Chapter 2: One-Dimensional Kinematics

Introduction

This chapter explores the fundamental concepts of motion in one dimension, focusing on the relationships between displacement, distance, velocity, speed, and acceleration. It provides the mathematical framework for analyzing motion with both constant and non-constant acceleration, including graphical interpretations and real-world examples.

Goals of the Chapter

• Understand definitions, similarities, differences, and relationships between displacement, distance, average and instantaneous velocity, speed, average and instantaneous acceleration.

• Solve general problems of motion with constant acceleration.

• Solve equations of motion with non-constant acceleration.

• Analyze freefall problems.

• Interpret velocity, acceleration, displacement, and change in velocity from graphical data.

Key Concepts in One-Dimensional Kinematics

Displacement, Distance, Speed, and Velocity

Understanding the distinctions between these terms is crucial for analyzing motion:

• Distance: The total length of the path traveled, regardless of direction. It is a scalar quantity.

• Displacement: The change in position of an object; it is a vector quantity and includes direction.

• Speed: The rate at which distance is covered; a scalar quantity.

• Velocity: The rate at which displacement changes; a vector quantity.

• Instantaneous speed: The magnitude of instantaneous velocity.

• Average speed: Defined as the total distance traveled divided by the total time taken.

• Average velocity: Defined as the total displacement divided by the total time taken.

Formula for Average Speed:

Formula for Average Velocity:

Important Note: Average speed is not the magnitude of average velocity.

Example: Speed vs Velocity

Consider a swimmer who swims 50 m out and 50 m back in 49.82 s:

• Average speed:

• Average velocity: (since displacement is zero)

Calculus and Kinematics

Velocity is the derivative of position with respect to time, and acceleration is the derivative of velocity:

The definite integral of velocity gives displacement:

The definite integral of acceleration gives change in velocity:

Typical Average Velocities

Common speeds for reference:

Equations of Motion with Constant Acceleration

Fundamental Equations

These equations allow you to solve for any unknown variable when the others are known, assuming constant acceleration.

Example Problem: Constant Acceleration

A car accelerates from rest at for :

• Time to travel :

• Speed after :

Example Problem: Cop and Van

Analyzing pursuit problems involves setting up equations for both objects and solving for the time and distance at which they meet.

• Cop:

• Van:

• Set equations equal and solve for .

Freefall Motion

Conditions for Freefall

• Distance traveled is small relative to Earth's radius.

• Air resistance is negligible.

• Effects of Earth's rotation can be ignored.

• Acceleration due to gravity:

Example Problem: Ball Thrown Upward

• Equations of motion:

• Use quadratic formula to solve for time when (ball hits ground).

Nonconstant Acceleration

Solving with Integrals

• Given , integrate to find and .

• Use initial conditions to solve for constants of integration.

Example:

Graphical Analysis of Motion

Interpreting Graphs

• Velocity vs Time: Area under the curve gives displacement.

• Acceleration vs Time: Area under the curve gives change in velocity.

• Position vs Time: Slope gives velocity.

Example Problems: Graphical Interpretation

• Calculate displacement and distance by integrating velocity over time.

• Determine intervals of increasing speed and acceleration from graph shape.

• Average acceleration:

Summary Table: Scalar vs Vector Quantities

Key Takeaways

• Always distinguish between scalar and vector quantities.

• Use calculus for non-constant acceleration problems.

• Graphical analysis is a powerful tool for understanding motion.

• Apply kinematic equations for constant acceleration scenarios.

Additional info: These notes expand on the original slides by providing full definitions, formulas, and context for each concept, ensuring a self-contained study guide for exam preparation.