Chapter 1: Representing Motion – College Physics Study Notes

Chapter 1: Representing Motion

Introduction

This chapter introduces the fundamental concepts of motion, including how to represent and describe motion using diagrams, mathematical principles, and scientific notation. It also reviews the basic mathematical tools and units used in physics.

Section 1.1: Motion – A First Look

Motion is defined as the change of an object’s position or orientation with time. The path along which an object moves is called its trajectory.

• Types of Motion: Includes constant speed, speeding up, and slowing down.

• Motion Diagram: A sequence of images showing an object's position at equal time intervals, useful for visualizing motion.

Section 1.2: Models and Modeling

Models are simplified representations of nature that capture essential features for study. Two main types:

• Descriptive Models: Describe properties in the simplest terms.

• Explanatory Models: Use laws of physics to predict behavior.

• Particle Model: Treats a moving object as if all its mass is concentrated at a single point, simplifying motion analysis.

Section 1.3: Position and Time – Putting Numbers on Nature

To specify position, a reference point (origin), distance, and direction are needed. The combination forms a coordinate system.

• Position: Represented by a coordinate along an axis.

• Time: Each frame in a motion diagram is labeled with time (symbol t).

• Displacement: The difference between final and initial position: $\Delta x = x_{f} - x_{i}$

• Time Interval: $\Delta t = t_{f} - t_{i}$, always positive.

Example: Bicycle Ride Displacement

Emily rides from 3 mi east to 2 mi west of a water tower. Her displacement is:

• Initial position: $x_{i} = +3$ mi

• Final position: $x_{f} = -2$ mi

• Displacement: $\Delta x = x_{f} - x_{i} = -2 - 3 = -5$ mi (westward)

Problem-solving process: Strategize, Prepare, Solve, Assess.

Section 1.4: Velocity

Motion at constant speed in a straight line is called uniform motion. Speed measures how fast an object moves; velocity includes direction.

• Average Velocity: $v_{avg} = \frac{\Delta x}{\Delta t}$

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

• Velocity to the left is negative; to the right is positive.

Example: Albatross Flight

• Initial position: 60 mi east

• Final position: 80 mi east

• Time interval: 0.25 h

• Velocity: $v = \frac{80 - 60}{0.25} = 80$ mi/h (eastward)

Section 1.5: Significant Figures, Scientific Notation, and Units

Measurements are limited by precision, expressed through significant figures.

• Multiplication/Division: Answer matches the least precise number.

• Addition/Subtraction: Answer matches the smallest number of decimal places.

• Scientific Notation: Used for very large or small numbers, clarifies significant figures.

• SI Units: Standard units in science: meters (m), seconds (s), kilograms (kg).

Table: Common SI Units

Quantity	SI Unit
Length	meter (m)
Time	second (s)
Mass	kilogram (kg)

Example: Walking Speed Estimate

• Distance: 1 mile

• Time: 30 min

• Speed: $\frac{1 \text{ mile}}{0.5 \text{ hour}} = 2 \text{ mi/h}$

• Convert to m/s: $\approx 1 \text{ m/s}$

Section 1.6: Vectors and Motion – A First Look

Physical quantities are classified as scalars (magnitude only) or vectors (magnitude and direction).

• Vector: Represented as an arrow; magnitude is length.

• Displacement Vector: From initial to final position.

• Vector Addition: Place tail of one at tip of another; resultant is from tail of first to tip of last.

Vectors and Trigonometry

• Pythagorean Theorem: $a^2 + b^2 = c^2$

• Sine, Cosine, Tangent: $\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}}$, $\cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}}$, $\tan \theta = \frac{\text{opposite}}{\text{adjacent}}$

• Inverse trig functions find angles from side lengths.

Example: Anna’s Displacement

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

• Magnitude: $\sqrt{90^2 + 50^2} = 100$ m

• Direction: $\theta = \tan^{-1}(\frac{50}{90}) \approx 29^{\circ}$ north of east

Velocity Vectors

Velocity vectors point in the direction of motion, with magnitude equal to speed. In motion diagrams, velocity vectors connect successive positions.

Section 1.7: Summary and Applications

• Motion Diagrams: Use the particle model to represent motion with dots at equal time intervals.

• Scalars and Vectors: Scalars (temperature, time, mass); vectors (velocity, displacement).

• Describing Motion: Position, displacement, time interval, velocity.

• Units: SI units are standard in science.

• Working with Numbers: Scientific notation, unit conversions, significant figures, order-of-magnitude estimates.

Integrated Example: Goose Flight Speed

• Actual distance flown: sum of two legs

• Straight-line distance: hypotenuse of triangle

• Extra distance: difference between actual and straight-line

• Flight speed: $\frac{\text{total distance}}{\text{total time}}$

• Convert to km/h: $49 \frac{\text{mi}}{\text{h}} \times \frac{1.61 \text{km}}{1.00 \text{mi}} = 79 \frac{\text{km}}{\text{h}}$

Summary Table: Scalar vs Vector Quantities

Quantity	Type	Description
Temperature	Scalar	Magnitude only
Time	Scalar	Magnitude only
Mass	Scalar	Magnitude only
Velocity	Vector	Magnitude and direction
Displacement	Vector	Magnitude and direction

Additional info: These notes expand on brief points from the original slides and textbook, providing full academic context, definitions, examples, and relevant equations for college-level physics students.