Chapter 3: Motion and Kinematics

Introduction to Kinematics

Kinematics is the branch of physics that describes the motion of objects without considering the causes of motion. It involves the mathematical description of position, velocity, and acceleration as functions of time.

• Kinematics: The study of how objects move.

• Uniform motion: Motion along a straight line at a constant speed.

• Position-time graph: A graph that shows how an object's position changes over time.

Position-Time Graphs

Understanding Position-Time Graphs

Position-time graphs provide a visual representation of an object's motion. The slope of the graph at any point gives the velocity of the object at that instant.

• Slope of the position-time graph represents the object's velocity.

• Steeper slopes indicate faster speeds.

• Negative slopes indicate motion in the negative direction (e.g., moving left or down).

• The slope is calculated as the ratio of intervals: .

Key Equations

• Average velocity:

• Uniform motion equation:

Example: Interpreting Position-Time Graphs

• If the position-time graph is a straight line, the motion is uniform (constant velocity).

• If the graph curves, the velocity is changing (acceleration is present).

Velocity-Time Graphs

Relating Velocity and Position Graphs

Velocity-time graphs show how an object's velocity changes over time. The area under the velocity-time graph represents the displacement of the object.

• Velocity is the slope of the position-time graph.

• Area under the velocity-time graph gives the change in position (displacement).

Converting Between Graphs

• To obtain a velocity-time graph from a position-time graph, calculate the slope at each point.

• To obtain a position-time graph from a velocity-time graph, calculate the area under the curve up to each point in time.

Instantaneous Velocity and Calculus

Instantaneous Velocity

Instantaneous velocity is the velocity of an object at a specific instant in time. It is found by taking the derivative of the position function with respect to time.

• Definition:

• Graphically, it is the slope of the tangent to the position-time curve at a given point.

Using Calculus in Kinematics

• The derivative of a constant is zero:

• The derivative of a sum is the sum of the derivatives:

• Power rule for derivatives: If , then

Example: Calculating Velocity from Position Function

• Given (in meters), find the velocity at s.

• First, compute the derivative:

• At s: m/s

Worked Example: Meeting Point Problem

Application of Uniform Motion Equations

Two people start from different cities and travel towards each other at constant speeds. The meeting point can be found by equating their position functions.

• Let Bob start from Chicago at 9:00 AM, traveling east at 60 mph.

• Susan starts from Pittsburgh (400 miles east of Chicago) at the same time, traveling west at 40 mph.

• Let and be their positions at time .

• Set to find the meeting point.

Table: Summary of Uniform Motion Equations

Summary

• Position-time and velocity-time graphs are essential tools for analyzing motion.

• The slope of the position-time graph gives velocity; the area under the velocity-time graph gives displacement.

• Calculus allows for precise determination of instantaneous velocity and acceleration.

• Uniform motion equations are foundational for solving kinematics problems.