Vectors and Motion in Two Dimensions: Physics with Calculus Study Notes

Vectors and Motion in Two Dimensions

Introduction

This chapter explores the use of vectors to analyze motion in two dimensions, a foundational concept in Physics with Calculus. Topics include vector operations, coordinate systems, motion on ramps, projectile motion, and the decomposition of motion into components.

Recap: One-Dimensional Motion

Velocity and Acceleration

• Velocity is the rate of change of position: $v = \frac{\Delta x}{\Delta t}$ (units: m/s).

• Acceleration is the rate of change of velocity: $a = \frac{\Delta v}{\Delta t}$ (units: m/s2).

• An object is speeding up if velocity and acceleration have the same sign; slowing down if they have opposite signs.

Free Fall

• Free fall is a special case of constant acceleration, with $a = -g$ (downward, $g \approx 9.8\ \mathrm{m/s^2}$).

Vectors: Definitions and Properties

What is a Vector?

• A vector is a quantity with both magnitude and direction (e.g., displacement, velocity, acceleration).

• A scalar has only magnitude (e.g., mass, temperature).

Vector Addition

• Vectors are added using the tip-to-tail rule or the parallelogram rule.

• The resultant vector is the sum of two or more vectors.

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

Multiplication by a Scalar

• Multiplying a vector by a positive scalar changes its magnitude but not its direction: $\vec{B} = c\vec{A}$.

• Multiplying by a negative scalar reverses the direction: $-\vec{A}$ points opposite to $\vec{A}$.

• The zero vector has zero magnitude and no direction.

Vector Subtraction

• Subtracting vectors: $\vec{A} - \vec{B} = \vec{A} + (-\vec{B})$.

• Graphically, reverse the direction of $\vec{B}$ and add to $\vec{A}$.

Coordinate Systems and Vector Components

Cartesian Coordinate System

• A coordinate system is a grid used to specify positions and directions.

• Most problems use Cartesian coordinates (x, y axes at right angles).

Component Vectors

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

• The x-component is parallel to the x-axis; the y-component is parallel to the y-axis.

Finding Components from Magnitude and Angle

• Given $|\vec{A}|$ and angle $\theta$ from the x-axis:

$A_x = A \cos \theta$ $A_y = A \sin \theta$

Finding Magnitude and Direction from Components

• Given $A_x$ and $A_y$:

$A = \sqrt{A_x^2 + A_y^2}$ $\theta = \tan^{-1}\left(\frac{A_y}{A_x}\right)$

Example: Acceleration Vector Components

• For a vector $\vec{a}$ of magnitude 6.0 m/s2 at 30° below the negative x-axis:

• Both $a_x$ and $a_y$ are negative.

Adding Vectors Using Components

• To add vectors $\vec{A}$ and $\vec{B}$, add their components:

$C_x = A_x + B_x$ $C_y = A_y + B_y$

Tilted Axes

• For motion on a slope, align the x-axis with the slope for easier analysis.

• Decompose vectors relative to the tilted axes.

Motion on a Ramp

Constant-Velocity Motion on a Ramp

• When an object moves up a ramp at constant speed, the vertical height gained is determined by the vertical component of velocity.

Accelerated Motion on a Ramp

• For a frictionless ramp, acceleration is parallel to the ramp and is a component of gravity:

$a_x = g \sin \theta$

Example: Maximum Speed for a Skier

• Given a slope of length 360 m and vertical drop 170 m, the maximum speed is found using kinematics and trigonometry.

Motion in Two Dimensions

General Two-Dimensional Motion

• Objects can move in a plane, with displacement, velocity, and acceleration as vectors.

• Examples: skating, Ferris wheel, car rounding a curve.

Motion Diagrams and Vectors

• Displacement and velocity vectors can be drawn between successive positions in a motion diagram.

Acceleration in Two Dimensions

• Acceleration occurs when velocity changes in magnitude or direction.

• Vector definition: $\vec{a} = \frac{\Delta \vec{v}}{\Delta t}$

Projectile Motion

Definition and Characteristics

• A projectile moves in two dimensions under gravity alone (neglecting air resistance).

• The path is a parabola.

• Horizontal and vertical motions are independent.

Kinematic Equations for Projectile Motion

• Horizontal motion: $x = x_0 + v_{0x} t$ (constant velocity, $a_x = 0$)

• Vertical motion: $y = y_0 + v_{0y} t - \frac{1}{2} g t^2$ (constant acceleration, $a_y = -g$)

Problem-Solving Approach

• Separate the problem into horizontal and vertical components.

• Draw a diagram, define variables, and apply the appropriate kinematic equations.

• Time of flight is the same for both components.

Example: Dock Jumping Dog

• A dog runs off a dock at 8.5 m/s from a height of 0.61 m. Find the horizontal distance traveled before hitting the water.

• Solution involves finding the time to fall vertically, then using horizontal velocity to find range.

Range of a Projectile

• The range is the horizontal distance traveled.

• Maximum range occurs at a launch angle of 45° (without air resistance).

• Flight time depends on the vertical component of velocity.

Summary Table: Key Equations and Concepts

Concept	Equation	Description
Vector Addition	$\vec{C} = \vec{A} + \vec{B}$	Tip-to-tail or parallelogram rule
Components	$A_x = A \cos \theta$, $A_y = A \sin \theta$	From magnitude and angle
Magnitude	$A = \sqrt{A_x^2 + A_y^2}$	From components
Projectile Motion (horizontal)	$x = x_0 + v_{0x} t$	Constant velocity
Projectile Motion (vertical)	$y = y_0 + v_{0y} t - \frac{1}{2} g t^2$	Constant acceleration
Ramp Acceleration	$a_x = g \sin \theta$	Component of gravity along ramp

Study Tips

• Practice decomposing vectors into components and recombining them.

• Draw diagrams for all two-dimensional motion problems.

• Always separate horizontal and vertical motions in projectile problems.

• Check units and reasonableness of answers.