Chapter 1: Vector Addition with Components

Unit Vectors

Unit vectors are essential tools in physics for representing direction in space. They have a magnitude of 1 and no units, and point in the positive direction of a given axis.

• Definition: A unit vector is a vector with magnitude 1, used to indicate direction along an axis.

• Cartesian Coordinate System:

  • i points in the +x direction

  • j points in the +y direction

  • k points in the +z direction

• Notation: Any vector can be expressed as a sum of its components multiplied by the corresponding unit vectors.

Components of Vectors

Vectors can be broken down into components along chosen axes, which simplifies calculations and analysis.

• Vector Component: The projection of a vector along a chosen axis.

• Vector Components: and are the vector components of .

• Unit Vector Notation:

• Scalar Components: and are the scalar components (just the numbers, not vectors).

Vector-Component Relationship

In a right triangle formed by a vector and its components:

Vector Addition by Components

Vectors can be added by summing their respective components.

• Addition:

• Component Addition: ,

• Magnitude:

• Direction:

Example

Given vectors and , the vector in component form is:

Exercise

A hiker walks 2.50 km at 45° southeast, then 4.00 km at 60° north of east. To find the total displacement:

• Break each leg into x and y components using trigonometry.

• Add the components to get the resultant vector.

• Calculate magnitude and direction using the formulas above.

Chapter 2: Quantities of Linear Motion

Kinematics and Linear Motion

Kinematics is the study of motion, focusing on how position changes over time. Linear motion refers to movement along a straight line (1D motion).

• Key Quantities: Position, velocity, and acceleration

Position and Displacement

Position is the location of a particle along an axis, given by its coordinate . Displacement is the change in position and is a vector quantity.

• Displacement Formula:

• For 1D motion, direction is indicated by positive or negative signs.

• Example: If m and m, m (right); if m and m, m (left).

Velocity & Acceleration

Velocity describes how position changes with time, while acceleration describes how velocity changes with time.

• Average Velocity:

• Instantaneous Velocity:

• Average Acceleration:

• Instantaneous Acceleration:

Representing Linear Motion Graphically

Graphs of position vs. time and velocity vs. time are used to analyze motion.

• Position vs. Time:

  • Horizontal line: object is stationary

  • Straight line: constant velocity

  • Curved line: changing velocity (acceleration)

• Velocity vs. Time:

  • Horizontal line: constant velocity

  • Sloped line: constant acceleration

  • Curved line: changing acceleration

Average and Instantaneous Velocity

Average velocity is calculated over a time interval, while instantaneous velocity is the value at a specific instant.

• Average Velocity:

• Instantaneous Velocity:

• Instantaneous velocity is found from the slope of the tangent to the position vs. time graph.

Average and Instantaneous Acceleration

Average acceleration is the change in velocity over a time interval, while instantaneous acceleration is the rate of change at a specific instant.

• Average Acceleration:

• Instantaneous Acceleration:

• Instantaneous acceleration is found from the slope of the tangent to the velocity vs. time graph.

Speeding Up and Slowing Down

The relationship between velocity and acceleration determines whether an object speeds up or slows down.

• If velocity and acceleration are in the same direction, the object is speeding up.

• If velocity and acceleration are in opposite directions, the object is slowing down.

Summary Table: Key Quantities in Linear Motion

Example: Average Acceleration Calculation

• A car traveling in the negative x-direction at speeds up to in $5$ seconds.

• Equation:

• Numerical substitution:

• Negative sign indicates acceleration is in the negative x-direction.

Additional Info

• Next topics include constant acceleration motion and free fall motion.