Motion in One Dimension: Position, Displacement, and Velocity (PHYS 2110 Mechanics)

Motion in One Dimension: Position, Displacement, and Velocity

Preview and Learning Objectives

This section introduces the fundamental concepts required to describe motion along a straight line, focusing on position, displacement, and velocity. Students will learn to use precise language and mathematical representations to analyze motion, distinguish between scalar and vector quantities, and apply vector algebra in one dimension.

• Kinematics: The study of motion without considering its causes.

• Visual Representations: Use of motion diagrams and graphs to model real-world situations.

• Physical Quantities: Classification into scalars (magnitude only) and vectors (magnitude and direction).

• Learning Objectives:

  • Describe the motion of a point particle using correct terminology.

  • Distinguish between position, distance, and displacement.

  • Represent vectors and perform vector algebra in one dimension.

Lecture Outline

• How can motion be described as a function of time? (Motion graphs)

• What are the basic quantities that allow us to describe motion with words? (Position, velocity, acceleration)

• What is a vector quantity?

• How can we describe the direction of a vector?

• How can we add vectors?

Visual Representations of Motion

Motion Diagrams and Motion Graphs

Visual tools such as motion diagrams and motion graphs are essential for understanding and analyzing motion.

• Motion Diagram: Shows the position of an object at equal time intervals, often using a particle model (object represented by a single point).

• Motion Graph: Plots position as a function of time, providing information about the object's movement and changes in speed.

• Example: A ball rolling on a smooth surface is recorded at equal time intervals. The motion diagram and graph together reveal changes in speed and direction.

Position, Displacement, and Velocity

Definitions and Distinctions

Understanding the differences between position, displacement, distance, average speed, and average velocity is crucial for analyzing motion.

• Position (x): The location of an object at a particular instant, often measured from a chosen origin.

• Displacement (Δx): The change in position of an object, defined as $\Delta x = x_f - x_i$ (final position minus initial position). Displacement is a vector quantity and can be positive or negative.

• Distance: The total length of the path traveled, always positive and a scalar quantity.

• Average Speed: The distance traveled divided by the time interval required to travel that distance. $\text{Average speed} = \frac{\text{distance}}{\text{time interval}}$

• Average Velocity: The displacement divided by the time interval. $\text{Average velocity} = \frac{\Delta x}{\Delta t}$

• Example: In a strobe photograph, the position of a ball is recorded at two times. The displacement is the change in position, while the distance is the total path length.

Working with Displacements

Calculating displacement involves subtracting the initial position from the final position. The sign of the displacement indicates direction.

• Example Calculations:

  • (a) $x_i = 1.4\,\text{m}$, $x_f = 3.1\,\text{m}$, $\Delta x = 3.1 - 1.4 = 1.7\,\text{m}$

  • (b) $x_i = 2.3\,\text{m}$, $x_f = 0.9\,\text{m}$, $\Delta x = 0.9 - 2.3 = -1.4\,\text{m}$

  • (c) $x_i = -0.4\,\text{m}$, $x_f = -1.3\,\text{m}$, $\Delta x = -1.3 - (-0.4) = -0.9\,\text{m}$

Distance Traveled

The distance traveled is the total length of the path covered by a moving object, regardless of direction.

• Formula: $\text{Distance traveled} = |x_f - x_i|$

• Example: If a ball moves from $x_0$ to $x_2$ and back to $x_0$, the total distance is the sum of the absolute values of each segment.

Average Speed and Average Velocity

Average speed and average velocity are related but distinct concepts. Average speed is a scalar and does not indicate direction, while average velocity is a vector and does.

• Average Speed: $\text{Average speed} = \frac{\text{distance}}{\text{time interval}}$

• Average Velocity: $\text{Average velocity} = \frac{\Delta x}{\Delta t}$

• Graphical Interpretation: On a position vs. time graph, the slope of the line represents velocity. A steeper slope indicates a higher speed.

• Example: If the position graph is a straight line, the speed is constant. If the slope changes, the speed changes.

Scalars and Vectors

Definitions

Physical quantities are classified as scalars or vectors based on whether they have direction.

• Scalars: Quantities described by magnitude (number and unit) only. Examples: mass, time, speed.

• Vectors: Quantities described by both magnitude and direction. Examples: displacement, velocity, force.

• Vector Representation: Vectors are represented by arrows; the length indicates magnitude, and the direction indicates the vector's direction.

Vector Algebra in One Dimension

Vectors in one dimension can be added or subtracted using ordinary algebra, taking care to include the correct sign for direction.

• Adding Vectors: $A + B$ (if both are in the same direction)

• Subtracting Vectors: $A - B$ (if in opposite directions)

• Multiplying by a Scalar: $kA$ (where $k$ is a scalar)

The Nature of Physical Quantities and Laws of Physics

Physical Quantities

Physical quantities are measured and expressed using units and can be either scalar or vector.

• Scalars: Only magnitude (e.g., mass, time, temperature).

• Vectors: Magnitude and direction (e.g., displacement, velocity, force).

Laws of Physics

Physical laws are expressed as mathematical equations and are valid in all coordinate systems. They describe the behavior of physical phenomena in the universe.

• Example: Newton's laws of motion, which apply equally in all coordinate systems.

Coordinate Systems

Defining a Coordinate System

A coordinate system is an artificial framework imposed to analyze a problem. The choice of origin and axes is arbitrary but should be consistent throughout the analysis.

• Right-Handed Cartesian Coordinate System: Commonly used in physics, with axes perpendicular to each other.

• Labeling Axes: Positive and negative directions must be clearly defined.

• Example: Choosing the origin at the starting point of motion and the positive x-axis in the direction of motion.

Representation of a Vector

A unit vector defines a direction in space and has a magnitude of one. Vectors can be expressed in terms of their components along coordinate axes.

• Unit Vector: $\hat{i}$ (x-direction), $\hat{j}$ (y-direction)

• Vector Components: $\vec{A} = A_x \hat{i} + A_y \hat{j}$

• Multiplying by a Scalar: $k \vec{A}$ changes the magnitude but not the direction.

Summary Table: Scalars vs. Vectors

Quantity Type	Definition	Examples
Scalar	Described by magnitude only	Mass, time, speed, temperature
Vector	Described by magnitude and direction	Displacement, velocity, force

Additional info:

• Graphs and diagrams are essential for visualizing motion and interpreting physical quantities.

• Understanding the distinction between distance and displacement is critical for solving kinematics problems.

• Unit vectors are used to specify direction in multi-dimensional problems, but in one dimension, direction is indicated by sign (+/-).