One-Dimensional Kinematics

Position, Velocity, and Acceleration

One-dimensional kinematics describes the motion of objects along a straight line, focusing on position, velocity, and acceleration as functions of time.

• Position (x): The location of an object along a coordinate axis at a given time.

• Velocity (v): The rate of change of position with respect to time. It can be positive or negative depending on direction.

• Acceleration (a): The rate of change of velocity with respect to time.

Key Equations:

• Average velocity:

• Instantaneous velocity:

• Average acceleration:

• Instantaneous acceleration:

Example: If a dog moves from to m in 5 s, its average velocity is .

Two-Dimensional Kinematics

Vector Displacement and Motion in a Plane

Two-dimensional kinematics involves motion in a plane, requiring vector analysis for displacement, velocity, and acceleration.

• Displacement Vector: Represents change in position in two dimensions, often using components .

• Vector Addition: Displacements are added using the Pythagorean theorem and trigonometry.

• Unit Vector Notation: Displacement and velocity can be expressed as .

Key Equations:

• Magnitude of displacement:

• Direction:

Example: If a dog moves 100 m east and then 200 m north, the total displacement is m at an angle north of east.

Projectile Motion

Analysis of Objects Thrown or Launched

Projectile motion describes the two-dimensional motion of objects under the influence of gravity, following a parabolic trajectory.

• Horizontal Motion: Constant velocity,

• Vertical Motion: Constant acceleration due to gravity,

• Trajectory Equation:

Key Equations:

• Horizontal displacement:

• Vertical displacement:

• Time to reach maximum height:

• Maximum height:

Example: A ball is thrown at m/s at above horizontal. The horizontal and vertical components are , .

Graphical Analysis of Motion

Interpreting Position, Velocity, and Acceleration Graphs

Graphs are essential tools for visualizing and analyzing motion. They can show how position, velocity, or acceleration change over time.

• Position-Time Graph: Slope gives velocity.

• Velocity-Time Graph: Slope gives acceleration; area under curve gives displacement.

• Acceleration-Time Graph: Area under curve gives change in velocity.

Example: If the velocity-time graph is a straight line, acceleration is constant. If the graph is curved, acceleration is changing.

Vector Components and Unit Vector Notation

Expressing Vectors in Terms of Components

Vectors can be broken down into components along the x and y axes, which simplifies calculations in two-dimensional motion.

• Unit Vectors: (x-direction), (y-direction)

• Vector Expression:

• Magnitude:

Example: A velocity vector m/s has magnitude $5$ m/s.

Relative Motion

Analyzing Motion from Different Reference Frames

Relative motion considers how the velocity of an object appears from different reference frames, such as a moving train or observer.

• Relative Velocity:

• Application: Used to analyze collisions, moving vehicles, and objects in motion relative to each other.

Example: If a train moves at 80 km/h and a passenger walks at 5 km/h in the same direction, the passenger's velocity relative to the ground is $85$ km/h.

Projectile Motion: Graphical Representation

Horizontal and Vertical Velocity Components

Projectile motion graphs can show how the horizontal and vertical components of velocity change over time.

• Horizontal Component (): Remains constant throughout the flight.

• Vertical Component (): Changes linearly due to gravity;

Example: On a graph, is a horizontal line, while is a straight line with negative slope.

Applications and Problem Solving

Solving Kinematics Problems

Physics problems often require applying kinematic equations, vector analysis, and graphical interpretation to real-world scenarios.

• Step 1: Identify knowns and unknowns.

• Step 2: Choose appropriate equations.

• Step 3: Solve algebraically, then numerically.

• Step 4: Check units and reasonableness of answer.

Example: A soccer ball is kicked; use projectile motion equations to find time in air and range.

HTML Table: Comparison of Motion Types

Additional info: These notes expand on the exam questions by providing definitions, equations, and examples for all major kinematics topics covered in the practice exam. The table summarizes the differences between motion types for quick reference.