position Vector ($\mathbf{r}$):

  • Specifies the location of a particle or object relative to a chosen coordinate origin in a coordinate plane or space.

• Displacement Vector ($\Delta \mathbf{r}$):

  • Represents the straight-line vector change in position from an initial coordinate location to a final coordinate location:         $\Delta \mathbf{r} = \mathbf{r}_{f} - \mathbf{r}_{i}$

• Velocity Vector ($\mathbf{v}$):

  • Measures the rate of change of position per unit time.

  • Points in the instantaneous direction of motion.

• Average Acceleration Vector ($\mathbf{a}_{\text{av}}$):

  • Measures the rate of change of velocity per unit time:         $\mathbf{a}_{\text{av}} = \frac{\Delta \mathbf{v}}{\Delta t}$

  • Points in the vector direction of the change in velocity $\Delta \mathbf{v}$.

Unit Vector Notation and Scalar Multipliers

• Definition of Unit Vectors:

  • Unit vectors are dimensionless vectors having a standardized magnitude of exactly $1$ (unit length).

  • They serve exclusively to specify directional orientations along coordinate axes.

  • Standard Cartesian unit vectors:

    • $\hat{\mathbf{i}}$: Unit vector directed along the positive x-axis (horizontal).

    • $\hat{\mathbf{j}}$: Unit vector directed along the positive y-axis (vertical).

    • $\hat{\mathbf{k}}$: Unit vector directed along the positive z-axis (depth).

• Scalar Multiplication of Unit Vectors:

  • Multiplying a unit vector by a scalar magnitude scales the vector's physical length without altering its fundamental directional line of action.

  • A positive scalar multiplier scales length while maintaining original orientation.

  • A negative scalar multiplier scales length and reverses the directional sense by $180^\circ$.

  • Example: Vector $\mathbf{a} = 2.5\,\hat{a}$ possesses a overall magnitude of $2.5$ units aligned precisely in the direction of the unit vector $\hat{a}$.

Geographic Direction Convention and Bearings

• Standard Reference Angle System:

  • Directional bearings in physics use acute reference angles strictly less than or equal to $90^\circ$ measured from the primary cardinal axes (North, South, East, West).

• Notation Guidelines and Examples:

  • $\text{N}30^\circ\text{E}$: $30\text{ degrees}$ East of North (start pointing North, rotate $30^\circ$ toward East).

  • $\text{N}30^\circ\text{W}$: $30\text{ degrees}$ West of North.

  • $\text{S}30^\circ\text{E}$: $30\text{ degrees}$ East of South.

  • $\text{S}30^\circ\text{W}$: $30\text{ degrees}$ West of South.

• Axis-Aligned Exceptional Cases:

  • When vectors align directly along coordinate system axes, standard bearing angular designations simplify to: Due North, Due South, Due East, and Due West.

Graphical Vector Addition and Subtraction

• The Tip-to-Tail Addition Rule:

  • To evaluate vector sums $\mathbf{A} + \mathbf{B} = \mathbf{R}$ geometrically:

    1. Draw vector $\mathbf{A}$ to scale in its specified directional angle.

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

    3. Draw the resultant vector $\mathbf{R}$ starting from the original tail of $\mathbf{A}$ extending straight to the final tip of $\mathbf{B}$.

• Vector Subtraction Dynamics:

  • Vector subtraction is defined in terms of vector addition involving a negated vector:         $\mathbf{a} - \mathbf{b} = \mathbf{a} + (-\mathbf{b}) = \mathbf{R}$

  • The negative vector $-\mathbf{b}$ retains identical magnitude to $\mathbf{b}$, but points in the exact opposite direction ($180^\circ$ rotated).

• Step-by-Step Graphical Vector Addition Procedure:

  1. Obtain a ruler and protractor; define a consistent conversion scale factor (e.g., $1\,\text{cm} = 1.0\,\text{N}$ or $1.0\,\text{m} = 0.5\,\text{cm}$).

  2. Measure and construct the angular direction of the first vector relative to chosen reference axes.

  3. Draw the scaled length of the first vector.

  4. Construct a secondary reference coordinate origin at the arrow tip of the current vector.

  5. Measure the orientation angle and scaled length of the second vector from this newly constructed origin.

  6. Iterate steps 4 and 5 for any additional vectors.

  7. Draw the final resultant vector $\mathbf{R}$ directly connecting the initial primary origin to the final arrow tip.

  8. Measure $\mathbf{R}$'s physical length with the ruler and scale back to original physical units; measure its final angle using the protractor.

Trigonometric Principles and Vector Decompositions

• Right-Triangle Trigonometric Definitions (SOH CAH TOA):

  • \sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}}}

  • \cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}}}

  • \tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}}}

• Pythagorean Relationship:

  • For any right triangle with perpendicular sides $A_x$ and $A_y$ and hypotenuse $A$:         $A_x^2 + A_y^2 = A^2$         $A = \sqrt{A_x^2 + A_y^2}$

• Orthogonal Vector Components:

  • A two-dimensional vector $\mathbf{A}$ oriented at angle $\theta$ above the horizontal axis resolves into perpendicular components:         $A_x = A \cos(\theta)$         $A_y = A \sin(\theta)$

• Quadrant Component Signs (ASTC Rule):

  • Quadrant I ($0^\circ < \theta < 90^\circ$): $A_x > 0$, $A_y > 0$ (All trigonometric functions positive).

