Comprehensive Study Notes on Vector Algebra and Two-Dimensional Projectile Kinematics

Fundamentals of Vectors and Trigonometry

• Definition of a Vector:

  • A vector is a specialized quantity (a "super number") that possesses both an intrinsic magnitude (size) and a specific direction in space.
  • Anything that does not move perfectly vertically, perfectly horizontally, or along a single straight line requires a vector representation.

• Right-Triangle Trigonometry and Vector Components:

  • Consider a right-angled triangle formed by a vector $\mathbf{A}$ oriented at an angle $\theta$ relative to the horizontal axis:
  • Hypotenuse ($h$): The length or magnitude of the vector, denoted as $A$
  • Adjacent side ($a$): The horizontal component of the vector, denoted as $A_x$
  • Opposite side ($b$): The vertical component of the vector, denoted as $A_y$
  • Trigonometric Ratios:
  • $\sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}} = \frac{A_y}{A}$
  • $\cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}} = \frac{A_x}{A}$
  • Component Expressions:
  • Vertical component: $A_y = A \sin(\theta)$
  • Horizontal component: $A_x = A \cos(\theta)$

• Calculating Magnitude and Direction from Components:

  • Direction angle ($\theta$) derived via inverse tangent:
  • $\theta = \arctan\left(\frac{A_y}{A_x}\right)$
  • Vector magnitude ($A$) derived via the Pythagorean Theorem:
  • $A = \sqrt{A_x^2 + A_y^2}$
  • Fundamental Pythagorean trigonometric identity:
  • $\sin^2(\theta) + \cos^2(\theta) = 1$

• One-Dimensional vs. Two-Dimensional Motion:

  • One-Dimensional (1D) Motion: Motion restricted along a single axis (e.g., an object falling purely straight down the vertical $y$-axis).
  • Two-Dimensional (2D) Motion: Motion occurring simultaneously along the horizontal $x$-axis and vertical $y$-axis.
  • At every point in 2D trajectory, an object possesses both horizontal ($v_x$) and vertical ($v_y$) velocity components.
  • Moving upward corresponds to a positive vertical component ($v_y > 0$), while moving downward corresponds to a negative vertical component.

Vector Notation, Components, and Magnitude

• Distinction in Vector Symbology:

  • $\vec{A}$ (Letter with overhead arrow): Refers to the full vector entity (encompassing both magnitude and spatial direction).
  • $A$ (Letter without overhead arrow): Refers exclusively to the scalar magnitude (the non-negative scalar length or size of the arrow).

• Unit Vector Representation and Magnitude Calculation Example:

  • Given a vector with horizontal component $A_x = 3$ and vertical component $A_y = 4$:
  • Vector representation: $\vec{A} = 3\hat{i} + 4\hat{j}$
  • Magnitude calculation: $A = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5$

Graphical and Algebraic Vector Addition

• Translational Invariance of Vectors:

  • A vector is defined entirely by its length and pointing direction, independent of its origin in space.
  • Translating a vector $\vec{B}$ anywhere on a coordinate plane does not alter its properties, provided its length and direction remain unchanged.

• Tip-to-Tail Graphical Method:

  • To perform graphical addition of vector $\vec{B}$ to vector $\vec{A}$ ($\vec{C} = \vec{A} + \vec{B}$):
  1. Place the tail (start) of vector $\vec{B}$ at the tip (arrowhead) of vector $\vec{A}$.
  2. Draw the resultant vector $\vec{C}$ extending from the tail of vector $\vec{A}$ to the tip of vector $\vec{B}$.

• Component-Wise Algebraic Addition:

  • Graphical methods are useful for quick qualitative sketching, but quantitative calculations require resolving vectors into orthogonal components.
  • Adding two-dimensional vectors amounts to adding their corresponding scalar components independently:
  • Horizontal component of resultant: $C_x = A_x + B_x$
  • Vertical component of resultant: $C_y = A_y + B_y$
  • Component resolution is performed using standard trigonometric relationships ($\sin(\theta)$ and $\cos(\theta)$).

Vector Subtraction and Average Acceleration

• Definition of Average Acceleration Vector:

  • Average acceleration is defined as the change in velocity vector per unit time:
  • $\vec{a}_{\text{avg}} = \frac{\Delta \vec{v}}{\Delta t} = \frac{\vec{v}_2 - \vec{v}_1}{\Delta t}$
  • Physical units: meters per second squared ($m\,s^{-2}$ or $m/s^2$).

