Describing Motion: Kinematics in One Dimension

Introduction

Kinematics is the branch of physics that describes the motion of objects without considering the causes of motion. In one dimension, kinematics focuses on motion along a straight line, using concepts such as displacement, velocity, and acceleration. This study guide summarizes the foundational principles and definitions necessary for understanding motion in one dimension.

Glossary of Key Terms

• Scalar: A quantity described by magnitude only (e.g., mass, speed, time).

• Vector: A quantity described by both magnitude and direction (e.g., displacement, velocity, acceleration).

• Displacement: The change in position of an object; a vector quantity.

• Distance: The total length of the path traveled; a scalar quantity.

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

• Average Speed: Total distance divided by total time elapsed.

• Average Velocity: Displacement divided by total time elapsed; a vector quantity.

• Instantaneous Velocity: The velocity of an object at a specific instant.

• Average Acceleration: Change in velocity divided by time interval.

• Instantaneous Acceleration: The acceleration at a specific instant.

• Motion Diagram: A graphical representation of an object's motion at successive times, showing velocity and acceleration vectors.

• Kinematic Equations: Equations that describe motion under constant acceleration.

• Free Fall: Motion under the influence of gravity alone.

• Air Resistance: The frictional effect of air on moving objects.

Two Branches of Mechanics

Overview

• Mechanics: The branch of physics that studies forces and motion.

• Statics: Deals with forces in equilibrium among material bodies.

• Dynamics: Involves the motion of objects and the relationship between motion and other physical concepts.

• Kinematics: A sub-branch of dynamics focused on describing motion without regard to its causes.

Reference Frames and Displacement

Position and Types of Motion

• Position: The location of an object with respect to a chosen reference frame.

• Translational Motion: Movement along a path (e.g., a car on a highway).

• Rotational Motion: Movement around an axis (e.g., Earth's spin).

• Displacement: Change in position, .

• Motion Diagram: Shows the object's position at successive times with velocity and acceleration vectors.

Definition of Reference Frame

Explanation

• All measurements of position, distance, or speed must be made with respect to a reference frame.

• Reference Frame: A coordinate system with a specified origin, orientation, and time, used to measure motion.

• Example: A person walking inside a moving train has a different speed relative to the train than to the ground.

Distance vs. Displacement

Comparison

• Distance: Total path length traveled; always positive.

• Displacement: Change in position; can be positive or negative depending on direction.

• Example: Walking 70 m east and then 30 m west: total distance = 100 m, displacement = 40 m east.

Vector and Scalar Quantities

Definitions and Examples

• Vector Quantities: Require both magnitude and direction (e.g., velocity, acceleration).

• Scalar Quantities: Require only magnitude (e.g., speed, mass).

• In one-dimensional motion, direction can be indicated by sign (+ or -).

Speed and Average Speed

Definitions

• Speed: How fast something is moving, regardless of direction.

• Average Speed:

• Units: meters per second (m/s).

• Always positive.

Average Velocity

Definition and Calculation

• Average Velocity:

• Direction matches displacement.

• Units: meters per second (m/s).

• Graphically, average velocity is the slope of the line connecting two points on a position-time graph.

Instantaneous Velocity

Definition and Graphical Interpretation

• Instantaneous Velocity: The limit of average velocity as the time interval approaches zero.

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

• Magnitude is what a speedometer reads at an instant.

Acceleration

Average and Instantaneous Acceleration

• Average Acceleration:

• Instantaneous Acceleration:

• Units: meters per second squared (m/s2).

• Acceleration is positive if velocity increases in the positive direction; negative if velocity increases in the negative direction.

• "Deceleration" refers to a decrease in speed, not necessarily negative acceleration.

Graphical Interpretation of Velocity and Acceleration

Position-Time and Velocity-Time Graphs

• On a position-time graph, the slope gives velocity.

• On a velocity-time graph, the slope gives acceleration.

• Area under the velocity-time graph gives displacement.

Kinematic Equations for Constant Acceleration

Equations

• Use these equations only when acceleration is constant.

Free Fall and Acceleration Due to Gravity

Concepts

• Free Fall: Motion under gravity alone, with acceleration downward.

• All objects fall with the same acceleration, regardless of mass (neglecting air resistance).

• At maximum height, instantaneous velocity is zero.

• Galileo's experiments contradicted Aristotle's view that heavier objects fall faster.

Air Resistance

Effects

• Air resistance is a frictional force that opposes motion through the atmosphere.

• Objects with large surface areas and small masses (e.g., feathers) are more affected.

• In a vacuum, all objects fall at the same rate.

Derivation of Kinematic Equations (Integral Calculus)

Using Calculus

• For constant acceleration :

• For time-varying acceleration, integrate to find and :

Summary Table: Scalar vs. Vector Quantities

Historical Context: Galileo vs. Aristotle

• Aristotle: Believed heavier objects fall faster than lighter ones.

• Galileo: Demonstrated that all objects fall at the same rate in the absence of air resistance.

• Galileo's inclined plane experiments allowed precise measurement of acceleration due to gravity.