Chapter 1: Representing Motion – Study Notes

Representing Motion

Introduction to Motion

Motion is a fundamental concept in physics, describing the change in position or orientation of an object over time. Understanding motion requires both qualitative and quantitative approaches, including diagrams, mathematical relationships, and units.

Types of Motion

There are several types of motion, each characterized by the path an object follows:

• Straight-line motion: Movement along a single direction.

• Circular motion: Movement along a circular path.

• Projectile motion: Movement under the influence of gravity, typically following a parabolic trajectory.

• Rotational motion: Movement around a fixed axis.

Motion Diagrams

Motion diagrams are visual representations of an object's position at successive times. They help simplify complex motion and are often used with the particle model, which treats the object as a single point mass.

• Constant speed: Equal spacing between positions.

• Speeding up: Increasing spacing between positions.

• Slowing down: Decreasing spacing between positions.

Models and Modeling

Models are simplified representations of physical systems, used to describe and predict behavior. Two main types are:

• Descriptive models: Describe properties in simple terms.

• Explanatory models: Use laws of physics to predict outcomes.

The particle model is a key simplification, treating all mass as concentrated at a single point.

Position, Time, and Displacement

Position and Coordinate Systems

To specify an object's position, a reference point (origin), distance, and direction are needed. A coordinate system consists of an origin and an axis with positive and negative directions.

Time and Motion Diagrams

Each frame in a motion diagram is labeled with its corresponding time, denoted by the symbol t.

Displacement

Displacement is the change in position, calculated as the difference between final and initial positions:

• Formula: $\Delta x = x_f - x_i$

Time Intervals

A time interval $\Delta t$ measures the elapsed time between two events:

• Formula: $\Delta t = t_f - t_i$

Example: Displacement Calculation

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

• Formula: $\Delta x = x_f - x_i = (-2\,\text{mi}) - (3\,\text{mi}) = -5\,\text{mi}$

Distance vs. Displacement

Distance is the total path length traveled, while displacement is the straight-line change in position. For example, an ant zig-zagging covers a distance of 50 cm but a displacement of -30 cm.

Velocity and Speed

Definitions

Speed is the rate at which an object moves, regardless of direction. Velocity includes both speed and direction.

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

• Velocity formula: $v = \frac{\Delta x}{\Delta t}$

Example: Albatross Flight

An albatross moves from 60 mi to 80 mi east of its roost in 0.25 h:

• Velocity: $v = \frac{80\,\text{mi} - 60\,\text{mi}}{0.25\,\text{h}} = 80\,\text{mi/h}$

Significant Figures, Scientific Notation, and Units

Measurements and Significant Figures

Measurements have limited precision, indicated by significant figures—digits that are reliably known.

Rules for Significant Figures

• 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

Scientific notation expresses very large or small numbers as a decimal multiplied by a power of ten, clarifying significant figures.

SI Units and Unit Conversion

The SI system is the standard in science. Common units include:

Quantity	Unit	Abbreviation
Time	Second	s
Length	Meter	m
Mass	Kilogram	kg

Unit conversions use conversion factors to switch between systems.

Order-of-Magnitude Estimates

Order-of-magnitude estimates use rough values and are accurate to about one significant figure, indicated by the symbol $\sim$.

Quantity	SI Unit	Approximate Conversion
Mass	kg	1 kg $\approx$ 2 lb
Length	m	1 m $\approx$ 3 ft
Length	cm	3 cm $\approx$ 1 in
Length	km	5 km $\approx$ 3 mi
Speed	m/s	1 m/s $\approx$ 2 mph
Speed	km/h	10 km/h $\approx$ 6 mph

Vectors and Motion: A First Look

Scalars and Vectors

Physical quantities are classified as:

• Scalars: Have only magnitude (e.g., temperature, mass).

• Vectors: Have both magnitude and direction (e.g., velocity, displacement).

Displacement Vectors

The displacement vector shows the distance and direction from an object's initial to final position, regardless of the path taken.

Vector Addition and Subtraction

Vectors are added graphically by placing the tail of one at the tip of another. The resultant vector is drawn from the tail of the first to the tip of the last.

• Net displacement: Sum of individual displacement vectors.

Trigonometry and Vectors

Trigonometry is used to relate the sides and angles of triangles, essential for vector calculations.

• Pythagorean theorem: $H^2 = A^2 + O^2$

• Sine: $\sin \theta = \frac{O}{H}$

• Cosine: $\cos \theta = \frac{A}{H}$

• Tangent: $\tan \theta = \frac{O}{A}$

Example: Net Displacement Calculation

Anna walks 90 m east and 50 m north. Her net displacement is:

• Magnitude: $d_{net} = \sqrt{(90\,\text{m})^2 + (50\,\text{m})^2} \approx 100\,\text{m}$

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

Summary of Key Concepts

• Motion diagrams and the particle model simplify the study of motion.

• Scalars have magnitude only; vectors have magnitude and direction.

• Position is specified relative to a coordinate system; displacement is the change in position.

• Velocity is displacement divided by time interval.

• SI units are standard in science; unit conversions are essential.

• Significant figures indicate measurement precision; scientific notation clarifies large/small numbers.

• Order-of-magnitude estimates are rough calculations used for quick approximations.