Physics I Chapter 2

Chapter 2: Kinematics: Motion in One Dimension

Goals

  • Students will learn to describe motion using sketches, motion diagrams, graphs, and algebraic equations.

  • Look for consistency between different representations of motion.

2.1 What Is Motion?

Describing Motion

  • Definition of Motion: Motion is a change in an object’s position relative to a given observer during a certain change in time.

  • Importantly, one must identify the observer to determine if an object moved.

  • Relativity of Motion: Motion is relative; the motion of any object of interest depends on the point of view of the observer.

    • Example: An observer in a spaceship perceives the motion of the Sun differently than an Earth-bound observer.

    • Critical elements: The "object of interest" and the "observer" must be specified when describing motion.

Reference Frames

  • Importance of Reference Frames: Specifying the observer is crucial before describing the motion of an object.

  • Definition of a Reference Frame: Includes:

    • An object of reference (often stationary).

    • A coordinate system with a scale for measuring distances.

    • A clock to measure time.

Modeling Motion

  • Modeling Motion: Simplified assumptions are made to analyze complex situations.

  • Types of Motion:

    • The simplest type of motion analyzed is linear motion.

    • Definition of Linear Motion: A model of motion that assumes an object, treated as point-like, moves along a straight line.

2.2 A Conceptual Description of Motion

Visual Representation

  • Visual diagrams represent motion, such as a bowling ball rolling on a smooth surface.

Observational Experiment: Using Dots to Represent Motion

  • Experiment 1: Push a bowling ball to roll on a smooth floor, placing evenly spaced beanbags beside it each second.

  • Experiment 2: Repeat Experiment 1 but push the ball harder. The spacing of beanbags increases but remains even.

  • Experiment 3: Roll the bowling ball on a carpet; the distance between beanbags decreases as it rolls over the carpet.

  • Experiment 4: Roll the ball while continually pushing it with a board, which increases the spacing between the beanbags as it rolls.

Patterns Observed:

  • Dots visualize the motion: evenly spaced dots indicate consistent speed, closer dots indicate slowing down, and farther dots indicate increasing speed.

Motion Diagrams

  • Definition of Motion Diagrams: Motion diagrams contain more information, represented by dots and arrows. Each dot indicates position, and arrows indicate velocity direction and magnitude.

  • Velocity Arrows: Their lengths show how fast the cart is moving at each dot position.

2.3 Operations with Vectors

Vector Basics

  • Vector Definition: A vector has both magnitude (length) and direction (orientation).

  • Components of a Vector:

    • Tail (origin) and Head (tip of the arrow).

    • A negative vector has the same magnitude as its positive counterpart but points in the opposite direction.

Adding Vectors

  • Tail to Head Method: To add vectors, position the tail of one vector at the head of another. This method applies to any number of vectors.

Subtracting Vectors

  • Operation of Subtraction: Vector subtraction can also utilize the tail-to-head approach by reversing the direction of the vector to be subtracted.

Multiplying Vectors by Scalars

  • When a vector is multiplied by a scalar, the resulting vector is parallel or antiparallel, with a magnitude based on the product of the magnitude of the vector and scalar.

2.4 Quantities for Describing Motion

  • Qualitative vs Quantitative Descriptions: Motion diagrams qualitatively represent motion; quantitative descriptions are required for analysis.

  • Quantities for Linear Motion:

    • Time, Position, Displacement, Distance, Path Length.

Time and Time Interval

  • Time (t): Clock reading, a scalar quantity. Units: seconds (s).

  • Time Interval ($ riangle t$): Defined as (t2 − t1); represents a change in clock readings, also a scalar quantity.

Position, Displacement, Distance, and Path Length

  • Position: The location of an object with respect to a coordinate system.

  • Displacement ($ extbf{d}$): A vector from the initial position to the final position.

  • Distance: The magnitude of the displacement, always positive.

  • Path Length: The total length traveled by the object along its actual route.

Example Illustration

  • Scenario: A car backs up toward the origin (x=0) then moves forward.

    • Initial position: $xi = -3.0$ m, Final position: $xf = 5.0$ m.

    • Displacement calculation: $ ext{dx} = xf - xi = 5.0 - (-3.0) = 8.0$ m.

    • Distance traveled: Always positive; total distance = 8.0 m.

Important Statements

  • True Statements:

    • Displacement and distance are not always the same.

    • Path length equals distance.

2.5 Representing Motion with Data Tables and Graphs

Data Collection and Graphing

  • Data Tables: Record position and time data collected during motion observation.

  • Position-Versus-Time Graphs: Help in identifying patterns; kinematics refers to the description of motion.

Graphing Fundamentals

  • Axes Orientation: Typically, time (t) is on the horizontal (independent) axis, and position (x) is on the vertical (dependent) axis.

  • Trendline: A best-fit curve that represents the motion data.

Motion Diagrams vs. Graphs

  • Motion Diagrams: Dots correspond to points on the position axis; they provide less precise information than kinematics graphs.

  • Position Graph: Combines position data with timing of motion.

2.6 Constant Velocity Linear Motion

Fundamental Concepts

  • Mathematical Expression: $x(t) = kt + b$, where $k$ is the slope (rate of change of position) and $b$ is the initial position when $t = 0$.

  • Slope Interpretation: The slope indicates the speed of the object, positive or negative directionality.

Displacement Calculation

  • Utilizing velocity graphs, displacement ($ ext{x} - ext{x_0}$) can be evaluated as the area between the velocity graph line and the time axis during a specified interval.

2.7 Motion at Constant Acceleration

Constant Acceleration Characteristics

  • Definition: Acceleration is the rate at which velocity changes.

  • Types of Motion: Linear motion with constant acceleration allows the use of specific equations for determining position and velocity during intervals.

Equations of Motion

  • Relevant equations for motion under constant acceleration include:

    • Displacement equation: x = x0 + v0 t + rac{1}{2} a t^2

    • Velocity equation: v = v_0 + a t

2.8 Displacement of an Object Moving at Constant Acceleration

Position Determination

  • For any initial position $x0$ at clock reading $t0=0$, one can determine the position $x$ at any later time $t$, given the initial velocity $v{0x}$ and constant acceleration $ax$.

Graph Representation

  • The graph depicting position vs. time under constant acceleration reveals a parabolic curve implying the acceleration's effect on linear motion over time.

Summary

  • Reference Frame: Comprised of an object of reference, a point of reference, a coordinate system, and a clock.

  • Quantities in Motion: Include time (scalar), position (scalar), displacement (vector), distance (scalar), speed (scalar), and acceleration (vector).

  • Key Functional Equations:

    • Displacement: d = |x - x_0|.

    • Velocity: v = rac{ riangle x}{ riangle t}.

    • Acceleration: a = rac{ riangle v}{ riangle t}.