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

Chapter 3: Vectors and Motion in Two Dimensions

Introduction

This chapter explores the use of vectors to analyze motion in two dimensions, including projectile and circular motion. Understanding vectors and their components is essential for describing physical quantities such as displacement, velocity, and acceleration in multidimensional contexts.

Vectors: Definition and Properties

• Vector: A quantity with both magnitude (size) and direction. Examples include displacement, velocity, and acceleration.

• Magnitude: The length or size of a vector, always a positive scalar.

• Direction: Specifies the orientation of the vector in space.

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

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

Vector Addition and Subtraction

• Resultant Vector: The sum of two or more vectors.

• Tip-to-Tail Rule: Place the tail of one vector at the tip of another to add them.

• Parallelogram Rule: Vectors can be added by constructing a parallelogram.

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

• Subtraction: To subtract vector $\vec{B}$ from $\vec{A}$, add $-\vec{B}$ to $\vec{A}$.

Multiplication by a Scalar

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

• Multiplying by a negative scalar reverses the direction.

• Multiplying by zero yields the zero vector.

Coordinate Systems and Vector Components

• Coordinate System: A grid (usually Cartesian) used to measure positions and directions.

• Component Vectors: Any vector $\vec{A}$ can be decomposed into x- and y-components: $\vec{A} = \vec{A}_x + \vec{A}_y$

• Finding Components: Use trigonometry: $A_x = A \cos \theta$ $A_y = A \sin \theta$

• Pythagorean Theorem: To find magnitude from components: $A = \sqrt{A_x^2 + A_y^2}$

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

Motion on a Ramp

• For motion on an incline, align the x-axis with the slope.

• Vertical displacement is found using the vertical component of velocity.

• Acceleration parallel to the ramp is a component of gravitational acceleration: $a = g \sin \theta$

• Example: A car moving up a slope at constant speed gains height according to its vertical velocity component.

Motion in Two Dimensions

• Displacement, velocity, and acceleration are vectors that change in both magnitude and direction.

• Velocity in two dimensions: $\vec{v} = \frac{\Delta \vec{r}}{\Delta t}$

• Acceleration in two dimensions: $\vec{a} = \frac{\Delta \vec{v}}{\Delta t}$

• Acceleration occurs if speed or direction changes.

Projectile Motion

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

• Path is a parabola.

• Horizontal and vertical motions are independent:

• Horizontal motion: constant velocity ($a_x = 0$)

• Vertical motion: constant acceleration ($a_y = -g$)

• Kinematic equations for projectile motion: $x = x_0 + v_{0x} t$ $y = y_0 + v_{0y} t - \frac{1}{2} g t^2$ $v_{y} = v_{0y} - g t$

• Range depends on initial speed and launch angle.

Circular Motion

• Uniform circular motion: constant speed, changing direction.

• Velocity is tangent to the circle.

• Acceleration (centripetal) points toward the center: $a_c = \frac{v^2}{r}$

• Doubling speed quadruples centripetal acceleration.

Relative Motion

• Velocity depends on the observer's frame of reference.

• Relative velocity equation: $\vec{v}_{A/B} = \vec{v}_{A/C} - \vec{v}_{B/C}$

• Example: A plane flying east with a south wind has a ground speed found by vector addition.

Summary Table: Key Concepts

Concept	Equation	Description
Vector Addition	$\vec{A} + \vec{B}$	Tip-to-tail or parallelogram rule
Components	$A_x = A \cos \theta$, $A_y = A \sin \theta$	Decompose vector into axes
Projectile Motion	$x = x_0 + v_{0x} t$, $y = y_0 + v_{0y} t - \frac{1}{2} g t^2$	Horizontal and vertical equations
Circular Motion	$a_c = \frac{v^2}{r}$	Centripetal acceleration
Relative Velocity	$\vec{v}_{A/B} = \vec{v}_{A/C} - \vec{v}_{B/C}$	Velocity relative to different observers

Example Applications

• Dock Jumping: Calculating the horizontal distance a dog travels using projectile motion equations.

• Speed Skaters: Estimating centripetal acceleration in tight turns.

• Airplane Ground Speed: Using vector addition to find resultant velocity.

Important Points for Exam Prep

• Vectors must be decomposed into components for problem-solving.

• Projectile motion problems require separate analysis of horizontal and vertical motions.

• Circular motion involves centripetal acceleration, even at constant speed.

• Relative motion is essential for understanding velocities in different frames.