Chapter 1: Representing Motion

Chapter Overview

This chapter introduces the fundamental concepts of motion, including how to describe, represent, and analyze motion using diagrams, mathematical models, and vectors. It also reviews essential mathematical tools such as trigonometry, significant figures, and scientific notation, which are foundational for further study in physics with calculus.

Section 1.1: Motion – A First Look

Types of Motion

• Motion is the change of an object’s position or orientation with time.

• The trajectory is the path along which an object moves.

• Common types of motion include straight-line, circular, projectile, and rotational motion.

Motion Diagrams

• A motion diagram is a sequence of images showing an object’s position at equal time intervals.

• Motion diagrams help visualize whether an object moves at constant speed, speeds up, or slows down.

• For two-dimensional motion, diagrams show changes in both speed and direction.

Section 1.2: Models and Modeling

Models in Physics

• Models are simplified representations of physical systems that capture essential features.

• Two main types: Descriptive models (describe properties) and Explanatory models (predict behavior based on physical laws).

The Particle Model

• The particle model treats a moving object as if all its mass is concentrated at a single point.

• This model is useful for simplifying motion analysis, especially when the object's size and shape are not important.

Section 1.3: Position and Time – Putting Numbers on Nature

Position and Coordinate Systems

• To specify an object’s position, define a reference point (origin), a distance from the origin, and a direction.

• A coordinate system consists of an origin and an axis with positive and negative directions.

Time and Motion Diagrams

• Each position in a motion diagram is labeled with a time (t), measured from a clock.

Displacement

• Displacement is the change in position:

• Displacement is a vector quantity (has magnitude and direction).

Time Interval

• The time interval is the elapsed time:

• Time intervals are always positive.

Worked Example: Displacement Calculation

• Example: Emily rides from 3 miles east to 2 miles west of a water tower in 0.5 hours. Her displacement is miles (to the west).

Section 1.4: Velocity

Velocity and Speed

• Uniform motion is motion at a constant speed in a straight line.

• Speed is how fast an object moves (scalar), while velocity includes both speed and direction (vector).

• Average velocity:

Worked Example: Calculating Velocity

• Example: An albatross moves from 60 miles to 80 miles east of its roost in 0.25 hours. Its velocity is mph east.

Section 1.5: Significant Figures, Scientific Notation, and Units

Significant Figures

• Significant figures reflect the precision of a measurement.

• Multiplication/division: answer has the same number of significant figures as the least precise value.

• Addition/subtraction: answer has the same number of decimal places as the least precise value.

Scientific Notation

• Used to express very large or small numbers clearly and to show significant figures.

• Example: m = m

Units and Unit Conversion

• The SI system is the standard in science: meters (m), kilograms (kg), seconds (s).

• Unit conversions use conversion factors (e.g., ).

Order-of-Magnitude Estimates

• An order-of-magnitude estimate is a rough calculation, accurate to about one significant figure.

Section 1.6: Vectors and Motion – A First Look

Scalars and Vectors

• Scalar quantities have only magnitude (e.g., mass, time).

• Vector quantities have both magnitude and direction (e.g., displacement, velocity).

• Vectors are represented as arrows; the length shows magnitude, and the arrow points in the direction.

Displacement Vectors

• The displacement vector points from the initial to the final position, regardless of the path taken.

Vector Addition and Subtraction

• To add vectors, place the tail of the second at the tip of the first; the resultant vector goes from the tail of the first to the tip of the second.

• Subtraction is adding the negative of a vector.

Vectors and Trigonometry

• Trigonometry is used to relate the sides and angles of triangles formed by vectors.

• Pythagorean theorem: for right triangles.

• Sine, cosine, and tangent functions relate angles to side lengths.

Worked Example: Displacement Using Vectors

• Example: Anna walks 90 m east, then 50 m north. Her net displacement is the hypotenuse of a right triangle.

• Magnitude: m

• Direction: north of east

Velocity Vectors

• Velocity vectors point in the direction of motion; their length represents speed.

Worked Example: Motion Diagram with Velocity Vectors

• Example: A ball is hit at a 60° angle and caught. The motion diagram shows velocity vectors changing in both magnitude and direction.

Section 1.7: Summary and Applications

Key Concepts

• Motion diagrams and the particle model are foundational tools for representing and analyzing motion.

• Scalars and vectors are essential for distinguishing between quantities with and without direction.

• Position, displacement, velocity, and time are the primary quantities for describing motion.

• Proper use of significant figures, scientific notation, and units ensures clarity and accuracy in calculations.

Table: Common SI Units

Table: Rules for Significant Figures

Summary

• Understanding and representing motion is the foundation for all further study in physics.

• Mastery of vectors, units, and estimation techniques is essential for problem-solving in physics with calculus.