  • Quadrant II ($90^\circ < \theta < 180^\circ$): $A_x < 0$, $A_y > 0$ (Sine positive).

  • Quadrant III ($180^\circ < \theta < 270^\circ$): $A_x < 0$, $A_y < 0$ (Tangent positive).

  • Quadrant IV ($270^\circ < \theta < 360^\circ$): $A_x > 0$, $A_y < 0$ (Cosine positive).

• Special Right Triangle Ratios:

  • $30^\circ - 60^\circ - 90^\circ$ Triangle: Side lengths follow proportions $x : x\sqrt{3} : 2x$.

    • Side opposite $30^\circ$: $x$

    • Side opposite $60^\circ$: $x\sqrt{3}$

    • Hypotenuse opposite $90^\circ$: $2x$

  • $45^\circ - 45^\circ - 90^\circ$ Triangle: Side lengths follow proportions $x : x : x\sqrt{2}$.

    • Legs opposite $45^\circ$: $x$

    • Hypotenuse opposite $90^\circ$: $x\sqrt{2}$

• Unit Circle Values in Quadrant I:

  • $0^\circ$ ($0\,\text{rad}$): $\cos(0^\circ) = 1$, $\sin(0^\circ) = 0$

  • $30^\circ$ ($\frac{\pi}{6}\,\text{rad}$): $\cos(30^\circ) = \frac{\sqrt{3}}{2}$, $\sin(30^\circ) = \frac{1}{2}$

  • $45^\circ$ ($\frac{\pi}{4}\,\text{rad}$): $\cos(45^\circ) = \frac{\sqrt{2}}{2}$, $\sin(45^\circ) = \frac{\sqrt{2}}{2}$

  • $60^\circ$ ($\frac{\pi}{3}\,\text{rad}$): $\cos(60^\circ) = \frac{1}{2}$, $\sin(60^\circ) = \frac{\sqrt{3}}{2}$

  • $90^\circ$ ($\frac{\pi}{2}\,\text{rad}$): $\cos(90^\circ) = 0$, $\sin(90^\circ) = 1$

Analytical Component Method of Vector Addition

• Mathematical Addition Strategy:

  1. Break down every constituent vector into its algebraic horizontal ($x$) and vertical ($y$) components using directional cosines and sines.

  2. Sum all horizontal components to obtain the net horizontal resultant component:         $R_x = \sum A_x = A_x + B_x + C_x + \dots$

  3. Sum all vertical components to obtain the net vertical resultant component:         $R_y = \sum A_y = A_y + B_y + C_y + \dots$

  4. Evaluate total resultant vector magnitude via Pythagorean theorem:         $R = \sqrt{R_x^2 + R_y^2}$

  5. Evaluate resultant vector orientation angle via inverse tangent function:         $\theta = \tan^{-1}\left(\left|\frac{R_y}{R_x}\right|\right)$

  6. Adjust angle $\theta$ to absolute coordinate frame depending on algebraic sign signs of $R_x$ and $R_y$

Detailed Numerical Vector Examples

• Two-Vector Addition Worked Problem:

  • Given Vector $\mathbf{A}$: Magnitude $20.\,\text{m}$ oriented at $30.\circ$ North of East.

    • $A_x = 20.\cos(30.\circ) = 17\,\text{m}$

    • $A_y = 20.\sin(30.\circ) = 10.\,\text{m}$

  • Given Vector $\mathbf{B}$: Magnitude $12\,\text{m}$ oriented at $60.\circ$ North of East.

    • $B_x = 12\cos(60.\circ) = 6.0\,\text{m}$

    • $B_y = 12\sin(60.\circ) = 10.\,\text{m}$

  • Resultant Vector Calculations:

    • $R_x = A_x + B_x = 17\,\text{m} + 6.0\,\text{m} = 23\,\text{m}$

    • $R_y = A_y + B_y = 10.\,\text{m} + 10.\,\text{m} = 20.\,\text{m}$

    • $R = \sqrt{23^2 + 20.^2} = \sqrt{529 + 400} = \sqrt{929} \approx 30.\,\text{m}$

    • $\theta = \tan^{-1}\left(\frac{20.}{23}\right) = 41^\circ$

  • Final Vector Resultant: $\mathbf{R} = 30.\,\text{m}$ at $41^\circ\text{ North of East}$.