• Methodology of Vector Subtraction:

  • Vector subtraction is evaluated as the addition of a negated vector:
  • $\vec{v}_2 - \vec{v}_1 = \vec{v}_2 + (-\vec{v}_1)$
  • Negating a Vector ($-\vec{v}_1$): Reverses the arrow's direction by $180^\circ$ while preserving its length.
  • If initial velocity $\vec{v}_1$ points upward and to the right, then negated vector $-\vec{v}_1$ points downward and to the left.

• Determining Acceleration Direction for a Launched Ball:

  • Consider a ball tossed upward and rightward:
  • Initial velocity at $t = 0\,s$ ($\vec{v}_1$): Points upward and to the right.
  • Later velocity at $t = 1\,s$ ($\vec{v}_2$): Points downward and to the right.
  • Time interval: $\Delta t = 1\,s$
  • Vector evaluation of $\Delta \vec{v} = \vec{v}_2 + (-\vec{v}_1)$.
  • The rightward horizontal component of $\vec{v}_2$ and the leftward horizontal component of $-\vec{v}_1$ cancel out.
  • The downward vertical components of both $\vec{v}_2$ and $-\vec{v}_1$ reinforce each other.
  • The resulting average acceleration vector $\vec{a}_{\text{avg}}$ points directed straight downward.

• Tip-to-Tip Direct Subtraction Method:

  • To construct $\vec{v}_2 - \vec{v}_1$ directly, draw an arrow going directly from the tip of the initial vector $\vec{v}_1$ to the tip of the final vector $\vec{v}_2$

Historical Context and Military Applications of Kinematics

• Origins of Kinematic Theory:
  • The practical mathematical formulation of 2D projectile kinematics was driven historically (300 to 400 years ago) by military requirements to predict artillery round trajectories.
  • Objective: Calculate ballistics so cannonballs hit intended targets (such as military barracks) while missing non-targeted structures (such as churches).
• Early Technological Challenges:
  • Early ballistics calculations struggled because fast-moving spherical cannonballs were affected by air resistance, which initial simplistic vacuum models neglected.
  • Historical military funding and requirements consistently drove physics developments, continuing through modern developments such as mid-20th-century nuclear weaponry.

2D Kinematics and Projectile Motion Equations

• Dimensional Decomposition:

  • General spatial vectors possess three orthogonal components: $A_x$, $A_y$, and $A_z$
  • Motions along orthogonal spatial axes are completely independent of one another.
  • A two-dimensional trajectory is analyzed as two concurrent, independent one-dimensional problems.

• Horizontal ($x$-axis) Kinematics in Ideal Projectile Motion:

  • Assuming negligible air resistance and no horizontal forces:
  • Horizontal acceleration: $a_x = 0$
  • Horizontal velocity: $v_x = v_{0x} = \text{constant}$
  • Horizontal position equation: $x = x_0 + v_{0x} t$

• Vertical ($y$-axis) Kinematics in Ideal Projectile Motion:

  • Gravitational force acts downward continuously throughout flight:
  • Vertical acceleration: $a_y = -g$
  • Vertical velocity equation: $v_y = v_{0y} - g t$
  • Vertical position equation: $y = y_0 + v_{0y} t - \frac{1}{2} g t^2$

Gravitational Acceleration Variations and Validity Bounds

• Standard Working Values for Gravitational Acceleration ($g$):

  • Magnitude approximations: $g = 9.8\,m/s^2$, $g = 9.81\,m/s^2$, or roughly $10\,m/s^2$ for rough estimation.
  • In vertical kinematic equations, vector direction is explicitly down: $a_y = -9.8\,m/s^2$

• Geophysical Variations in $g$:

  • Gravity is not strictly constant over Earth's surface due to shape irregularities, variable continental crust thickness, ocean basins, and planet rotation.
  • Measured local values of $g$:
  • Sea level at Singapore: $\approx 9.79\,m/s^2$
  • Summit of Mount Everest: $\approx 9.82\,m/s^2$
  • Scale comparison: Earth's mean radius is nearly $7000\,km$ (specifically around $6371\,km$). Topographical surface variations spanning a few kilometers cause local variations in $g$ of less than 1%.

• Domain of Validity for Projectile Kinematics:

  • Projectile motion equations are valid strictly from the instant after an object leaves the launcher/hand until the instant before it makes contact with the ground or target.
  • Force interactions occurring during launch or during ground impact involve entirely different physical models and equations.