Vectors and Motion in Two Dimensions: Study Notes

Vectors and Motion in Two Dimensions

Introduction

This chapter covers the fundamental concepts of vectors, their mathematical treatment, and the analysis of motion in two dimensions, including projectile and circular motion. Mastery of these topics is essential for understanding a wide range of problems in introductory physics.

Vectors

Definition and Properties

• Vector: A quantity that has both magnitude and direction (e.g., displacement, velocity, acceleration).

• Scalar: A quantity with magnitude only (e.g., speed, mass, temperature).

• Notation: Vectors are typically denoted with an arrow, such as $\vec{A}$, or in boldface (A).

Example: A velocity of 5 m/s east is a vector; 5 m/s is a scalar.

Vector Representation

• Vectors can be represented graphically by arrows, where the length indicates magnitude and the arrowhead indicates direction.

• The magnitude of a vector $\vec{v}$ is written as $v$.

Vector Addition and Subtraction

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

• Parallelogram Rule: Place vectors so they start at the same point; the diagonal of the parallelogram represents the sum.

• Mathematical Expression: $\vec{C} = \vec{A} + \vec{B}$

• Vector Subtraction: $\vec{E} = \vec{C} - \vec{A}$

Multiplication by a Scalar

• Multiplying a vector $\vec{A}$ by a scalar $n$ changes its magnitude but not its direction (unless $n$ is negative).

• $\vec{C} = n\vec{A}$

• If $n = 0$, the result is the zero vector; if $n = -1$, the vector reverses direction.

Coordinate Systems and Components

Cartesian Coordinate System

• Vectors are often described in terms of their components along the x- and y-axes.

• A point in the plane is given by coordinates $(x, y)$.

Vector Components

• Any vector $\vec{A}$ can be resolved into components: $A_x$ (x-direction), $A_y$ (y-direction).

• Notation: $A_x$ and $A_y$ are the scalar components; $\vec{A}_x$ and $\vec{A}_y$ are the vector components.

• Positive or negative signs indicate direction along the axes.

Finding Resultant Vectors

• Sum the components of all vectors along each axis:

Component	Formula
x-component	$D_x = A_x + B_x + \ldots$
y-component	$D_y = A_y + B_y + \ldots$

• The magnitude of the resultant vector $\vec{D}$ is $D = \sqrt{D_x^2 + D_y^2}$

• The direction is $\tan \phi = \frac{D_y}{D_x}$

Motion in Two Dimensions

Displacement and Velocity

• Displacement: The change in position, a vector quantity.

• Average Velocity: $\vec{v} = \frac{\vec{d}}{\Delta t}$

Acceleration in Two Dimensions

• Average Acceleration: $\vec{a} = \frac{\Delta \vec{v}}{\Delta t} = \frac{\vec{v}_f - \vec{v}_i}{\Delta t}$

• Both velocity and acceleration are vectors and can change in magnitude and/or direction.

Kinematic Equations for Constant Acceleration

• For each direction (x and y), the kinematic equations apply:

Equation	x-direction	y-direction
Final velocity	$(v_x)_f = (v_x)_i + a_x \Delta t$	$(v_y)_f = (v_y)_i + a_y \Delta t$
Displacement	$x_f = x_i + (v_x)_i \Delta t + \frac{1}{2} a_x \Delta t^2$	$y_f = y_i + (v_y)_i \Delta t + \frac{1}{2} a_y \Delta t^2$
Velocity squared	$(v_x)_f^2 = (v_x)_i^2 + 2 a_x \Delta x$	$(v_y)_f^2 = (v_y)_i^2 + 2 a_y \Delta y$

Projectile Motion

Characteristics of Projectile Motion

• A projectile is an object moving in two dimensions under the influence of gravity only (assuming air resistance is negligible).

• The horizontal and vertical motions are independent except for sharing the same time interval.

• Vertical acceleration: $a_y = -g = -9.8\ \mathrm{m/s}^2$ (downward)

• Horizontal acceleration: $a_x = 0$ (no horizontal force)

Equations for Projectile Motion

• Initial velocity components:

  • $v_{0x} = v_0 \cos \theta$

  • $v_{0y} = v_0 \sin \theta$

• Horizontal motion (constant velocity):

  • $x = x_0 + v_{0x} t$

• Vertical motion (constant acceleration):

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

  • Vertical velocity: $v_y = v_{0y} - g t$

Trajectory and Range

• The path of a projectile is a parabola.

• The range (horizontal distance) is maximized at a launch angle of 45° for a given initial speed.

• For two launch angles that add up to 90°, the range is the same (e.g., 30° and 60°).

Example: Basketball Shot

• A basketball is launched at an angle $\theta$ with initial speed $v_0$.

• Horizontal component: $v_{0x} = v_0 \cos \theta$

• Vertical component: $v_{0y} = v_0 \sin \theta$

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

Circular Motion

Uniform Circular Motion

• An object moves in a circle at constant speed but with continuously changing direction.

• Although speed is constant, velocity is not (because direction changes).

• There is always an acceleration directed toward the center of the circle (centripetal acceleration).

Centripetal Acceleration

• Given by $a_c = \frac{v^2}{r}$, where $v$ is the speed and $r$ is the radius of the circle.

• This acceleration is always perpendicular to the velocity vector and points toward the center of the circle.

Problem-Solving Strategies for Projectile Motion

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

• Draw a diagram showing initial and final positions, and indicate velocity components.

• Establish a coordinate system (usually x-axis horizontal, y-axis vertical).

• Resolve the initial velocity into x and y components.

• List known values and identify what is to be found.

• Use kinematic equations for each direction separately, linked by the common time variable.

• Check units and reasonableness of the answer.

Summary Table: Kinematic Equations for 2D Motion

Equation	x-direction	y-direction
Final velocity	$(v_x)_f = (v_x)_i + a_x \Delta t$	$(v_y)_f = (v_y)_i + a_y \Delta t$
Displacement	$x_f = x_i + (v_x)_i \Delta t + \frac{1}{2} a_x \Delta t^2$	$y_f = y_i + (v_y)_i \Delta t + \frac{1}{2} a_y \Delta t^2$
Velocity squared	$(v_x)_f^2 = (v_x)_i^2 + 2 a_x \Delta x$	$(v_y)_f^2 = (v_y)_i^2 + 2 a_y \Delta y$

Additional info: These notes are based on standard introductory physics content and expand upon the provided slides for clarity and completeness.