• Three-Vector Component Addition Worked Problem:

  • Given Vector $\mathbf{A} = 50.0\,\text{cm East}$

    • $A_x = 50.0\,\text{cm}$, $A_y = 0.0\,\text{cm}$

  • Given Vector $\mathbf{B} = 40.0\,\text{cm}$ at $35^\circ\text{ North of East}$

    • $B_x = 40.0\cos(35^\circ) = 32.766\,\text{cm}$

    • $B_y = 40.0\sin(35^\circ) = 22.943\,\text{cm}$

  • Given Vector $\mathbf{C} = 30.0\,\text{cm North}$

    • $C_x = 0.0\,\text{cm}$, $C_y = 30.0\,\text{cm}$

  • Resultant Summations:

    • $R_x = 50.0 + 32.766 + 0 = 82.766\,\text{cm}$

    • $R_y = 0 + 22.943 + 30.0 = 52.943\,\text{cm}$

    • $R = \sqrt{82.766^2 + 52.943^2} \approx 98\,\text{cm}$

    • $\theta = \tan^{-1}\left(\frac{52.943}{82.766}\right) \approx 33^\circ$

  • Final Resultant: $\mathbf{R} \approx 98\,\text{cm}$ at $33^\circ\text{ North of East}$.

• Multi-Vector Displacement Walk Example:

  • John walks $50\,\text{m}$ East ($\mathbf{A} = 50\,\hat{\mathbf{i}} + 0\,\hat{\mathbf{j}}$).

  • John walks $50\,\text{m}$ at $30^\circ\text{ North of East}$ ($\mathbf{B} = 50\cos(30^\circ)\,\hat{\mathbf{i}} + 50\sin(30^\circ)\,\hat{\mathbf{j}}$).

  • John walks $50\,\text{m}$ West ($\mathbf{C} = -50\,\hat{\mathbf{i}} + 0\,\hat{\mathbf{j}}$).

  • John walks $50\,\text{m}$ South ($\mathbf{D} = 0\,\hat{\mathbf{i}} - 50\,\hat{\mathbf{j}}$).

Relative Motion Analysis

• Relative Velocity Formula:

  • For three reference frames or objects (1, 2, and 3), the relative velocity relationship is expressed as:         $\mathbf{v}_{13} = \mathbf{v}_{12} + \mathbf{v}_{23}$

  • $\mathbf{v}_{13}$ represents the velocity of object 1 relative to frame 3.

  • $\mathbf{v}_{12}$ represents the velocity of object 1 relative to frame 2.

  • $\mathbf{v}_{23}$ represents the velocity of frame 2 relative to frame 3.

• One-Dimensional Relative Motion (Train Boxcar Example):

  • A train travels groundward at velocity $v_{\text{tg}} = 15.0\,\text{m/s}$ to the right (+x direction).

  • A passenger walks inside the train boxcar at speed $v_{\text{pt}} = 1.2\,\text{m/s}$.

  • Case A (Passenger walking forward, same direction as train):         $v_{\text{pg}} = v_{\text{pt}} + v_{\text{tg}} = 1.2\,\text{m/s} + 15.0\,\text{m/s} = 16.2\,\text{m/s}\text{ (East)}$

  • Case B (Passenger walking backward, opposite direction to train):         $v_{\text{pg}} = -v_{\text{pt}} + v_{\text{tg}} = -1.2\,\text{m/s} + 15.0\,\text{m/s} = 13.8\,\text{m/s}\text{ (East)}$

• Two-Dimensional Relative Motion (Worker Climbing Ladder on Moving Train):

  • A train moves horizontally relative to the ground with velocity $\mathbf{v}_{\text{tg}}$.

  • A worker climbs a ladder vertically relative to the train with velocity $\mathbf{v}_{\text{pt}}$.

  • The worker's resultant velocity relative to ground $\mathbf{v}_{\text{pg}}$ forms a vector right triangle:         $\mathbf{v}_{\text{pg}} = \mathbf{v}_{\text{pt}} + \mathbf{v}_{\text{tg}}$         $v_{\text{pg}} = \sqrt{v_{\text{pt}}^2 + v_{\text{tg}}^2}$         $\theta = \tan^{-1}\left(\frac{v_{\text{pt}}}{v_{\text{tg}}}\right)$

• Real-World Application Scenarios:

  • Ball Fired Backward from Moving Vehicle:

    • A cannon mounted on a truck moving forward at speed $v_{\text{truck}}$ fires a projectile backward at speed $v_{\text{ball/truck}} = -v_{\text{truck}}$.

    • Velocity relative to ground: $v_{\text{ball/ground}} = v_{\text{ball/truck}} + v_{\text{truck}} = -v_{\text{truck}} + v_{\text{truck}} = 0\,\text{m/s}$.

    • The projectile drops straight down vertically relative to an observer on the ground.

*   **Apparent Rain Angle in Moving Vehicle & Wind Effects**:
    *   When a car is stationary, vertically falling rain drops appear to fall straight down ($\phi = 0^\circ$ relative to vertical).
    *   As the car moves forward at speed $v_{\text{car}}$, the relative velocity of the rain with respect to the car gains a horizontal component, causing the rain drops to appear slanted from inside the vehicle.
    *   Higher forward vehicle speeds increase the slant angle $\phi$.
    *   When falling rain interacts with horizontal wind, holding an umbrella effectively requires tilting the umbrella handle at an angle matching the resultant rain angle (e.g., tilted $18^\circ$ relative to vertical).