Motion in Two and Three Dimensions: Vectors, Position, Velocity, and Acceleration

Vectors and Coordinate Systems

Introduction to Vectors

Understanding the motion of objects in two and three dimensions requires a solid grasp of vectors and coordinate systems. Vectors are essential for describing both the position and movement of objects in space.

• Cartesian Grid: A two-dimensional (x-y) grid used to specify locations in space. Each point is defined by two numbers (coordinates).

• Why 'Cartesian'? The system is named after René Descartes, who developed the coordinate system.

• Defining Position: The position of an object (e.g., a blue square) is given by its coordinates (x, y) on the grid.

• Distance Between Points: The distance between two points (e.g., blue and red squares) can be calculated using the Pythagorean theorem.

GPS-Location and Coordinate Choices

On a global scale, positions are often described using latitude and longitude, which are analogous to a grid wrapped around a sphere. Any two independent coordinates can be used to specify a location on a surface.

• Examples of Coordinate Pairs: N and W (latitude and longitude), S and E, or X and Y.

• Application: GPS devices use such coordinate systems to specify locations on Earth.

Scalars vs. Vectors

Definitions and Properties

Physical quantities can be classified as either scalars or vectors, depending on whether they have direction as well as magnitude.

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

• Vector: A quantity with both magnitude and direction (e.g., displacement, velocity, force).

• Unit Vector: A vector with a magnitude of one, used to indicate direction. The unit vector in the direction of r is defined as:

$\hat{r} = \vec{r} / |\vec{r}|$

• Notation: A 'hat' symbol (^) denotes a unit vector.

Position and Velocity Vectors

Average Velocity

The average velocity between two points is defined as the displacement vector divided by the time interval between those points. The direction of the average velocity is the same as that of the displacement.

• Displacement Vector: $\Delta \vec{r} = \vec{r}_2 - \vec{r}_1$

• Average Velocity:

$\vec{v}_{\text{avg}} = \frac{\Delta \vec{r}}{\Delta t}$

• Direction: Same as the displacement vector.

Instantaneous Velocity

The instantaneous velocity is the rate of change of the position vector with respect to time. It is always tangent to the path of the particle.

• Component Form:

$v_x = \frac{dx}{dt}$ $v_y = \frac{dy}{dt}$ $v_z = \frac{dz}{dt}$

• Direction: Tangent to the particle's path at any instant.

Vector Components

Breaking Vectors into Components

Vectors in two or three dimensions can be expressed in terms of their components along the coordinate axes. This is essential for calculations involving vector addition, subtraction, and resolving motion.

• 1D System: Only one number (with sign) is needed.

• 2D System: Two numbers (e.g., $\hat{x}$ and $\hat{y}$ components).

• 3D System: Three numbers (e.g., $\hat{x}$, $\hat{y}$, $\hat{z}$ or $\hat{i}$, $\hat{j}$, $\hat{k}$).

• General Vector:

$\vec{r} = r_x \hat{i} + r_y \hat{j} + r_z \hat{k}$

• Angle Convention: Angles are typically measured from the positive x-axis, counterclockwise.

Component Equations for Projectile Motion

For motion with an initial velocity $v_0$ at an angle $\alpha_0$ from the x-axis, the position and velocity components as functions of time are:

• $x = (v_0 \cos \alpha_0)t$

• $y = (v_0 \sin \alpha_0)t - \frac{1}{2}gt^2$

• $v_x = v_0 \cos \alpha_0$

• $v_y = v_0 \sin \alpha_0 - gt$

Note: The assignment of sine and cosine depends on how the angle is defined in the problem. Use SOH-CAH-TOA to determine the correct components.

Acceleration Vector

Average and Instantaneous Acceleration

The acceleration vector describes how the velocity of an object changes with time. In two or three dimensions, both the magnitude and direction of velocity can change.

• Average Acceleration:

$\vec{a}_{\text{avg}} = \frac{\Delta \vec{v}}{\Delta t}$

• Instantaneous Acceleration:

$\vec{a} = \lim_{\Delta t \to 0} \frac{\Delta \vec{v}}{\Delta t} = \frac{d\vec{v}}{dt}$

• Direction: The acceleration vector points in the direction of the change in velocity, not necessarily along the path of motion.

Summary Table: Scalars vs. Vectors

Quantity	Scalar	Vector
Definition	Magnitude only	Magnitude and direction
Examples	Speed, mass, temperature	Displacement, velocity, force
Notation	Italic (e.g., m)	Bold or arrow (e.g., v or $\vec{v}$)

Examples and Applications

• Finding Displacement: Given two position vectors, subtract to find the displacement.

• Calculating Average Velocity: Divide displacement by the time interval.

• Projectile Motion: Use component equations to determine position and velocity at any time.

• Unit Vectors: Used to specify direction in vector equations.

Additional info: These notes provide foundational concepts for analyzing motion in two and three dimensions, which are essential for further study in physics, including kinematics, dynamics, and applications such as projectile motion and circular motion.