Chapter 3: Vectors and Motion in Two Dimensions - Study Notes

Vectors and Motion in Two Dimensions

Introduction

This chapter explores the fundamental concepts of vectors and their application in analyzing motion in two dimensions. Understanding vectors is essential for describing displacement, velocity, and acceleration in physics, especially when objects move in a plane rather than along a single line.

Section 3.1: Using Vectors

Definition and Properties of Vectors

• Vector: A quantity with both magnitude (size) and direction.

• Magnitude: The length or size of the vector, always a non-negative scalar.

• Direction: Indicates where the vector points in space.

• Equality: Two vectors are equal if they have the same magnitude and direction, regardless of their initial position.

• Displacement Vector: Represents the straight-line distance from initial to final position.

Vector Addition and Subtraction

• Tip-to-Tail Rule: Place the tail of the second vector at the tip of the first; the resultant vector is drawn from the tail of the first to the tip of the second.

• Parallelogram Method: Both vectors start at the same point; the resultant is the diagonal of the parallelogram formed.

• Commutativity: Vector addition is commutative: $\vec{A} + \vec{B} = \vec{B} + \vec{A}$

• Subtraction: To subtract $\vec{B}$ from $\vec{A}$, add $-\vec{B}$ (reverse direction of $\vec{B}$).

Multiplication by a Scalar

• Multiplying a vector by a positive scalar changes its magnitude but not its direction.

• Multiplying by zero yields the zero vector (no magnitude or direction).

• Multiplying by a negative scalar reverses the direction of the vector.

Section 3.2: Coordinate Systems and Vector Components

Coordinate Systems

• Cartesian Coordinates: Most commonly used; axes are perpendicular and intersect at the origin.

• Each axis has positive and negative directions, separated by zero at the origin.

Vector Components

• Any vector $\vec{A}$ can be decomposed into components along the x and y axes: $\vec{A} = \vec{A}_x + \vec{A}_y$

• Component: The projection of the vector onto an axis; can be positive or negative depending on direction.

• Magnitude of components found using trigonometry: $A_x = A \cos \theta$ $A_y = A \sin \theta$

• Magnitude of vector from components: $A = \sqrt{A_x^2 + A_y^2}$

• Direction (angle): $\theta = \tan^{-1}\left(\frac{A_y}{A_x}\right)$

Working with Components

• Vectors can be added by summing their respective components: $\vec{C} = \vec{A} + \vec{B}$ $C_x = A_x + B_x$ $C_y = A_y + B_y$

• Vector equations are shorthand for simultaneous equations involving components.

Tilted Axes

• For motion on a slope, axes can be rotated to align with the slope.

• Components are found using the same trigonometric methods, but relative to the tilted axes.

Section 3.3: Motion on a Ramp

Constant-Velocity and Accelerated Motion

• Motion on a ramp is analyzed by decomposing vectors into components parallel and perpendicular to the ramp.

• For constant velocity, vertical displacement is found using the vertical component of velocity.

• For accelerated motion, acceleration parallel to the ramp is a component of gravitational acceleration: $a_x = g \sin \theta$

Example: Height Gained by a Car

• Vertical displacement: $\Delta y = v_y \cdot t$

• Application: A car moving up a slope at constant speed gains height based on the vertical component of its velocity.

Section 3.4: Motion in Two Dimensions

General Two-Dimensional Motion

• Objects move in a plane; displacement, velocity, and acceleration are all vectors.

• Velocity vector points in the direction of displacement.

• Acceleration occurs when velocity changes in magnitude or direction.

• Acceleration vector: $\vec{a} = \frac{\Delta \vec{v}}{\Delta t}$

Section 3.5: Projectile Motion

Definition and Characteristics

• Projectile: An object moving in two dimensions under the influence of gravity alone.

• Horizontal and vertical motions are independent.

• Vertical acceleration: $a_y = -g$

• Horizontal acceleration: $a_x = 0$

Kinematic Equations for Projectile Motion

• Horizontal motion: $x = x_0 + v_{0x} t$

• Vertical motion: $y = y_0 + v_{0y} t - \frac{1}{2} g t^2$

• Time of flight, range, and maximum height can be calculated using these equations.

Example: Dock Jumping

• Dog runs horizontally off a dock; vertical motion is free fall.

• Time in air determined by vertical displacement: $y = y_0 - \frac{1}{2} g t^2$

• Horizontal distance: $x = v_{0x} t$

Section 3.6: Problem-Solving Approach for Projectile Motion

Steps for Solving Projectile Motion Problems

1. Make simplifying assumptions (e.g., neglect air resistance).

2. Draw a visual overview and establish a coordinate system.

3. Find initial velocity components using launch angle.

4. Write down known values and define symbols.

5. Solve using horizontal and vertical kinematic equations.

6. Assess the result for reasonableness and correct units.

Section 3.7: Circular Motion

Uniform Circular Motion

• Object moves at constant speed in a circle; velocity direction changes continuously.

• Acceleration points toward the center (centripetal acceleration).

• Centripetal acceleration: $a_c = \frac{v^2}{r}$

• Doubling speed increases centripetal acceleration by a factor of four.

Example: Speed Skaters

• Estimate centripetal acceleration using track radius and speed.

• Speed: $v = \frac{\text{distance}}{\text{time}}$

• Acceleration: $a_c = \frac{v^2}{r}$

Summary Table: Vector Operations

Operation	Result	Equation
Addition	Resultant vector	$\vec{A} + \vec{B}$
Subtraction	Difference vector	$\vec{A} - \vec{B}$
Scalar Multiplication	Scaled vector	$k \vec{A}$
Component Decomposition	x and y components	$A_x = A \cos \theta$, $A_y = A \sin \theta$
Magnitude from Components	Length of vector	$A = \sqrt{A_x^2 + A_y^2}$
Direction from Components	Angle	$\theta = \tan^{-1}\left(\frac{A_y}{A_x}\right)$

Key Equations

• Vector addition: $\vec{A} + \vec{B}$

• Magnitude: $A = \sqrt{A_x^2 + A_y^2}$

• Direction: $\theta = \tan^{-1}\left(\frac{A_y}{A_x}\right)$

• Projectile motion (vertical): $y = y_0 + v_{0y} t - \frac{1}{2} g t^2$

• Projectile motion (horizontal): $x = x_0 + v_{0x} t$

• Centripetal acceleration: $a_c = \frac{v^2}{r}$

Additional info: These notes expand on brief points from the original slides, providing full academic context, definitions, and examples for clarity and completeness.